![1 Introduction
1.1 Introduction
Wireless networks are becoming more and more ubiquitous in the modern world, and
more and more essential to today’s society. In 30 years they have progressed from the
province of a tiny minority of the world’s population in only the most developed nations,
to the point where there are very nearly as many wireless subscriptions as people in the
world [24]. The services offered have extended from very limited speech services at the
introduction of first-generation mobile systems in 1985, to broadband Internet access
and full motion video today. Moreover, we are at the point where wireless networks will
extend beyond connecting people (of whom there are a limited number), to connecting
their devices – an effectively unlimited number. Some believe that there are already
more devices than people connected to the Internet, and predictions that 50 billion or
more devices will be connected by 2020 are circulating widely [60]. Of course, that is
only the start.
All this implies that the density of wireless networks will inevitably increase. To pro-
vide telecommunication services to the human populations of our cities, at continually
increasing data rates, will require increasing numbers of access points, for which back-
haul will become an increasing problem, and require more widespread use of wireless
backhaul. The devices will also form a network many times as dense as any current
wireless networks, also likely to require connection to the core network. In both cases it
is likely that the current point-to-multipoint architecture of wireless networks, exempli-
fied by both cellular and WiFi systems, will be replaced by a multi-hop mesh network
architecture.
The concept of the mobile ad-hoc network (MANET), one of the best-established
concepts in wireless mesh networking, has been in existence for many years [9], yet
has not really fulfilled its predicted potential. There are very few wireless networks in
use today that implement a truly multi-hop networking approach. There seems to be a
barrier to the practical implementation of multi-hop wireless networking that will surely
have to be overcome in order to implement the ultra-dense wireless networks that are
likely to be required in the near future.
Perhaps the most fundamental basis for such a barrier is that described by Gupta and
Kumar in their well-known paper [20]. They show that for a conventional approach to
wireless networking, in which transmissions from other nodes in the network are treated
as interference, the total capacity of the network scales as the square root of the number
02
19:28:49](https://image.slidesharecdn.com/22267127-250619075953-3759a6f1/85/Wireless-Physical-Layer-Network-Coding-Jan-Sykora-Alister-Burr-16-320.jpg)
![4 Introduction
of nodes – that is, the capacity per node decreases as the size of the network increases.
Hence as networks become denser, and more hops are required, the capacity available
to each terminal will decrease.
This interference problem has become widely recognized as the most significant prob-
lem limiting the performance of future wireless networks, including point-to-multipoint
networks as well as multi-hop. Traditionally it has been mitigated by means of the cel-
lular paradigm, which limits interference by ensuring that a certain re-use distance is
respected. Increased density is accommodated by using smaller and smaller cells with
greatly reduced transmit power, but this approach is now reaching its limit, both because
of the large numbers of radio access points it requires and the resulting backhaul prob-
lem, and because cell sizes are becoming comparable in size with buildings and other
city features.
All this suggests that it is time for a completely new paradigm in wireless networking,
and a major objective of this book is to lay the foundations for such a paradigm, which
we call the “Network-Aware Physical Layer.”
1.2 The “Network-Aware Physical Layer”
Since the 1970s the design of communications networks has been based upon a lay-
ered paradigm, in which network functions are divided between protocol layers, each
assumed to be transparent to the ones above it. The original layered model, dating from
the late 1970s, was of course the OSI seven-layer model [2], but recently the layers
implicitly defined in the TCP-IP protocol suite [1] have been more influential. In either
case, the lower layers – the network layer, the link layer, and the physical layer – are of
most interest to us here, since they provide the most basic functions of a communication
network, namely routing, multiple access and error control, and modulation and coding,
respectively.
Of these layers, the physical layer is the one that handles the signals which are actu-
ally transmitted over the communication medium: in our case these are related to the
electromagnetic fields that form the radio waves. In the traditional layered paradigm the
physical layer receives a signal from the medium and converts it to a bit stream, which is
then passed to the link layer. However, this has the fundamental disadvantage that infor-
mation is lost in the process that might improve the performance of functions which
are located in higher layers. For example, it is well known that error correction is less
efficient when operating on a bit stream (corresponding to hard decision decoding) than
when it has access to a soft decision metric, which is usually obtained from the signal.
Moreover, it also means that signals from nodes other than the transmitter of inter-
est must be treated as interference, which conveys no useful information but degrades
the performance of the receiver in decoding the wanted signal. This arises because the
traditional physical layer is assumed to operate on only one point-to-point link, which
means signals on other links are interference (and vice versa). This is illustrated in Fig-
ure 1.1. The figure illustrates a multi-hop network in which data can travel from source
to destination via two routes. We focus on the link of interest marked: in the traditional
02
19:28:49](https://image.slidesharecdn.com/22267127-250619075953-3759a6f1/85/Wireless-Physical-Layer-Network-Coding-Jan-Sykora-Alister-Burr-17-320.jpg)
![6 Introduction
Figure 1.2 A simple cooperative communication system.
Access), or else is scheduled by using time-division or frequency-division multiple
access (TDMA or FDMA). However code-division multiple access (CDMA), widely
used in third-generation (3G) mobile systems, uses channels (corresponding to spread-
ing codes) that are typically not fully orthogonal, and hence requires processing of the
received mixed signal, which must be carried out at the physical layer, to separate the
data. Similarly error control: while forward error correction (FEC) coding is conven-
tionally regarded as part of the physical layer, retransmission protocols such as ARQ
(Automatic Repeat reQuest) have traditionally been implemented at the link layer. How-
ever, recently hybrid FEC/ARQ schemes have somewhat blurred this distinction, since
they require combining of signals transmitted in the course of multiple retransmissions.
Until recently, however, the functions of the network layer, specifically routing, have
been excluded from the physical layer. This began to change about a decade ago with
the introduction of cooperative communications [32]. Cooperative systems involve at
least one relay node as well as the source and destination nodes (Figure 1.2), to assist the
transmission of the source’s data. Typically it receives the source signal in one time-slot,
and retransmits it in some form in a subsequent slot. In most cases the processing within
the relay is entirely at the physical layer, and frequently it is the original signal or some
function of it that is retransmitted, without being converted to bits first. This is perhaps
the simplest example of the physical layer being extended over a network involving
multiple hops, beyond the simple link between one transmitter and one receiver.
This is, however, a very rudimentary version of routing. In this book we consider a
much more general scenario involving multiple sources and multiple destinations, and
multi-hop relaying between them. Thus routing is an essential element. The approach
we will use, however, differs from routing in the conventional layered paradigm in two
respects. The first is that it resembles cooperative communications in that processing
within the relay takes place at the physical layer, involving signals directly. Unlike a
bridge or a router in a conventional network, the relay does not decode the source data
and transfer it to the link or network layer, but rather processes the received signals
and forwards some function of them. The second is that what it forwards may not be
a representation of data from a single source, but rather some function of data from
several sources – a “mixture” of data from multiple sources to be separated at a later
stage and delivered to the required destination. Thus it may no longer be possible to
identify distinct routes for individual data streams, as is conventionally assumed.
This latter aspect can also be applied at the network layer of a multi-hop network,
and corresponds to a technique introduced at the beginning of this century, known as
network coding, which we will now discuss.
02
19:28:49](https://image.slidesharecdn.com/22267127-250619075953-3759a6f1/85/Wireless-Physical-Layer-Network-Coding-Jan-Sykora-Alister-Burr-19-320.jpg)
![1.3 Network Coding at the Network Layer 7
1.3 Network Coding at the Network Layer
Network layer network coding (NC) [5] addresses a network modeled as a directed
graph connecting source nodes to destination nodes via a set of relaying nodes. In gen-
eral there may be multiple sources and multiple destinations. The edges of the graph
represent discrete links between pairs of nodes. This is clearly a good model of a data
communications network with wired connections, such as the Internet, though we will
see later that it does not represent a wireless network so well. For a unicast network, in
which there is only one source and one destination, it can be proven that the maximum
data flow rate is given by the max-flow, min-cut theorem [14]. However, Ahlswede et al.
[5] showed that in the multicast case, where multiple destinations wish to receive the
same data, the maximum flow rate cannot be achieved if relaying nodes operate simply
as switches, connecting data flows on incoming links to outgoing links. Instead nodes
should apply network coding, in which the symbols on an outgoing link are generated by
some function of the symbols on two or more incoming links. This may be illustrated by
the network shown in Figure 1.3, known as the butterfly network. The figure shows two
versions of a network, in which two data sources each wish to send their data to both
of two destinations, over a network in which all links have unit capacity. Figure 1.3a
represents a conventional network in which the nodes can only switch a data stream
from an incoming link onto an outgoing edge, or duplicate it and send on more than one
outgoing edge. Thus the upper of the two relay nodes (which are marked as circles) can
only select either stream A or stream B to send on its outgoing link (here it selects A).
This is duplicated by the lower relay node, and hence the right-hand destination node
can receive both streams, but the left-hand one receives only A. Figure 1.3b shows a
network employing network coding. Here the upper relay node computes the exclusive
OR (XOR) function (or modulo-2 sum) of the symbols in the data streams, and for-
wards the result. The lower relay node duplicates this to both destinations, and they can
each recover both streams, because one is directly available, and the other can be recon-
structed by reversing the network coding function applied at the relay node with the
aid of the directly available stream. Thus the left-hand destination can now reconstruct
stream B by applying A ⊕ (A ⊕ B) = B.
Figure 1.3 Butterfly network.
02
19:28:49](https://image.slidesharecdn.com/22267127-250619075953-3759a6f1/85/Wireless-Physical-Layer-Network-Coding-Jan-Sykora-Alister-Burr-20-320.jpg)
![8 Introduction
Figure 1.4 Linear network coding.
We will revisit this network topology later, in a slightly different context, but of course
this principle also applies to much more complex networks, including networks con-
taining cycles. Also in this case very simple coding is applied at the relay – simply the
bit-by-bit XOR – but in general more complex encoding is required. There exists a wide
variety of forms of coding, but [27] showed that linear coding over the finite field F2m
is effective: in fact [34] had already shown that linear coding can achieve the maximum
flow in a multicast network. Figure 1.4 illustrates this coding applied to a node: the out-
put symbol Y is given by the formula in the diagram, in two different notations. In the
first form ⊗ and ⊕ represent multiplication and addition within F2m ; in the second this
is simply represented as a summation. The symbols on the incoming links are symbols
in F2m : they are drawn from an alphabet whose size is a power of 2, and can in fact be
represented as length m binary strings. The coefficients Ai, i = 1 . . . n are also elements
of F2m , and again can be represented as length m binary strings. The addition operation
is in fact simple bit-by-bit modulo-2 addition, but multiplication is more complicated: it
is usually defined using primitive element operations on finite field (see Section A.2.1
or [8]).
It is clear that if all nodes apply a linear function of this sort, with symbols and
coefficients from the same field, then the vector of output symbols across all relay nodes
may be related to the vector of source symbols by a matrix. Equally clearly, for the
destination nodes to reconstruct the source data this matrix must be full rank. We will
revisit this model more rigorously later in the book.
1.4 Wireless Physical Layer Network Coding
The network model implicit in the conception of network coding, as illustrated in Fig-
ures 1.3 and 1.4, has one important deficiency as a representation of a wireless network.
It assumes that the incoming links are discrete, and the symbols they carry are sepa-
rately available to the network coding function in the node. This is a valid model of
a wired network, but a wireless network does not have defined, discrete connections
between nodes in the same way. Rather the electromagnetic fields due to signals trans-
mitted simultaneously from two nodes will add together at the antenna of a receiving
node, resulting in a superimposition of the two signals. Moreover they may be attenu-
ated and/or phase shifted due to the wireless channel in largely unpredictable ways. In
02
19:28:49](https://image.slidesharecdn.com/22267127-250619075953-3759a6f1/85/Wireless-Physical-Layer-Network-Coding-Jan-Sykora-Alister-Burr-21-320.jpg)
![1.5 Historical Perspective 11
Figure 1.7 Illustration of PNC operation.
correspond to (network coded) 0, while the black circle corresponds to 1. This clearly
increases capacity compared to both the joint decoding approach and the network coding
approach.
1.5 Historical Perspective
At this point we will take a break from the technical details of WPNC to discuss how
we reached this point, and the initial development of WPNC up to the present. We have
already discussed some of the information theoretic background, and have mentioned
the development of network coding. It is worth noting, however, that many of the theo-
retical foundations of multi-user information theory were laid in the 1970s – including
analysis of the multiple access channel [4], [35], of the broadcast channel [11], and
of the relay channel [12]. However, there has been little practical implementation of
these concepts even up to today, although that is now changing, notably because of the
pressures on wireless networks noted above, and also because multiple antenna systems
have important synergies with the MAC and broadcast channels, which have led to the
introduction of multi-user MIMO (MU-MIMO) systems in fourth-generation wireless
systems. Multi-user information theory can now be seen as an important step towards
the development of network information theory in the past decade or so, extending
these concepts beyond single-hop multi-user networks. Both network coding and WPNC
occupy the field of network information theory, and many concepts from it underlie the
work in this book.
WPNC itself was discovered independently by three research groups, who
approached it from slightly different angles, resulting in distinct approaches that, how-
ever, are clearly based on the same principles. Zhang, Liew, and Lam [64], of the
Chinese University of Hong Kong, were probably motivated by concepts from network
coding. They introduced the application of WPNC to the two-way relay channel, which
we will review in the next chapter but which is quite similar to the butterfly network we
have already seen. They also generalized it to a multi-hop chain network.
Popovski and colleagues at the University of Aalborg introduced an analog version
of WPNC at the same time [49], based on earlier work applying network coding to the
two-way relay channel [33]. They subsequently extended this to a scheme they refer
02
19:28:49](https://image.slidesharecdn.com/22267127-250619075953-3759a6f1/85/Wireless-Physical-Layer-Network-Coding-Jan-Sykora-Alister-Burr-24-320.jpg)
![12 Introduction
to as denoise and forward [28]. Around the same time other work, e.g. [50], discussed
other strategies for the two-way relay channel (though without proposing the full WPNC
concept), and this was also the emphasis of the work by Popovski et al.
The third group was Nazer and Gastpar, then both at University of California Berke-
ley. Their earliest published work dates from 2005 [43], and was framed more as an
approach to a new information-theoretic problem: that of decoding functions of symbols
from multiple sources, rather than the sources themselves. However, this is evidently
directly relevant to WPNC if the functions are those required in network coding, and
leads to an approach called compute and forward. Their subsequent work and the work
of other workers inspired by it has moved into the area of lattice coding, as a useful basis
for the functions, and has retained a strong algebraic flavor.
Lattice coding is itself a field with a long history. It is based on the mathematical the-
ory of lattice constructions, especially in more than three dimensions, but is connected
with group theory as well as the physics and chemistry of crystals, going back to the
middle of the nineteenth century. Its application to coding theory was extensively dis-
cussed in the 1980s in the classic reference on the topic, [10]. However, more recently
it has undergone something of a renaissance, especially since it has been demonstrated
that lattice codes with lattice decoding can also approach the Shannon capacity [16]. The
work of Nazer and Gastpar [45] also used it to establish achievable regions for compute
and forward.
Since this fundamental work the field has remained very active. Much of the early
work continued to focus on the two-way relay channel, but recently this has been
extended to other topologies, such as the multi-way relay channel, multiple relay net-
works, and multi-hop networks. Early work also focussed on information theoretic
aspects, with little attention to practical implementation, but more recently more prac-
tical aspects have been investigated, such as the use of practical coding schemes,
synchronization, performance on realistic wireless channels, etc. Recently also prac-
tical test-beds for the concept have been implemented [3, 38]. Of course, much of this
work will feature in the remainder of this book.
1.6 Practical Usage Scenarios
We have already described the developments in wireless communications that provide
the practical drivers for the move toward the network-aware physical layer in general,
and the implementation of WPNC in particular. Here we will look in a little more detail
at some specific scenarios in which it might be applied. The drivers we have considered
include both conventional wireless broadband services via cellular and WiFi networks,
and machine-type communications, including the “Internet of Things.” However, these
two different application areas may give rise to different network topologies, so we will
discuss them separately here.
As mentioned above, access networks for cellular mobile networks are becoming
denser in order to support rapidly increasing capacity density requirements arising from
both increasing numbers of users and increasing data rate demand per user. To mitigate
02
19:28:49](https://image.slidesharecdn.com/22267127-250619075953-3759a6f1/85/Wireless-Physical-Layer-Network-Coding-Jan-Sykora-Alister-Burr-25-320.jpg)
![2.5 Achievable Rates of HDF and JDF 25
of clarity, we will not state them explicitly now and we urge the reader to check them
carefully in order to avoid misinterpretations. We also consider the simplistic case of
two BPSK sources in a real-valued AWGN hierarchical MAC channel. Even though
this example has very little practical relevance, and it still does not allow closed-form
mathematical results (they must be determined numerically), the treatment is relatively
simple and prepares the ground for the more complex expositions used later in the
book.
2.5.1 Two-Source BPSK Hierarchical MAC
We assume two coded sources with messages bA ∈ [1 : 2NRA ], bB ∈ [1 : 2NRB ], where
N is the codeword length, and source codebooks CA, CB with identical code rates RA =
RB. In information theory, the message is frequently described by a scalar index drawn
from some discrete value range. It stresses the fact that the form of the information is
irrelevant and the only important aspect is the total number of message values. It also
has a nice interpretation as line index numbers of the codebook. The total number of
codebook lines is Mi = 2NRi , i ∈ {A, B}, where Ri is the so-called rate. The codeword
length N is the length of the line in the codebook. The rate of the code is the binary-base
logarithm of the codebook size per codesymbol, Ri = lg Mi/N, i.e. how many binary
symbols are represented by one codesymbol.
The codesymbols cA,n, cB,n ∈ {0, 1} use the BPSK channel alphabet sA,n, sB,n ∈ {±1},
with size M = 2, mapped symbol-wise to the codesymbols. The observation model is a
real-valued AWGN channel
xn = sA,n(cA,n) + sB,n(cB,n) + wn (2.5)
where the noise has σ2
w variance per dimension and its probability density function
(PDF) is
pw(w) =
1
2πσ2
w
exp −
w2
2σ2
w
. (2.6)
The SNR is defined as
γ =
E[|si|2]
σ2
w
. (2.7)
The hierarchical mapping function is XOR
cn = χc(cA,n, cB,n) = cA,n ⊕ cB,n (2.8)
and thus it has the minimal cardinality cn ∈ {0, 1}. Under a number of specific assump-
tions treated later (e.g. isomorphic layered code, regular and symbol-wise independent
and identically distributed (IID) perfect random codebooks, etc.; see Sections 5.7, 5.7.3,
5.7.4, and Chapter 4), we can assess the coded system performance using single chan-
nel symbol information-theoretic properties. The isomorphic assumption implies that we
can uniquely decode the hierarchical map of the information data messages. The hier-
archical data map is b = χ(bA, bB) and b ∈ [1 : 2NR] where R is the hierarchical data
rate.
03
19:31:33](https://image.slidesharecdn.com/22267127-250619075953-3759a6f1/85/Wireless-Physical-Layer-Network-Coding-Jan-Sykora-Alister-Burr-38-320.jpg)
![26 Wireless Physical Layer Network Coding: a Gentle Introduction
2.5.2 JDF Strategy
The JDF strategy is limited by a classical multi-user rate region. Both data streams must
be first reliably individually decoded before they can be used in the network-level NC.
The achievable rates are given in terms of mutual information expressions
RA I(CA; X|CB), (2.9)
RB I(CB; X|CA), (2.10)
RA + RB I(CA, CB; X). (2.11)
We dropped the sequence index n from the notation. All following statements refer to a
single symbol.
The mutual information between a pair of random variables describes how much the
outcome uncertainty of one of them is reduced after observing the other one. The condi-
tional mutual information assumes that the stochastic behavior of the variables involved
is conditioned by the knowledge of the conditioning variable (e.g. the codesymbol is
known). In the case where it is not clear from the context, or when we need to distin-
guish it explicitly, we use capital letters to denote the random variables and lower-case
letters to denote their particular values.
The achievable rate is the rate of some given codebook construction that can be
decoded with some given decoding strategy with error probability approaching zero
for N → ∞. The achievable rate is typically determined by some function containing
mutual information expressions. Under common memoryless channel and so-called IID
random codebook assumptions, the involved mutual information expressions are related
to individual symbols. The random IID codebook is an abstraction in constructing the
hypothetical idealized codebook that makes the information theoretic proofs of coding
theorems possible; see Section A.4 for details.
Owing to the symmetry of the channel and the symmetry of the codebooks, the
achievable rates have the first-order limit
RA = RB I1 = I(CA; X|CB) = I(CB; X|CA) (2.12)
and the second-order limit
RA = RB I2/2 (2.13)
where
I2 = I(CA, CB; X). (2.14)
Thy symmetry of the system and the minimal cardinality map then implies R= RA =RB.
The first-order limits are essentially the single-user rates
I1 = H[X
] − H[X
|C]
= H[X
] − H[W] (2.15)
where
H[W] =
1
2
lg(2π e σ2
w) (2.16)
03
19:31:33](https://image.slidesharecdn.com/22267127-250619075953-3759a6f1/85/Wireless-Physical-Layer-Network-Coding-Jan-Sykora-Alister-Burr-39-320.jpg)
![2.5 Achievable Rates of HDF and JDF 27
is the AWGN entropy and X = SA(CA) + W is the effective single-user channel model
with the second source removed to equivalently model the conditioning in the mutual
information. The effective observation has PDF
p(x
) =
cA
p(x
|sA(cA))p(cA)
=
cA
pw(x
− sA(cA))p(cA) (2.17)
and entropy
H[X
] = − Ep(x)
lg p(x
)
. (2.18)
The second-order limit is
I2 = H[X] − H[X|CA, CB]
= H[X] − H[W] (2.19)
where the observation entropy is
H[X] = − Ep(x)
lg p(x)
(2.20)
and
p(x) =
cA,cB
p(x|sA(cA), sB(cB))p(cA)p(cB)
=
cA,cB
pw(x − sA(cA) − sB(cB))p(cA)p(cB). (2.21)
All codesymbols have uniform a priori probability mass function (PMF) p(cA) = 1/M
and p(cB) = 1/M.
2.5.3 HDF Strategy
The HDF strategy, in contrast to JDF, directly decodes the hierarchical data map. The
achievable hierarchical rate, under some conditions (e.g. using regular isomorphic lay-
ered NCM, etc.; see Sections 5.7 and 5.7.4 for details), is given by the hierarchical
mutual information
R I(C; X) = IH. (2.22)
Notice that we do not need to explicitly decode the individual source data streams,
and that the hierarchical data rate is directly given by the single-symbol information-
theoretic limit. The symmetry of the scenario and the minimal cardinality map again
implies
R = RA = RB. (2.23)
The hierarchical mutual information I(C; X) evaluation requires the knowledge of the
hierarchical channel symbol conditional PDF. It describes the observed received signal
from the perspective of the hierarchical channel symbol which is, in turn, mapped to
03
19:31:33](https://image.slidesharecdn.com/22267127-250619075953-3759a6f1/85/Wireless-Physical-Layer-Network-Coding-Jan-Sykora-Alister-Burr-40-320.jpg)
![28 Wireless Physical Layer Network Coding: a Gentle Introduction
the hierarchical codebook encoded message b. The conditional PDF for our minimal
cardinality map with uniformly distributed symbols is (see details in Section 4.4)
p(x|c) =
1
M
cA,cB:c
p (x|sA(cA), sB(cB)) , (2.24)
where the summation set cA, cB : c is the summation over all cA, cB consistent with
hierarchical symbol c, i.e. such that c = χc(cA, cB). The hierarchical mutual information
is then
IH = H[X] − H[X|C]. (2.25)
Unlike the JDF case, the conditioning in H[X|C] still leaves some ambiguity because of
the many-to-one hierarchical mapping function property. The conditional entropy thus
needs to be explicitly evaluated using
H[X|C] = − Ep(x,c)
lg p(x|c)
(2.26)
where p(x|c) is given above and p(x, c) = p(x|c)p(c) where hierarchical symbols have
uniform PMF p(c) = 1/M.
2.5.4 Achievable Rates
The achievable rates for JDF and HDF strategies are now evaluated numerically. The
integrals of the expectations in the entropies do not have closed-form solutions; however,
it is a relatively easy numerical task. We first visualize them by plotting (in Figure 2.4)
the achievable hierarchical rate for JDF, which is given by the bottleneck of the first-
and the second-order rate limits
–10 –5 0 5 10 15 20
0.0
0.2
0.4
0.6
0.8
1.0
1.2
γ [dB]
R
HDF
,
R
JDF
Hierarchical rate in H–MAC
HDF
(solid),
JDF
(dashed)
Figure 2.4 Hierarchical rate achievable for two BPSK sources and relay with JDF (dashed line)
and HDF (solid line) strategies.
03
19:31:33](https://image.slidesharecdn.com/22267127-250619075953-3759a6f1/85/Wireless-Physical-Layer-Network-Coding-Jan-Sykora-Alister-Burr-41-320.jpg)
![2.6 2WRC with QPSK: the Problem of Channel Parametrization 29
RJDF min I1,
I2
2
, (2.27)
and for HDF, which is given by the hierarchical mutual information
RHDF IH. (2.28)
As we see, the performance is clearly alphabet-limited at high SNR values, where it
saturates at the fixed ceiling. This ceiling is, however, higher for the HDF strategy,
where it is given by lg M, and it provides the single-user level performance as if there
was no interference at all. It is, however, in contrast to JDF, where the performance is
given by the interference limited regime of the second-order rate region condition. Even
with zero noise, the JDF cannot support the lg M hierarchical rate. This comparison
exactly shows the performance advantage where WPNC (HDF in this example) tech-
nique demonstrates its supremacy and it also justifies our aim of turning the interference
into a “friendly” form.
The low SNR region is dominated by the influence of the noise and the actual inter-
action of the coded signals remains less significant. We call this region the noise-limited
region. The advantage of HDF, which can effectively cope with the interference by turn-
ing it into a “friendly” interaction that reduces the cardinality of codewords that need
to be distinguished, does not help now. The specific hierarchical constellation shape,
namely the fact that two points (±2) belong to the same codesymbol map, now makes
the situation slightly worse for low SNR. This will be explicitly treated in Section 4.5.
In the noise limited region, JDF outperforms HDF.
Figure 2.5 shows the rate region from the perspective of both sources. The HDF strat-
egy has a rectangular region since both the rates RA and RB are equal, provided that
they are less than IH. In contrast with that, the JDF strategy has the classical multi-user
MAC shaped region. The region has close-to-rectangular shape for low SNR – the noise-
limited regime. The interference limited regime for high SNR makes the second-order
limit the dominant one. The symmetric rate RA = RB is thus limited by the second-order
limit I2/2. The pair of lines for γ = 5 [dB] nicely demonstrates that the “corner” point
of HDF can be outside the JDF region, while the JDF itself provides slightly greater
first-order rate limits.
The trade-off between noise limitation and interference limitation can be nicely
seen when evaluating the ratio (I2/2)/I1 (Figure 2.6). It describes how much the
second-order limit influences the symmetric rate. The second-order limit captures
how the performance is affected by the presence of the other user. The first-order
limit captures the stand-alone single-user behavior and thus captures the noise-related
performance.
2.6 2WRC with QPSK: the Problem of Channel Parametrization
So far our examples have been restricted to the binary case: BPSK modulation. For
most practical applications it will be necessary to extend to higher-order modulation. In
03
19:31:33](https://image.slidesharecdn.com/22267127-250619075953-3759a6f1/85/Wireless-Physical-Layer-Network-Coding-Jan-Sykora-Alister-Burr-42-320.jpg)
![30 Wireless Physical Layer Network Coding: a Gentle Introduction
0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
RA
R
B
Rate region, γ[dB]∈{–5,0,5,10}
HDF
(solid),
JDF
(dashed)
Figure 2.5 Achievable rate regions for two BPSK sources and relay with JDF and HDF strategies.
Each pair of solid (HDF) and dashed (JDF) lines corresponds to one SNR value γ . High SNR
values correspond to outer pairs.
–10 –5 0 5 10 15 20
0.0
0.2
0.4
0.6
0.8
1.0
1.2
(I
2
/2)/I
1
JDF 2nd order vs. 1st order rate limit ratio
γ [dB]
Figure 2.6 Second-order vs. first-order limit ratio (I2/2)/I1 for BPSK sources and relay with JDF
strategy.
this section we will consider QPSK modulation, and we will see that this raises further
issues about the choice of mapping function. First, it allows more options: for linear
functions because there are more coefficients to choose from, and for the more general
table representation because the table is larger and allows more permutations. Secondly,
03
19:31:33](https://image.slidesharecdn.com/22267127-250619075953-3759a6f1/85/Wireless-Physical-Layer-Network-Coding-Jan-Sykora-Alister-Burr-43-320.jpg)
![2.6 2WRC with QPSK: the Problem of Channel Parametrization 33
(a) Re[u]
–3 –2 –1 0 1 2 3
Re[u]
–3 –2 –1 0 1 2 3
Im[u]
–3
–2
–1
0
1
2
3
Im[u]
–3
–2
–1
0
1
2
3
0000
1000
0100
1100
0010
1010
0110
1110
0001
1001
0101
1101
0011
1011
0111
1111
(b)
0000
1000
0100
1100
0010
1010
0110
1110
0001
1001
0101
1101
0011
1011
0111
1111
Re[u]
–3 –2 –1 0 1 2 3
(c)
Im[u]
–3
–2
–1
1
0
2
3
0000
1000 0100
1100
0010
1010 0110
1110
0001
1001 0101
1101
0011
1011 0111
1111
Figure 2.9 (Nearly) singular fading for (a) h ≈ 1, (b) h ≈ j, (c) h ≈ 1 + j.
terms of the labels shown in Figure 2.9 this means the network coded label is formed
by two XOR functions, first of bits 1 and 3 and second of bits 2 and 4). For h = 1, as
in Figure 2.9a, we can see that this results in the same two-bit binary label for all the
(nearly) coincident points in the constellation. However, for h = j, as in Figure 2.9b,
we observe that the clashes are not resolved: for example of the four coincident points
around the origin two will be labelled “10,” and the other two “01.” But if instead the
two functions XOR first bits 1 and 4, and secondly bits 2 and 3 of the composite label,
we will label all four of these points “01,” and similarly the other four clashes in the
constellation will also be resolved.
This highlights an important general issue that arises with any modulation scheme
with more than two points in its constellation. Unlike the BPSK case, where we observed
that the XOR function resolved both singular fade states and therefore could be used
for any fading, for QPSK (and any other non-binary modulation), different mapping
functions are required in order to resolve all fade states. Hence adaptive mapping is
required at the relay.
03
19:31:33](https://image.slidesharecdn.com/22267127-250619075953-3759a6f1/85/Wireless-Physical-Layer-Network-Coding-Jan-Sykora-Alister-Burr-46-320.jpg)
![36 Wireless Physical Layer Network Coding: a Gentle Introduction
Table 2.3 Table for joint mapping function from two relays in HWN.
bB
bA
0 1 . . . M − 1
0 (0, 0) (1, M − 1) . . . (M − 1, 1)
1 (1, 1) (2, 0) . . . (0, 2)
.
.
.
.
.
.
...
.
.
.
M − 1 (M − 1, M − 1) (0, M − 2) . . . (M − 2, 0)
It is also clear that the functions at the two relays must be different. In fact a stronger
condition is required: whenever two different pairs of source symbols give the same
output for one function, they must produce a different result for the other function:
∀(bA, bB), (b
A, b
B) : (bA, bB) = (b
A, b
B), χi (bA, bB) = χi
b
A, b
B ,
it must hold that χī (bA, bB) = χī
b
A, b
B (2.38)
where i ∈ {1, 2}, ī = 3 − i. The table formulation, as before, can be used for any
arbitrary pair of discrete functions. If we restrict ourselves to linear functions, then the
pair of output symbols from the relays can be written
(b1, b2) = (a1A ⊗ bA ⊕ a1B ⊗ bB, a2A ⊗ bA ⊕ a2B ⊗ bB) . (2.39)
This may also be written in matrix form, as
br
= Abs
(2.40)
where bs = [bA, bB]T, br = [b1, b2]T, and
A =
a1A a1B
a2A a2B
. (2.41)
Then the condition for unambiguous decodability becomes simply that A is invertible,
that is, that its rows and columns be linearly independent. This, of course, also implies
that the functions at the two relays are different.
Provided the cardinality of the sources is a power of 2, we can also use the binary
matrix representation of a linear mapping function. Using the same notation as before,
the relay mapping functions can then be written (notice that vectors b are now modified
to reflect the binary representation)
b1 = G1
bA
bB
, (2.42)
b2 = G2
bA
bB
, (2.43)
br
=
b1
b2
=
G1
G2
bA
bB
= Gbs
. (2.44)
03
19:31:33](https://image.slidesharecdn.com/22267127-250619075953-3759a6f1/85/Wireless-Physical-Layer-Network-Coding-Jan-Sykora-Alister-Burr-49-320.jpg)
![Mr. Owens. That's the man that had run across Jefferson and run
into the basement of the library, so I went back to the scene of the
shooting of Officer Tippit and another call had come and some of my
men yelled to me that they had a suspect in the Texas Theatre, and
everyone left there, but nobody was left to help guard the scene
except the crime lab man, so I remained at the scene, and
everybody else went to the Texas Theatre.
Mr. Ely. Do you remember who the crime lab man was who was
there?
Mr. Owens. At the time I thought it was Captain Doughty
[spelling] D-o-u-g-h-t-y. They finished up taking the pictures and I
left the scene and went to Methodist Hospital where Officer Tippit
had been taken, and I was taken back to the room where he was
taken, and in just a brief examination of the body I saw where one
bullet had entered his right chest about the pocket and went
through a package of cigarettes. Another one hit him about the
center of the chest and hit a button, and another one, I believe, was
in his right temple, I'm not sure which temple it was, but those three
wounds, I did see. I don't know whether he was shot any more or
not. I remained at the hospital for quite a time, and then I went
back to the Oak Cliff substation where I was assigned.
Mr. Ely. And because you were assigned to the Oak Cliff
substation, you at no time during these 2 days or so went into the
main police headquarters; is that correct?
Mr. Owens. What, now?
Mr. Ely. You didn't go to the main police headquarters because
you were assigned to the Oak Cliff substation?
Mr. Owens. No; that's right.
Mr. Ely. Now, I show you a map which is labeled Putnam
Deposition Exhibit No. 1. Could you tell us what sort of a map this
is?](https://image.slidesharecdn.com/22267127-250619075953-3759a6f1/85/Wireless-Physical-Layer-Network-Coding-Jan-Sykora-Alister-Burr-57-320.jpg)
![Mr. Ball. Outside of that you haven't had any trouble?
Mr. Applin. No, sir.
Mr. Ball. Now, November 22, 1963, were you in Dallas?
Mr. Applin. Yes; I believe I was.
Mr. Ball. What were you doing here?
Mr. Applin. Well, I was working for the Rollform Corp.
Mr. Ball. How do you spell it?
Mr. Applin. Well, I have got one of their checks—check stubs here
in my pocket, I believe. At least I think I have. Here it is [indicating].
Mr. Ball. What were you doing in Dallas?
Mr. Applin. Working.
Mr. Ball. Working here in Dallas?
Mr. Applin. Yes, sir.
Mr. Ball. What kind of work?
Mr. Applin. Well, I was working as, open-head crane operator,
and painter and front-end loader.
Mr. Ball. Did you go to the picture show that afternoon?
Mr. Applin. Yes, sir; I did.
Mr. Ball. How did you happen to be off duty that day?
Mr. Applin. They was installing a new cutting press for the rollers,
and they did not need me, so, they let me off for 2 days.
Mr. Ball. For 2 days?
Mr. Applin. For 2 days.
Mr. Ball. What did you do? Go to the picture show?
Mr. Applin. Yes, sir; I did.
Mr. Ball. What time of day did you go there?](https://image.slidesharecdn.com/22267127-250619075953-3759a6f1/85/Wireless-Physical-Layer-Network-Coding-Jan-Sykora-Alister-Burr-75-320.jpg)
![Mr. Applin. Well, he stopped and asked two boys sitting down in
the front, asked them to stand up and——
Mr. Ball. Did he search them?
Mr. Applin. Yes, sir; they shuffled them down.
Mr. Ball. Did he search you?
Mr. Applin. No, sir; they came on up to Oswald, where he was
sitting.
Mr. Ball. Where was he sitting?
Mr. Applin. I—he was sitting, I guess, about 3 or 4 rows down.
Mr. Ball. You mean from the rear of the theatre?
Mr. Applin. From the rear.
Mr. Ball. And how far over from the aisle?
Mr. Applin. I guess that would be about three seats. They was
sitting about two or three seats.
Mr. Ball. What did you see him do?
Mr. Applin. He—started off, the officer said, Will you stand up,
please? And he stood up.
Mr. Ball. How close were you to the officer and this man when
you heard the officer say, Stand up?
Mr. Applin. I guess it was about—it was not over four seats down
from the back, rear.
Mr. Ball. Were you at the rear?
Mr. Applin. Yes, sir; I was at the rear of the show.
Mr. Ball. You were at the rear of the show?
Mr. Applin. Yes, sir; well, there was a partition here. A partition
here [indicating], and there was about, oh, I guess about four rows
down from me.](https://image.slidesharecdn.com/22267127-250619075953-3759a6f1/85/Wireless-Physical-Layer-Network-Coding-Jan-Sykora-Alister-Burr-78-320.jpg)
![Mr. Ball. Took a swing at him and what happened then?
Mr. Applin. Well, the officer, I heard him say, Here he is. And
during the proceeding of that, I guess about 5 or 10 seconds later,
there was another—I think it was two officers, or one, passed me
and ran down there to him.
Mr. Ball. Did you see a gun?
Mr. Applin. Well, the gun didn't come into view until after about
four or five officers were there.
Mr. Ball. Then did you see a gun?
Mr. Applin. Yes, sir; but only—there was one gun. The pistol. It
came into view before any of the other officers got there.
Mr. Ball. That is what I mean. What do you say happened about
that? Who pulled a gun?
Mr. Applin. Well, anyhow, the officer was facing this way
[indicating] and Oswald was facing this way [indicating]. And then
the gun was pointed out that way [indicating].
Mr. Ball. Wait a minute. I can't follow you when you say it was
this way, and this way, sir. You told me that this officer asked
Oswald to stand up?
Mr. Applin. Yes, sir.
Mr. Ball. Did he stand up?
Mr. Applin. Yes, sir; he did.
Mr. Ball. Then did he put his hand some place on Oswald?
Mr. Applin. Yes, sir; along about——
Mr. Ball. Where?
Mr. Applin. I guess about his hips.
Mr. Ball. Then what did Oswald do?
Mr. Applin. He took a right-hand swing at him.](https://image.slidesharecdn.com/22267127-250619075953-3759a6f1/85/Wireless-Physical-Layer-Network-Coding-Jan-Sykora-Alister-Burr-80-320.jpg)
Wireless Physical Layer Network Coding Jan Sykora Alister Burr
- 1. Wireless Physical Layer Network Coding Jan Sykora Alister Burr download https://ebookbell.com/product/wireless-physical-layer-network- coding-jan-sykora-alister-burr-44534254 Explore and download more ebooks at ebookbell.com
- 2. Here are some recommended products that we believe you will be interested in. You can click the link to download. Wireless Networks From The Physical Layer To Communication Computing Sensing And Control Franceschetti https://ebookbell.com/product/wireless-networks-from-the-physical- layer-to-communication-computing-sensing-and-control- franceschetti-4109984 Physical Layer Security In Wireless Cooperative Networks Wang https://ebookbell.com/product/physical-layer-security-in-wireless- cooperative-networks-wang-6752862 Eee Standard For Information Technologytelecommunications And Information Exchange Between Systems Local And Metropolitan Area Networks Specific Requirements Part 11 Wireless Medium Access Control Mac And Physical Layer Phy Specifications Ieee https://ebookbell.com/product/eee-standard-for-information- technologytelecommunications-and-information-exchange-between-systems- local-and-metropolitan-area-networks-specific-requirements- part-11-wireless-medium-access-control-mac-and-physical-layer-phy- specifications-ieee-32735296 Physical Layer Security For Wireless Sensing And Communication Hseyin Arslan https://ebookbell.com/product/physical-layer-security-for-wireless- sensing-and-communication-hseyin-arslan-48004208
- 3. Physical Layer Security In Wireless Communications Xiangyun Zhou https://ebookbell.com/product/physical-layer-security-in-wireless- communications-xiangyun-zhou-4748224 Securing Wireless Communications At The Physical Layer 1st Edition Zang Li https://ebookbell.com/product/securing-wireless-communications-at-the- physical-layer-1st-edition-zang-li-4194234 Physical Layer Approaches For Securing Wireless Communication Systems 1st Edition Hong Wen Auth https://ebookbell.com/product/physical-layer-approaches-for-securing- wireless-communication-systems-1st-edition-hong-wen-auth-4395394 Key 5g Physical Layer Technologies Enabling Mobile And Fixed Wireless Access 1st Ed Douglas H Morais https://ebookbell.com/product/key-5g-physical-layer-technologies- enabling-mobile-and-fixed-wireless-access-1st-ed-douglas-h- morais-22502848 Key 5g Physical Layer Technologies Enabling Mobile And Fixed Wireless Access 2nd Edition 2nd Duglas Moralis https://ebookbell.com/product/key-5g-physical-layer-technologies- enabling-mobile-and-fixed-wireless-access-2nd-edition-2nd-duglas- moralis-36346680
- 5. Wireless Physical Layer Network Coding Discover a fresh approach for designing more efficient and cooperative wireless commu- nications networks with this systematic guide. Covering everything from fundamental theory to current research topics, leading researchers describe a new, network-aware coding strategy that exploits the signal interactions that occur in dense wireless net- works directly at the waveform level. Using an easy-to-follow, layered structure, this unique text begins with a gentle introduction for those new to the subject, before moving on to explain key information-theoretic principles and establish a consistent framework for wireless physical layer network coding (WPNC) strategies. It provides a detailed treatment of Network Coded Modulation, covers a range of WPNC techniques such as Noisy Network Coding, Compute and Forward, and Hierarchical Decode and Forward, and explains how WPNC can be applied to parametric fading channels, frequency selec- tive channels, and complex stochastic networks. This is essential reading whether you are a researcher, graduate student, or professional engineer. Jan Sykora is a professor in the Faculty of Electrical Engineering at the Czech Technical University in Prague, and a consultant for the communications industry in the fields of advanced coding and signal processing. Alister Burr is Professor of Communications in the Department of Electronic Engineer- ing at the University of York. 13:38:33
- 6. 13:38:33
- 7. Wireless Physical Layer Network Coding JAN SYKORA Czech Technical University in Prague ALISTER BURR University of York 13:38:33
- 8. University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107096110 DOI: 10.1017/9781316156162 c Cambridge University Press 2018 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2018 Printed in the United Kingdom by Clays, St Ives plc A catalogue record for this publication is available from the British Library. ISBN 978-1-107-09611-0 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. 13:38:33
- 9. Preface About the Book The book addresses strategies and principles of physical layer coding and signal pro- cessing that fully respect and utilize knowledge of the structure of a wireless network. This technique substantially increases the overall network throughput, efficiency, and reliability. Wireless Physical Layer Network Coding (WPNC) (a.k.a. Physical Layer Network Coding (PLNC)) is a general framework for physical (PHY) layer coding and processing strategies in which PHY behavior at a given node depends on its position in the network topology, and the signal-level processing/decoding exploits multiple paths between source and destination. We introduce the concept of Network Coded Modulation (NCM) as a network-structure-aware signal space code, which processes a (hierarchical) joint function of source data. At intermediate nodes NCM utilizes hier- archical decoding, and it is also designed to allow unambiguous decoding at the final destination using multiple hierarchical observations, arriving via different routes. The book addresses the fundamental principles of WPNC in the context of network informa- tion theory, and provides a comprehensive classification of the strategies. It also covers advanced design and techniques, including particular coding and processing designs and their respective properties. We also address selected hot research topics and open problems. Motivation for the Book It is becoming widely accepted that the most significant future developments in the physical layer of wireless communication systems will not take place in the PHY layer of individual communication links, but rather in the context of complete wireless net- works, especially as the density of wireless networks continues to increase. Over the past decade or so there have been significant developments in network information theory; these have shown that very significant overall performance gains are available compared with the conventional paradigm in which PHY techniques are applied to individual links only, leaving network aspects to be dealt with only at higher layers of the protocol stack. One such new research field is network coding, in which coding techniques are applied to multiple data streams at intermediate nodes in a network, rather than only to individ- ual streams on single links. This can exploit network topology to significantly improve 01 20:04:08
- 10. xii Preface throughput in multi-user networks. However, in its original form it operates at the level of data streams, rather than signal waveforms, and hence is not well suited to the inher- ently broadcast nature of wireless networks. Wireless physical layer network coding (WPNC) allows it to be applied directly to wireless networks, with a further significant improvement in efficiency. The key advance on conventional PHY techniques is that the nodes are aware of the network topology and their place within it, and both signal- ing waveforms and node signal processing exploit this knowledge to improve overall network throughput. Book Scope and Organization The book is carefully balanced, being divided into several “layers” giving different depths of information for audiences with various levels of background knowledge. Part I gives a gentle introduction to the key concept with the explanation kept in acces- sible form. Part II presents fundamental principles in more detail, but still using a “big picture” global perspective. Part III addresses a mosaic of various particular design tech- niques and principles that can practically fulfill the general principles of Part II. The Appendix provides some background material for readers with a weaker background in communication, signal processing, and information theory. Throughout the book, we maintain a strong emphasis on the proper classification and structuring of the problems, techniques, and particular coding, processing, and decod- ing schemes under discussion. This will help readers to properly orient themselves in the complex landscape of the different individual approaches. In the currently available literature these frequently overlap, and suffer from rather “fuzzy” terminology. This may lead to incorrect comparisons due to the high complexity of the field and the ambigu- ity and inconsistency of the terminology. (Terminology also changes rapidly due to the rapid progress of the research community.) The book is not primarily intended as a university course textbook but rather as a ref- erence source for researchers, PhD students, and engineers who would like to understand the principles of WPNC in the context of other techniques or would like to start their own research work in this field. Therefore the book is a highly structured set of Parts– Chapters–Sections, which are intended, as far as possible, to be read in a self-contained manner. Jan Sykora and Alister Burr 01 20:04:08
- 11. Abbreviations 2WRC 2-Way Relay Channel AF Amplify and Forward AWGN Additive White Gaussian Noise BC Broadcast Channel BPSK Binary Phase Shift Keying CF Compute and Forward CpsF Compress and Forward CRLB Cramer–Rao Lower Bound CSE Channel State Estimation DF Decode and Forward DFT Discrete Fourier Transform GF Galois Field H- Hierarchical H-BC Hierarchical BC H-constellation Hierarchical Constellation H-decoding Hierarchical Decoding HDF Hierarchical Decode and Forward HI Hierarchical Information H-Ifc Hierarchical Interference H-MAC Hierarchical MAC HNC map Hierarchical Network Code map H-NTF Hierarchical Network Transfer Function H-NTM Hierarchical Network Transfer Matrix H-PEP Hierarchical Pairwise Error Probability H-SCFD Hierarchical Successive CF Decoding HSI Hierarchical Side-Information H-SODEM Hierarchical Soft-Output Demodulator Ifc Interference iff if and only if IH-codebook Isomorphic H-codebook IID Independent and Identically Distributed JDF Joint Decode and Forward LHS left-hand side MAC Multiple Access Channel MAP Maximum A posteriori Probability 13:38:34
- 12. Abbreviations xvii MIMO Multiple-Input Multiple-Output ML Maximum Likelihood MMSE Minimum Mean Square Error MPSK M-ary Phase Shift Keying MSE Mean Square Error NCM Network Coded Modulation NC Network Coding NC-JDF Network Coding over JDF NNC Noisy Network Coding OFDM Orthogonal Frequency Division Multiplexing PDF Probability Density Function PMF Probability Mass Function PSK Phase Shift Keying QAM Quadrature Amplitude Modulation QF Quantize and Forward QPSK Quadriphase Phase Shift Keying RHS right-hand side Rx Receiver SF Soft Forward SNR Signal-to-Noise Ratio SODEM Soft-Output Demodulator UMP Uniformly Most Powerful WPNC Wireless Physical Layer Network Coding w.r.t. with respect to Tx Transmitter WCC Wireless Cloud Coding XOR eXclusive OR operation 13:38:34
- 13. 13:38:34
- 14. Part I Motivation and Gentle Introduction 13:28:16
- 15. 13:28:16
- 16. 1 Introduction 1.1 Introduction Wireless networks are becoming more and more ubiquitous in the modern world, and more and more essential to today’s society. In 30 years they have progressed from the province of a tiny minority of the world’s population in only the most developed nations, to the point where there are very nearly as many wireless subscriptions as people in the world [24]. The services offered have extended from very limited speech services at the introduction of first-generation mobile systems in 1985, to broadband Internet access and full motion video today. Moreover, we are at the point where wireless networks will extend beyond connecting people (of whom there are a limited number), to connecting their devices – an effectively unlimited number. Some believe that there are already more devices than people connected to the Internet, and predictions that 50 billion or more devices will be connected by 2020 are circulating widely [60]. Of course, that is only the start. All this implies that the density of wireless networks will inevitably increase. To pro- vide telecommunication services to the human populations of our cities, at continually increasing data rates, will require increasing numbers of access points, for which back- haul will become an increasing problem, and require more widespread use of wireless backhaul. The devices will also form a network many times as dense as any current wireless networks, also likely to require connection to the core network. In both cases it is likely that the current point-to-multipoint architecture of wireless networks, exempli- fied by both cellular and WiFi systems, will be replaced by a multi-hop mesh network architecture. The concept of the mobile ad-hoc network (MANET), one of the best-established concepts in wireless mesh networking, has been in existence for many years [9], yet has not really fulfilled its predicted potential. There are very few wireless networks in use today that implement a truly multi-hop networking approach. There seems to be a barrier to the practical implementation of multi-hop wireless networking that will surely have to be overcome in order to implement the ultra-dense wireless networks that are likely to be required in the near future. Perhaps the most fundamental basis for such a barrier is that described by Gupta and Kumar in their well-known paper [20]. They show that for a conventional approach to wireless networking, in which transmissions from other nodes in the network are treated as interference, the total capacity of the network scales as the square root of the number 02 19:28:49
- 17. 4 Introduction of nodes – that is, the capacity per node decreases as the size of the network increases. Hence as networks become denser, and more hops are required, the capacity available to each terminal will decrease. This interference problem has become widely recognized as the most significant prob- lem limiting the performance of future wireless networks, including point-to-multipoint networks as well as multi-hop. Traditionally it has been mitigated by means of the cel- lular paradigm, which limits interference by ensuring that a certain re-use distance is respected. Increased density is accommodated by using smaller and smaller cells with greatly reduced transmit power, but this approach is now reaching its limit, both because of the large numbers of radio access points it requires and the resulting backhaul prob- lem, and because cell sizes are becoming comparable in size with buildings and other city features. All this suggests that it is time for a completely new paradigm in wireless networking, and a major objective of this book is to lay the foundations for such a paradigm, which we call the “Network-Aware Physical Layer.” 1.2 The “Network-Aware Physical Layer” Since the 1970s the design of communications networks has been based upon a lay- ered paradigm, in which network functions are divided between protocol layers, each assumed to be transparent to the ones above it. The original layered model, dating from the late 1970s, was of course the OSI seven-layer model [2], but recently the layers implicitly defined in the TCP-IP protocol suite [1] have been more influential. In either case, the lower layers – the network layer, the link layer, and the physical layer – are of most interest to us here, since they provide the most basic functions of a communication network, namely routing, multiple access and error control, and modulation and coding, respectively. Of these layers, the physical layer is the one that handles the signals which are actu- ally transmitted over the communication medium: in our case these are related to the electromagnetic fields that form the radio waves. In the traditional layered paradigm the physical layer receives a signal from the medium and converts it to a bit stream, which is then passed to the link layer. However, this has the fundamental disadvantage that infor- mation is lost in the process that might improve the performance of functions which are located in higher layers. For example, it is well known that error correction is less efficient when operating on a bit stream (corresponding to hard decision decoding) than when it has access to a soft decision metric, which is usually obtained from the signal. Moreover, it also means that signals from nodes other than the transmitter of inter- est must be treated as interference, which conveys no useful information but degrades the performance of the receiver in decoding the wanted signal. This arises because the traditional physical layer is assumed to operate on only one point-to-point link, which means signals on other links are interference (and vice versa). This is illustrated in Fig- ure 1.1. The figure illustrates a multi-hop network in which data can travel from source to destination via two routes. We focus on the link of interest marked: in the traditional 02 19:28:49
- 18. 1.2 The “Network-Aware Physical Layer” 5 Figure 1.1 Traditional physical layer in a network. paradigm the physical layer consists of the modulator at the transmitting node, the radio link between them, and the demodulator in the receiving node: that is, it relates to that link only, in isolation from the rest of the network. Thus a signal from another transmit- ter must be treated as interference (as shown), even though it carries information from the same original source, and could in principle be exploited to improve the reception of the data of interest. Because interference is deleterious, it must usually be avoided wherever possible in traditional networks. This means that each node must transmit as far as possible on a channel orthogonal to the channel assigned to every other node – typically in a dif- ferent time-slot or at a different frequency. This divides the resource available to the network and greatly reduces its efficiency. Again, information theory teaches us that greater capacity can often be achieved when multiple sources are allowed to transmit at the same time in non-orthogonal channels: for example, the capacity region of the mul- tiple access channel (MAC) is achieved when the sources transmit simultaneously in the same channel, and is greater than the rate achieved by time-sharing of the channel. The “network-aware” physical layer, on the other hand, does not need to nominate one node as transmitter of interest and hence treat all other signals but this one as inter- ference. A network-aware receiver is aware – at the physical layer – of its location in the network, and what signals it may expect to receive in a given channel or time-slot. It is therefore able to determine what processing to apply to the composite signal formed by the mixture of all these signals. Similarly a network-aware transmitter is aware what effect its transmitted signals will have on other receivers, and can tailor the transmission in such a way that the received combination can also be processed as required. Simply, if multiple interacting signals are unavoidable (e.g. due to the physical density of the network), it is better to make them useful one to each other as much as possible, instead of avoiding them. We do that directly on the signal level by properly constructing the transmitted coded signals and properly processing and decoding the received signals. This allows multiple nodes to transmit on the same channel, and avoids the division of resources. A receiver may even benefit from receiving combined signals rather than sep- arate signals. It means that fewer signals have to be treated as deleterious interference, and any that do are typically weaker signals that have little effect. This paradigm is not entirely novel: some functions which might be regarded as belonging to the link layer have already been implemented in the physical layer. One example is multiple access, which in computer networks is commonly implemented at the link layer by using protocols such as ALOHA or CSMA (Carrier Sense Multiple 02 19:28:49
- 19. 6 Introduction Figure 1.2 A simple cooperative communication system. Access), or else is scheduled by using time-division or frequency-division multiple access (TDMA or FDMA). However code-division multiple access (CDMA), widely used in third-generation (3G) mobile systems, uses channels (corresponding to spread- ing codes) that are typically not fully orthogonal, and hence requires processing of the received mixed signal, which must be carried out at the physical layer, to separate the data. Similarly error control: while forward error correction (FEC) coding is conven- tionally regarded as part of the physical layer, retransmission protocols such as ARQ (Automatic Repeat reQuest) have traditionally been implemented at the link layer. How- ever, recently hybrid FEC/ARQ schemes have somewhat blurred this distinction, since they require combining of signals transmitted in the course of multiple retransmissions. Until recently, however, the functions of the network layer, specifically routing, have been excluded from the physical layer. This began to change about a decade ago with the introduction of cooperative communications [32]. Cooperative systems involve at least one relay node as well as the source and destination nodes (Figure 1.2), to assist the transmission of the source’s data. Typically it receives the source signal in one time-slot, and retransmits it in some form in a subsequent slot. In most cases the processing within the relay is entirely at the physical layer, and frequently it is the original signal or some function of it that is retransmitted, without being converted to bits first. This is perhaps the simplest example of the physical layer being extended over a network involving multiple hops, beyond the simple link between one transmitter and one receiver. This is, however, a very rudimentary version of routing. In this book we consider a much more general scenario involving multiple sources and multiple destinations, and multi-hop relaying between them. Thus routing is an essential element. The approach we will use, however, differs from routing in the conventional layered paradigm in two respects. The first is that it resembles cooperative communications in that processing within the relay takes place at the physical layer, involving signals directly. Unlike a bridge or a router in a conventional network, the relay does not decode the source data and transfer it to the link or network layer, but rather processes the received signals and forwards some function of them. The second is that what it forwards may not be a representation of data from a single source, but rather some function of data from several sources – a “mixture” of data from multiple sources to be separated at a later stage and delivered to the required destination. Thus it may no longer be possible to identify distinct routes for individual data streams, as is conventionally assumed. This latter aspect can also be applied at the network layer of a multi-hop network, and corresponds to a technique introduced at the beginning of this century, known as network coding, which we will now discuss. 02 19:28:49
- 20. 1.3 Network Coding at the Network Layer 7 1.3 Network Coding at the Network Layer Network layer network coding (NC) [5] addresses a network modeled as a directed graph connecting source nodes to destination nodes via a set of relaying nodes. In gen- eral there may be multiple sources and multiple destinations. The edges of the graph represent discrete links between pairs of nodes. This is clearly a good model of a data communications network with wired connections, such as the Internet, though we will see later that it does not represent a wireless network so well. For a unicast network, in which there is only one source and one destination, it can be proven that the maximum data flow rate is given by the max-flow, min-cut theorem [14]. However, Ahlswede et al. [5] showed that in the multicast case, where multiple destinations wish to receive the same data, the maximum flow rate cannot be achieved if relaying nodes operate simply as switches, connecting data flows on incoming links to outgoing links. Instead nodes should apply network coding, in which the symbols on an outgoing link are generated by some function of the symbols on two or more incoming links. This may be illustrated by the network shown in Figure 1.3, known as the butterfly network. The figure shows two versions of a network, in which two data sources each wish to send their data to both of two destinations, over a network in which all links have unit capacity. Figure 1.3a represents a conventional network in which the nodes can only switch a data stream from an incoming link onto an outgoing edge, or duplicate it and send on more than one outgoing edge. Thus the upper of the two relay nodes (which are marked as circles) can only select either stream A or stream B to send on its outgoing link (here it selects A). This is duplicated by the lower relay node, and hence the right-hand destination node can receive both streams, but the left-hand one receives only A. Figure 1.3b shows a network employing network coding. Here the upper relay node computes the exclusive OR (XOR) function (or modulo-2 sum) of the symbols in the data streams, and for- wards the result. The lower relay node duplicates this to both destinations, and they can each recover both streams, because one is directly available, and the other can be recon- structed by reversing the network coding function applied at the relay node with the aid of the directly available stream. Thus the left-hand destination can now reconstruct stream B by applying A ⊕ (A ⊕ B) = B. Figure 1.3 Butterfly network. 02 19:28:49
- 21. 8 Introduction Figure 1.4 Linear network coding. We will revisit this network topology later, in a slightly different context, but of course this principle also applies to much more complex networks, including networks con- taining cycles. Also in this case very simple coding is applied at the relay – simply the bit-by-bit XOR – but in general more complex encoding is required. There exists a wide variety of forms of coding, but [27] showed that linear coding over the finite field F2m is effective: in fact [34] had already shown that linear coding can achieve the maximum flow in a multicast network. Figure 1.4 illustrates this coding applied to a node: the out- put symbol Y is given by the formula in the diagram, in two different notations. In the first form ⊗ and ⊕ represent multiplication and addition within F2m ; in the second this is simply represented as a summation. The symbols on the incoming links are symbols in F2m : they are drawn from an alphabet whose size is a power of 2, and can in fact be represented as length m binary strings. The coefficients Ai, i = 1 . . . n are also elements of F2m , and again can be represented as length m binary strings. The addition operation is in fact simple bit-by-bit modulo-2 addition, but multiplication is more complicated: it is usually defined using primitive element operations on finite field (see Section A.2.1 or [8]). It is clear that if all nodes apply a linear function of this sort, with symbols and coefficients from the same field, then the vector of output symbols across all relay nodes may be related to the vector of source symbols by a matrix. Equally clearly, for the destination nodes to reconstruct the source data this matrix must be full rank. We will revisit this model more rigorously later in the book. 1.4 Wireless Physical Layer Network Coding The network model implicit in the conception of network coding, as illustrated in Fig- ures 1.3 and 1.4, has one important deficiency as a representation of a wireless network. It assumes that the incoming links are discrete, and the symbols they carry are sepa- rately available to the network coding function in the node. This is a valid model of a wired network, but a wireless network does not have defined, discrete connections between nodes in the same way. Rather the electromagnetic fields due to signals trans- mitted simultaneously from two nodes will add together at the antenna of a receiving node, resulting in a superimposition of the two signals. Moreover they may be attenu- ated and/or phase shifted due to the wireless channel in largely unpredictable ways. In 02 19:28:49
- 22. 1.4 Wireless Physical Layer Network Coding 9 Figure 1.5 Network coded butterfly network with schedule. the classical paradigm they are subject to fading and cause mutual interference to one another. However, there are two approaches by which such discrete links can be emulated in a wireless network. The first is straightforward: separate orthogonal channels are provided for each link. In principle any means of orthogonalization could be used: differ- ent time-slots, different frequency channels, or different orthogonal bearer waveforms. For simplicity we will here assume that different time-slots are used: that the links are orthogonal in the time domain. Considering the network coded butterfly network in Fig- ure 1.3b, this would require four time-slots per pair of source symbols to deliver the data to both destinations, as shown in Figure 1.5. This clearly reduces the efficiency of the network. This also illustrates a general point about wireless networks that will be important in this book. Wireless devices are typically subject to the half-duplex constraint: that is, they cannot transmit and receive simultaneously on the same channel or in the same time-slot. There has been recent work on the implementation of full duplex wire- less nodes, but that is beyond the scope of this book, in which for the most part we will assume the half-duplex constraint must be respected. This constraint immediately implies that a relay node can transmit in at most half of the time-slots. As mentioned previously, information theory shows that transmission on orthogonal channels is not the optimum way of signaling from multiple source nodes to a single destination or relay node. In information theoretic terms this is known as the multiple access channel (MAC). The capacity of a MAC is defined by its rate region, as illus- trated in Figure 1.6, for a two-user MAC. The left of the diagram illustrates the scenario: two sources, S1 and S2, transmit at rates R1 and R2 respectively to a common destination. The region within the solid line in the graph on the right denotes the rate region: the set of rate pairs that can be achieved with low error rate. Note that it implies that three limits operate: a limit on the rates R1 and R2 that each source can transmit independently plus a limit on the sum rate R1 + R2. Note, however, that a conventional system using TDMA (i.e. using orthogonal time- slots) would be restricted to the triangular region shown by the dashed line — since any increase in the rate from one source would always have to be exactly balanced by 02 19:28:49
- 23. 10 Introduction Figure 1.6 Rate region for two-user MAC. a reduction in the rate from the other. The system can do better than time-sharing by allowing both sources to transmit simultaneously, and at the receiver to first decode one, then cancel the interference it causes and decode the other. This allows an increase in the sum rate significantly above the time-sharing rate. Thus in the network coded butterfly network we could allow sources A and B to transmit simultaneously, merging time slots 1 and 2 in the schedule shown in Figure 1.5, and increasing the network throughput. However, this still constitutes a bottleneck in the network, because it requires symbols from both sources to be decoded even though what is required is only the one symbol formed by combining them with the network code function. Taking this into account, it is possible (as we will see later) to establish what we will call the WPNC region, which is the set of source rates which allows this symbol to be decoded. This is shown by the dash-dotted lines in Figure 1.6, and allows rates outside the conventional two-user MAC region. It is achievable e.g. by the use of nested lattice codes, as will be discussed in Chapter 5. To achieve a rate outside the MAC region requires that rather than being obtained by decoding the two sources separately, and then applying network coding at the network layer (a strategy we will call joint decoding), the network coded symbol must be decoded directly from the received signal at the physical layer – in other words by physical layer network coding (PLNC). In this book we refer to the technique as wireless physical layer network coding (WPNC), and it is the main topic of the book. The term “wireless” is used here because the inherent superposition of wireless signals mentioned above means that this form of network coding is essential in wireless systems to obtain all the information available. There will of course be much more detail to come, and in particular there will be a “gentle” introduction to the main principles in the next chapter, so here we will restrict ourselves to a very simple example of how this might work and how it can enhance capacity. Figure 1.7 shows the scenario. Two terminals transmit uncoded BPSK, taking signal values ±1 over channels with the same attenuation and phase shift to a relay. We assume that the relay applies network coding using the XOR function. At the relay the signals add, resulting in the values ±2 and 0. A joint detection strategy would need to decode the two sources separately, and this is clearly not possible if the value 0 is received, since it might represent the data 01 or 10. WPNC, on the other hand, has only to detect which network coded symbol the received signal corresponds to. This avoids the problem, since 01 and 10 both correspond to the network coded symbol 1. Thus the received signal can be interpreted as a constellation in which both the signals marked with white circles 02 19:28:49
- 24. 1.5 Historical Perspective 11 Figure 1.7 Illustration of PNC operation. correspond to (network coded) 0, while the black circle corresponds to 1. This clearly increases capacity compared to both the joint decoding approach and the network coding approach. 1.5 Historical Perspective At this point we will take a break from the technical details of WPNC to discuss how we reached this point, and the initial development of WPNC up to the present. We have already discussed some of the information theoretic background, and have mentioned the development of network coding. It is worth noting, however, that many of the theo- retical foundations of multi-user information theory were laid in the 1970s – including analysis of the multiple access channel [4], [35], of the broadcast channel [11], and of the relay channel [12]. However, there has been little practical implementation of these concepts even up to today, although that is now changing, notably because of the pressures on wireless networks noted above, and also because multiple antenna systems have important synergies with the MAC and broadcast channels, which have led to the introduction of multi-user MIMO (MU-MIMO) systems in fourth-generation wireless systems. Multi-user information theory can now be seen as an important step towards the development of network information theory in the past decade or so, extending these concepts beyond single-hop multi-user networks. Both network coding and WPNC occupy the field of network information theory, and many concepts from it underlie the work in this book. WPNC itself was discovered independently by three research groups, who approached it from slightly different angles, resulting in distinct approaches that, how- ever, are clearly based on the same principles. Zhang, Liew, and Lam [64], of the Chinese University of Hong Kong, were probably motivated by concepts from network coding. They introduced the application of WPNC to the two-way relay channel, which we will review in the next chapter but which is quite similar to the butterfly network we have already seen. They also generalized it to a multi-hop chain network. Popovski and colleagues at the University of Aalborg introduced an analog version of WPNC at the same time [49], based on earlier work applying network coding to the two-way relay channel [33]. They subsequently extended this to a scheme they refer 02 19:28:49
- 25. 12 Introduction to as denoise and forward [28]. Around the same time other work, e.g. [50], discussed other strategies for the two-way relay channel (though without proposing the full WPNC concept), and this was also the emphasis of the work by Popovski et al. The third group was Nazer and Gastpar, then both at University of California Berke- ley. Their earliest published work dates from 2005 [43], and was framed more as an approach to a new information-theoretic problem: that of decoding functions of symbols from multiple sources, rather than the sources themselves. However, this is evidently directly relevant to WPNC if the functions are those required in network coding, and leads to an approach called compute and forward. Their subsequent work and the work of other workers inspired by it has moved into the area of lattice coding, as a useful basis for the functions, and has retained a strong algebraic flavor. Lattice coding is itself a field with a long history. It is based on the mathematical the- ory of lattice constructions, especially in more than three dimensions, but is connected with group theory as well as the physics and chemistry of crystals, going back to the middle of the nineteenth century. Its application to coding theory was extensively dis- cussed in the 1980s in the classic reference on the topic, [10]. However, more recently it has undergone something of a renaissance, especially since it has been demonstrated that lattice codes with lattice decoding can also approach the Shannon capacity [16]. The work of Nazer and Gastpar [45] also used it to establish achievable regions for compute and forward. Since this fundamental work the field has remained very active. Much of the early work continued to focus on the two-way relay channel, but recently this has been extended to other topologies, such as the multi-way relay channel, multiple relay net- works, and multi-hop networks. Early work also focussed on information theoretic aspects, with little attention to practical implementation, but more recently more prac- tical aspects have been investigated, such as the use of practical coding schemes, synchronization, performance on realistic wireless channels, etc. Recently also prac- tical test-beds for the concept have been implemented [3, 38]. Of course, much of this work will feature in the remainder of this book. 1.6 Practical Usage Scenarios We have already described the developments in wireless communications that provide the practical drivers for the move toward the network-aware physical layer in general, and the implementation of WPNC in particular. Here we will look in a little more detail at some specific scenarios in which it might be applied. The drivers we have considered include both conventional wireless broadband services via cellular and WiFi networks, and machine-type communications, including the “Internet of Things.” However, these two different application areas may give rise to different network topologies, so we will discuss them separately here. As mentioned above, access networks for cellular mobile networks are becoming denser in order to support rapidly increasing capacity density requirements arising from both increasing numbers of users and increasing data rate demand per user. To mitigate 02 19:28:49
- 26. 1.6 Practical Usage Scenarios 13 the interference problems this causes, the concept of network MIMO or cooperative multipoint (CoMP) has been introduced. In this approach several base stations cooper- ate to serve a user terminal, instead of each one being served by a single base station, so that signals received by another base station must be treated as interference. The net- work then exploits signals that would otherwise be interference, which can then enhance performance rather than degrading it. However, this requires that signals are no longer decoded only in one base station, and also implies that digitized signals rather than only user data should be transmitted between base stations and the core network. More recently the cloud radio access network (C-RAN) concept has been introduced, in which base station sites, containing baseband signal processing and higher-layer networking functions, are replaced by remote radio units (RRU) containing only the antennas, RF processing, and signal sampling and digitization. Baseband processing and all higher- layer functions for a large number of these RRUs are then concentrated in centralized baseband units (BBU). This clearly enables base station cooperation of the sort required by network MIMO to be more readily implemented. The connection between the RRU and the BBU is then known as fronthaul rather than backhaul, because it carries signal information rather than user data. The concept is illustrated in Figure 1.8. The major disadvantage of C-RAN is that the capacity required for the fronthaul is typically many times the total user data being transmitted, since it is a digitized signal rather than the actual data, and therefore typically requires longer sample words than the num- ber of information bits per symbol to represent the signal at sufficient precision. It has therefore usually been assumed that optical fiber would need to be used to provide fron- thaul connections (as opposed to wireless), which would greatly increase the cost of the network. WPNC provides a potential alternative, which greatly reduces fronthaul load, poten- tially allowing it to be implemented over wireless. As in the example illustrated in Figure 1.7 above, a base station receiving signals simultaneously from two terminals might decode a network coded function of the two, rather than attempting to decode one in the presence of interference from the other. Thus it exploits all signals received from a terminal just as network MIMO does, and it achieves a performance that is similar in the sense that it provides the same diversity order, albeit typically with a small degradation in terms of required signal to noise ratio. However, because the network coded signal in principle contains the same number of symbols as each of the user data streams, it Figure 1.8 Cloud Radio Access Network. 02 19:28:49
- 27. 14 Introduction Figure 1.9 Mesh network for “Internet of Things” applications. requires no expansion of the fronthaul load compared to the total user data rate. This might well allow wireless links to be used, with the potential to reduce network costs. Machine-type communications, on the other hand, are likely to call for a different network structure. Potential applications include sensor networks, industrial process control, “smart grid” and “smart city” systems, to name just a few. These have in com- mon that they are likely to involve very large numbers of devices, widely distributed across a service area, with very low power. This may mean that it is not feasible to provide a dense enough access network to serve all these devices directly, so these applications are likely to lead to a mesh network topology based on device-to-device communications and low-cost relay nodes to provide links back to the core network, as illustrated in Figure 1.9. In many cases the data rate per device is relatively small and occurs in the form of small packets, but there are large numbers of devices and large numbers of packets in total. In addition many applications are highly time-critical and require very low latency. We have already reviewed the limitations of multi-hop mesh network topologies when the conventional network paradigm is used, especially the capacity bottleneck that results from interference between links, and this will clearly apply in many of these applications. Moreover, the conventional paradigm tends to result in packet colli- sions, requiring retransmission at intermediate hops that potentially increases end-to-end delay. Thus WPNC is very well suited to these applications, since its exploitation of otherwise interfering signals has the potential to overcome the capacity bottleneck in multi-hop networks. Similarly it can exploit colliding packets to extract information that can be further forwarded through a network, minimizing the need for retransmissions. Both of these application areas are examples of the current developments in wire- less communications towards ultra-dense networks, in which it is no longer feasible to avoid interference between different links within the same network. The paradigm of the “network-aware physical layer,” which we have introduced in this chapter, and will explore in the remainder of this book, is therefore extremely timely. 02 19:28:49
- 28. 2 Wireless Physical Layer Network Coding: a Gentle Introduction 2.1 The 2-Way Relay Channel In this chapter we begin to describe the principles of WPNC, taking a more “gentle” approach than we do in the remainder of the book, minimizing the use of mathematics in favor of more descriptive and graphical approaches as a means to explain these prin- ciples. We will see that the simple example described in Section 1.4 already captures some of the important issues, but we will begin the process of generalizing it and setting it in the context of a complete network, albeit a very simple one. Accordingly we focus on the 2-way relay channel (2WRC)1 as a very simple example of a complete network (in fact the simplest possible, as we will see in a moment) in which WPNC can be applied. The 2WRC is illustrated in Figure 2.1. The basic idea is that two terminals each have data to exchange with the other, but (perhaps because the distance between them is too great for a direct link) they wish to use an intermediate relay node for the purpose. The reason for focussing on the 2WRC is that it provides a simple example of a multi- hop wireless network supporting multiple data flows, as well as being an example that demonstrates the benefits of WPNC particularly clearly and one that is of some practical interest. In fact, as mentioned in Section 1.5, a large proportion of the work in the field in the past decade has exclusively addressed this network. We emphasize here, following on from Section 1.2, that WPNC applies to wireless networks, not to individual point-to-point links – this is the essence of the “network- aware physical layer.” Such networks must necessarily involve more than one wireless “hop” between transmitter and receiver, and hence must include a relay node as well as source and destination terminal nodes. They must also necessarily involve more than one data source, leading to multiple data flows through the network that also interact at some point within it. On this basis the 2WRC, containing two terminal nodes and one relay and involving two flows each originating at one of the terminals, is in fact the simplest possible example. We will begin by comparing the WPNC approach to the 2WRC with two previous approaches: the conventional one and one based on network coding at the network layer, showing the potential benefits of WPNC over both of these. We will then describe and compare some alternative schemes which can all in some sense be labeled as WPNC. 1 Sometimes, it is also abbreviated as TWRC (Two-Way Relay Channel). 03 19:31:33
- 29. 16 Wireless Physical Layer Network Coding: a Gentle Introduction Figure 2.1 2-way relay channel. This will lead us to one of the common requirements of these schemes: the need for unambiguous decodability, that is, that the original data can be recovered at the destina- tion without ambiguity and therefore with certainty. We will also introduce the concept of hierarchical side information, and describe its role in unambiguous decoding. Up to this point we will assume that BPSK modulation is used, as in our example in Section 1.4, but we will next extend our consideration to QPSK modulation. As we will then see, this introduces additional problems that do not arise with BPSK as a result of the unpredictable parameters of the channel – primarily the effect of fading. This causes phase shifts and amplitude variations in the signal that in general are unknown to the transmitter. It is in particular the relative values of these parameters between the channels from the two sources and the relay that influence the behavior of the network. Finally we will extend our consideration to other example network topologies, and in particular to what we refer to as the hierarchical wireless network, where a set of source nodes are connected to the destination via one or more layers of relays. We will see how similar considerations apply in such networks as in the 2WRC. Note that in this chapter, for simplicity in explaining the basic principles of WPNC, we assume uncoded transmission in most of the text (with the exception of Section 2.5). In later chapters an important theme will be how forward error correction (FEC) coding can be incorporated into the scheme. 2.2 Conventional, Network-Layer Network Coding, and WPNC Approaches The 2WRC can be operated in a variety of modes, involving different schedules for the activation of the nodes that comprise it. These are illustrated in Figure 2.2. The conventional approach using a point-to-point physical layer would require four time- slots, or phases, for a complete cycle of transmissions. First terminal A transmits to the relay R, then R retransmits A’s data to terminal B. Next B transmits to R, and R retransmits B’s data to A. In the conventional paradigm none of these phases can take place concurrently, either because the transmissions would then interfere at the relay, or because of the half-duplex constraint on the relay. In the network-layer network coding (NC) approach, illustrated in Figure 2.2b, the relay is no longer restricted to simply forwarding data it has received. Instead it cal- culates a function of the data of both A and B, which we refer to as the network code function or mapping. In our present example, because the data are binary, the func- tion is the exclusive OR (XOR) function, but in the general case a wide range of other options are possible, as we will see. This then allows a three-phase schedule, as shown in the figure. Terminal A transmits its data to the relay in the first phase, then terminal B transmits its data in the second phase. The relay then forms the function A ⊕ B, and transmits this simultaneously to terminals A and B in the third phase. This procedure 03 19:31:33
- 30. 2.2 Conventional, Network-Layer Network Coding, and WPNC Approaches 17 Figure 2.2 Activation schedules for the 2-way relay channel: (a) conventional, four-phase; (b) network-layer NC, three-phase; (c) WPNC, two-phase. works because each terminal has available the data it originally transmitted, and can decode the data from the other terminal by applying a second XOR function, as we will see in Section 2.4 below. We will refer to information like data B in this case, which assists a terminal in recovering its data of interest even though it does not itself depend on that data, as hierarchical side-information (HSI). The rationale for this terminol- ogy will be explained in Chapter 3. Of course terminal A can perform an equivalent process. We may note that in terms of data flows the 2WRC is equivalent to the “butterfly network” discussed in Section 1.3 above, illustrated in Figure 1.3b. Here the upper of the two nodes in the center of the diagram represents the application of the XOR function to the incoming data, while the lower represents the broadcast of the network coded (i.e. XORed) data. The links directly joining sources and destinations represent the HSI which the source in each terminal makes available to the network decoding function, carried out in the nodes at the bottom of the diagram. This diagram has the advantage of making the transfer of the HSI explicit. Note that the 2WRC is equivalent to a butterfly network in which the HSI transfer is perfect, because the source and destination reside in the same terminal. Later in the book we will consider another example in which these links may not be perfect, because source and destination may be separated. This clearly has implications for the operation of the network, as we will see. We noted in Section 1.3 when considering the application of NC (at the network layer) to the butterfly network that the NC model effectively assumes that the data flows 03 19:31:33
- 31. 18 Wireless Physical Layer Network Coding: a Gentle Introduction from the two sources arrive over discrete links, which we noted was not naturally the case in wireless networks. However, the schedule shown in Figure 2.2b overcomes this by separating the two links in two time-slots, in other words by applying time-division multiple access (TDMA) over the wireless medium to provide orthogonal channels for the links. The data on these links can then be decoded separately before the network code function is applied. For this reason it must be treated as a form of network-layer NC, rather than WPNC. The approach shown in Figure 2.2c, however, reduces the schedule to two phases. Now terminals A and B transmit simultaneously in the same time-slot (and in the same frequency channel). Thus their signals arrive at the relay as a superposition of the electromagnetic waves of the two wireless signals, so that the signals are no longer readily separable at the relay, and so it will not be easy (unless using coded signals and multi-user decoding which, however, imposes some limitations on the rates as will be described later) to decode their data separately. However, the relay does not neces- sarily need to do so: all it requires to do is to extract the network code function from the superposed received signal. Since the output of the function has less entropy (that is, contains less information) than the combined information of the original data sequences, in general this may be an easier process than separate decoding. This question will be addressed much more rigorously in later chapters of this book. However, the very simple example of WPNC that we gave in Section 1.4 shows how in some circumstances it may be impossible to regenerate the original data sequences but still readily possible to obtain the network coded data. The example is illustrated in Figure 1.7, where it is assumed that both sources transmit BPSK to the relay over channels that happen to have the same phase shift and attenuation. Thus the signals combine to give a constellation with three signal points rather than four, which we have labelled −2, 0 and 2. Note that −2 and +2 correspond to the case where the two sources transmit (0,0) and (1, 1), respectively, while 0 occurs with either (0, 1) or (1, 0). Hence if this point is received at the relay it cannot with certainty decide which of these two pairs of data symbols was received. However, since these two pairs both result in the same network coded symbol, namely 1 (since 1 ⊕ 0 = 0, 0 ⊕ 1 = 1), it is able to decode this symbol with certainty. And of course if either −2 or +2 is received, this will be decoded as network coded 0, since 1 ⊕ 1 = 1, 0 ⊕ 0 = 0. (Note that while it is very unlikely that the two channels will be exactly the same, as required by this example, nevertheless if they are close, so that the pairs (0, 1) and (1, 0) produce very similar signals, in the presence of noise it will still be very difficult to distinguish them, but remain easy to obtain the network coded symbol.) Note, however, that this direct extraction of the network code function must neces- sarily take place at the physical layer, since the information must be obtained from the received signal, which is only available at the physical layer. It cannot in general be separated into decoding of source data symbols followed by network coding applied at the network layer. However, it must be a physical layer that is aware of the nature of the superposed signals it will receive: both their statistical characteristics (especially the combined constellation they may form) and their significance as a representation of 03 19:31:33
- 32. 2.3 WPNC Relay Strategies 19 different combinations of source data. In this sense the physical layer must be “network aware,” as discussed in Section 1.2. The example discussed above provides only one of several ways of processing the received signal and retransmitting some function of it. In the next section we com- pare it with some alternative strategies. In the remainder of the chapter (and indeed the remainder of the book) we will for the most part focus on the two-phase protocol of Figure 2.2. We will often refer to the first phase (sources to relay) as the multiple access channel (MAC) phase, and the second (relay to destinations) as the broadcast chan- nel (BC) phase, because the phases involve many-to-one and one-to-many transmission, respectively, like the corresponding channels. 2.3 WPNC Relay Strategies Here we consider the case of WPNC as applied to the 2WRC (that is, where a two- phase schedule is applied, as illustrated in Figure 2.2c), and especially some alternative strategies available to the relay. The fundamental requirement that the relay must fulfill is to transmit some function of the two data symbols which is such that the required data can be unambiguously decoded at the destination, given the appropriate HSI. In the next section we will consider in more detail the requirements placed on the relay function by this unambiguous decodability criterion, but here we will consider some simple functions and strategies to obtain them. The simplest such strategy is for the relay to directly store the received signal, amplify it and retransmit it. This is known as amplify and forward (AF). The destination in each of the two terminals can recover the required data, assuming that its own data and information about the channels between both terminals and the relay are available to it, by subtracting the interference at the relay due to those data. The disadvantage of AF is that the noise at the relay receiver is also amplified, and adds to the noise on the relay– destination link. However, provided both channels and data are perfectly known, the effect of the second signal at the relay can be completely eliminated. In terms of the rate region illustrated in Figure 2.3, this means that the rate region is rectangular, since the data flow from one source to destination is completely unaffected by the flow from the other. Once the interference has been removed, the end-to-end link can be represented by a single equivalent channel whose noise is given by the sum of the noise at the final destination and the noise at the relay amplified and transmitted over the relay– destination link. Therefore the capacity of each user and hence the size of the region is reduced because noise is greater than on either of the channels on their own. The rate region is shown by the solid line in Figure 2.3. Note that the regions shown in this diagram are intended to be illustrative only, not exact results for any specific channel. The second strategy is to apply multiple access techniques at the relay to first decode each source separately, then apply the network code function to the decoded symbols, and broadcast the resulting network coded symbol to both destinations. As previously mentioned, the classical way to do this is to first decode the lower-rate source, which is able to use a more powerful error correction code, estimate the interference this causes and subtract it, so that the higher-rate source is able to decode as if it were operating on 03 19:31:33
- 33. 20 Wireless Physical Layer Network Coding: a Gentle Introduction Figure 2.3 Rate regions for 2WRC: AF, JDF, HDF, and BC. an interference-free channel. The network code function which then operates on the two decoded symbols must be chosen in such a way that it can be decoded at the destination given its own source data. This will be discussed in the next section, but we will note here that an advantage of this joint decode and forward (JDF) strategy, if it turns out to be possible, is that we are free to choose any network code function that fulfills this requirement. It also has the advantage compared with AF that each node decodes in the presence only of its own noise: we do not encounter the cascade combination of noise from successive links that occurs in AF. The rate region now has to be evaluated for the two phases of the network’s schedule separately, whereupon the overall rate is the smaller of the two, since the phase with lower capacity will act as a bottleneck. Here the rate region of the MAC phase is just the expected rate region for a MAC (as shown in Figure 2.3, where the dashed line is the JDF rate region, and also previously discussed in Figure 1.6), because as for the MAC both sources have to be decoded at the relay. The individual rate limits for R1 and R2 that bound the rate region arise from the cancellation of interference due to the other flow, and thus are simply the capacity bound for the corresponding point-to-point link. In the broadcast phase also the two links each function like a point-to-point link, in which the rate of one does not affect the other, and so the rate region is rectangular. Figure 2.3 addresses the case where the channels between the two terminals and the relay are balanced in terms of propagation, and hence the rates for the two users for the broadcast channel are the same as the individual rate limits for the MAC channel. Hence in this case the MAC rate region lies within the BC rate region, and hence in this and many other cases it is the MAC phase that gives rise to a bottleneck and defines the overall rate region for the network. Moreover, because the MAC rate region is pentagonal rather than rectangular, it may also mean that the corner point of the AF region extends outside it, as shown in Figure 2.3, so that it is possible to achieve higher rates for the two users simultaneously by using AF than JDF, although its individual rate limits are lower than in JDF. The third strategy is the one we have already described above, and illustrated in the previous chapter. The relay decodes the network coded function directly from the received signal. Thus it does not necessarily need to decode the two source symbols separately, but only determine which value the network coded function should take. This in general is an easier decoding task than the joint decoding described above, because the function is a many-to-one function, and it requires fewer values to be distinguished. For example, in the case discussed in Section 2.2 above, and illustrated in Figure 1.7, 03 19:31:33
- 34. 2.3 WPNC Relay Strategies 21 Table 2.1 Summary of multi-source/node PHY techniques – classical single-user point-to-point (P2P), classical multi-user (MU), network-level NC, and native WPNC. P2P PHY MU PHY NC WPNC Topology: direct neighbors signal interaction − + − + Topology: full network structure − − + + Signal structure: constellation (signal) space level + + − + Relay Tx signal codeword map: a function of data − − + + Relay Rx signal codeword map: a function of data − − − + the received signal could in principle take four values, but the network code function (the XOR function) takes only two values. Thus the decoder at the relay needs only dis- tinguish between two pairs of signals. In the example given in Figure 1.7 one of these pairs contains two fully coincident points, and so, as already mentioned, it would be impossible to decode both sources separately,2 but nevertheless the network code can be decoded. We refer to the sets of points from the full received constellation that corre- spond to the same network code value as clusters: in the example illustrated, the distance between the nearest points in the two clusters is in fact the same as it would be in the received constellation from a single source without interference but, in general, if points do not coincide the inter-cluster distance will be smaller than in the interference-free constellation. For this reason the limitations on the rates of the individual sources are a little lower, and hence the rate region is smaller than the BC region, although again, because it is rectangular, its corner may project beyond the MAC rate region, as shown in Figure 2.3. It is, however, larger than for AF, because the noise is smaller. We refer to this strategy as hierarchical decode and forward (HDF), because what is decoded is a hierarchical function of the source symbols, although in this very simple example the hierarchy contains only a single level (see Chapter 3, which explains the hierarchical principle in detail). Similarly the constellation of the received signal is a hierarchical constellation, consisting of a hierarchy of clusters and points. In terms of relative performance, Figure 2.3 shows the comparison between the three approaches we have discussed. As mentioned, the figure is of course only illustrative, and the exact numerical comparison depends on the details of the channels and signal-to- noise ratios involved. However, it is clear that at least potentially HDF can outperform the other schemes in terms of overall sum rate, even if JDF can achieve a higher rate for the individual sources. In terms of complexity, AF is undoubtedly the simplest to implement, especially at the relay, since very little baseband processing except storage is required. In principle JDF may require a maximum likelihood (ML) or maximum a posteriori probability (MAP) detector, with complexity proportional to the received constellation size, and therefore exponential with the rate. The implementation of HDF, and conditions under which it may be simplified, will be an important theme of this book. Table 2.1 shows a summary of processing aspects for various classes of PHY techniques used in multi-node and multi-source networks. 2 For simplicity we refer here to uncoded transmission. In coded systems the codebook structure might help to distinguish these points. 03 19:31:33
- 35. 22 Wireless Physical Layer Network Coding: a Gentle Introduction 2.4 Unambiguous Decoding and Hierarchical Side-Information If the relay transmits a function of the source symbols back to the terminals (rather than the symbols themselves), it is clearly essential that the terminals are able to recover the original data symbols that are of interest to them: in other words to decode the network code function applied at the relay. More formally, we say that the symbol received at the destination must allow unambiguous decoding of the source of interest. Unambiguous decoding is possible provided the combination of network coded symbols received at a given destination corresponds only to one possible symbol from the source of inter- est. Otherwise an ambiguity remains about the source symbol after the network coded symbol has been received, and information is lost. As we will see, however, the destina- tion terminals require additional information to allow them to decode; we have already referred to this as hierarchical side-information (HSI). We must ensure that unambigu- ous decoding is possible when the HSI and the network coded symbol, which we call hierarchical information (HI), are both available at the destination. In our example using the 2WRC unambiguous decoding is very easy to achieve. As we have seen, the relay obtains the XOR function A ⊕ B of the two source data symbols, and forwards it to both destinations, where it provides HI about the source symbol of interest. In this case the destinations also have as HSI the data symbol transmitted in the previous time-slot by the source collocated in the same terminal. This does not itself contain any information about the source symbol of interest (that from the other terminal), but it does help to decode that symbol. For example termi- nal B combines the data A ⊕ B received from the relay with its own data, forming (A ⊕ B) ⊕ B = A ⊕ (B ⊕ B) = A ⊕ 0 = A, and thus recovers the data A that it requires. To generalize this somewhat, let us suppose that the data symbols from the two sources, which we will denote as bA and bB, are drawn from an alphabet A of size M (we say that they have cardinality M). The network code or mapping function applied at the relay is denoted as χ (bA, bB). In order unambiguously to decode data symbol bA at terminal B we require that the combination of the network coded symbol χ (bA, bB) and the source symbol bB should uniquely define the symbol bA from source A, for all possible bA and bB. This requires that the combination is different if bA is different, that is, that {χ (bA, bB) , bB} = χ b A, bB , bB , ∀bB, bA, b A = bA (2.1) or, more simply, χ (bA, bB) = χ b A, bB , ∀bB, bA, b A = bA. (2.2) This is commonly called the exclusive law. Conversely, for unambiguous decoding of bB at terminal A we require χ (bA, bB) = χ bA, b B , ∀bA, bB, b B = bB. (2.3) Note that this form of the requirement for unambiguous decoding applies specifically to the 2WRC: for other topologies it should be modified, as we will see in Section 2.7 of this chapter. 03 19:31:33
- 36. 2.4 Unambiguous Decoding and Hierarchical Side-Information 23 Table 2.2 Table to define mapping function. bB bA 0 1 . . . M − 1 0 0 1 . . . M − 1 1 M − 1 0 . . . M − 2 . . . . . . ... . . . M − 1 1 2 . . . 0 This requirement in its turn imposes requirements on the mapping function. These requirements can be expressed in various ways, just as the mapping function can be defined in different ways. A general way to define the mapping, at least for small num- bers of arguments, is by means of a table, as illustrated in Table 2.2. Once again, this table is intended to illustrate principles: except as discussed below the particular content of the table is not intended to be prescriptive. This table exhaustively lists the output value of the function bAB = χ (bA, bB) for all combinations of input, and thus allows us to define an arbitrary (discrete) function of the two arguments. The approach can also be extended, in principle, to functions of more than two arguments by increasing the number of dimensions of the table, but this clearly is not necessary for the 2WRC. Note that the cardinality of the output alphabet of the function, MAB = |AAB|, bAB ∈ AAB, need not be the same as that of its arguments, and indeed the cardinalities of the two inputs, MA = |AA| and MB = |AB|, bA ∈ AA, bB ∈ AB do not need to be the same. We observe that if the output cardinality of the function is equal to the total size of the table, i.e. MAB = MAMB, then the function may be unambiguously decodable even without any HSI, since each entry can be mapped unambiguously to the corresponding pair of source symbols, provided no symbol is repeated within the table. This is referred to as full cardinality. However, in many ways it would nullify the benefits of the 2WRC, so for HDF we prefer a function with lower cardinality than this. We may observe from the table illustrated in Table 2.2 that symbol bA can be unambiguously decoded pro- vided any symbol occurs only once on any given column of the table, so that if bB is known (which defines the column), the coded symbol unambiguously defines the row, and hence bA. This requires that MAB ≥ MA. Similarly bB can be decoded if any symbol occurs only once in a row, which requires that MAB ≥ MB. Hence correct operation of the 2WRC requires that MAB ≥ max (MA, MB). The equality in this expression defines what is known as minimal cardinality. Any value between this minimum and full cardinality will be referred to as extended cardinality. There are other, less general ways of defining the function. In particular we have already noted that network coding functions which are linear on some algebraic field are used. We note that linearity may also be defined on a ring as well as a field, but for brevity we refer here primarily to the field. The function may then be defined in the form χ (bA, bB, . . .) = aA ⊗ bA ⊕ aB ⊗ bB ⊕ · · · (2.4) 03 19:31:33
- 37. 24 Wireless Physical Layer Network Coding: a Gentle Introduction where the symbols bA, bB, . . . and the coefficients aA, aB, . . . belong to the same field, and ⊕ and ⊗ denote addition and multiplication in the field, respectively. If such a function is applied in the 2WRC, it is easy to see that bA can be unambigu- ously decoded provided the corresponding coefficient aA has a unique inverse in the field, since at destination B the term aBbB can be subtracted and the residue multiplied by the inverse of aA (and conversely for bB). Because in a field all elements except 0 have unique inverses, this is always possible provided both coefficients are non-zero (that is, the function depends on both its arguments). In the binary case we have been considering so far the table definition of the function as described above is 2×2, and its entries are 1s and 0s. Since there must be one “1” and one “0” on each row and each column, the table must take the form of the XOR function (or its inverse). It is therefore also a linear function, whose symbols and coefficients are in F2. The argument above also shows that both coefficients must be “1”; thus our binary 2WRC example leaves us no options in the choice of network code function. In Section 2.7 of this chapter we will extend these concepts to a more general network topology, but at this point it is worth noting that the considerations we have dealt with here create conditions on the design of the network code functions for a WPNC network that apply to the whole network. In the next section, on the other hand, we will encounter conditions on the function that apply at an individual relay node. 2.5 Achievable Rates of HDF and JDF Among all strategies for multi-user and multi-node wireless networks, the HDF (as one particular example of a PHY-native WPNC technique) and JDF (as a more traditional approach) are the ones sharing some important commonalities, namely in processing a hierarchical many-to-one function of the data streams at the relay. The JDF does that by concatenating the traditional multi-user decoding of all individual data streams, where- upon the discrete network-level NC is subsequently applied. In contrast, HDF decodes the mapping function directly using the signal space observation. The example cases treated so far, have assumed uncoded transmission or kept the statements at a quite generic qualitative level for the sake of simplicity. However, the performance comparison of HDF and JDF is of such importance that we now expose the coded case in a slightly more exact form. More elaborate mathe- matical treatment will serve as a gentle introduction to the information-theoretic style of analyzing WPNC systems used in the rest of the book. Particular numerical results will also serve as a justification of the HDF-based approach and as a motivation for the rest of the book. We will consider a very simple scenario for the hierarchical MAC channel where two sources communicate at the same time and frequency (with mutually interfering signals) with one relay that aims to decode a hierarchical many-to-one data mapping function. We will assume coded transmission and compare the achievable rates of HDF and JDF. There are many additional conditions and constraints under which the follow- ing statements hold and these are treated in detail in the rest of the book. For the sake 03 19:31:33
- 38. 2.5 Achievable Rates of HDF and JDF 25 of clarity, we will not state them explicitly now and we urge the reader to check them carefully in order to avoid misinterpretations. We also consider the simplistic case of two BPSK sources in a real-valued AWGN hierarchical MAC channel. Even though this example has very little practical relevance, and it still does not allow closed-form mathematical results (they must be determined numerically), the treatment is relatively simple and prepares the ground for the more complex expositions used later in the book. 2.5.1 Two-Source BPSK Hierarchical MAC We assume two coded sources with messages bA ∈ [1 : 2NRA ], bB ∈ [1 : 2NRB ], where N is the codeword length, and source codebooks CA, CB with identical code rates RA = RB. In information theory, the message is frequently described by a scalar index drawn from some discrete value range. It stresses the fact that the form of the information is irrelevant and the only important aspect is the total number of message values. It also has a nice interpretation as line index numbers of the codebook. The total number of codebook lines is Mi = 2NRi , i ∈ {A, B}, where Ri is the so-called rate. The codeword length N is the length of the line in the codebook. The rate of the code is the binary-base logarithm of the codebook size per codesymbol, Ri = lg Mi/N, i.e. how many binary symbols are represented by one codesymbol. The codesymbols cA,n, cB,n ∈ {0, 1} use the BPSK channel alphabet sA,n, sB,n ∈ {±1}, with size M = 2, mapped symbol-wise to the codesymbols. The observation model is a real-valued AWGN channel xn = sA,n(cA,n) + sB,n(cB,n) + wn (2.5) where the noise has σ2 w variance per dimension and its probability density function (PDF) is pw(w) = 1 2πσ2 w exp − w2 2σ2 w . (2.6) The SNR is defined as γ = E[|si|2] σ2 w . (2.7) The hierarchical mapping function is XOR cn = χc(cA,n, cB,n) = cA,n ⊕ cB,n (2.8) and thus it has the minimal cardinality cn ∈ {0, 1}. Under a number of specific assump- tions treated later (e.g. isomorphic layered code, regular and symbol-wise independent and identically distributed (IID) perfect random codebooks, etc.; see Sections 5.7, 5.7.3, 5.7.4, and Chapter 4), we can assess the coded system performance using single chan- nel symbol information-theoretic properties. The isomorphic assumption implies that we can uniquely decode the hierarchical map of the information data messages. The hier- archical data map is b = χ(bA, bB) and b ∈ [1 : 2NR] where R is the hierarchical data rate. 03 19:31:33
- 39. 26 Wireless Physical Layer Network Coding: a Gentle Introduction 2.5.2 JDF Strategy The JDF strategy is limited by a classical multi-user rate region. Both data streams must be first reliably individually decoded before they can be used in the network-level NC. The achievable rates are given in terms of mutual information expressions RA I(CA; X|CB), (2.9) RB I(CB; X|CA), (2.10) RA + RB I(CA, CB; X). (2.11) We dropped the sequence index n from the notation. All following statements refer to a single symbol. The mutual information between a pair of random variables describes how much the outcome uncertainty of one of them is reduced after observing the other one. The condi- tional mutual information assumes that the stochastic behavior of the variables involved is conditioned by the knowledge of the conditioning variable (e.g. the codesymbol is known). In the case where it is not clear from the context, or when we need to distin- guish it explicitly, we use capital letters to denote the random variables and lower-case letters to denote their particular values. The achievable rate is the rate of some given codebook construction that can be decoded with some given decoding strategy with error probability approaching zero for N → ∞. The achievable rate is typically determined by some function containing mutual information expressions. Under common memoryless channel and so-called IID random codebook assumptions, the involved mutual information expressions are related to individual symbols. The random IID codebook is an abstraction in constructing the hypothetical idealized codebook that makes the information theoretic proofs of coding theorems possible; see Section A.4 for details. Owing to the symmetry of the channel and the symmetry of the codebooks, the achievable rates have the first-order limit RA = RB I1 = I(CA; X|CB) = I(CB; X|CA) (2.12) and the second-order limit RA = RB I2/2 (2.13) where I2 = I(CA, CB; X). (2.14) Thy symmetry of the system and the minimal cardinality map then implies R= RA =RB. The first-order limits are essentially the single-user rates I1 = H[X ] − H[X |C] = H[X ] − H[W] (2.15) where H[W] = 1 2 lg(2π e σ2 w) (2.16) 03 19:31:33
- 40. 2.5 Achievable Rates of HDF and JDF 27 is the AWGN entropy and X = SA(CA) + W is the effective single-user channel model with the second source removed to equivalently model the conditioning in the mutual information. The effective observation has PDF p(x ) = cA p(x |sA(cA))p(cA) = cA pw(x − sA(cA))p(cA) (2.17) and entropy H[X ] = − Ep(x) lg p(x ) . (2.18) The second-order limit is I2 = H[X] − H[X|CA, CB] = H[X] − H[W] (2.19) where the observation entropy is H[X] = − Ep(x) lg p(x) (2.20) and p(x) = cA,cB p(x|sA(cA), sB(cB))p(cA)p(cB) = cA,cB pw(x − sA(cA) − sB(cB))p(cA)p(cB). (2.21) All codesymbols have uniform a priori probability mass function (PMF) p(cA) = 1/M and p(cB) = 1/M. 2.5.3 HDF Strategy The HDF strategy, in contrast to JDF, directly decodes the hierarchical data map. The achievable hierarchical rate, under some conditions (e.g. using regular isomorphic lay- ered NCM, etc.; see Sections 5.7 and 5.7.4 for details), is given by the hierarchical mutual information R I(C; X) = IH. (2.22) Notice that we do not need to explicitly decode the individual source data streams, and that the hierarchical data rate is directly given by the single-symbol information- theoretic limit. The symmetry of the scenario and the minimal cardinality map again implies R = RA = RB. (2.23) The hierarchical mutual information I(C; X) evaluation requires the knowledge of the hierarchical channel symbol conditional PDF. It describes the observed received signal from the perspective of the hierarchical channel symbol which is, in turn, mapped to 03 19:31:33
- 41. 28 Wireless Physical Layer Network Coding: a Gentle Introduction the hierarchical codebook encoded message b. The conditional PDF for our minimal cardinality map with uniformly distributed symbols is (see details in Section 4.4) p(x|c) = 1 M cA,cB:c p (x|sA(cA), sB(cB)) , (2.24) where the summation set cA, cB : c is the summation over all cA, cB consistent with hierarchical symbol c, i.e. such that c = χc(cA, cB). The hierarchical mutual information is then IH = H[X] − H[X|C]. (2.25) Unlike the JDF case, the conditioning in H[X|C] still leaves some ambiguity because of the many-to-one hierarchical mapping function property. The conditional entropy thus needs to be explicitly evaluated using H[X|C] = − Ep(x,c) lg p(x|c) (2.26) where p(x|c) is given above and p(x, c) = p(x|c)p(c) where hierarchical symbols have uniform PMF p(c) = 1/M. 2.5.4 Achievable Rates The achievable rates for JDF and HDF strategies are now evaluated numerically. The integrals of the expectations in the entropies do not have closed-form solutions; however, it is a relatively easy numerical task. We first visualize them by plotting (in Figure 2.4) the achievable hierarchical rate for JDF, which is given by the bottleneck of the first- and the second-order rate limits –10 –5 0 5 10 15 20 0.0 0.2 0.4 0.6 0.8 1.0 1.2 γ [dB] R HDF , R JDF Hierarchical rate in H–MAC HDF (solid), JDF (dashed) Figure 2.4 Hierarchical rate achievable for two BPSK sources and relay with JDF (dashed line) and HDF (solid line) strategies. 03 19:31:33
- 42. 2.6 2WRC with QPSK: the Problem of Channel Parametrization 29 RJDF min I1, I2 2 , (2.27) and for HDF, which is given by the hierarchical mutual information RHDF IH. (2.28) As we see, the performance is clearly alphabet-limited at high SNR values, where it saturates at the fixed ceiling. This ceiling is, however, higher for the HDF strategy, where it is given by lg M, and it provides the single-user level performance as if there was no interference at all. It is, however, in contrast to JDF, where the performance is given by the interference limited regime of the second-order rate region condition. Even with zero noise, the JDF cannot support the lg M hierarchical rate. This comparison exactly shows the performance advantage where WPNC (HDF in this example) tech- nique demonstrates its supremacy and it also justifies our aim of turning the interference into a “friendly” form. The low SNR region is dominated by the influence of the noise and the actual inter- action of the coded signals remains less significant. We call this region the noise-limited region. The advantage of HDF, which can effectively cope with the interference by turn- ing it into a “friendly” interaction that reduces the cardinality of codewords that need to be distinguished, does not help now. The specific hierarchical constellation shape, namely the fact that two points (±2) belong to the same codesymbol map, now makes the situation slightly worse for low SNR. This will be explicitly treated in Section 4.5. In the noise limited region, JDF outperforms HDF. Figure 2.5 shows the rate region from the perspective of both sources. The HDF strat- egy has a rectangular region since both the rates RA and RB are equal, provided that they are less than IH. In contrast with that, the JDF strategy has the classical multi-user MAC shaped region. The region has close-to-rectangular shape for low SNR – the noise- limited regime. The interference limited regime for high SNR makes the second-order limit the dominant one. The symmetric rate RA = RB is thus limited by the second-order limit I2/2. The pair of lines for γ = 5 [dB] nicely demonstrates that the “corner” point of HDF can be outside the JDF region, while the JDF itself provides slightly greater first-order rate limits. The trade-off between noise limitation and interference limitation can be nicely seen when evaluating the ratio (I2/2)/I1 (Figure 2.6). It describes how much the second-order limit influences the symmetric rate. The second-order limit captures how the performance is affected by the presence of the other user. The first-order limit captures the stand-alone single-user behavior and thus captures the noise-related performance. 2.6 2WRC with QPSK: the Problem of Channel Parametrization So far our examples have been restricted to the binary case: BPSK modulation. For most practical applications it will be necessary to extend to higher-order modulation. In 03 19:31:33
- 43. 30 Wireless Physical Layer Network Coding: a Gentle Introduction 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 RA R B Rate region, γ[dB]∈{–5,0,5,10} HDF (solid), JDF (dashed) Figure 2.5 Achievable rate regions for two BPSK sources and relay with JDF and HDF strategies. Each pair of solid (HDF) and dashed (JDF) lines corresponds to one SNR value γ . High SNR values correspond to outer pairs. –10 –5 0 5 10 15 20 0.0 0.2 0.4 0.6 0.8 1.0 1.2 (I 2 /2)/I 1 JDF 2nd order vs. 1st order rate limit ratio γ [dB] Figure 2.6 Second-order vs. first-order limit ratio (I2/2)/I1 for BPSK sources and relay with JDF strategy. this section we will consider QPSK modulation, and we will see that this raises further issues about the choice of mapping function. First, it allows more options: for linear functions because there are more coefficients to choose from, and for the more general table representation because the table is larger and allows more permutations. Secondly, 03 19:31:33
- 44. 2.6 2WRC with QPSK: the Problem of Channel Parametrization 31 we find that problems arise for some values of channel parameters (that is, amplitude and phase of channel fading) that do not arise with BPSK. Note here that for most of this book we will assume that channels are subject to quasi-static flat fading, and thus that a wireless channel can be defined by its amplitude and phase, usually defined by the complex coefficient h. In the simple example of WPNC with BPSK described in Figure 1.7 in Section 1.4 we have assumed that the two channels have the same parameters: they are subject to exactly the same fading. This of course is unlikely in practice, but it results in the received constellation shown, which contains only three points since two of the combinations of source data symbols (“01” and “10”) result in the same signal at the receiver – namely zero. This is a state we describe as singular fading, defined as follows for the case of two source nodes received at one relay. The full details will be given in Section 3.5.3. Here we present only a simplified case for two source nodes and uncoded signals. Singular fading occurs if the channel fading parameters are such that two different combinations of source symbols transmitted from two nodes result in the same received signal at a relay, neglecting the effect of noise. Mathematically it means ∃(sA, sB) = (s A, s B) : uAB = hAsA + hBsB = u AB = hAs A + hBs B. (2.29) That is hA sA − s A = hB s B − sB (2.30) and (s B − sB)h = (sA − s A), h = hB hA (2.31) for some (sA, sB) = s A, s B , where sA, sB, s A, s B are transmitted signals correspond- ing to symbols bA, bB, b A, b B, and h denotes the relative fading of the two channels. It will already have been obvious that the shape of the constellation depends only on the ratio of the two channel coefficients, since any common factor of the two will result only in a phase/amplitude shift of the whole received constellation. We refer to symbol combinations that result in the same signal as clashes. In the case of BPSK, since the ss take only two possible values, there are only two values of h that give rise to singular fading: +1 and −1. We might say there also exist two further such values, 0 and ∞, in which one channel or the other is completely faded, but these are not of interest for the 2WRC since they would in any case prevent the network from operating, in the same way they would in a conventional relay network. All other relative fade coefficients will yield a constellation with four distinct points, as shown in Figure 2.7. Singular fade states such as these are important for two reasons: firstly because they represent channel conditions under which joint decoding will not operate, and secondly because if WPNC is to operate correctly clashing symbol combinations should encode to the same network coded symbol.3 If this is the case we say that the clash is resolved, and if all clashes corresponding to a singular fade state are resolved, 3 Both these points hold for a simple uncoded case. For the coded case with properly constructed codebooks (e.g. the idealized abstraction of random IID codebook), the unresolved singular fading only reduces the achievable rates (see Part III for more details). 03 19:31:33
- 45. 32 Wireless Physical Layer Network Coding: a Gentle Introduction Figure 2.7 Receive constellations at relay for BPSK in 2WRC with different fade states: (a) non-singular fading; (b) singular fade with h = 1; (c) singular fade with h = −1. Figure 2.8 Receive constellation for QPSK. we say that the singular fade state is itself resolved. An unresolved clash will mean that the relay is unable to decode the corresponding network coded symbol, since the received signal will correspond with equal probability to (at least) two network coded symbols. We note that for the binary 2WRC both singular fade states are resolved by the XOR function, which is fortunate, because we have also shown above that this too is the only function that will allow unambiguous decoding. As we have already noted, it is vanishingly improbable that an exactly singular fade state will occur. However, in the presence of noise, fading that is close to singular will also prevent reliable decoding, since the noise will result in a high error rate in distinguishing between the two network coded symbols. For QPSK, however, the situation becomes more complex. The four transmitted sig- nals now take the values ±1 ± j, resulting (for general non-singular fading) in 16 points in the received constellation, as illustrated in Figure 2.8. Excluding the values 0 and ∞, singular fading now occurs for h ∈ {±1, ± j, ±1 ± j, (±1 ± j)/2}. (2.32) Figure 2.9 shows the received constellations for three representative cases from these (or rather for fading close to these states, so the separate labels can more easily be seen). Note that the binary labels shown for the constellation points are formed by concatenat- ing the two-bit binary labels (using conventional Gray code labeling) for the symbols from the two sources. We now consider mapping functions that can resolve the clashes that occur in these fade states. We will look for linear functions for this purpose. Perhaps the obvious approach is to apply the XOR function to each bit of the binary label separately (in 03 19:31:33
- 46. 2.6 2WRC with QPSK: the Problem of Channel Parametrization 33 (a) Re[u] –3 –2 –1 0 1 2 3 Re[u] –3 –2 –1 0 1 2 3 Im[u] –3 –2 –1 0 1 2 3 Im[u] –3 –2 –1 0 1 2 3 0000 1000 0100 1100 0010 1010 0110 1110 0001 1001 0101 1101 0011 1011 0111 1111 (b) 0000 1000 0100 1100 0010 1010 0110 1110 0001 1001 0101 1101 0011 1011 0111 1111 Re[u] –3 –2 –1 0 1 2 3 (c) Im[u] –3 –2 –1 1 0 2 3 0000 1000 0100 1100 0010 1010 0110 1110 0001 1001 0101 1101 0011 1011 0111 1111 Figure 2.9 (Nearly) singular fading for (a) h ≈ 1, (b) h ≈ j, (c) h ≈ 1 + j. terms of the labels shown in Figure 2.9 this means the network coded label is formed by two XOR functions, first of bits 1 and 3 and second of bits 2 and 4). For h = 1, as in Figure 2.9a, we can see that this results in the same two-bit binary label for all the (nearly) coincident points in the constellation. However, for h = j, as in Figure 2.9b, we observe that the clashes are not resolved: for example of the four coincident points around the origin two will be labelled “10,” and the other two “01.” But if instead the two functions XOR first bits 1 and 4, and secondly bits 2 and 3 of the composite label, we will label all four of these points “01,” and similarly the other four clashes in the constellation will also be resolved. This highlights an important general issue that arises with any modulation scheme with more than two points in its constellation. Unlike the BPSK case, where we observed that the XOR function resolved both singular fade states and therefore could be used for any fading, for QPSK (and any other non-binary modulation), different mapping functions are required in order to resolve all fade states. Hence adaptive mapping is required at the relay. 03 19:31:33
- 47. 34 Wireless Physical Layer Network Coding: a Gentle Introduction Incidentally these mapping functions can also be represented by using a binary matrix notation. We represent the M-ary source symbols bA, bB as length m vectors bA, bB, where M = 2m and concatenate them. The linear function may then be represented as multiplication by a binary matrix bAB = χb (bA, bB) = G bA bB . (2.33) The mapping function we invoked for Figure 2.9a can then be represented by the matrix G = 1 0 1 0 0 1 0 1 , (2.34) and for Figure 2.9b by G = 1 0 0 1 0 1 1 0 . (2.35) Considering the third singular fade state, h = 1 + j, for which the received constel- lation is illustrated in Figure 2.9c, we observe that neither of the two functions so far discussed will resolve any of the four clashes. There is, however, a similar function that will resolve this state (and others like it). We may use a pair of XOR functions which combine both bits of one symbol label with one each of the bits of the other symbol – that is, the first function XORs bits 1, 2, and 3 of the composite label, while the second XORs bits 1, 2, and 4. We observe that this resolves the four clashes in the constella- tion shown, and similar functions can resolve clashes in the other equivalent fade states (namely h = ±1 ± j and h = (±1 ± j)/2). In this case the mapping matrix is G = 1 1 1 0 1 1 0 1 . (2.36) However unfortunately this function fails the exclusive law mentioned in the previous section, and hence does not ensure unambiguous decodability at both destinations. Since both XOR functions combine both bits of the label of bA, there are at least two different bAs (e.g. “01” and “10”) which yield the same network code for given bB, and hence the destination is unable to distinguish them. In fact it can be shown that there is no quaternary network code function (i.e. giving a four-level result, equivalent to two bits) that resolves the fade states h = ±1 ± j and h = (±1 ± j)/2 and allows unambiguous decoding at both destinations in the 2WRC. This underlines the point that the end-to- end performance of a network using WPNC needs to be considered: it is not sufficient to choose mapping functions at relays only on the basis that they resolve the singular fading encountered there. 2.7 Hierarchical Wireless Network Example Finally we consider a second, a little more complicated, example network. We describe it as hierarchical wireless network (HWN) because it models a hierarchy of nodes from 03 19:31:33
- 48. 2.7 Hierarchical Wireless Network Example 35 Figure 2.10 Hierarchical wireless network example. the source terminals at the lowest level, via a series of layers of relay nodes, to a hub node, which is the final destination for all data. This could be a model for the uplink of a radio access network in which terminals communicate with a series of small, local access points, which then forward data via backhaul links to a concentrator node that is connected to the core network. Here we consider the simplest possible example of such a network (illustrated in Figure 2.10), consisting of two sources SA and SB transmitting symbols bA and bB to two relays R1 and R2, which then forward network coded symbols b1 = χ1 (bA, bB) and b2 = χ2 (bA, bB) to one destination D. We assume that both relays receive signals from both sources, via channels subject to independent fading. Note that in this network there is no HSI: the data from both relays constitute HI, since they both depend on source data which is of interest to the destination. The same issues of singular fading and unambiguous decodability arise in this net- work. At each relay the mapping function should adapt to the fading of the channels to resolve as far as possible any singular fade states. But the resulting mapping functions should combine at the destination to enable the destination to deduce unambiguously which combination of symbols was sent. We can define a new version of the exclusive law to cover this case (see Section 3.4 for a general treatment) (b1, b2) = (χ1 (bA, bB) , χ2 (bA, bB)) = (b 1, b 2) = χ1 b A, b B , χ2 b A, b B , ∀(bA, bB), (b A, b B) : (bA, bB) = (b A, b B). (2.37) That is, any two different pairs of source symbols must result in a different pair of network coded symbols at the relays. We can treat the pair of mapping functions at the two relays as a single joint mapping function, which can be tabulated in the same way as in Table 2.2. Table 2.3 illustrates such a table – again the content of the table is illustrative only. In this case the entries of the table are pairs of symbols from the two relays, and each pair must be distinct, corresponding unambiguously to a pair of source symbols. The number of distinct pairs of symbols (b1, b2) must be at least as great as the number of pairs (bA, bB), that is M1M2 ≥ MAMB (where M1 = |A1|, b1 ∈ A1, M2 = |A2|, b2 ∈ A2). Once again, the cardinality of the outputs of the mapping functions do not need to have the same cardinality as the inputs. In this network there is in fact no lower limit on the cardinality of one relay, provided that of the other is sufficient to compensate. If one relay has full cardinality (i.e. MAMB), then the other is not needed at all (although we may treat it as having cardinality 1). 03 19:31:33
- 49. 36 Wireless Physical Layer Network Coding: a Gentle Introduction Table 2.3 Table for joint mapping function from two relays in HWN. bB bA 0 1 . . . M − 1 0 (0, 0) (1, M − 1) . . . (M − 1, 1) 1 (1, 1) (2, 0) . . . (0, 2) . . . . . . ... . . . M − 1 (M − 1, M − 1) (0, M − 2) . . . (M − 2, 0) It is also clear that the functions at the two relays must be different. In fact a stronger condition is required: whenever two different pairs of source symbols give the same output for one function, they must produce a different result for the other function: ∀(bA, bB), (b A, b B) : (bA, bB) = (b A, b B), χi (bA, bB) = χi b A, b B , it must hold that χī (bA, bB) = χī b A, b B (2.38) where i ∈ {1, 2}, ī = 3 − i. The table formulation, as before, can be used for any arbitrary pair of discrete functions. If we restrict ourselves to linear functions, then the pair of output symbols from the relays can be written (b1, b2) = (a1A ⊗ bA ⊕ a1B ⊗ bB, a2A ⊗ bA ⊕ a2B ⊗ bB) . (2.39) This may also be written in matrix form, as br = Abs (2.40) where bs = [bA, bB]T, br = [b1, b2]T, and A = a1A a1B a2A a2B . (2.41) Then the condition for unambiguous decodability becomes simply that A is invertible, that is, that its rows and columns be linearly independent. This, of course, also implies that the functions at the two relays are different. Provided the cardinality of the sources is a power of 2, we can also use the binary matrix representation of a linear mapping function. Using the same notation as before, the relay mapping functions can then be written (notice that vectors b are now modified to reflect the binary representation) b1 = G1 bA bB , (2.42) b2 = G2 bA bB , (2.43) br = b1 b2 = G1 G2 bA bB = Gbs . (2.44) 03 19:31:33
- 50. 2.7 Hierarchical Wireless Network Example 37 In this case it is the matrix G that must be invertible (i.e. non-singular): again, all its columns must be linearly independent. This will not be the case if the two functions are the same. Because there are two relays in this network, there are more options for joint mapping functions, and this means more flexibility in resolving singular fade states. For example in the QPSK case we find that the singular fade states h = ±1±j and h = (±1±j)/2 may now be resolved without necessarily compromising unambiguous decodability, though of course since the functions must be different this will not be possible if both relays are in the same singular fade state. For these singular fade states we may use the mapping matrix Gi = 1 1 1 0 1 1 0 1 . (2.45) This may be combined with various mapping matrices in the second relay, provided the combination is not singular. For example, the combined matrix might be G = ⎡ ⎢ ⎢ ⎣ 1 1 1 0 1 1 0 1 1 0 0 1 0 1 1 0 ⎤ ⎥ ⎥ ⎦ . (2.46) Some care is, however, needed when extending this bit-wise mapping to a coded case where the isomorphism of the hierarchical codeword is needed; see Section 6.3.4 for details. 03 19:31:33
- 51. 03 19:31:33
- 52. Part II Fundamental Principles of WPNC 13:31:09
- 53. 13:31:09
- 54. 3 Fundamental Principles and System Model 3.1 Introduction This chapter is essentially all about basic definitions and classifications of various sce- narios based on them. It is a bit tedious but necessary in order to develop a clear understanding of the terms. Also, the terms could have rather wide interpretations and we need to define them precisely. Proper classification of the techniques also helps to understand how they are mutually related, what they have in common, and how they differ. We attempt to present a highly modular view of the roles and functions of indi- vidual nodes in the WPNC network. This will help us later to develop various techniques (NCM design, decoding technique) that are universally usable in nodes serving a variety of roles. We start with scenarios and models where we describe the roles of nodes, the con- straints imposed by their radio interfaces, and various issues related to the topology of the network. Then we continue with the core hierarchical principle. It describes how data functions flowing through the network are encapsulated hierarchically. We also show how a direct neighborhood of the node affects the overall end-to-end description of the network. Then we turn our attention back to the individual node. We show how its opera- tion can, under very general conditions, be decomposed into front-end processing, node processing, and back-end processing operations. We show how the node processing operation is related to the many-to-one function of the source nodes’ data, which will lead to the definition of the hierarchical symbol, and we also define a form of infor- mation measure that is used to represent it to the rest of the network. Depending on a node’s predecessor path in the network graph, this form of information measure can have various forms of usefulness from the point of view of the given node. We will define hierarchical information, hierarchical side-information (friendly interference), and classical interference. Previous definitions help us to classify various strategies of the node. The node pro- cessing operation will be classified (e.g. amplify and forward, decode and forward, soft forward, compress and forward). Depending on the properties of the hierarchical sym- bol and its associated hierarchical NC map we will introduce full, minimal, extended, and lossy hierarchical maps. Then we classify back-end strategies from the source-encoding viewpoint (direct and analog hierarchical broadcast, and source-encoded NC broadcast). Even more 04 19:41:26
- 55. Another Random Document on Scribd Without Any Related Topics
- 56. might have taken them out to the Oak Cliff substation and put them in our property room—I don't know. Mr. Ely. Now, you were back at the stage where somebody had given you the gun, and let's go on from there. Mr. Owens. Yes—we were informed by a man whom I do not know, that the suspect that shot Officer Tippit had run across a vacant lot toward Jefferson, and thrown down his jacket, I think he said, white, I'm not sure. Not finding anybody that had seen him come out of that area, we blocked off that square block. Mr. Ely. Can you tell us specifically what block you blocked off? Mr. Owens. I believe it was the 400 block of East Jefferson—the 400 or 500 block. It was this block bound by Jefferson, 10th, Patton, and Denver—I believe that was the area. Then we started searching the buildings and houses—there are some old two-story houses there used as businesses. Mr. Ely. What was the nature of your search of these buildings? Did you just look through the halls? Mr. Owens. Well, I didn't go in. I was standing on the outside and the other officers were going in. I was covering off. Then, we heard over the radio that some officer, who by the number, I took to be a three-wheeler motorcycle officer had seen someone answering the description, go into the basement of the library, which is on the corner of Marsalis and Jefferson, which was about two blocks away. Quite a few of us left that area we were at and proceeded to the library, covered it off, and they brought out the one that they thought was the suspect, but he fit the general description, but he was not the one we were looking for. He was an employee of the library that heard the President had gotten shot and he had been to lunch and he was running over there to tell them that the President got shot. Mr. Ely. In other words, someone saw this employee run into the library, and that's the reason you came in. He had just run into the library?
- 57. Mr. Owens. That's the man that had run across Jefferson and run into the basement of the library, so I went back to the scene of the shooting of Officer Tippit and another call had come and some of my men yelled to me that they had a suspect in the Texas Theatre, and everyone left there, but nobody was left to help guard the scene except the crime lab man, so I remained at the scene, and everybody else went to the Texas Theatre. Mr. Ely. Do you remember who the crime lab man was who was there? Mr. Owens. At the time I thought it was Captain Doughty [spelling] D-o-u-g-h-t-y. They finished up taking the pictures and I left the scene and went to Methodist Hospital where Officer Tippit had been taken, and I was taken back to the room where he was taken, and in just a brief examination of the body I saw where one bullet had entered his right chest about the pocket and went through a package of cigarettes. Another one hit him about the center of the chest and hit a button, and another one, I believe, was in his right temple, I'm not sure which temple it was, but those three wounds, I did see. I don't know whether he was shot any more or not. I remained at the hospital for quite a time, and then I went back to the Oak Cliff substation where I was assigned. Mr. Ely. And because you were assigned to the Oak Cliff substation, you at no time during these 2 days or so went into the main police headquarters; is that correct? Mr. Owens. What, now? Mr. Ely. You didn't go to the main police headquarters because you were assigned to the Oak Cliff substation? Mr. Owens. No; that's right. Mr. Ely. Now, I show you a map which is labeled Putnam Deposition Exhibit No. 1. Could you tell us what sort of a map this is?
- 58. Mr. Owens. It is what we call a district map of the various districts of the city of Dallas. Mr. Ely. The various districts to which patrolmen are assigned, is that correct? Mr. Owens. It is what it was set up for. Now, there isn't a squad for each numbered district. Some squads have two or more numbers. I mean, the districts cover that. Mr. Ely. And could you tell us to which district or districts on that map Officer Tippit was assigned on November 22, 1963? Mr. Owens. He was assigned to district 78. Now, I don't know whether we were short any squads that day or not, and if we were, he would be assigned to cover another district also. His call number would still be 78. Mr. Ely. Would his call number be 78 even if he were outside the district? Mr. Owens. Oh, yes. Mr. Ely. I show you now one of the radio logs which is designated Sawyer Deposition Exhibit A. Am I correct in saying that at 12:54 p.m., according to this log, Officer Tippit reported by radio that he was then at the corner of Lancaster and Eighth? Mr. Owens. That's right. Mr. Ely. Now, in which district on this map would the corner of Lancaster and Eighth fall? Mr. Owens. In district 109. Mr. Ely. That would be district 109. In which district on the map was Officer Tippit shot? Mr. Owens. In district 91. Mr. Ely. Now, we would like to have your opinion as to why Officer Tippit, who was assigned to district 78, would have been in district 109 at 12:54 p.m. and then later in district 91? In giving us
- 59. your answer, please feel free to refer to both of these radio logs, which are Sawyer Deposition Exhibits A and B, and also draw upon your experience with the Dallas Police Department and the common procedure for reacting to an emergency. Mr. Owens. It says here on channel 1, this is Sawyer Deposition Exhibit B, Attention all squads in the downtown area, code 3, to Elm and Houston with caution, and knowing that the President's parade was going to be down in that area and also at 12:44 this: attention all squads, the suspect in the shooting, Elm and Houston, is reported to be an unknown white male, approximately 30, slender build, height, 5 feet 6 inches, weight, 165 pounds, reported to be armed with what is thought to be a .30 caliber rifle, no further description or information at this time; and then it recites at 12:45 signal 19 involving the President—that was at 12:45—— Mr. Ely. And signal 19 means what? Mr. Owens. A shooting—anything of that magnitude in the shooting of the President is one of the greatest magnitudes, and any officer would proceed as near that location as possible to try to apprehend whoever had done it. Mr. Ely. Well, would somebody in an outlying district head for Elm and Houston itself, or would he just come in closer? Mr. Owens. He would move in that direction, and when they had ordered all downtown squads to proceed to Elm and Houston, knowing that he was going to have to answer calls in the downtown area while they are there, and if you know that in all probability you may get called in, and—instead of the district you are in, you are going to head down there so it won't take you near as long, and also you can still be in the area if the suspect comes your way, you will have a better chance of apprehending him. Mr. Ely. So, you think Tippit might have been filling in for the people whom he knew had been pulled in to Elm and Houston? Mr. Owens. That's what I think—not only filling in, but also looking for the suspect, because he heard about the shooting and
- 60. the general description of the suspect, and not knowing which way he went, but he could have gone any way, then he is going to head downtown as soon as possible so if he sees someone answering that description, he can apprehend him. Mr. Ely. You would say it would be normal procedure for an officer in district 78, which is located out in the outlying districts, to head downtown in any emergency? Mr. Owens. That's true. Mr. Ely. Could you perhaps give us an explanation of why he headed over toward 109 and 91? That doesn't seem to be the most direct route. Mr. Owens. According to this map—it doesn't show all the things on there—it looks like you would have to zigzag quite a bit, but you wouldn't. You could go down Corinth Street and go across the viaduct, but that would get him down on Industrial, which would still be a lot of traffic to go through. He could go down Clarendon to Marsalis and go North Ewing and then get over to Lancaster, and a would give him a straight shoot to the Houston Street viaduct, which would take him right to Elm and Houston. Mr. Ely. So that you think a path of going from 78 to 109 to 91 would be a more or less logical route for getting into the center of town? Mr. Owens. Yes; I do. Mr. Ely. On the 22d of November, did you, yourself, have an area which you were patroling? Mr. Owens. I was supervising all of the Oak Cliff area, and since I was acting lieutenant, and I made the assignments for that day, I was at the station at 4020 West Illinois at the time. Mr. Ely. In which numbered area is that located? Mr. Owens. That would be on district 97, and no one sent me, but when I heard all of this—so many squads getting called to report
- 61. there, then I went. Mr. Ely. You headed toward the downtown area yourself? Mr. Owens. Yes; I went to Elm and Houston myself. Mr. Ely. Even though you didn't have a specific order to go in there either? Mr. Owens. That's right—that's true. Mr. Ely. Officer McDonald, who testified before the Commission, told us that he went to the corner of Elm and Houston, do you know which numbered area on this map he was assigned to? Mr. Owens. He was working district 95, which covers district 95 and 96. Mr. Ely. Off the record. (Discussion off the record between Counsel Ely and the witness Owens.) Mr. Owens. I don't know what district Officer J. L. Angel was working, but it was my understanding that he also went to Elm and Houston. Mr. Ely. Well, he was working somewhere in the Oak Cliff area, was he? Mr. Owens. Yes; he was working in the Oak Cliff area under the same sergeant that Officer Tippit was working under, so he would be in the same general area which covers these districts in here. Mr. Ely. That would be districts 82 and 85? Mr. Owens. No—81, 82, 85, 86, 87, or 76, 77, 78, or 79—that's that sergeant's district. Mr. Ely. All right, thank you very much, sergeant. Mr. Owens. I don't know of anything else—as I say, I couldn't remember where they handed me the gun. I knew it was at the scene because my wife said she saw it on television and I had his
- 62. gun, and when I asked her about it she said it wasn't the suspect's gun she knew because she has been a policeman's wife long enough to know I wouldn't be handling a gun like that if it was the suspect's. Mr. Ely. All right, Sergeant, thank you very much. Mr. Owens. All right, thank you.
- 63. TESTIMONY OF WILLIAM ARTHUR SMITH The testimony of William Arthur Smith was taken at 4:25 p.m., on April 2, 1964, in the office of the U.S. attorney, 301 Post Office Building, Bryan and Ervay Streets, Dallas, Tex., by Mr. Joseph A. Ball, assistant counsel of the President's Commission. Mr. Ball. Mr. Smith, stand up and raise your right hand. Do you solemnly swear that the evidence you are about to give before the Commission shall be the truth, the whole truth, and nothing but the truth, so help you God? Mr. Smith. Yes, sir. Mr. Ball. Sit down. Mr. Ball. State your name, please. Mr. Smith. William Arthur Smith. Mr. Ball. And where do you live? Mr. Smith. 328½ East Davis. Mr. Ball. What is your age? Mr. Smith. Twenty. Mr. Ball. You live with whom? Whom do you live with? Mr. Smith. My mother. Mr. Ball. At this address?
- 64. Mr. Smith. Yes, sir. Mr. Ball. Tell me something about yourself, where you were born and where you went to school. Mr. Smith. I was born in Pine Bluff, Ark., and went to school Wason Chapel. Mr. Ball. How far through school did you go? Mr. Smith. Three months into the 12th grade. Mr. Ball. Three months into the 12th grade? Mr. Smith. Yes, sir. Mr. Ball. What did you do after that? Mr. Smith. Been working ever since, most of the time. Mr. Ball. What kind of work do you do? Have you done? Mr. Smith. Corrugated box. Mr. Ball. Beg your pardon? Mr. Smith. Corrugated box. Mr. Ball. That is where you are working now? Mr. Smith. No, sir; working at a metal shop. Mr. Ball. Any metal shop? Mr. Smith. Yes. Mr. Ball. Have you ever been in trouble with the police? Mr. Smith. Yes, sir. Mr. Ball. What kind of trouble did you get in? Mr. Smith. Auto theft. Mr. Ball. You're on probation now, aren't you? Mr. Smith. Two years. Mr. Ball. Two years? Ever have any other trouble?
- 65. Mr. Smith. Tickets. Mr. Ball. Just tickets? Traffic tickets? Mr. Smith. Two right now. Mr. Ball. You ever have any trouble as a juvenile? Mr. Smith. No, sir. Mr. Ball. Now, on November 22, 1963, were you working any place? Mr. Smith. No, sir. Mr. Ball. Didn't have a job? Mr. Smith. No, sir. Mr. Ball. Where did you spend the day that day? Mr. Smith. 505 East 10th. Mr. Ball. Why were you there? Mr. Smith. Visiting a friend. Mr. Ball. What is his name? Mr. Smith. Jimmy Burt. Mr. Ball. When did you go over there that day? Mr. Smith. In the morning. In the morning. Mr. Ball. In the morning? Mr. Smith. Yes, sir. Mr. Ball. What time did you leave there that day? Mr. Smith. In the evening. Mr. Ball. So, you spent the whole day there? Mr. Smith. Yes. Mr. Ball. Did something happen a little after 1 o'clock there that day that you noticed?
- 66. Mr. Smith. Yes, sir; policeman got shot. Mr. Ball. Now, at the time the policeman was shot, where were you? Mr. Smith. In the front yard, at 505 East 10th. Mr. Ball. Who was with you? Mr. Smith. Jimmy Burt. Mr. Ball. That was about how far from where the policeman got shot? Mr. Smith. One block. Mr. Ball. That would be about a block east, wouldn't it? Mr. Smith. Yes, sir. Mr. Ball. Policeman was shot in the 400 block? Mr. Smith. Yes, sir. Mr. Ball. And you were in the 500 block? Mr. Smith. Yes, sir. Mr. Ball. What called your attention to this incident? Mr. Smith. I heard some shots. Mr. Ball. And what? You looked down that way? Mr. Smith. Yes, sir. Mr. Ball. What did you see? Mr. Smith. Saw Oswald running and policeman falling. Mr. Ball. Did you see his face, or just his back? Mr. Smith. Saw the side of him, the side and back of him when he was running. Mr. Ball. Did you see him before he ran? Mr. Smith. Yes.
- 67. Mr. Ball. Saw the side of his face? Mr. Smith. Yes. Mr. Ball. And he ran in what direction? Mr. Smith. West. Mr. Ball. Did you follow him? Mr. Smith. No, sir. Mr. Ball. Did you go down to where the policeman was shot? Mr. Smith. Yes. Mr. Ball. What did you see? Mr. Smith. Saw the policeman lying on the ground. I mean on the street. Mr. Ball. And did a crowd gather around there? Mr. Smith. Yes, sir. Mr. Ball. How long did you stay there? Mr. Smith. About 45 minutes. Mr. Ball. Did you give your name to the police? Mr. Smith. No, sir. Mr. Ball. Why? Mr. Smith. Because I was on probation. I thought it might hurt my probation record. Mr. Ball. All right; you did tell someone you had seen it, didn't you? Mr. Smith. Yes. Mr. Ball. Who? Mr. Smith. This boy I ran around with. Mr. Ball. What's his name?
- 68. Mr. Smith. James Markham. Mr. Ball. Is he the son of Helen Markham? Mr. Smith. Yes, sir. Mr. Ball. Did you talk to her? Mr. Smith. No, sir; she talks to me. Mr. Ball. Mrs. Markham talked to you? Mr. Smith. Yes. Mr. Ball. And did you tell Mrs. Markham? Mr. Smith. I told her what I saw and that is the reason I am here, I a—— Mr. Ball. Did the police come out and see you? Mr. Smith. The FBI. Mr. Ball. The FBI did? Did you tell them the same story you told me? Mr. Smith. Yes, sir. Mr. Ball. Now, did you see Oswald on television? Mr. Smith. Yes, sir. Mr. Ball. On the night of the shooting? Mr. Smith. Yes, sir. Mr. Ball. Did it appear to you to be the same man you had seen? Mr. Smith. He had lighter hair than he did when I saw him. Mr. Ball. Well, now, wait a minute. You mean the man you saw on television—— Mr. Smith. Had lighter hair. Mr. Ball. Mr. Smith—than the man you saw running away? Mr. Smith. Yes.
- 69. Mr. Ball. Is that right? Mr. Smith. Yes, sir. Mr. Ball. What color hair did the man have that you saw running away? Mr. Smith. Brown, brownish-black. It was dark. Mr. Ball. How did the hair appear on television? Mr. Smith. Looked blond. Mr. Ball. Were you later shown a picture of Oswald? Mr. Smith. Yes, sir. Mr. Ball. By whom? Mr. Smith. FBI agent. Mr. Ball. What was the color of the hair in the picture? Mr. Smith. Brown. Mr. Ball. What did you see? What did you tell the FBI agent about the appearance of the man in the picture? Mr. Smith. I said it looked more like him than it did on television. Mr. Ball. And did you think when he showed you the picture that it looked anything like the man you had seen running away? Mr. Smith. What I saw of him; yes. Mr. Ball. First time you ever saw this man was after you heard these shots? Mr. Smith. Yes, sir. Mr. Ball. Is that right? You had never seen him walking? Mr. Smith. No. Mr. Ball. You hadn't seen him walking in front of the house—— Mr. Smith. No, sir.
- 70. Mr. Ball. Where you were standing? Mr. Smith. No, sir. Mr. Ball. What kind of clothes did he have on when he shot the officer? Mr. Smith. He had on dark pants—just a minute. He had on dark pants and a sport coat of some kind. I can't really remember very well. Mr. Ball. I will show you a coat—— Mr. Smith. This looks like it. Mr. Ball. This is Commission's Exhibit 162, a grey, zippered jacket. Have you ever seen this before? Mr. Smith. Yes, sir; that looks like what he had on. A jacket. Mr. Ball. That is the jacket he had on? Mr. Smith. Yes. Mr. Ball. Now, when the deposition is completed it will be written up and you will have a right to look it over and sign it, or if you want to you can waive your signature. They will accept your waiver and send it on to the Commission without it. Do you have any choice on that? Mr. Smith. I will sign it. It don't make any difference to me. Mr. Ball. Would you just as leave waive your signature? Mr. Smith. Ever what that means. Mr. Ball. That means you don't have to sign it. Mr. Smith. I will sign it. Mr. Ball. Do you want to sign it? Mr. Smith. Yes; I will sign it. Mr. Ball. Okay. Do you have a telephone number?
- 71. Mr. Smith. No, sir. Mr. Ball. Well, the young lady will notify you when you can come in and sign it. I thank you very much.
- 72. TESTIMONY OF GEORGE JEFFERSON APPLIN, JR. The testimony of George Jefferson Applin, Jr. was taken at 4:05 p.m., on April 2, 1964, in the office of the U.S. attorney, 301 Post Office Building, Bryan and Ervay Streets, Dallas, Tex., by Mr. Joseph A. Ball, assistant counsel of the President's Commission. Mr. Ball. Will you stand up, Mr. Applin, and we—raise your right hand to be sworn, please. Mr. Applin. Yes. Mr. Ball. Do you solemnly swear that the testimony you are about to give for this Commission will be the truth, the whole truth and nothing but the truth, so help you God? Mr. Applin. I do. Mr. Ball. Will you be seated, please, and state your name for the record. Mr. Applin. George Jefferson Applin, Jr. Mr. Ball. Where do you live? Mr. Applin. 714 East Hull, Denison, Tex. Mr. Ball. What is your occupation? Mr. Applin. Well, my occupation, common laborer, but I am working for Phillips 66 there in Denison, service station. Mr. Ball. You have come into Dallas from Denison, haven't you?
- 73. Mr. Applin. Yes, sir. Mr. Ball. Well, that is about 68 miles? Mr. Applin. Yes, sir. Mr. Ball. And you are entitled to get compensation for your transportation? Mr. Applin. Yes. Mr. Ball. And we'll have your name and address in the record, and I will try to make arrangements for that information to take care of your expenses. You came in when? This morning? Mr. Applin. No; it was about 15 minutes after 2 o'clock, when I came in here. Mr. Ball. Came into Dallas? Mr. Applin. Yes. Mr. Ball. And—— Mr. Applin. No; I was here at 2 o'clock, but I had a flat and my car stalled on me about three or four blocks over. Mr. Ball. And you intend to return home tonight, do you? Mr. Applin. Yes, sir. Mr. Ball. So, you won't have any hotel expense, will you? Mr. Applin. No, sir. Mr. Ball. Now, tell me something about yourself, where you were born and where you went to school, and how far in school, what you have done since then? Mr. Applin. Well, I was born in Madona Hospital in Denison, and lived there pretty near all my life. Mr. Ball. How old are you? Mr. Applin. Twenty-two.
- 74. Mr. Ball. Did you go to school? Mr. Applin. Yes, sir; I went to LaMar School and junior high. Mr. Ball. And how far did you go? Finished junior high? Mr. Applin. No, sir; I went to the eighth grade. Mr. Ball. Have you been beyond the eighth grade? Mr. Applin. No, sir. Mr. Ball. What did you do after that? Mr. Applin. Well, I helped my daddy some, and got odd jobs and stuff. Mr. Ball. Live with your mother now? Mr. Applin. Yes, sir; I do. I live with my parents. Mr. Ball. Your mother and father? Mr. Applin. Yes, sir. Mr. Ball. You have been doing mostly common labor, have you? Mr. Applin. Yes, sir; mostly common labor. Mr. Ball. Ever been in trouble with the law of any sort? Mr. Applin. Yes, sir; I have. Mr. Ball. What kind of trouble? Mr. Applin. Burglary. Mr. Ball. When was that? Mr. Applin. In 1963. Mr. Ball. Did you do any time? Mr. Applin. No, sir; I got a probated sentence for it. Mr. Ball. That is the only trouble you have ever had? Mr. Applin. Well, for—except for minor traffic violations.
- 75. Mr. Ball. Outside of that you haven't had any trouble? Mr. Applin. No, sir. Mr. Ball. Now, November 22, 1963, were you in Dallas? Mr. Applin. Yes; I believe I was. Mr. Ball. What were you doing here? Mr. Applin. Well, I was working for the Rollform Corp. Mr. Ball. How do you spell it? Mr. Applin. Well, I have got one of their checks—check stubs here in my pocket, I believe. At least I think I have. Here it is [indicating]. Mr. Ball. What were you doing in Dallas? Mr. Applin. Working. Mr. Ball. Working here in Dallas? Mr. Applin. Yes, sir. Mr. Ball. What kind of work? Mr. Applin. Well, I was working as, open-head crane operator, and painter and front-end loader. Mr. Ball. Did you go to the picture show that afternoon? Mr. Applin. Yes, sir; I did. Mr. Ball. How did you happen to be off duty that day? Mr. Applin. They was installing a new cutting press for the rollers, and they did not need me, so, they let me off for 2 days. Mr. Ball. For 2 days? Mr. Applin. For 2 days. Mr. Ball. What did you do? Go to the picture show? Mr. Applin. Yes, sir; I did. Mr. Ball. What time of day did you go there?
- 76. Mr. Applin. Well, actually, I went to—I was over in Oak Cliff, around about, I guess, about 12 o'clock, I imagine is what time it was. I was there and the show hadn't opened up, so, I was sitting in my car listening to the radio up until the time that the show opened. Mr. Ball. You went in the show when it opened? Mr. Applin. Yes, sir. Mr. Ball. Paid your way? Mr. Applin. Yes, sir. Mr. Ball. And where did you take your seat? What part of the theatre? Mr. Applin. About six rows down, I got in the middle aisle, about the middle of the chairs. Mr. Ball. Middle aisle, six rows from the rear? Mr. Applin. Yes, sir. Mr. Ball. And you were how far from the middle aisle into the row of seats? Mr. Applin. Well, about—seemed quite a little while since I thought about this. I guess I was about four or five seats over from the aisle. Mr. Ball. From the aisle. Now, did something happen there during that showing of that picture that you remember? Mr. Applin. Well, I know this much, Audie Murphy introduced the picture. Mr. Ball. Then some police officers came in there? Mr. Applin. No, sir; the lights came on. Mr. Ball. Then what do you remember happening? Mr. Applin. I seen the officers come down the right-hand aisle. Mr. Ball. From the rear, or from the front?
- 77. Mr. Applin. From the rear. Mr. Ball. Come in from the screen side, or the place you enter? Mr. Applin. Where you enter it. Mr. Ball. From your rear? Mr. Applin. Yes, sir; came in on the right-hand aisle over against the wall. Mr. Ball. Did he have anything in his hands? Mr. Applin. Yes; I believe he had a shotgun. Might have been a rifle. Mr. Ball. What else did you see? Mr. Applin. Well, when I seen him, I was wondering what was the matter and what about the lights. Mr. Ball. You got up and ran up to the front? Mr. Applin. Went to the front to find out what was happened— was happened—happening. As I was going up an officer passed me going down and I stopped to find out. Mr Ball. Did you ask him? Mr. Applin. No, sir; he passed me before I got a chance to ask him. Mr. Ball. What did he do? Mr. Applin. Went to the front and turned around and started back up. Mr. Ball. Started back up the aisle? Mr. Applin. Yes, sir. Mr. Ball. Towards you? Mr. Applin. Yes, sir. Mr. Ball. And what did you see him do?
- 78. Mr. Applin. Well, he stopped and asked two boys sitting down in the front, asked them to stand up and—— Mr. Ball. Did he search them? Mr. Applin. Yes, sir; they shuffled them down. Mr. Ball. Did he search you? Mr. Applin. No, sir; they came on up to Oswald, where he was sitting. Mr. Ball. Where was he sitting? Mr. Applin. I—he was sitting, I guess, about 3 or 4 rows down. Mr. Ball. You mean from the rear of the theatre? Mr. Applin. From the rear. Mr. Ball. And how far over from the aisle? Mr. Applin. I guess that would be about three seats. They was sitting about two or three seats. Mr. Ball. What did you see him do? Mr. Applin. He—started off, the officer said, Will you stand up, please? And he stood up. Mr. Ball. How close were you to the officer and this man when you heard the officer say, Stand up? Mr. Applin. I guess it was about—it was not over four seats down from the back, rear. Mr. Ball. Were you at the rear? Mr. Applin. Yes, sir; I was at the rear of the show. Mr. Ball. You were at the rear of the show? Mr. Applin. Yes, sir; well, there was a partition here. A partition here [indicating], and there was about, oh, I guess about four rows down from me.
- 79. Mr. Ball. All right. In other words, the officer hadn't reached you yet, when he asked Oswald to stand up? Mr. Applin. No, sir. Mr. Ball. You stood up and went toward the rear of the theatre, did you? Mr. Applin. Yes. Mr. Ball. And going to ask the officer what was going on? Mr. Applin. Yes, sir. Mr. Ball. Then, you were about four rows away from where Oswald was—— Mr. Applin. Apprehended. Mr. Ball. And did you hear the officer, what he said? Mr. Applin. Yes, sir; heard mainly what both of them said. Mr. Ball. What did the officer say? Mr. Applin. The officer said, Will you stand up, please. Mr. Ball. What did the man say? Mr. Applin. Well, he just stood up. Mr. Ball. Did he say anything? Mr. Applin. No, sir; I didn't hear him say anything at that time. Mr. Ball. And what happened then? Mr. Applin. Well, when he stood up, the officer stepped over to search him down. The officer, Oswald, or the man, took a swing at him. When he did, the officer grabbed him. Mr. Ball. Took a swing at him with his fist? Mr. Applin. Yes, sir; he did. Mr. Ball. With his left or right? Mr. Applin. Right fist.
- 80. Mr. Ball. Took a swing at him and what happened then? Mr. Applin. Well, the officer, I heard him say, Here he is. And during the proceeding of that, I guess about 5 or 10 seconds later, there was another—I think it was two officers, or one, passed me and ran down there to him. Mr. Ball. Did you see a gun? Mr. Applin. Well, the gun didn't come into view until after about four or five officers were there. Mr. Ball. Then did you see a gun? Mr. Applin. Yes, sir; but only—there was one gun. The pistol. It came into view before any of the other officers got there. Mr. Ball. That is what I mean. What do you say happened about that? Who pulled a gun? Mr. Applin. Well, anyhow, the officer was facing this way [indicating] and Oswald was facing this way [indicating]. And then the gun was pointed out that way [indicating]. Mr. Ball. Wait a minute. I can't follow you when you say it was this way, and this way, sir. You told me that this officer asked Oswald to stand up? Mr. Applin. Yes, sir. Mr. Ball. Did he stand up? Mr. Applin. Yes, sir; he did. Mr. Ball. Then did he put his hand some place on Oswald? Mr. Applin. Yes, sir; along about—— Mr. Ball. Where? Mr. Applin. I guess about his hips. Mr. Ball. Then what did Oswald do? Mr. Applin. He took a right-hand swing at him.
- 81. Welcome to our website – the perfect destination for book lovers and knowledge seekers. We believe that every book holds a new world, offering opportunities for learning, discovery, and personal growth. That’s why we are dedicated to bringing you a diverse collection of books, ranging from classic literature and specialized publications to self-development guides and children's books. More than just a book-buying platform, we strive to be a bridge connecting you with timeless cultural and intellectual values. With an elegant, user-friendly interface and a smart search system, you can quickly find the books that best suit your interests. Additionally, our special promotions and home delivery services help you save time and fully enjoy the joy of reading. Join us on a journey of knowledge exploration, passion nurturing, and personal growth every day! ebookbell.com