![1.3 Wireless channels 5
Fig. 1.1. An example of different paths in a wireless channel.
to the topic, as it is needed in our discussion of space-time codes, and refer the
interested reader to other books that concentrate on the subject [57, 103, 111, 123].
1.3 Wireless channels
One of the distinguishing characteristics of wireless channels is the fact that there
are many different paths between the transmitter and the receiver. The existence of
various paths results in receiving different versions of the transmitted signal at the
receiver. These separate versions experience different path loss and phases. At the
receiver all received signals are accumulated together creating a non-additive white
Gaussian noise (AWGN) model for the wireless channels. Since an AWGN model
does not describe the wireless channels, it is important to find other models that
represent the channels. To portray such a model, first we study different possible
paths for the received signals. Figure 1.1 demonstrates the trajectory of different
paths in a typical example.
If there is a direct path between the transmitter and the receiver, it is called the
line of sight (LOS). A LOS does not exist when large objects obstruct the line
between the transmitter and the receiver. If LOS exists, the corresponding signal
received through the LOS is usually the strongest and the dominant signal. At least,
the signal from the LOS is more deterministic. While its strength and phase may
change due to mobility, it is a more predictable change that is usually just a function
of the distance and not many other random factors.
A LOS is not the only path that an electromagnetic wave can take from a trans-
mitter to a receiver. An electromagnetic wave may reflect when it meets an object
that is much larger than the wavelength. Through reflection from many surfaces,
the wave may find its path to the receiver. Of course, such paths go through longer
distances resulting in power strengths and phases other than those of the LOS path.
Another way that electromagnetic waves propagate is diffraction. Diffraction
occurs when the electromagnetic wave hits a surface with irregularities like sharp
edges.](https://image.slidesharecdn.com/2351772-250527055320-4826c58a/85/Spacetime-Coding-Theory-And-Practice-Hamid-Jafarkhani-27-320.jpg)
![1.4 Statistical models for fading channels 13
and the receiver. Later, we will consider the case where such a LOS path exists.
In a multipath channel with I multiple paths, transmitting a signal over the carrier
frequency fc results in receiving the sum of I components from different paths plus
a Gaussian noise as follows:
r(t) =
I
i=1
ai cos(2π fct + φi ) + η(t), (1.11)
where ai and φi are the amplitude and phase of the ith component, respectively,
and η(t) is the Gaussian noise. Expanding the cos(·) term in (1.11) results in
r(t) = cos(2π fct)
I
i=1
ai cos(φi ) − sin(2π fct)
I
i=1
ai sin(φi ) + η(t). (1.12)
It is customary in digital communications to call the first and second summations
“in phase” and “quadrature” terms, respectively. The terms A =
I
i=1ai cos(φi )
and B =
I
i=1ai sin(φi ) are the summation of I random variables since the objects
in the environment are randomly located. For a large value of I, as is usually the
case, using the central limit theorem, the random variables A and B are indepen-
dent identically distributed (iid) Gaussian random variables. The envelope of the
received signal is R =
√
A2 + B2. Since A and B are iid zero-mean Gaussian ran-
dom variables, the envelope follows a Rayleigh distribution. The probability density
function (pdf) of a Rayleigh random variable is
fR(r) =
r
σ2
exp
−r2
2σ2
, r ≥ 0, (1.13)
where σ2
is the variance of the random variables A and B. The received power, is
an exponential random variable with a pdf:
f (x) =
1
2σ2
exp
−x
2σ2
, x ≥ 0. (1.14)
Note that the average power, the average of the above exponential random variable,
is E[R2
] = 2σ2
which is the sum of the variances of A and B.
The received signals in (1.11) or (1.12) represent the analog signal at the first
stage of the receiver. We usually deal with the baseband digital signal after the
matched filter and the sample and hold block. With a small abuse of the notation,
we denote such a baseband discrete-time signal by rt . In fact, rt is the output of the
matched filter after demodulation when the input of the receiver is r(t). Similarly,
st and ηt are the discrete-time versions of s(t) and η(t), the transmitted signal
and the noise, respectively. Note that in the above analysis the transmitted signal
was implicit. Then, using the above arguments, one can show that the relationship](https://image.slidesharecdn.com/2351772-250527055320-4826c58a/85/Spacetime-Coding-Theory-And-Practice-Hamid-Jafarkhani-35-320.jpg)
![16 Introduction
scale. There are two important issues related to the concept of diversity. One is
how to provide the replicas of the transmitted signal at the receiver with the lowest
possible consumption of the power, bandwidth, decoding complexity and other
resources. The second issue is how to use these replicas of the transmitted signal
at the receiver in order to have the highest reduction in the probability of error. We
study some of the methods to achieve these two goals in what follows.
1.5.1 Diversity methods
The replica of the transmitted signal can be sent through different means [99].
For example, it can be transmitted in a different time slot, a different frequency,
a different polarization, or a different antenna. The goal is to send two or more
copies of the signal through independent fades. Then, since it is less likely to have
all the independent paths in deep fades, using appropriate combining methods, the
probability of error will be lower.
When different time slots are used for diversity, it is called temporal diversity
[129]. Two time intervals separated for more than the coherence time of the channel
go through independent fades. Therefore, we may send copies of the transmitted
signal from these separated time slots. Error-correcting codes can be utilized to
reduce the amount of redundancy. In other words, sending a copy of the signal from
different time slots is equivalent to using a repetition code. More efficient error-
correcting codes may be used as well. If the fading is slow, that is the coherence time
of the channel is large, the separation between time slots used for temporal diversity
is high. In this case, the receiver suffers from a huge delay before it can start the
process of decoding. The coded symbols are interleaved before sending through the
channel. While interleaving increases the delay, it converts a slow fading channel
to a fast fading channel that is more appropriate for temporal diversity. Temporal
diversity is not bandwidth efficient because of the underlying redundancy.
Another method of diversity is frequency diversity [8]. Frequency diversity uses
different carrier frequencies to achieve diversity. The signal copies are transmit-
ted from different carrier frequencies. To achieve diversity, the carrier frequencies
should be separated by more than the coherence bandwidth of the channel. In this
case, different replicas of the signal experience independent fades. Similar to tem-
poral diversity, frequency diversity suffers from bandwidth deficiency. Also the
receiver needs to tune to different carrier frequencies.
One method of diversity that may not suffer from bandwidth deficiency is spa-
tial diversity or antenna diversity [152]. Spatial diversity uses multiple antennas to
achieve diversity. Multiple antennas may be used at the receiver or transmitter. If
the antennas are separate enough, more than half of the wavelength, signals cor-
responding to different antennas fade independently. The use of multiple antennas](https://image.slidesharecdn.com/2351772-250527055320-4826c58a/85/Spacetime-Coding-Theory-And-Practice-Hamid-Jafarkhani-38-320.jpg)
![1.5 Diversity 19
where N0/2 is the variance of the real and imaginary parts of the complex Gaussian
noise. To maximize this likelihood function, the receiver needs to find the optimal
transmitted signal, ŝ, that minimizes
M
m=1|rm − sαm|2
. Note that with no diversity,
M = 1, the cost function to minimize is |r1 − sα1|2
or |r − sα|2
for simplicity. This
is equivalent to finding ŝ, among all possible transmitted signals, that is closest to
rα∗
. For a constellation with equal energy symbols, for example PSK, we have
ŝ = arg min
s
M
m=1
|rm − sαm|2
= arg min
s
−s
M
m=1
αmr∗
m − s∗
M
m=1
α∗
mrm
= arg min
s
M
m=1
rmα∗
m − s
2
. (1.22)
Therefore, the ML decoding is similar to that of the system with no diversity if
instead of rα∗
we use a weighted average of the received signals,
M
m=1rmα∗
m. This
is called maximum ratio combining (MRC). To summarize, MRC uses a matched
filter, that is optimum receiver, for each received signal and using the optimal
weights ωm = α∗
m combines the outputs of the matched filters. If the average power
of the transmitted symbol is Es, the SNR of the mth receiver is γm = |αm|2
(Es/N0).
To derive the SNR of the output of the maximum ratio combiner, first we calculate
M
m=1
rmα∗
m =
M
m=1
(αms + ηm)α∗
m =
M
m=1
|αm|2
s +
M
m=1
ηmα∗
m. (1.23)
Then, the SNR at the output of the maximum ratio combiner is
γ =
M
m=1
|αm|2
2
Es
M
m=1
|αm|2 N0
=
M
m=1
|αm|2 Es
N0
=
M
m=1
γm. (1.24)
Therefore, the effective receive SNR of a system with diversity M is equivalent to
the sum of the receive SNRs for M different paths. The importance of this M-fold
increase in SNR is in the relationship between the average error probability and the
average receive SNR. Let us assume that all different paths have the same average
SNR, that is E[γm] = A. Then, using (1.24), the average SNR at the output of the
maximum ratio combiner is
γ = M A. (1.25)
This M-fold increase in the receive average SNR results in a diversity gain of M. It
can be shown that this is the maximum possible diversity gain when M copies of
the signal are available in a Rayleigh fading channel.](https://image.slidesharecdn.com/2351772-250527055320-4826c58a/85/Spacetime-Coding-Theory-And-Practice-Hamid-Jafarkhani-41-320.jpg)
![20 Introduction
Increasing the effective receive SNR using MRC affects the probability of error
at the receiver. For a system with no diversity, the average error probability is
proportional to the inverse of the SNR, A−1
, at high SNRs [100]. Since each of
the M paths follows a Rayleigh fading distribution, the average error probabil-
ity of a system with M independent Rayleigh paths is proportional to A−M
. As
we defined before, the ratio M in the exponent of the receive SNR is called the
diversity gain. Therefore, using MRC we achieve a diversity gain equal to the
number of available independent paths. We provide a rigorous proof of this fact in
Chapter 4.
Equal gain combining is a special case of maximum ratio combining with equal
weights. In equal gain combining, co-phased signals are utilized with unit weights.
The average SNR at the output of the equal gain combiner is
γ =
1 +
π
4
(M − 1)
A. (1.26)
1.5.2.2 Selection combining
Using MRC, when the source of M independent signals is the receive antennas,
the receiver needs to demodulate all M receive signals. In other words, M radio
frequency (RF) chains are required at the receiver to provide the baseband signals.
Since most RF chains are implemented by analog circuits, usually their physical
size and price are high. In some applications, there is not enough room for several
RF chains or their price does not justify the gain achieved by MRC. Therefore, it
may be beneficial to design a combiner that uses only one RF chain.
Selection combining or antenna selection picks the signal with the highest SNR
and uses it for decoding. Picking the signal is equivalent to choosing the correspond-
ing antenna among all receive antennas. Equivalently, it is the same as selecting
the best polarization in the case of polarization diversity. As before, let us assume
M replicas of the transmitted signal, for example through M receive antennas. As
shown before, if the fading is Rayleigh, the random variable γm, the SNR of the mth
antenna, follows an exponential distribution. With a slight abuse of the notation,
the pdf of γm for m = 1, 2, . . . , M is
fγm
(γ ) =
1
E[γm]
e− γ
E[γm] , γ 0, (1.27)
where E[γm] is the average SNR of the mth receive signal. Let us assume that all
receivesignalshavethesameaverageSNR,thatis E[γm] = A.Then,thecumulative
distribution function (CDF) of the mth receive SNR is
Fγm
(γ ) = P[γm ≤ γ ] = 1 − e− γ
A , m = 1, 2, . . . , M. (1.28)](https://image.slidesharecdn.com/2351772-250527055320-4826c58a/85/Spacetime-Coding-Theory-And-Practice-Hamid-Jafarkhani-42-320.jpg)
![1.5 Diversity 21
Since different receive signals are independent from each other, the probability
that all of them have an SNR smaller than γ is
P [γ1, γ2, . . . , γM ≤ γ ] =
1 − e− γ
A
M
. (1.29)
On the other hand, the probability that at least one receive signal achieves an SNR
greater than γ , denoted by PM (γ ), is
PM (γ ) = 1 −
1 − e− γ
A
M
. (1.30)
The corresponding pdf is
f (γ ) =
M
A
1 − e− γ
A
M−1
e− γ
A . (1.31)
Therefore, the average SNR at the output of the selection combiner, γ , is
γ =
∞
0
γ f (γ ) dγ = A
M
m=1
1
m
. (1.32)
As a result, without increasing the transmission power, selection combining offers
M
m=1
1
m
times improvement in the average SNR. This is less than the maximum
improvementratioof M.Thereforeselectioncombiningdoesnotprovideanoptimal
diversity gain and as a result an optimal performance enhancement. However, its
complexity is low since it only requires one RF chain. In other words, selection
combining provides a trade-off between complexity and performance.
In selection combining, the receiver needs to find the strongest signal at each
time instant. To avoid the monitoring of the received SNRs, one may use scanning
selection combining which is a special case of selection combining. In scanning
selection combining, first, the M receive signals are scanned to find the highest
SNR. The corresponding signal is used until its SNR is below a predetermined
threshold.Then,anewselectionisdoneandtheprocessiscontinued.Inotherwords,
a new selection is needed only if the selected signal goes through a deep fade.
Figure 1.8 compares the SNR gains of different combining methods using 1 to
10 receive antennas. As expected, MRC provides the highest gain while requiring
the highest complexity. For a higher number of receive antennas, the gap between
the MRC and selection combining grows. This is in a trade-off with the lower
complexity of selection combining with only one RF chain. It is possible to use a
number of RF chains that is neither one nor M for more than two receive antennas.
Let us assume that the receiver contains J RF chains where 1 J M and M 2.
Then, the receiver chooses J receive signals with the highest SNR and combines
them using MRC. The block diagram of such a hybrid combining method is shown
in Figure 1.9. The instantaneous SNR at the output of the hybrid selection/maximal](https://image.slidesharecdn.com/2351772-250527055320-4826c58a/85/Spacetime-Coding-Theory-And-Practice-Hamid-Jafarkhani-43-320.jpg)
![22 Introduction
1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
Number of receive antennas
SNR
gain
(dB)
Maximum Ratio Combining
Equal Gain Combining
Selection Combining
Hybrid Combining with 2 RF chains
Fig. 1.8. Comparing the gain of different combining methods.
Fading
signals
RF
chain Maximum
ratio
combiner
Select
RF
chain
Fig. 1.9. Block diagram of hybrid selection/maximum ratio combining.
ratio combining is
γ =
J
j=1
γj , (1.33)
where γj is the SNR of the jth selected signal [150]. The average SNR at the output
of the hybrid selection/maximal ratio combiner, γ , is
γ = AJ 1 +
M
m=J+1
1
m
. (1.34)
In Figure 1.8, we also show the SNR gain for a hybrid selection/maximal ratio
combiner with J = 2 RF chains. As depicted in the figure, using more RF chains
results in a higher gain. For a small number of receive antennas, a hybrid combiner](https://image.slidesharecdn.com/2351772-250527055320-4826c58a/85/Spacetime-Coding-Theory-And-Practice-Hamid-Jafarkhani-44-320.jpg)
![1.6 Spatial multiplexing gain and its trade-off with diversity 23
with only J = 2 RF chains provides most of the gain. For a higher number of
receive antennas, the gap between the hybrid combiner and MRC increases and
more RF chains are needed to close the gap.
1.6 Spatial multiplexing gain and its trade-off with diversity
In the last section, we mostly concentrated on diversity gains that are achieved
by using multiple receive antennas. In a multiple-input multiple-output (MIMO)
channel,diversitygainmaybeachievedbyusingbothtransmitandreceiveantennas.
In the rest of this book, we will investigate how to use multiple transmit antennas
to achieve high levels of diversity. However, multiple transmit antennas can be
utilized to achieve other goals as well. For example, a higher capacity and as a
result a higher transmission rate is possible by increasing the number of transmit
antennas. Let us, for the sake of simplicity, assume a MIMO channel with equal
number of transmit and receive antennas. Then, as we will show in Chapter 2, in
a rich scattering environment the capacity increases linearly with the number of
antennas without increasing the transmission power. This results in the possibility
of transmitting at a higher rate, for example by using spatial multiplexing. If the
number of transmit antennas is not the same as the number of receive antennas, in
general one can transmit up to min{N, M} symbols per time slot, where N is the
number of transmit antennas and M is the number of receive antennas. For example,
if M ≥ N, one can send N symbols and achieve a diversity gain of M − N + 1.
Note that for equal number of transmit and receive antennas, the diversity gain in
this case is one. On the other hand, the maximum spatial diversity while transmitting
only one symbol per time slot is M N. Therefore, the advantage of a MIMO channel
can be utilized in two ways: (i) to increase the diversity of the system and (ii) to
increase the number of transmitted symbols. For the general case of more than
one transmit antenna, M ≥ N 1, there is a theoretical trade-off between the
number of transmit symbols and the diversity of the system. For one transmit
antenna, in both cases one symbol per time slot is transmitted and the two systems
coincide.
Inmanycases,spatialmultiplexinggainreferstothefactthatonecanusemultiple
antennas to transmit at a higher rate compared to the case of one antenna. As we
will show in Chapter 2, the capacity of a MIMO channel increases by raising the
SNR. Since the transmission rate relates to capacity, it is reasonable to hope that
the rate can be increased as the SNR increases. This argument has resulted in the
following definition for spatial multiplexing gain in [159]:
SMG = lim
γ →∞
r
log(γ )
, (1.35)](https://image.slidesharecdn.com/2351772-250527055320-4826c58a/85/Spacetime-Coding-Theory-And-Practice-Hamid-Jafarkhani-45-320.jpg)
![24 Introduction
0 1 2 3
0
2
4
6
8
10
12
Spatial multiplexing gain
Diversity
Optimal trade-off
Fig. 1.10. Optimal trade-off between spatial multiplexing gain (an indication of
rate) and diversity.
where r is the rate of the code, at the transmitter, in bits/channel use and is a
function of SNR. Note that the relationship of the above spatial multiplexing gain
to the transmission rate is similar to that of the diversity gain to the probability
of error in (1.19). The main rationale behind such a rate normalization is the fact
that SMG measures how far the rate r is from the capacity. Note that the range
of the spatial multiplexing gain is from 0 to min{N, M}. The following theorem
from [159] derives the optimal trade-off between spatial multiplexing gain, which
is an indication of the rate, and diversity, which is an indication of the probability
of error
Theorem 1.6.1 Let us assume a code at the transmitter of a MIMO channel with N
transmit antennas and M receive antennas. For a given spatial multiplexing gain
SMG = i, where i = 0, 1, . . . , min{N, M} is an integer, the maximum diversity
gain Gd(i) is given by Gd(i) = (N − i)(M − i) if the block length of the code is
greater than or equal to N + M − 1. The optimal trade-off curve is achieved by
connecting the points (i, Gd(i)) by lines.
An example of the optimal trade-off for N = 4 transmit antennas and M = 3
receive antennas is depicted in Figure 1.10.
1.7 Closed-loop versus open-loop systems
When the transmitter does not have any information about the channel, the system
is an “open-loop” system. In this case, the receiver may estimate the channel and](https://image.slidesharecdn.com/2351772-250527055320-4826c58a/85/Spacetime-Coding-Theory-And-Practice-Hamid-Jafarkhani-46-320.jpg)
![1.7 Closed-loop versus open-loop systems 25
Fig. 1.11. Block diagram of (a) open-loop and (b) closed-loop multiple-input
multiple-output systems.
use the channel state information (CSI) for decoding. However, the transmitter
does not have access to the channel state information. On the contrary, in some sys-
tems, the receiver sends some information about the channel back to the transmitter
through a feedback channel. This is called a “closed-loop” system and the trans-
mitter can use this information to improve the performance. The block diagrams of
open-loop and closed-loop systems are depicted in Figure 1.11.
In a time division duplexing (TDD) system, the radio channel is shared between
the mobile and base station. In other words, the same frequency, the same chan-
nel, is utilized to transmit information in both directions using time sharing. As
a result, the characteristics of the uplink channel, from mobile to base station,
and the downlink channel, from base station to mobile, are similar. One possi-
ble assumption in such a TDD system is to consider the uplink and downlink
channels as the reciprocal of each other. If this is a valid assumption, the channel
estimation at the receiver can be utilized when transmitting back in a TDD sys-
tem. Therefore, a closed-loop system can be used without sending any feedback
information.
In a closed-loop system, the gain achieved by processing at the transmitter and
the receiver is sometimes called “array gain.” Similar to the diversity gain, the array
gain results in an increase in the average receive SNR. If the transmitter knows the
channel perfectly, beamforming [86] is the best solution. In fact, in such a case
there is no need for spatial diversity. On the other hand, in most practical situations,
there is a limited feedback information available which may not be perfect due
to the quantization error or other factors. With such limited channel information,
spatial diversity is still useful. The transmitted codeword can be tuned to better
fit the channel based on the received information in the closed-loop system. To](https://image.slidesharecdn.com/2351772-250527055320-4826c58a/85/Spacetime-Coding-Theory-And-Practice-Hamid-Jafarkhani-47-320.jpg)
![26 Introduction
maintain low implementation complexity, a simple linear beamforming scheme at
the transmitter is preferred. If non-perfect channel state information is available, a
combined space-time coding and beamforming system is utilized [76, 87].
1.8 Historical notes on transmit diversity
Transmit diversity, a form of spatial diversity, has been studied extensively as a
method of combating detrimental effects in wireless fading channels [4, 42, 51,
96, 102, 110, 135, 136, 139, 140, 153, 154]. The use of multiple transmit anten-
nas for diversity provides better performance without increasing the bandwidth or
transmission power. The first bandwidth-efficient transmit diversity scheme was
proposed in [153]. It includes the delay diversity scheme of [110] as a special case.
It is proved that the diversity advantage of this scheme is equal to the number of
transmit antennas which is optimal [151]. Later, a multilayered space-time architec-
ture was introduced in [43]. The scheme proposed in [43] uses spatial multiplexing
to increase the data rate and not necessarily provides transmit diversity. Simple
iterative decoding algorithms that have been proposed in conjunction with spatial
multiplexing can achieve spatial diversity, mostly receive diversity. The criterion to
achieve the maximum transmit diversity was derived in [51]. A complete study of
design criteria for maximum diversity and coding gains in addition to the design of
space-time trellis codes was proposed in [139]. It includes the delay diversity as a
special case. Information theoretical results in [140] and [42] showed that there is a
huge advantage in utilizing spatial diversity. The introduction of space-time block
coding in [136] provided a theoretical framework that started an increasing interest
on the subject.
The best use of multiple transmit antennas depends on the amount of channel
state information that is available to the encoder and decoder. In a flat fading chan-
nel, the channel has a constant gain and linear phase response over a bandwidth
which is greater than the bandwidth of the transmitted signal. Let us assume that
the channel does not change rapidly during the transmission of a frame of data.
The combination of the above two assumptions results in a quasi-static flat fading
channel model [42]. In most practical cases, the system estimates the channel at
the receiver by transmitting some known pilot signals. The receiver utilizes the cor-
responding received signals to estimate the channel. In such a system, a coherent
detection is utilized in which the decoder uses the value of the path gains estimated
at the receiver [138]. In all references that we have mentioned so far in this section,
the codes are designed for the case that the receiver knows the channel. In some
other cases, such an estimation of the channel is not available at the receiver or
the channel changes rapidly such that the channel estimation is not useful. Then, a
noncoherent detection needs to be employed. For one transmit antenna, differential](https://image.slidesharecdn.com/2351772-250527055320-4826c58a/85/Spacetime-Coding-Theory-And-Practice-Hamid-Jafarkhani-48-320.jpg)
![1.9 Summary 27
detection schemes exist that neither require the knowledge of the channel nor
employpilotsymboltransmission.Apartialsolutiontogeneralizedifferentialdetec-
tion schemes to the case of multiple transmit antennas was proposed in [132]. This
was a joint channel- and data-estimation scheme that can lead to error propagation.
Noncoherent detection schemes based on unitary space-time codes were proposed
in [59, 60]. The first differential decoding modulation scheme for multiple antennas
that provides simple encoding and decoding based on orthogonal designs was pro-
posed in [73, 133]. Another construction based on group codes followed in [61, 63].
1.9 Summary
r Line of Sight: A direct path between the transmitter and the receiver.
r Reflection: When the electromagnetic wave meets an object that is much larger than the
wavelength.
r Diffraction: When the electromagnetic wave hits a surface with irregularities like sharp
edges.
r Scattering: When the medium through which the electromagnetic wave propagates con-
tains a large number of objects smaller than the wavelength.
r Attenuation or path loss (sometimes called large-scale fading) is due to propagation losses,
filter losses, antenna losses, and so on.
r Fading is used to describe the rapid fluctuation of the amplitude of a radio signal over a
short period of time or travel distance, so that large-scale path loss effects may be ignored.
Fading is a time-varying phenomenon.
r Multipath: The presence of reflecting objects and scatterers creates a constantly changing
environment that dissipates the signal energy in amplitude, phase, and time. A multipath
channel can be modeled as a linear time-varying channel.
r Flat Slow Fading or Frequency Non-Selective Slow Fading: When the bandwidth of the
signal is smaller than the coherence bandwidth of the channel and the signal duration is
smaller than the coherence time of the channel.
r Flat Fast Fading or Frequency Non-Selective Fast Fading: When the bandwidth of the
signal is smaller than the coherence bandwidth of the channel and the signal duration is
larger than the coherence time of the channel.
r Frequency Selective Slow Fading: When the bandwidth of the signal is larger than the
coherence bandwidth of the channel and the signal duration is smaller than the coherence
time of the channel.
r Frequency Selective Fast Fading: When the bandwidth of the signal is larger than the
coherence bandwidth of the channel and the signal duration is larger than the coherence
time of the channel.
r In a Rayleigh fading model, the input–output relationship between the baseband signals
is
rt = αst + ηt ,](https://image.slidesharecdn.com/2351772-250527055320-4826c58a/85/Spacetime-Coding-Theory-And-Practice-Hamid-Jafarkhani-49-320.jpg)
![28 Introduction
where α is a complex Gaussian random variable. In other words, the real and imagi-
nary parts of the fade coefficient α are real zero-mean Gaussian random variables. The
amplitude of the fade coefficient, |α|, is a Rayleigh random variable.
r Diversity provides a less-attenuated replica of the transmitted signal to the receiver. Diver-
sity is defined as
Gd = − lim
γ →∞
log(Pe)
log(γ )
,
where Pe is the error probability at a signal-to-noise ratio equal to γ .
r Diversity methods include temporal diversity, frequency diversity, spatial diversity, angle
diversity, polarization diversity, and so on.
r The multiple versions of the signals created by different diversity schemes need to be
combined to improve the performance. Two main combining methods that are utilized at
the receiver are maximum ratio combining and selection combining.
r For a given spatial multiplexing gain SMG = i, where i = 0, 1, . . . , min{N, M} is an
integer, the maximum diversity gain Gd(i) is given by Gd(i) = (N − i)(M − i) if the
block length of the code is greater than or equal to N + M − 1.
1.10 Problems
1 A multipath channel includes five paths with powers 0.1, 0.01, 0.02, 0.001, and 0.5 W.
The corresponding delays are 0.1, 0.2, 0.3, 0.4, and 0.5 s, respectively. If the signal
bandwidth is 200 kHz, is the channel flat or frequency selective?
2 A mobile is moving at a speed of 100 km/h. The transmitted frequency is 900 MHz while
the signal bandwidth is 300 kHz. Is the channel slow or fast?
3 Consider a Rayleigh random variable, R, that is constructed as the envelope of two iid
zero-mean unit-variance Gaussian random variables.
(a) What is the probability that R 0.1?
(b) If two independent random variables R1 and R2 have the Rayleigh distribution of R,
what is the probability that R1 0.1 and R2 0.1?
4 Another popular model for the distribution of the envelope random variable, R, is a
Nakagami distribution with the following pdf:
fR(r) =
2mm
r2m−1
(m)Am
exp
−mr2
A
, r ≥ 0, m ≥ 0.5,
where A = E[R2
] and (·) is the gamma function.
(a) Show that for m = 1, the Nakagami distribution converges to a Rayleigh distribution.
(b) Draw the pdf of a Nakagami distribution for A = 1 and different values of m =
0.5, 1, 1.5, 2, 10.
5 Consider a system that receives M replicas of the transmitted signal through M inde-
pendent paths. The pdf of γm for m = 1, 2, . . . , M follows (1.27) and all receive signals
have the same average SNR, E[γm] = 1. Draw the pdf of the SNR at the output of the
selection combiner and the maximum ratio combiner for M = 1, 2, 3, 4.](https://image.slidesharecdn.com/2351772-250527055320-4826c58a/85/Spacetime-Coding-Theory-And-Practice-Hamid-Jafarkhani-50-320.jpg)
![1.10 Problems 29
6 For BPSK, the probability of error for a SNR of γ is 0.5 erfc(
√
γ ), where erfc is the
complementary error function:
erfc(x) =
2
√
π
∞
x
exp(−t2
) dt.
Let us assume M = 2 independent paths that have the same average SNR, E[γm] = 1
such that the pdf of γm for m = 1, 2 follows (1.27). What is the probability of error at
the output of the selection combiner and the maximum ratio combiner?](https://image.slidesharecdn.com/2351772-250527055320-4826c58a/85/Spacetime-Coding-Theory-And-Practice-Hamid-Jafarkhani-51-320.jpg)
![2.1 Transmission model for multiple-input multiple-output channels 31
1
@
@
@
@
@
@
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
2
A
A
A
A
A
A
A
A
A
A
A
A
N-1
α1,1 α1,2 · · · α1,M
α2,1 α2,2 · · · α2,M
.
.
.
αN−1,1 αN−1,2 · · · αN−1,M
N
αN,1 αN,2 · · · αN,M
1
2
.
.
.
M
Fig. 2.1. A multiple-input multiple-output (MIMO) channel.
correspond to two practical scenarios. First, we assume a quasi-static channel,
where the path gains are constant over a frame of length T and change from frame
to frame. In most cases, we assume that the path gains vary independently from
one frame to another. Another assumption is to consider a correlation between the
fades in adjacent time samples. One popular example of such a second-order model
is the Jakes model [74].
The value of T dictates the slow or fast nature of the fading. If a block of data
is transmitted over a time frame T that is smaller than T , the fading is slow. In this
case, the fades do not change during the transmission of one block of data and the
values of path gains in (2.1) are constant for every frame. On the other hand, in a
fast fading model, the path gains may change during the transmission of one frame
of data, T T .
To form a more compact input-output relationship, we collect the signals that
are transmitted from N transmit antennas during T time slots in a T × N matrix,
C, as follows:
C =
⎛
⎜
⎜
⎜
⎝
C1,1 C1,2 · · · C1,N
C2,1 C2,2 · · · C2,N
.
.
.
.
.
.
...
.
.
.
CT,1 CT,2 · · · CT,N
⎞
⎟
⎟
⎟
⎠
. (2.2)](https://image.slidesharecdn.com/2351772-250527055320-4826c58a/85/Spacetime-Coding-Theory-And-Practice-Hamid-Jafarkhani-53-320.jpg)
![part of it into a State, while the remaining section was not made a
State, without the consent of the Territory; he conceived that
Congress must, in such event, form this section also into a State.
He, therefore, was of opinion that Congress must consult the people
of the Territory before they shall divide the Territory.
As to the expediency of the resolution, he thought it very expedient
to make the division therein marked out. The effect of it would be
that the whole of Lake Erie would be thrown out of the State to be
formed, and the inconvenience to the section of the Territory not
incorporated in the new State would be very great, if it should be
attached to the Indiana Territory, from its great distance, which he
understood was contemplated.
Mr. Giles said that the committee who reported these resolutions, so
far from entertaining a disposition to change the ordinance, had
strictly observed the conditions therein prescribed. [Mr. G. here
quoted the ordinance.] It appeared therefrom that Congress was
under an obligation, after laying off one State, to form the remainder
into a State. But when? Hereafter, whenever they shall think it
expedient to do so.
Mr. Bayard agreed that there was no obligation imposed upon
Congress to decide definitively the boundary of a State. If the
ultimate right of Congress, after the formation of a new State, to
alter the boundary be doubted, they have a right to remove all
doubts by so declaring at this time. It is certain that at present great
inconvenience would arise from drawing the boundary as fixed in the
resolution.
The population of the Territory does not amount to that which is
sufficient to give it admission into the Union. He had, however, no
disposition to oppose its admission, notwithstanding this
circumstance. The population in the Eastern State does not exceed
forty-five thousand. We are now about to pare off five or six
thousand inhabitants, which will bring it down to thirty-nine
thousand. A population of forty-five thousand is quite small enough
for an independent State. It is a smaller population than exists in](https://image.slidesharecdn.com/2351772-250527055320-4826c58a/85/Spacetime-Coding-Theory-And-Practice-Hamid-Jafarkhani-56-320.jpg)
![Charles Johnson, Samuel L. Mitchill, Thomas Moore, Anthony New,
Thomas Newton, jr., Joseph H. Nicholson, John Smilie, Israel Smith,
John Smith, (of Virginia,) Samuel Smith, John Taliaferro, jr., David
Thomas, Philip R. Thompson, Abram Trigg, Isaac Van Horne, and
Robert Williams.
The bill was then ordered to be engrossed for a third reading to-
morrow.
Friday, April 9.
A message from the Senate informed the House that the Senate
have passed a bill, entitled An act to amend the Judicial System of
the United States; to which they desire the concurrence of this
House.
[The chief alterations made from the old system consist in the
holding the Supreme Court only once a year by four justices, and the
establishment of six circuits, within each district of which circuit
courts are to be holden twice a year, composed of one justice of the
Supreme Court and the judge of the district, in which said court is
held.]
The bill was read twice, and referred to a select committee.
Ohio State Government.
An engrossed bill to enable the people of the Eastern division of the
Territory north-west of the river Ohio to form a constitution and
State Government, and for the admission of such State into the
Union on an equal footing with the original States, and for other
purposes, was read the third time, and the blanks therein filled up:
And, on the question that the same do pass, it was resolved in the
affirmative—yeas 47, nays 29, as follows:
Yeas.—Willis Alston, John Archer, John Bacon, Theodorus Bailey,
Phanuel Bishop, Richard Brent, Robert Brown, William Butler, Samuel
J. Cabell, Thomas Claiborne, Matthew Clay, John Clopton, John
Condit, Thomas T. Davis, John Dawson, William Dickson, Lucas](https://image.slidesharecdn.com/2351772-250527055320-4826c58a/85/Spacetime-Coding-Theory-And-Practice-Hamid-Jafarkhani-72-320.jpg)
Spacetime Coding Theory And Practice Hamid Jafarkhani
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- 7. SPACE-TIME CODING: THEORY AND PRACTICE This book covers the fundamental principles of space-time coding for wireless communications over multiple-input multiple-output (MIMO) channels, and sets out practical coding methods for achieving the performance improvements pre- dicted by the theory. Starting with background material on wireless communications and the capacity of MIMO channels, the book then reviews design criteria for space-time codes. A detailedtreatmentofthetheorybehindspace-timeblockcodesthenleadsontoanin- depth discussion of space-time trellis codes. The book continues with discussion of differential space-time modulation, BLAST and some other space-time processing methods. The final chapter addresses additional topics in space-time coding. Written by one of the inventors of space-time block coding, this book is ideal for a graduate student familiar with the basics of digital communications, and for engineers implementing the theory in real systems. The theory and practice sections can be used independently of each other. Exer- cises can be found at the end of the chapters. Hamid Jafarkhani is an associate professor in the Department of Electrical Engineering and Computer Science at the University of California, Irvine, where he is also the Deputy Director of the Center for Pervasive Communications and Computing. Before this, he worked for many years at AT&T Labs-Research where he was one of the co-inventors of space-time block coding.
- 9. SPACE-TIME CODING THEORY AND PRACTICE HAMID JAFARKHANI University of California, Irvine
- 10. cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge cb2 2ru, UK First published in print format isbn-13 978-0-521-84291-4 isbn-13 978-0-511-11562-2 © Cambridge University Press 2005 2005 Information on this title: www.cambridge.org/9780521842914 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. isbn-10 0-511-11562-8 isbn-10 0-521-84291-3 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. Published in the United States of America by Cambridge University Press, New York www.cambridge.org hardback eBook (NetLibrary) eBook (NetLibrary) hardback
- 11. Contents Preface page ix Standard notation xi Space-time coding notation xii List of abbreviations xiv 1 Introduction 1 1.1 Introduction to the book 1 1.2 Wireless applications 2 1.3 Wireless channels 5 1.4 Statistical models for fading channels 12 1.5 Diversity 15 1.6 Spatial multiplexing gain and its trade-off with diversity 23 1.7 Closed-loop versus open-loop systems 24 1.8 Historical notes on transmit diversity 26 1.9 Summary 27 1.10 Problems 28 2 Capacity of multiple-input multiple-output channels 30 2.1 Transmission model for multiple-input multiple-output channels 30 2.2 Capacity of MIMO channels 34 2.3 Outage capacity 39 2.4 Summary of important results 43 2.5 Problems 44 3 Space-time code design criteria 45 3.1 Background 45 3.2 Rank and determinant criteria 46 3.3 Trace criterion 50 3.4 Maximum mutual information criterion 52 v
- 12. vi Contents 3.5 Summary of important results 53 3.6 Problems 53 4 Orthogonal space-time block codes 55 4.1 Introduction 55 4.2 Alamouti code 55 4.3 Maximum-likelihood decoding and maximum ratio combining 59 4.4 Real orthogonal designs 60 4.5 Generalized real orthogonal designs 70 4.6 Complex orthogonal designs 76 4.7 Generalized complex orthogonal designs 77 4.8 Pseudo-orthogonal space-time block codes 90 4.9 Performance analysis 93 4.10 Simulation results 101 4.11 Summary of important results 107 4.12 Problems 108 5 Quasi-orthogonal space-time block codes 110 5.1 Pairwise decoding 110 5.2 Rotated QOSTBCs 112 5.3 Optimal rotation and performance of QOSTBCs 115 5.4 Other examples of QOSTBCs 120 5.5 QOSTBCs for other than four transmit antennas 122 5.6 Summary of important results 125 5.7 Problems 125 6 Space-time trellis codes 126 6.1 Introduction 126 6.2 Space-time trellis coding 130 6.3 Improved STTCs 136 6.4 Performance of STTCs 143 6.5 Simulation results 145 6.6 Summary of important results 149 6.7 Problems 149 7 Super-orthogonal space-time trellis codes 151 7.1 Motivation 151 7.2 Super-orthogonal codes 153 7.3 CGD analysis 167 7.4 Encoding and decoding 173 7.5 Performance analysis 179 7.6 Extension to more than two antennas 181 7.7 Simulation results 187
- 13. Contents vii 7.8 Summary of important results 192 7.9 Problems 193 8 Differential space-time modulation 195 8.1 Introduction 195 8.2 Differential encoding 198 8.3 Differential decoding 203 8.4 Extension to more than two antennas 207 8.5 Simulation results 214 8.6 Summary of important results 218 8.7 Problems 218 9 Spatial multiplexing and receiver design 221 9.1 Introduction 221 9.2 Spatial multiplexing 222 9.3 Sphere decoding 223 9.4 Using equalization techniques in receiver design 227 9.5 V-BLAST 230 9.6 D-BLAST 235 9.7 Turbo-BLAST 238 9.8 Combined spatial multiplexing and space-time coding 241 9.9 Simulation results 246 9.10 Summary of important results 249 9.11 Problems 249 10 Non-orthogonal space-time block codes 251 10.1 Introduction 251 10.2 Linear dispersion space-time block codes 252 10.3 Space-time block codes using number theory 259 10.4 Threaded algebraic space-time codes 260 10.5 Simulation results 267 10.6 Summary of important results 270 10.7 Problems 270 11 Additional topics in space-time coding 272 11.1 MIMO-OFDM 272 11.2 Implementation issues in MIMO-OFDM 278 11.3 Space-time turbo codes 283 11.4 Beamforming and space-time coding 286 11.5 Problems 290 References 291 Index 300
- 15. Preface The use of multiple antennas in most future wireless communication systems seems to be inevitable. Today, the main question is how to include multiple antennas and what are the appropriate methods for specific applications. The academic interest in space-time coding and multiple-input multiple-output (MIMO) systems has been growing for the last few years. Recently, the industry has shown a lot of interest as well. It is amazing how fast the topic has emerged from a theoretical curiosity to the practice of every engineer in the field. It was just a few years ago, when I started working at AT&T Labs – Research, that many would ask “who would use more than one antenna in a real system?” Today, such skepticism is gone. The fast growth of the interest and activities on space-time coding has resulted in aspectrumofpeoplewhoactivelyfollowthefield.Therangespansfrommathemati- cians who are only curious about the interesting mathematics behind space-time coding to engineers who want to build it. There is a need for covering both the theory and practice of space-time coding in depth. This book hopes to fulfill this need. The book has been written as a textbook for first-year graduate students and as a reference for the engineers who want to learn the subject from scratch. An early version of the book has been used as a textbook to teach a course in space- time coding at the University of California, Irvine. The goal of such a course is the introduction of space-time coding to anyone with some basic knowledge of dig- ital communications. In most cases, we start with common ideas for single-input single-output (SISO) channels and extend them to MIMO channels. Therefore, stu- dents or engineers with no knowledge of MIMO systems should be able to learn all the concepts. While graduate students might be interested in all the details and the proofs of theorems and lemmas, engineers may skip the proofs and concentrate on the results without sacrificing the continuity of the presentation. A typical course on space-time coding may start with some background material onwirelesscommunicationsandcapacityofMIMOchannelsascoveredinChapters 1 and 2. A review of design criteria for space-time codes is covered in Chapter 3. ix
- 16. x Preface Chapters 4 and 5 provide a detailed treatment of the theory behind space-time block codes. A practitioner who is only interested in the structure of the codes may bypass all the proofs in these chapters and concentrate only on the examples. Chapters 6 and 7 discuss space-time trellis codes in depth. Each chapter includes discussions on the performance analysis of the codes and simulation results. For those who are more interested in the practical aspects of the topic, simulation results are sufficient and the sections on performance analysis may be skipped. The practitioners may continue with Chapter 11 and its discussion on MIMO-OFDM and Chapter 9 on receiver design. On the other hand, those who are more interested in the theory of space-time codes can follow with Chapter 8 and its treatment of differential space-time modulation. Finally, for the sake of completeness, we discuss BLAST and some other space-time processing methods in Chapters 9 and 10. Homework problems have been included at the end of each chapter. The book includes the contribution of many researchers. I am grateful to all of them for generating excitement in the field of space-time coding. My special thanks goes to my very good friend and former colleague Professor Vahid Tarokh who introduced space-time coding to me. Also, I should thank my other colleagues at AT&T Labs – Research who initiated most of the basic concepts and ideas in space-time coding. Without the support of my department head at AT&T Labs – Research, Dr. Behzad Shahraray, I would not be able to contribute to the topic and I am thankful to him for providing the opportunity. Also, Professor Rob Calderbank has been a big supporter of the effort. The early versions of this book have been read and reviewed by my students and others. Their comments and suggestions have improved the quality of the presentation. Especially, comments from Professor John Proakis, Professor Syed Jafar, Dr. Masoud Olfat, and Hooman Honary have resulted in many improvements. My Ph.D. students, Li Liu, Javad Kazemitabar, and Yun Zhu, have helped with the proofreading of a few chapters. Many of the presented simulation results have been double checked by Yun Zhu. I also thank the National Science Foundation (NSF) for giving me an NSF Career Award to support my research and educational goals related to this book. Last but not least, I would like to thank my wife, Paniz, for her support and love.
- 17. Standard notation || · || Euclidean norm || · ||F Frobenius norm ⊗ tensor product ∗ conjugate + Moore-Penrose generalized inverse (pseudo-inverse) det determinant of a matrix E expectation fX (x) pdf of X H Hermetian imaginary part I imaginary part IN N × N identity matrix j √ −1 KX covariance of vector X real part R real part T transpose Tr trace of a matrix Var variance xi
- 18. Space-time coding notation αn,m path gain from transmit antenna n to receive antenna m χk a chi-square RV with 2k degrees of freedom η noise φ rotation parameter for constellations γ SNR ρ(N) Radon function θ rotation parameter for STBCs b number of transmit bits per channel use C capacity Cout outage capacity C I set of complex numbers C T × N transmitted codeword C set of super-orthogonal codes dmin minimum distance Es average power of transmitted symbols fd Doppler shift Gc coding gain Gd diversity gain G generating matrix for a STTC G generator matrix for a STBC H N × M channel matrix I number of states in a trellis J number of delta functions in the impulse response of a frequency selective fading channel J number of groups in a combined spatial multiplexing and space-time cod- ing system 2l number of branches leaving each state of a trellis K number of transmitted symbols per block xii
- 19. Space-time coding notation xiii L number of orthogonal (data) blocks in a SOSTTC L size of IFFT and FFT blocks in OFDM L-PSK a PSK constellation with L = 2b symbols M number of receive antennas N number of transmit antennas N T × M noise matrix N0 noise samples have a variance of N0/2 per complex dimension P number of trellis transitions (two trellis paths differ in P transitions) Pout outage probability Q the memory of a trellis r transmission rate in bits/(s Hz) r received signal r T × M received matrix R rate of a STC IR set of real numbers s transmitted signal St the state of the encoder at time t x indeterminant variable Z Z set of integers
- 20. Abbreviations ADC Analog to Digital Converter AGC Automatic Gain Control AWGN Additive White Gaussian Noise BER Bit Error Rate BLAST Bell Labs Layered Space-Time BPSK Binary Phase Shift Keying BSC Binary Symmetric Channel CCDF Complementary Cumulative Distribution Function CDF Cumulative Distribution Function CDMA Code Division Multiple Access CSI Channel State Information CT Cordless Telephone DAST Diagonal Algebraic Space-Time DASTBC Diagonal Algebraic Space-Time Block Code D-BLAST Diagonal BLAST DECT Digital Cordless Telephone DFE Decision Feedback Equalization DPSK Differential Phase Shift Keying EDGE Enhanced Data for Global Evolution FER Frame Error Rate FFT Fast Fourier Transform FIR Finite Impulse Response GSM Global System for Mobile IFFT Inverse Fast Fourier Transform iid independent identically distributed IMT International Mobile Telephone ISI Intersymbol Interference xiv
- 21. Abbreviations xv LAN Local Area Network LDSTBC Linear Dispersion Space-Time Block Code LOS Line of Sight MGF Moment Generating Function MIMO Multiple-Input Multiple-Output MISO Multiple-Input Single-Output MMAC Multimedia Mobile Access Communication MMSE Minimum Mean Squared Error MRC Maximum Ratio Combining ML Maximum-Likelihood MTCM Multiple Trellis Coded Modulation OFDM Orthogonal Frequency Division Multiplexing OSTBC Orthogonal Space-Time Block Code PAM Pulse Amplitude Modulation PAN Personal Area Network PAPR Peak-to-Average Power Ratio PDA Personal Digital Assistant PDC Personal Digital Cellular pdf probability density function PEP Pairwise Error Probability PHS Personal Handyphone System PSK Phase Shift Keying QAM Quadrature Amplitude Modulation QOSTBC Quasi-Orthogonal Space-Time Block Code QPSK Quadrature Phase Shift Keying RF Radio Frequency RLST Random Layered Space-Time RV Random Variable SER Symbol Error Rate SISO Single-Input Single-Output SIMO Single-Input Multiple-Output SM Spatial Multiplexing SNR Signal to Noise Ratio SOSTTC Super-Orthogonal Space-Time Trellis Code SQOSTTC Super-Quasi-Orthogonal Space-Time Trellis Code STBC Space-Time Block Code STTC Space-Time Trellis Code TAST Threaded Algebraic Space-Time TASTBC Threaded Algebraic Space-Time Block Code
- 22. xvi Abbreviations TCM Trellis Coded Modulation TDD Time Division Duplexing TDMA Time Division Multiple Access V-BLAST Vertical BLAST ZF Zero Forcing
- 23. 1 Introduction 1.1 Introduction to the book Recent advances in wireless communication systems have increased the throughput over wireless channels and networks. At the same time, the reliability of wireless communication has been increased. As a result, customers use wireless systems more often. The main driving force behind wireless communication is the promise of portability, mobility, and accessibility. Although wired communication brings more stability, better performance, and higher reliability, it comes with the neces- sity of being restricted to a certain location or a bounded environment. Logically, people choose freedom versus confinement. Therefore, there is a natural tendency towards getting rid of wires if possible. The users are even ready to pay a reasonable price for such a trade-off. Such a price could be a lower quality, a higher risk of disconnection, or a lower throughput, as long as the overall performance is higher than some tolerable threshold. The main issue for wireless communication systems is to make the conversion from wired systems to wireless systems more reliable and if possible transparent. While freedom is the main driving force for users, the incredible number of challenges to achieve this goal is the main motivation for research in the field. In this chapter, we present some of these challenges. We study different wireless communication applications and the behavior of wireless channels in these applications. We provide different mathematical models to char- acterize the behavior of wireless channels. We also investigate the challenges that a wireless communication system faces. Throughout the book, we provide different solutions to some of the challenges in wireless communication by using multiple antennas. The main topic of this book is how to overturn the difficulties in wireless communication by employing mul- tiple antennas. We start with a study of the capacity increase due to the use of multiple antennas. Then, we show how to design a space-time architecture for multiple transmit antennas to improve the performance of a wireless system 1
- 24. 2 Introduction while keeping the transmission power intact. Most of the book discusses differ- ent space-time coding methods in detail. The detailed discussion of each method includes design, properties, encoding, decoding, performance analysis, and simu- lation results. We pay close attention to the complexity of encoding and decoding for each method and to different trade-offs in terms of throughput, complexity, and performance. Not only do we provide the theoretical details of each method, but also we present the details of the algorithm implementation. Our overall goal is to keep a balance between the theory and the practice of space-time coding. 1.2 Wireless applications There are many systems in which wireless communication is applicable. Radio broadcasting is perhaps one of the earliest successful common applications. Tele- vision broadcasting and satellite communications are other important examples. However, the recent interest in wireless communication is perhaps inspired mostly by the establishment of the first generation cellular phones in the early 1980s. The first generation of mobile systems used analog transmission. The second generation of cellular communication systems, using digital transmission, were introduced in the 1990s. Both of these two systems were primarily designed to transmit speech. The success of cellular mobile systems and their appeal to the public resulted in a growing attention to wireless communications in industry and academia. Many researchers concentrated on improving the performance of wireless communication systems and expanding it to other sources of information like images, video, and data. Also, the industry has been actively involved in establishing new standards. As a result, many new applications have been born and the performance of the old applications has been enhanced. Personal digital cellular (PDC), global system for mobile (GSM) communications, IS-54, IS-95, and IS-136 are some of the early examples of these standards. While they support data services up to 9.6 kbits/s, they are basically designed for speech. More advanced services for up to 100 kbits/s data transmission has been evolved from these standards and are called 2.5 generation. Recently, third generation mobile systems are being considered for high bit-rate services. With multimedia transmission in mind, the third generation systems are aiming towards the transmission of 144–384 kbits/s for fast moving users and up to 2.048 Mbits/s for slow moving users. The main body of the third generation standards is known as international mobile telephone (IMT-2000). It includes the enhanced data for global evolution (EDGE) standard, which is a time division multiple access (TDMA) system and an enhance- ment of GSM. It also includes two standards based on wideband code division
- 25. 1.2 Wireless applications 3 multiple access (CDMA). One is a synchronous system called CDMA2000 and the other one is an asynchronous system named WCDMA. In addition to applications demanding higher bit rates, one can use multiple services in the third-generation standards simultaneously. This means the need for improved spectral efficiency and increased flexibility to deploy new services. There are many challenges and opportunities in achieving these goals. Of course the demand for higher bit rates does not stop with the deployment of the third-generation wireless systems. Another important application that drives the demand for high bit rates and spectral efficiency is wireless local area networks (LANs). It is widely recognized that wireless connection to the network is an inevitable part of future communication networks and systems in the emerging mobile wireless Internet. Needless to say, the design of systems with such a high spectralefficiencyisaverychallengingtask.Perhapsthemostsuccessfulstandardin this area is the IEEE 802.11 class of standards. IEEE 802.11a is based on orthogonal frequency division multiplexing (OFDM) to transmit up to 54 Mbits/s of data. It transmits over the 5 GHz unlicensed frequency band. IEEE 802.11b provides up to 11 Mbits/s over the 2.45 GHz unlicensed frequency band. IEEE 802.11g uses OFDM over the 2.45 GHz unlicensed frequency band to provide a data rate of up to 54 Mbits/s. Other examples of wireless LAN standards include high performance LAN (HiperLAN) and multimedia mobile access communication (MMAC). Both HiperLAN and MMAC use OFDM. The main purpose of a wireless LAN is to provide high-speed network connections to stationary users in buildings. This is an important application of wireless communications as it provides freedom from being physically connected, portability, and flexibility to network users. There are many other applications of wireless communications. Cordless telephone systems and wireless local loops are two important examples. Cordless telephone standards include the personal handyphone system (PHS), digital cordless telephone (DECT), and cordless telephone (CT2). Wireless personal area network (PAN) systems are utilized in applications with short distance range. IEEE 802.15 works on developing such standards. Bluetooth is a good example of how to build an ad hoc wireless network among devices that are in the vicinity of each other. The Bluetooth standard is based on frequency hop CDMA and transmits over the 2.45 GHz unlicensed frequency band. The goal of wireless PANs is to connect different portable and mobile devices such as cell phones, cordless phones, personal computers, personal digital assistants (PDAs), pagers, peripherals, and so on. The wireless PANs let these devices communicate and operate cohesively. Also, wireless PANs can replace the wire connection between different consumer electronic appliances, for example among keyboard, mouse, and computers or between television sets and cable receivers.
- 26. 4 Introduction 1.2.1 Wireless challenges While various applications have different specifications and use different wireless technologies, most of them face similar challenges. The priority of the different challenges in wireless communications may not be the same for different applica- tions; however, the list applies to almost all applications. Some of the challenges in wireless communications are: r a need for high data rates; r quality of service; r mobility; r portability; r connectivity in wireless networks; r interference from other users; r privacy/security. Many of the demands, for example the need for high data rates and the quality of service, are not unique to wireless communications. But, some of the challenges are specific to wireless communication systems. For example, the portability require- ment results in the use of batteries and the limitation in the battery life creates a challenge for finding algorithms with low power consumptions. This requires spe- cial attention in the design of transmitters and receivers. Since the base station does not operate on batteries and does not have the same power limitations, it may be especially desirable to have asymmetric complexities in different ends. Another example of challenges in wireless communications is the connectivity in wireless networks. The power of the received signal depends on the distance between the transmitter and the receiver. Therefore, it is important to make sure that if, because of the mobility of the nodes, their distance increases, the nodes remain connected. Also, due to the rapidly changing nature of the wireless channel, mobility brings many new challenges into the picture. Another important challenge in a wireless channel is the interference from other users or other sources of electro- magnetic waves. In a wired system, the communication environment is more under control and the interference is less damaging. While the demand for data rates and the performance of the signal processors increase exponentially, the spectrum and bandwidth are limited. The limited band- width of the wireless channels adds increases impairment. Increases in battery power grows slowly and there is a growing demand for smaller size terminals and handset devices. On the other hand, the users want the quality of wire-line commu- nication and the wire-line data rates grow rapidly. Researchers face many challenges to satisfy such high expectations through the narrow pipeline of wireless channels. The first step to solve these problems is to understand the behavior of the wireless channel. This is the main topic of the next section. We provide a brief introduction
- 27. 1.3 Wireless channels 5 Fig. 1.1. An example of different paths in a wireless channel. to the topic, as it is needed in our discussion of space-time codes, and refer the interested reader to other books that concentrate on the subject [57, 103, 111, 123]. 1.3 Wireless channels One of the distinguishing characteristics of wireless channels is the fact that there are many different paths between the transmitter and the receiver. The existence of various paths results in receiving different versions of the transmitted signal at the receiver. These separate versions experience different path loss and phases. At the receiver all received signals are accumulated together creating a non-additive white Gaussian noise (AWGN) model for the wireless channels. Since an AWGN model does not describe the wireless channels, it is important to find other models that represent the channels. To portray such a model, first we study different possible paths for the received signals. Figure 1.1 demonstrates the trajectory of different paths in a typical example. If there is a direct path between the transmitter and the receiver, it is called the line of sight (LOS). A LOS does not exist when large objects obstruct the line between the transmitter and the receiver. If LOS exists, the corresponding signal received through the LOS is usually the strongest and the dominant signal. At least, the signal from the LOS is more deterministic. While its strength and phase may change due to mobility, it is a more predictable change that is usually just a function of the distance and not many other random factors. A LOS is not the only path that an electromagnetic wave can take from a trans- mitter to a receiver. An electromagnetic wave may reflect when it meets an object that is much larger than the wavelength. Through reflection from many surfaces, the wave may find its path to the receiver. Of course, such paths go through longer distances resulting in power strengths and phases other than those of the LOS path. Another way that electromagnetic waves propagate is diffraction. Diffraction occurs when the electromagnetic wave hits a surface with irregularities like sharp edges.
- 28. 6 Introduction Finally, scattering happens in the case where there are a large number of objects smaller than the wavelength between the transmitter and the receiver. Going through these objects, the wave scatters and many copies of the wave propagate in many different directions. There are also other phenomenona that affect the propagation of electromagnetic waves like absorbtion and refraction. The effects of the above propagation mechanisms and their combination result in many properties of the received signal that are unique to wireless channels. These effects may reduce the power of the signal in different ways. There are two general aspects of such a power reduction that require separate treatments. One aspect is the large-scale effect which corresponds to the characterization of the signal power over large distances or the time-average behaviors of the signal. This is called attenuation or path loss and sometimes large-scale fading. The other aspect is the rapid change in the amplitude and power of the signal and this is called small-scale fading, or just fading. It relates to the characterization of the signal over short distances or short time intervals. In what follows, we explain models that explain the behavior of large-scale and small-scale fading. 1.3.1 Attenuation Attenuation, or large-scale fading, is caused by many factors including propagation losses, antenna losses, and filter losses. The average received signal, or the large- scale fading factor, decreases logarithmically with distance. The logarithm factor, or the path gain exponent, depends on the propagation medium and the environment between the transmitter and the receiver. For example, for a free space environment, like that of satellite communications, the exponent is two. In other words, the average received power Pr is proportional to d−2 , where d is the distance between the transmitter and the receiver. For other propagation environments, like urban areas, the path loss exponent is usually greater than 2. In other words, if the average transmitted power is Pt, we have Pr = βd−ν Pt, (1.1) where ν is the path loss exponent and β is a parameter that depends on the frequency and other factors. This is sometimes called the log-distance path loss model as the path loss and the distance have a logarithmic relationship. Calculating (1.1) at a reference distance d0 and computing the relative loss at distance d with respect to the reference distance d0 in decibels (dB) results in Lpath = β0 + 10ν log10 d d0 , (1.2)
- 29. 1.3 Wireless channels 7 where Lpath is the path loss in dB and β0 is the measured path loss at distance d0 in dB. As we mentioned before, the path loss exponent, ν, is a function of the environment between the transmitter and receiver. Its value is usually calculated by measuring the break signal and fitting the resulting measurements to the model. Typically, based on the empirical measurements, ν is between 2 and 6. In many practical situations, the above simple model does not match with the measured data. Measurements in different locations at the same distance from the transmitter result in unequal outcomes. It has been shown empirically that many local environmental effects, such as building heights, affect the path loss. These local effects are usually random and are caused by shadowing. To model them, a Gaussian distribution around the value in (1.2) is utilized. In other words, the path loss is modeled by Lpath = β0 + 10ν log10 d d0 + X, (1.3) where X is a zero-mean Gaussian random variable in dB with typical standard deviations ranging form 5 to 12 dB. This is called log-normal shadowing as the logarithm in dB is a normal random variable. This log-normal model is utilized in practice for the design and analysis of the system as a tool to provide the received powers. Knowing the parameters of the model, that is ν, d0, and the variance of the Gaussian, from measured data, one can generate the received power values for random locations in the system. 1.3.2 Fading Fading, or equivalently small-scale fading, is caused by interference between two or more versions of the transmitted signal which arrive at the receiver at slightly different times. These signals, called multipath waves, combine at the receiver antenna and the corresponding matched filter and provide an effective combined signal. This resulting signal can vary widely in amplitude and phase. The rapid fluc- tuation of the amplitude of a radio signal over a short period of time, equivalently a short travel distance, is such that the large-scale path loss effects may be ignored. The randomness of multipath effects and fading results in the use of different statistical arguments to model the wireless channel. To understand the behavior and reasoning behind different models, we study the cause and properties of fading. First, we study the effects of mobility on these channel models. Let us assume that the objects in the environment between the transmitter and the receiver are static and only the receiver is moving. In this case, the fading is purely a spatial phenomenon and is described completely by the distance. On the other hand, as the receiver moves through the environment, the spatial variations of the resulting signal translate into temporal variations for the receiver. In other words, due to
- 30. 8 Introduction s(t) r(t) - h(t, τ) - Fig. 1.2. Modeling a multipath channel with a linear time-varying impulse response. the mobility, there is a relationship between time and distance that creates a time- varying fading channel. Therefore, we can use time and distance interchangeably and equivalently in such a scenario. The time-varying nature of the wireless channel is also applied to the case that the surrounding objects are moving. Similarly, the resulting fluctuations in the received signal are structurally random. As it is clear from the name, multipath fading is caused by the multiple paths that exist between the transmitter and the receiver. As we discussed before, reflection, diffraction, and scattering create several versions of the signal at the receiver. The effective combined signal is random in nature and its strength changes rapidly over a small period of time. A multipath channel can be modeled as a linear time-varying channel as depicted in Figure 1.2. The behavior of the linear time-varying impulse response depends on different parameters of the channel. For example, the speed of the mobile and surrounding objects affect the characteristic of the model. We study such behaviors in what follows. The presence of reflecting objects and scatterers creates a constantly changing environment. Multipath propagation increases the time required for the baseband portion of the signal to reach the receiver. The resulting dissipation of the signal energy in amplitude, phase, and time may cause intersymbol interference (ISI). If the channel has a constant gain and linear phase response over a bandwidth which is greater than the bandwidth of the transmitted signal, the impulse response h(t, τ) can be approximated by a delta function at τ = 0 that may have a time- varying amplitude. In other words, h(t, τ) = α(t)δ(τ), where δ(·) is the Dirac delta function. This is a narrowband channel in which the spectral characteristics of the transmitted signal are preserved at the receiver. It is called flat fading or frequency non-selective fading. An example of the impulse response for a flat fading channel is depicted in Figure 1.3. As can be seen from the figure, the narrowband nature of the channel can be checked from the time and frequency properties of the channel. In the frequency domain, the bandwidth of the signal is smaller than the bandwidth of the channel. In the time domain, the width of the channel impulse response is smaller than the symbol period. As a result, a channel might be flat for a given transmission rate, or correspondingly for a given symbol period, while the same channel is not flat for a higher transmission rate. Therefore, it is not meaningful to say a channel is flat without having some information about the transmitted signal.
- 31. 1.3 Wireless channels 9 ( , ) h t τ ( ) s t ( ) r t ( ) S f ( ) H f ( ) R f S T S T τ + τ , S S C T B B τ S B C B Fig. 1.3. Flat fading. Also, we need to define the bandwidth of the channel to be able to compare it with the bandwidth of the signal. Usually the bandwidth of the channel is defined using its delay spread. To define the delay spread, let us assume that the multipath channel includes I paths and the power and delay of the ith path are pi and τi , respectively. Then, the weighted average delay is τ = I i=1 pi τi I i=1 pi . (1.4) The delay spread is defined as στ = τ2 − τ2 , (1.5) where τ2 = I i=1 pi τ2 i I i=1 pi . (1.6) Finally, the channel “coherence bandwidth” is approximated by Bc = 1 5στ . (1.7) As we defined earlier, in a flat fading channel, the channel coherence bandwidth Bc is much larger than the signal bandwidth Bs. On the other hand, if the channel possesses a constant gain and linear phase over a bandwidth that is smaller than the signal bandwidth, ISI exists and the received
- 32. 10 Introduction ( , ) h t τ ( ) s t ( ) r t ( ) S f ( ) H f ( ) R f S T S T τ + τ , S S C T B B τ S B C B Fig. 1.4. Frequency selective fading. Fig. 1.5. An approximated impulse response for a frequency selective fading. signal is distorted. Such a wideband channel is called frequency selective fading. Figure 1.4 shows an example of the impulse response for a frequency selective fading channel. In this case, the impulse response h(t, τ) may be approximated by a number of delta functions as shown in Figure 1.5. In other words, h(t, τ) = J j=1 αj (t)δ(τ − τj ). (1.8) Each delta component fades independently, that is αj (t) are independent. To be more specific, for frequency selective fading, the bandwidth of the signal is larger than the coherence bandwidth of the channel. Equivalently, in the time domain, the width of the channel impulse response is larger than the symbol period. Again, the frequency selective nature of the channel depends on the transmission rate as well as the channel characteristics. In summary, based on multipath time delay, the fading channel is categorized into two types: flat and frequency selective. Another independent phenomenon caused by mobility is the Doppler shift in frequency. Let us assume a signal with a wavelength of λ and a mobile receiver
- 33. 1.3 Wireless channels 11 with a velocity of v. Also, we define θ as the angle between the direction of the motion of the mobile and the direction of the arrival of the wave. In this case, the frequency change of the wave, known as Doppler shift and denoted by fd, is given by fd = v λ cos θ. (1.9) Since different paths have different angles, a variety of Doppler shifts correspond- ing to different multipath signals are observed at the receiver. In fact, the fre- quency change is random as the angle θ is random. The relative motion between the transmitter and the receiver results in random frequency modulation due to dif- ferent Doppler shifts on each of the multipath components. Also, if the surrounding objects are moving, they create a time-varying Doppler shift on different multipath components. Such a time-varying frequency shift can be ignored if the mobile speed is much higher than that of the surrounding objects. Since the receiver observes a range of different Doppler shifts, any transmitted frequency results in a range of received frequencies. This results in a spectral broadening at the receiver. Doppler spread is a measure of such a spectral widening and is defined as the range of frequencies over which the received Doppler spectrum is not zero. If the maxi- mum Doppler shift is fs, the transmitted frequency, fc, is received with components in the range fc − fs to fc + fs. If the baseband signal bandwidth is much greater than the Doppler spread, the fading is called slow fading. In this case, the effects of Doppler spread are negligible. The channel impulse response changes at a rate much slower than the transmitted baseband signal and the channel is assumed to be static over one or several reciprocal bandwidth intervals. On the other hand, if the effects of the Doppler spread are not negligible, it is a fast-fading channel. The channel impulse response changes rapidly within the symbol duration in a fast-fading channel. In summary, based on Doppler spread, the fading channel is categorized into two types: slow and fast. Defining the slow versus fast nature of a fading channel in terms of the signal bandwidth and Doppler spread may sound a little bit strange. Equivalently, slow- and fast-fading channels can be defined based on time domain properties. Towards this goal, first, we need to define the coherence time of a channel denoted by Tc. Two samples of a fading channel that are separated in time by less than the coherencetimearehighlycorrelated.Thisisastatisticalmeasuresincethedefinition depends on how much “correlation” is considered highly correlated. Practically, the coherence time is the duration of time in which the channel impulse response is effectively invariant. If a correlation threshold of 0.5 is chosen, the coherence time is approximated by Tc = 9 16π fs , (1.10)
- 34. 12 Introduction where fs is the maximum Doppler shift. If the signal duration is smaller than the coherence time, the whole signal is affected similarly by the channel and the channel is a slow fading channel. On the other hand, if the signal duration is larger than the coherence time, the channel changes are fast enough such that in practice different parts of the transmitted signal experience different channels. This is called fast fading since its main cause is the fast motion of the receiver or transmitter. So far, we have classified fading channels based on their multipath time delay into flat and frequency selective and based on Doppler spread into slow and fast. These two phenomena are independent of each other and result in the following four types of fading channels. r Flat Slow Fading or Frequency Non-Selective Slow Fading: When the bandwidth of the signal is smaller than the coherence bandwidth of the channel and the signal duration is smaller than the coherence time of the channel. r Flat Fast Fading or Frequency Non-Selective Fast Fading: When the bandwidth of the signal is smaller than the coherence bandwidth of the channel and the signal duration is larger than the coherence time of the channel. r Frequency Selective Slow Fading: When the bandwidth of the signal is larger than the coherence bandwidth of the channel and the signal duration is smaller than the coherence time of the channel. r Frequency Selective Fast Fading: When the bandwidth of the signal is larger than the coherence bandwidth of the channel and the signal duration is larger than the coherence time of the channel. 1.4 Statistical models for fading channels So far, we have modeled the fading channel by a linear time-varying impulse response. The impulse response was approximated by one delta function in the case of flat fading and multiple delta functions in the case of frequency selective fading. As discussed before, the nature of the multipath channel is such that the amplitude of these delta functions are random. This randomness mainly originates from the multipath and the random location of objects in the environment. There- fore, statistical models are needed to investigate the behavior of the amplitude and power of the received signal. In this section, we study some of the important models in the literature. 1.4.1 Rayleigh fading model First, let us concentrate on the case of flat fading. The results for frequency selective channels are very similar since the amplitudes of different delta functions fade independently. We also assume that there is no LOS path between the transmitter
- 35. 1.4 Statistical models for fading channels 13 and the receiver. Later, we will consider the case where such a LOS path exists. In a multipath channel with I multiple paths, transmitting a signal over the carrier frequency fc results in receiving the sum of I components from different paths plus a Gaussian noise as follows: r(t) = I i=1 ai cos(2π fct + φi ) + η(t), (1.11) where ai and φi are the amplitude and phase of the ith component, respectively, and η(t) is the Gaussian noise. Expanding the cos(·) term in (1.11) results in r(t) = cos(2π fct) I i=1 ai cos(φi ) − sin(2π fct) I i=1 ai sin(φi ) + η(t). (1.12) It is customary in digital communications to call the first and second summations “in phase” and “quadrature” terms, respectively. The terms A = I i=1ai cos(φi ) and B = I i=1ai sin(φi ) are the summation of I random variables since the objects in the environment are randomly located. For a large value of I, as is usually the case, using the central limit theorem, the random variables A and B are indepen- dent identically distributed (iid) Gaussian random variables. The envelope of the received signal is R = √ A2 + B2. Since A and B are iid zero-mean Gaussian ran- dom variables, the envelope follows a Rayleigh distribution. The probability density function (pdf) of a Rayleigh random variable is fR(r) = r σ2 exp −r2 2σ2 , r ≥ 0, (1.13) where σ2 is the variance of the random variables A and B. The received power, is an exponential random variable with a pdf: f (x) = 1 2σ2 exp −x 2σ2 , x ≥ 0. (1.14) Note that the average power, the average of the above exponential random variable, is E[R2 ] = 2σ2 which is the sum of the variances of A and B. The received signals in (1.11) or (1.12) represent the analog signal at the first stage of the receiver. We usually deal with the baseband digital signal after the matched filter and the sample and hold block. With a small abuse of the notation, we denote such a baseband discrete-time signal by rt . In fact, rt is the output of the matched filter after demodulation when the input of the receiver is r(t). Similarly, st and ηt are the discrete-time versions of s(t) and η(t), the transmitted signal and the noise, respectively. Note that in the above analysis the transmitted signal was implicit. Then, using the above arguments, one can show that the relationship
- 36. 14 Introduction between the baseband signals is rt = αst + ηt , (1.15) where α is a complex Gaussian random variable. In other words, the real and imaginary parts of the fade coefficient α are zero-mean Gaussian random variables. The amplitude of the fade coefficient, |α|, is a Rayleigh random variable. The input- output relationship in (1.15) is called a fading channel model. The coefficient α is called the path gain and the additive noise component ηt is usually a Gaussian noise. 1.4.2 Ricean fading model In a flat fading channel, if in addition to random multiple paths, a dominant sta- tionary component exists, the Gaussian random variables A and B are not zero mean anymore. This, for example, happens when a LOS path exists between the transmitter and the receiver. In this case, the distribution of the envelope random variable, R, is a Ricean distribution with the following pdf: fR(r) = r σ2 exp −(r2 + D2 ) 2σ2 I0 Dr σ2 , r ≥ 0, D ≥ 0, (1.16) where D denotes the peak amplitude of the dominant signal and I0(.) is the modified Bessel function of the first kind and of zero-order. As expected, the Ricean distri- bution converges to a Rayleigh distribution when the dominant signal disappears, that is D → 0. Similarly to the case of Rayleigh fading model, the discrete-time input–output relationship in the case of a Ricean fading model is also governed by (1.15). The main difference is that the real and imaginary parts of the path gain α are Gaussian random variables with non-zero means. As a result, the distribution of the amplitude |α| is Ricean instead of Rayleigh. 1.4.3 Frequency selective fading models In general, as discussed before, frequency selective fading is modeled by intersym- bol interference. Therefore, the channel can be modeled by the sum of a few delta functions. In this case, the corresponding discrete-time input–output relationship is rt = J−1 j=0 α j st− j + ηt . (1.17)
- 37. 1.5 Diversity 15 The path gains, α j , are independent complex Gaussian distributions and ηt rep- resents the noise. In the case of Rayleigh fading, they are zero-mean iid complex Gaussian random variables. A special case that has been extensively utilized in the literature is the case of a two-ray Rayleigh fading model. For a two-ray Rayleigh fading model, we have rt = α0 st + α1 st−1 + ηt , (1.18) where the real and imaginary parts of α0 and α1 are iid zero-mean Gaussian random variables. 1.5 Diversity UnliketheGaussianchannel,thefadingchannelmodelin(1.15)suffersfromsudden declines in the power. As we discussed before, this is due to the destructive addition of multipath signals in the propagation media. It can also be due to interference from other users. The amount of change in the received power can be sometimes more than 20 to 30 dB. The power of the thermal noise is usually not changing that much at the receiver. Therefore, the effective signal-to-noise ratio (SNR) at the receiver can go through deep fades and be dropped dramatically. Usually there is a minimum received SNR for which the receiver can reliably detect and decode the transmitted signal. If the received SNR is lower than such a threshold, a reliable recovery of the transmitted signal is impossible. This is usually called an “outage.” The probability of outage can be calculated based on the statistical model that models the channel or based on the actual measurements of the channel. It is the probability of having a received power lower than the given threshold. Themainideabehind“diversity”istoprovidedifferentreplicasofthetransmitted signaltothereceiver.Ifthesedifferentreplicasfadeindependently,itislessprobable to have all copies of the transmitted signal in deep fade simultaneously. Therefore, the receiver can reliably decode the transmitted signal using these received signals. This can be done, for example, by picking the signal with the highest SNR or by combining the multiple received signals. As a result, the probability of outage will be lower in the case that we receive multiple replicas of the signal using diversity. To define diversity quantitatively, we use the relationship between the received SNR, denoted by γ , and the probability of error, denoted by Pe. A tractable definition of the diversity, or diversity gain, is Gd = − lim γ →∞ log(Pe) log(γ ) , (1.19) where Pe is the error probability at an SNR equal to γ . In other words, diversity is the slope of the error probability curve in terms of the received SNR in a log-log
- 38. 16 Introduction scale. There are two important issues related to the concept of diversity. One is how to provide the replicas of the transmitted signal at the receiver with the lowest possible consumption of the power, bandwidth, decoding complexity and other resources. The second issue is how to use these replicas of the transmitted signal at the receiver in order to have the highest reduction in the probability of error. We study some of the methods to achieve these two goals in what follows. 1.5.1 Diversity methods The replica of the transmitted signal can be sent through different means [99]. For example, it can be transmitted in a different time slot, a different frequency, a different polarization, or a different antenna. The goal is to send two or more copies of the signal through independent fades. Then, since it is less likely to have all the independent paths in deep fades, using appropriate combining methods, the probability of error will be lower. When different time slots are used for diversity, it is called temporal diversity [129]. Two time intervals separated for more than the coherence time of the channel go through independent fades. Therefore, we may send copies of the transmitted signal from these separated time slots. Error-correcting codes can be utilized to reduce the amount of redundancy. In other words, sending a copy of the signal from different time slots is equivalent to using a repetition code. More efficient error- correcting codes may be used as well. If the fading is slow, that is the coherence time of the channel is large, the separation between time slots used for temporal diversity is high. In this case, the receiver suffers from a huge delay before it can start the process of decoding. The coded symbols are interleaved before sending through the channel. While interleaving increases the delay, it converts a slow fading channel to a fast fading channel that is more appropriate for temporal diversity. Temporal diversity is not bandwidth efficient because of the underlying redundancy. Another method of diversity is frequency diversity [8]. Frequency diversity uses different carrier frequencies to achieve diversity. The signal copies are transmit- ted from different carrier frequencies. To achieve diversity, the carrier frequencies should be separated by more than the coherence bandwidth of the channel. In this case, different replicas of the signal experience independent fades. Similar to tem- poral diversity, frequency diversity suffers from bandwidth deficiency. Also the receiver needs to tune to different carrier frequencies. One method of diversity that may not suffer from bandwidth deficiency is spa- tial diversity or antenna diversity [152]. Spatial diversity uses multiple antennas to achieve diversity. Multiple antennas may be used at the receiver or transmitter. If the antennas are separate enough, more than half of the wavelength, signals cor- responding to different antennas fade independently. The use of multiple antennas
- 39. 1.5 Diversity 17 may not be possible in small handheld devices. This is because of the fact that a minimum physical separation is needed between different antennas to achieve spatial diversity. Spatial diversity is not the only way to use antennas for providing diversity. Angular diversity uses directional antennas to achieve diversity. Different copies of the transmitted signal are collected from different angular directions. Unlike multiple antennas, it does not need separate physical locations. Therefore, it is also good for small devices. Another diversity method is polarization diversity that uses vertically and horizontally polarized signals to achieve diversity. Because of the scattering, the arriving signal, which is not polarized, can be split into two orthog- onal polarizations. If the signal goes through random reflections, its polarization state can be independent of the transmitted polarization. Unlike spatial diversity, polarization diversity does not require separate physical locations for the antennas. However, polarization diversity can only provide a diversity order of two and not more. 1.5.2 Combining methods The multiple versions of the signals created by different diversity schemes need to be combined to improve the performance. In this section, we study different methods of combining at the receiver. We assume that multiple receive antennas are available and provide multiple replicas of the transmitted signal at the receiver. The use of multiple transmit antennas is the topic of the other chapters in this book. While we refer to multiple receive antennas, our discussion of combining methods is applicable to other forms of diversity as well. In fact, the source of diversity does not affect the method of combination with the exception of transmit antenna diversity. For example, receiving two versions of the transmitted signal through polarization diversity is the same as receiving two signals from two receive antennas for the purpose of combining. There are two main combining methods that are utilized at the receiver: r Maximum Ratio Combining (MRC) r Selection Combining Figures 1.6 and 1.7 show the block diagrams of the maximum ratio combiners and the selection combiner. A hybrid scheme that combines these two main methods is also possible. In what follows, we explain the details of these combining methods. 1.5.2.1 Maximum ratio combining We consider a system that receives M replicas of the transmitted signal through M independent paths. Let us assume rm, m = 1, 2, . . . , M, as the mth received signal
- 40. 18 Introduction RF chain Maximum ratio combiner RF chain RF chain Fading signals Fig. 1.6. Block diagram of maximum ratio combining. Fading signals RF chain Selection combiner Select Fig. 1.7. Block diagram of selection combining. is defined by rm = αms + ηm, (1.20) where ηm is a white Gaussian noise sample added to the mth copy of the signal. A maximum-likelihood (ML) decoder combines these M received signals to find the most likely transmitted signal. We consider a coherent detection scheme where the receiver knows the channel path gains, αm. Since the noise samples are independent Gaussian random variables, the received signals are also independent Gaussian random variables for the given channel path gains and transmitted signal. Therefore, the conditional joint density function of the received signals is f (r1,r2, . . . ,rM|s, α1, α2, . . . , αM ) = 1 (π N0) M 2 exp ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ − M m=1 |rm − sαm|2 N0 ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ , (1.21)
- 41. 1.5 Diversity 19 where N0/2 is the variance of the real and imaginary parts of the complex Gaussian noise. To maximize this likelihood function, the receiver needs to find the optimal transmitted signal, ŝ, that minimizes M m=1|rm − sαm|2 . Note that with no diversity, M = 1, the cost function to minimize is |r1 − sα1|2 or |r − sα|2 for simplicity. This is equivalent to finding ŝ, among all possible transmitted signals, that is closest to rα∗ . For a constellation with equal energy symbols, for example PSK, we have ŝ = arg min s M m=1 |rm − sαm|2 = arg min s −s M m=1 αmr∗ m − s∗ M m=1 α∗ mrm = arg min s M m=1 rmα∗ m − s 2 . (1.22) Therefore, the ML decoding is similar to that of the system with no diversity if instead of rα∗ we use a weighted average of the received signals, M m=1rmα∗ m. This is called maximum ratio combining (MRC). To summarize, MRC uses a matched filter, that is optimum receiver, for each received signal and using the optimal weights ωm = α∗ m combines the outputs of the matched filters. If the average power of the transmitted symbol is Es, the SNR of the mth receiver is γm = |αm|2 (Es/N0). To derive the SNR of the output of the maximum ratio combiner, first we calculate M m=1 rmα∗ m = M m=1 (αms + ηm)α∗ m = M m=1 |αm|2 s + M m=1 ηmα∗ m. (1.23) Then, the SNR at the output of the maximum ratio combiner is γ = M m=1 |αm|2 2 Es M m=1 |αm|2 N0 = M m=1 |αm|2 Es N0 = M m=1 γm. (1.24) Therefore, the effective receive SNR of a system with diversity M is equivalent to the sum of the receive SNRs for M different paths. The importance of this M-fold increase in SNR is in the relationship between the average error probability and the average receive SNR. Let us assume that all different paths have the same average SNR, that is E[γm] = A. Then, using (1.24), the average SNR at the output of the maximum ratio combiner is γ = M A. (1.25) This M-fold increase in the receive average SNR results in a diversity gain of M. It can be shown that this is the maximum possible diversity gain when M copies of the signal are available in a Rayleigh fading channel.
- 42. 20 Introduction Increasing the effective receive SNR using MRC affects the probability of error at the receiver. For a system with no diversity, the average error probability is proportional to the inverse of the SNR, A−1 , at high SNRs [100]. Since each of the M paths follows a Rayleigh fading distribution, the average error probabil- ity of a system with M independent Rayleigh paths is proportional to A−M . As we defined before, the ratio M in the exponent of the receive SNR is called the diversity gain. Therefore, using MRC we achieve a diversity gain equal to the number of available independent paths. We provide a rigorous proof of this fact in Chapter 4. Equal gain combining is a special case of maximum ratio combining with equal weights. In equal gain combining, co-phased signals are utilized with unit weights. The average SNR at the output of the equal gain combiner is γ = 1 + π 4 (M − 1) A. (1.26) 1.5.2.2 Selection combining Using MRC, when the source of M independent signals is the receive antennas, the receiver needs to demodulate all M receive signals. In other words, M radio frequency (RF) chains are required at the receiver to provide the baseband signals. Since most RF chains are implemented by analog circuits, usually their physical size and price are high. In some applications, there is not enough room for several RF chains or their price does not justify the gain achieved by MRC. Therefore, it may be beneficial to design a combiner that uses only one RF chain. Selection combining or antenna selection picks the signal with the highest SNR and uses it for decoding. Picking the signal is equivalent to choosing the correspond- ing antenna among all receive antennas. Equivalently, it is the same as selecting the best polarization in the case of polarization diversity. As before, let us assume M replicas of the transmitted signal, for example through M receive antennas. As shown before, if the fading is Rayleigh, the random variable γm, the SNR of the mth antenna, follows an exponential distribution. With a slight abuse of the notation, the pdf of γm for m = 1, 2, . . . , M is fγm (γ ) = 1 E[γm] e− γ E[γm] , γ 0, (1.27) where E[γm] is the average SNR of the mth receive signal. Let us assume that all receivesignalshavethesameaverageSNR,thatis E[γm] = A.Then,thecumulative distribution function (CDF) of the mth receive SNR is Fγm (γ ) = P[γm ≤ γ ] = 1 − e− γ A , m = 1, 2, . . . , M. (1.28)
- 43. 1.5 Diversity 21 Since different receive signals are independent from each other, the probability that all of them have an SNR smaller than γ is P [γ1, γ2, . . . , γM ≤ γ ] = 1 − e− γ A M . (1.29) On the other hand, the probability that at least one receive signal achieves an SNR greater than γ , denoted by PM (γ ), is PM (γ ) = 1 − 1 − e− γ A M . (1.30) The corresponding pdf is f (γ ) = M A 1 − e− γ A M−1 e− γ A . (1.31) Therefore, the average SNR at the output of the selection combiner, γ , is γ = ∞ 0 γ f (γ ) dγ = A M m=1 1 m . (1.32) As a result, without increasing the transmission power, selection combining offers M m=1 1 m times improvement in the average SNR. This is less than the maximum improvementratioof M.Thereforeselectioncombiningdoesnotprovideanoptimal diversity gain and as a result an optimal performance enhancement. However, its complexity is low since it only requires one RF chain. In other words, selection combining provides a trade-off between complexity and performance. In selection combining, the receiver needs to find the strongest signal at each time instant. To avoid the monitoring of the received SNRs, one may use scanning selection combining which is a special case of selection combining. In scanning selection combining, first, the M receive signals are scanned to find the highest SNR. The corresponding signal is used until its SNR is below a predetermined threshold.Then,anewselectionisdoneandtheprocessiscontinued.Inotherwords, a new selection is needed only if the selected signal goes through a deep fade. Figure 1.8 compares the SNR gains of different combining methods using 1 to 10 receive antennas. As expected, MRC provides the highest gain while requiring the highest complexity. For a higher number of receive antennas, the gap between the MRC and selection combining grows. This is in a trade-off with the lower complexity of selection combining with only one RF chain. It is possible to use a number of RF chains that is neither one nor M for more than two receive antennas. Let us assume that the receiver contains J RF chains where 1 J M and M 2. Then, the receiver chooses J receive signals with the highest SNR and combines them using MRC. The block diagram of such a hybrid combining method is shown in Figure 1.9. The instantaneous SNR at the output of the hybrid selection/maximal
- 44. 22 Introduction 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Number of receive antennas SNR gain (dB) Maximum Ratio Combining Equal Gain Combining Selection Combining Hybrid Combining with 2 RF chains Fig. 1.8. Comparing the gain of different combining methods. Fading signals RF chain Maximum ratio combiner Select RF chain Fig. 1.9. Block diagram of hybrid selection/maximum ratio combining. ratio combining is γ = J j=1 γj , (1.33) where γj is the SNR of the jth selected signal [150]. The average SNR at the output of the hybrid selection/maximal ratio combiner, γ , is γ = AJ 1 + M m=J+1 1 m . (1.34) In Figure 1.8, we also show the SNR gain for a hybrid selection/maximal ratio combiner with J = 2 RF chains. As depicted in the figure, using more RF chains results in a higher gain. For a small number of receive antennas, a hybrid combiner
- 45. 1.6 Spatial multiplexing gain and its trade-off with diversity 23 with only J = 2 RF chains provides most of the gain. For a higher number of receive antennas, the gap between the hybrid combiner and MRC increases and more RF chains are needed to close the gap. 1.6 Spatial multiplexing gain and its trade-off with diversity In the last section, we mostly concentrated on diversity gains that are achieved by using multiple receive antennas. In a multiple-input multiple-output (MIMO) channel,diversitygainmaybeachievedbyusingbothtransmitandreceiveantennas. In the rest of this book, we will investigate how to use multiple transmit antennas to achieve high levels of diversity. However, multiple transmit antennas can be utilized to achieve other goals as well. For example, a higher capacity and as a result a higher transmission rate is possible by increasing the number of transmit antennas. Let us, for the sake of simplicity, assume a MIMO channel with equal number of transmit and receive antennas. Then, as we will show in Chapter 2, in a rich scattering environment the capacity increases linearly with the number of antennas without increasing the transmission power. This results in the possibility of transmitting at a higher rate, for example by using spatial multiplexing. If the number of transmit antennas is not the same as the number of receive antennas, in general one can transmit up to min{N, M} symbols per time slot, where N is the number of transmit antennas and M is the number of receive antennas. For example, if M ≥ N, one can send N symbols and achieve a diversity gain of M − N + 1. Note that for equal number of transmit and receive antennas, the diversity gain in this case is one. On the other hand, the maximum spatial diversity while transmitting only one symbol per time slot is M N. Therefore, the advantage of a MIMO channel can be utilized in two ways: (i) to increase the diversity of the system and (ii) to increase the number of transmitted symbols. For the general case of more than one transmit antenna, M ≥ N 1, there is a theoretical trade-off between the number of transmit symbols and the diversity of the system. For one transmit antenna, in both cases one symbol per time slot is transmitted and the two systems coincide. Inmanycases,spatialmultiplexinggainreferstothefactthatonecanusemultiple antennas to transmit at a higher rate compared to the case of one antenna. As we will show in Chapter 2, the capacity of a MIMO channel increases by raising the SNR. Since the transmission rate relates to capacity, it is reasonable to hope that the rate can be increased as the SNR increases. This argument has resulted in the following definition for spatial multiplexing gain in [159]: SMG = lim γ →∞ r log(γ ) , (1.35)
- 46. 24 Introduction 0 1 2 3 0 2 4 6 8 10 12 Spatial multiplexing gain Diversity Optimal trade-off Fig. 1.10. Optimal trade-off between spatial multiplexing gain (an indication of rate) and diversity. where r is the rate of the code, at the transmitter, in bits/channel use and is a function of SNR. Note that the relationship of the above spatial multiplexing gain to the transmission rate is similar to that of the diversity gain to the probability of error in (1.19). The main rationale behind such a rate normalization is the fact that SMG measures how far the rate r is from the capacity. Note that the range of the spatial multiplexing gain is from 0 to min{N, M}. The following theorem from [159] derives the optimal trade-off between spatial multiplexing gain, which is an indication of the rate, and diversity, which is an indication of the probability of error Theorem 1.6.1 Let us assume a code at the transmitter of a MIMO channel with N transmit antennas and M receive antennas. For a given spatial multiplexing gain SMG = i, where i = 0, 1, . . . , min{N, M} is an integer, the maximum diversity gain Gd(i) is given by Gd(i) = (N − i)(M − i) if the block length of the code is greater than or equal to N + M − 1. The optimal trade-off curve is achieved by connecting the points (i, Gd(i)) by lines. An example of the optimal trade-off for N = 4 transmit antennas and M = 3 receive antennas is depicted in Figure 1.10. 1.7 Closed-loop versus open-loop systems When the transmitter does not have any information about the channel, the system is an “open-loop” system. In this case, the receiver may estimate the channel and
- 47. 1.7 Closed-loop versus open-loop systems 25 Fig. 1.11. Block diagram of (a) open-loop and (b) closed-loop multiple-input multiple-output systems. use the channel state information (CSI) for decoding. However, the transmitter does not have access to the channel state information. On the contrary, in some sys- tems, the receiver sends some information about the channel back to the transmitter through a feedback channel. This is called a “closed-loop” system and the trans- mitter can use this information to improve the performance. The block diagrams of open-loop and closed-loop systems are depicted in Figure 1.11. In a time division duplexing (TDD) system, the radio channel is shared between the mobile and base station. In other words, the same frequency, the same chan- nel, is utilized to transmit information in both directions using time sharing. As a result, the characteristics of the uplink channel, from mobile to base station, and the downlink channel, from base station to mobile, are similar. One possi- ble assumption in such a TDD system is to consider the uplink and downlink channels as the reciprocal of each other. If this is a valid assumption, the channel estimation at the receiver can be utilized when transmitting back in a TDD sys- tem. Therefore, a closed-loop system can be used without sending any feedback information. In a closed-loop system, the gain achieved by processing at the transmitter and the receiver is sometimes called “array gain.” Similar to the diversity gain, the array gain results in an increase in the average receive SNR. If the transmitter knows the channel perfectly, beamforming [86] is the best solution. In fact, in such a case there is no need for spatial diversity. On the other hand, in most practical situations, there is a limited feedback information available which may not be perfect due to the quantization error or other factors. With such limited channel information, spatial diversity is still useful. The transmitted codeword can be tuned to better fit the channel based on the received information in the closed-loop system. To
- 48. 26 Introduction maintain low implementation complexity, a simple linear beamforming scheme at the transmitter is preferred. If non-perfect channel state information is available, a combined space-time coding and beamforming system is utilized [76, 87]. 1.8 Historical notes on transmit diversity Transmit diversity, a form of spatial diversity, has been studied extensively as a method of combating detrimental effects in wireless fading channels [4, 42, 51, 96, 102, 110, 135, 136, 139, 140, 153, 154]. The use of multiple transmit anten- nas for diversity provides better performance without increasing the bandwidth or transmission power. The first bandwidth-efficient transmit diversity scheme was proposed in [153]. It includes the delay diversity scheme of [110] as a special case. It is proved that the diversity advantage of this scheme is equal to the number of transmit antennas which is optimal [151]. Later, a multilayered space-time architec- ture was introduced in [43]. The scheme proposed in [43] uses spatial multiplexing to increase the data rate and not necessarily provides transmit diversity. Simple iterative decoding algorithms that have been proposed in conjunction with spatial multiplexing can achieve spatial diversity, mostly receive diversity. The criterion to achieve the maximum transmit diversity was derived in [51]. A complete study of design criteria for maximum diversity and coding gains in addition to the design of space-time trellis codes was proposed in [139]. It includes the delay diversity as a special case. Information theoretical results in [140] and [42] showed that there is a huge advantage in utilizing spatial diversity. The introduction of space-time block coding in [136] provided a theoretical framework that started an increasing interest on the subject. The best use of multiple transmit antennas depends on the amount of channel state information that is available to the encoder and decoder. In a flat fading chan- nel, the channel has a constant gain and linear phase response over a bandwidth which is greater than the bandwidth of the transmitted signal. Let us assume that the channel does not change rapidly during the transmission of a frame of data. The combination of the above two assumptions results in a quasi-static flat fading channel model [42]. In most practical cases, the system estimates the channel at the receiver by transmitting some known pilot signals. The receiver utilizes the cor- responding received signals to estimate the channel. In such a system, a coherent detection is utilized in which the decoder uses the value of the path gains estimated at the receiver [138]. In all references that we have mentioned so far in this section, the codes are designed for the case that the receiver knows the channel. In some other cases, such an estimation of the channel is not available at the receiver or the channel changes rapidly such that the channel estimation is not useful. Then, a noncoherent detection needs to be employed. For one transmit antenna, differential
- 49. 1.9 Summary 27 detection schemes exist that neither require the knowledge of the channel nor employpilotsymboltransmission.Apartialsolutiontogeneralizedifferentialdetec- tion schemes to the case of multiple transmit antennas was proposed in [132]. This was a joint channel- and data-estimation scheme that can lead to error propagation. Noncoherent detection schemes based on unitary space-time codes were proposed in [59, 60]. The first differential decoding modulation scheme for multiple antennas that provides simple encoding and decoding based on orthogonal designs was pro- posed in [73, 133]. Another construction based on group codes followed in [61, 63]. 1.9 Summary r Line of Sight: A direct path between the transmitter and the receiver. r Reflection: When the electromagnetic wave meets an object that is much larger than the wavelength. r Diffraction: When the electromagnetic wave hits a surface with irregularities like sharp edges. r Scattering: When the medium through which the electromagnetic wave propagates con- tains a large number of objects smaller than the wavelength. r Attenuation or path loss (sometimes called large-scale fading) is due to propagation losses, filter losses, antenna losses, and so on. r Fading is used to describe the rapid fluctuation of the amplitude of a radio signal over a short period of time or travel distance, so that large-scale path loss effects may be ignored. Fading is a time-varying phenomenon. r Multipath: The presence of reflecting objects and scatterers creates a constantly changing environment that dissipates the signal energy in amplitude, phase, and time. A multipath channel can be modeled as a linear time-varying channel. r Flat Slow Fading or Frequency Non-Selective Slow Fading: When the bandwidth of the signal is smaller than the coherence bandwidth of the channel and the signal duration is smaller than the coherence time of the channel. r Flat Fast Fading or Frequency Non-Selective Fast Fading: When the bandwidth of the signal is smaller than the coherence bandwidth of the channel and the signal duration is larger than the coherence time of the channel. r Frequency Selective Slow Fading: When the bandwidth of the signal is larger than the coherence bandwidth of the channel and the signal duration is smaller than the coherence time of the channel. r Frequency Selective Fast Fading: When the bandwidth of the signal is larger than the coherence bandwidth of the channel and the signal duration is larger than the coherence time of the channel. r In a Rayleigh fading model, the input–output relationship between the baseband signals is rt = αst + ηt ,
- 50. 28 Introduction where α is a complex Gaussian random variable. In other words, the real and imagi- nary parts of the fade coefficient α are real zero-mean Gaussian random variables. The amplitude of the fade coefficient, |α|, is a Rayleigh random variable. r Diversity provides a less-attenuated replica of the transmitted signal to the receiver. Diver- sity is defined as Gd = − lim γ →∞ log(Pe) log(γ ) , where Pe is the error probability at a signal-to-noise ratio equal to γ . r Diversity methods include temporal diversity, frequency diversity, spatial diversity, angle diversity, polarization diversity, and so on. r The multiple versions of the signals created by different diversity schemes need to be combined to improve the performance. Two main combining methods that are utilized at the receiver are maximum ratio combining and selection combining. r For a given spatial multiplexing gain SMG = i, where i = 0, 1, . . . , min{N, M} is an integer, the maximum diversity gain Gd(i) is given by Gd(i) = (N − i)(M − i) if the block length of the code is greater than or equal to N + M − 1. 1.10 Problems 1 A multipath channel includes five paths with powers 0.1, 0.01, 0.02, 0.001, and 0.5 W. The corresponding delays are 0.1, 0.2, 0.3, 0.4, and 0.5 s, respectively. If the signal bandwidth is 200 kHz, is the channel flat or frequency selective? 2 A mobile is moving at a speed of 100 km/h. The transmitted frequency is 900 MHz while the signal bandwidth is 300 kHz. Is the channel slow or fast? 3 Consider a Rayleigh random variable, R, that is constructed as the envelope of two iid zero-mean unit-variance Gaussian random variables. (a) What is the probability that R 0.1? (b) If two independent random variables R1 and R2 have the Rayleigh distribution of R, what is the probability that R1 0.1 and R2 0.1? 4 Another popular model for the distribution of the envelope random variable, R, is a Nakagami distribution with the following pdf: fR(r) = 2mm r2m−1 (m)Am exp −mr2 A , r ≥ 0, m ≥ 0.5, where A = E[R2 ] and (·) is the gamma function. (a) Show that for m = 1, the Nakagami distribution converges to a Rayleigh distribution. (b) Draw the pdf of a Nakagami distribution for A = 1 and different values of m = 0.5, 1, 1.5, 2, 10. 5 Consider a system that receives M replicas of the transmitted signal through M inde- pendent paths. The pdf of γm for m = 1, 2, . . . , M follows (1.27) and all receive signals have the same average SNR, E[γm] = 1. Draw the pdf of the SNR at the output of the selection combiner and the maximum ratio combiner for M = 1, 2, 3, 4.
- 51. 1.10 Problems 29 6 For BPSK, the probability of error for a SNR of γ is 0.5 erfc( √ γ ), where erfc is the complementary error function: erfc(x) = 2 √ π ∞ x exp(−t2 ) dt. Let us assume M = 2 independent paths that have the same average SNR, E[γm] = 1 such that the pdf of γm for m = 1, 2 follows (1.27). What is the probability of error at the output of the selection combiner and the maximum ratio combiner?
- 52. 2 Capacity of multiple-input multiple-output channels 2.1 Transmission model for multiple-input multiple-output channels We consider a communication system, where N signals are transmitted from N transmitters simultaneously. For example, in a wireless communication system, at each time slot t, signals Ct,n, n = 1, 2, . . . , N are transmitted simultaneously from N transmit antennas. The signals are the inputs of a multiple-input multiple-output (MIMO) channel with M outputs. Each transmitted signal goes through the wireless channel to arrive at each of the M receivers. In a wireless communication system with M receive antennas, each output of the channel is a linear superposition of the faded versions of the inputs perturbed by noise. Each pair of transmit and receive antennas provides a signal path from the transmitter to the receiver. The coefficient αn,m is the path gain from transmit antenna n to receive antenna m. Figure 2.1 depicts a baseband discrete-time model for a flat fading MIMO channel. Based on this model, the signal rt,m, which is received at time t at antenna m, is given by rt,m = N n=1 αn,mCt,n + ηt,m, (2.1) where ηt,m is the noise sample of the receive antenna m at time t. Based on (2.1), a replica of the transmitted signal from each transmit antenna is added to the signal of each receive antenna. Although the faded versions of different signals are mixed at each receive antenna, the existence of the M copies of the transmitted signals at the receiver creates an opportunity to provide diversity gain. If the channel is not flat, the received signal at time t depends on the transmitted signals at times before t as well. The result is an extension to the case of one transmit and one receive antenna in (1.17). In this chapter, we only consider the case of narrowband signals for which the channel is a flat fading channel. Another important factor in the behavior of the channel is the correlation between different path gains at different time slots. There are two general assumptions that 30
- 53. 2.1 Transmission model for multiple-input multiple-output channels 31 1 @ @ @ @ @ @ B B B B B B B B B B B B B B B B B B 2 A A A A A A A A A A A A N-1 α1,1 α1,2 · · · α1,M α2,1 α2,2 · · · α2,M . . . αN−1,1 αN−1,2 · · · αN−1,M N αN,1 αN,2 · · · αN,M 1 2 . . . M Fig. 2.1. A multiple-input multiple-output (MIMO) channel. correspond to two practical scenarios. First, we assume a quasi-static channel, where the path gains are constant over a frame of length T and change from frame to frame. In most cases, we assume that the path gains vary independently from one frame to another. Another assumption is to consider a correlation between the fades in adjacent time samples. One popular example of such a second-order model is the Jakes model [74]. The value of T dictates the slow or fast nature of the fading. If a block of data is transmitted over a time frame T that is smaller than T , the fading is slow. In this case, the fades do not change during the transmission of one block of data and the values of path gains in (2.1) are constant for every frame. On the other hand, in a fast fading model, the path gains may change during the transmission of one frame of data, T T . To form a more compact input-output relationship, we collect the signals that are transmitted from N transmit antennas during T time slots in a T × N matrix, C, as follows: C = ⎛ ⎜ ⎜ ⎜ ⎝ C1,1 C1,2 · · · C1,N C2,1 C2,2 · · · C2,N . . . . . . ... . . . CT,1 CT,2 · · · CT,N ⎞ ⎟ ⎟ ⎟ ⎠ . (2.2)
- 54. 32 Capacity of multiple-input multiple-output channels Similarly, we construct a T × M received matrix r that includes all received signals during T time slots: r = ⎛ ⎜ ⎜ ⎜ ⎝ r1,1 r1,2 · · · r1,M r2,1 r2,2 · · · r2,M . . . . . . ... . . . rT,1 rT,2 · · · rT,M ⎞ ⎟ ⎟ ⎟ ⎠ . (2.3) Then, assuming T T , gathering the path gains in an N × M channel matrix H H = ⎛ ⎜ ⎜ ⎜ ⎝ α1,1 α1,2 · · · α1,M α2,1 α2,2 · · · α2,M . . . . . . ... . . . αN,1 αN,2 · · · αN,M ⎞ ⎟ ⎟ ⎟ ⎠ , (2.4) results in the following matrix form of (2.1): r = C · H + N, (2.5) where N is the T × M noise matrix defined by N = ⎛ ⎜ ⎜ ⎜ ⎝ η1,1 η1,2 · · · η1,M η2,1 η2,2 · · · η2,M . . . . . . ... . . . ηT,1 ηT,2 · · · ηT,M ⎞ ⎟ ⎟ ⎟ ⎠ . (2.6) Different path gains may be independent from each other, that is αn,m is independent from αn ,m for n = n or m = m . Note that the independence assumption is in the spatial domain and not necessarily in the time domain. Also, if the antennas are not far enough from each other, it is possible that some spatial correlation exists among the path gains. If the distance between two specific antennas is more than half of the wavelength, it is usually assumed that their path gains are independent of each other. We also assume a quasi-static slow fading model such that the noise samples ηt,m are independent samples of a zero-mean circularly symmetric complex Gaussian random variable. This is an additive white Gaussian noise (AWGN) assumption for a complex baseband transmission. The noise samples, channel path gains, and transmitted signals are independent from each other. Also, the bandwidth is narrow enough such that the channel is flat over a band of frequency. Such a channel is called frequency non-selective and the channel matrix is constant over the frequency band of interest. In the sequel, we use the quasi-static, non-frequency selective assumptions unless we mention otherwise.
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- 56. part of it into a State, while the remaining section was not made a State, without the consent of the Territory; he conceived that Congress must, in such event, form this section also into a State. He, therefore, was of opinion that Congress must consult the people of the Territory before they shall divide the Territory. As to the expediency of the resolution, he thought it very expedient to make the division therein marked out. The effect of it would be that the whole of Lake Erie would be thrown out of the State to be formed, and the inconvenience to the section of the Territory not incorporated in the new State would be very great, if it should be attached to the Indiana Territory, from its great distance, which he understood was contemplated. Mr. Giles said that the committee who reported these resolutions, so far from entertaining a disposition to change the ordinance, had strictly observed the conditions therein prescribed. [Mr. G. here quoted the ordinance.] It appeared therefrom that Congress was under an obligation, after laying off one State, to form the remainder into a State. But when? Hereafter, whenever they shall think it expedient to do so. Mr. Bayard agreed that there was no obligation imposed upon Congress to decide definitively the boundary of a State. If the ultimate right of Congress, after the formation of a new State, to alter the boundary be doubted, they have a right to remove all doubts by so declaring at this time. It is certain that at present great inconvenience would arise from drawing the boundary as fixed in the resolution. The population of the Territory does not amount to that which is sufficient to give it admission into the Union. He had, however, no disposition to oppose its admission, notwithstanding this circumstance. The population in the Eastern State does not exceed forty-five thousand. We are now about to pare off five or six thousand inhabitants, which will bring it down to thirty-nine thousand. A population of forty-five thousand is quite small enough for an independent State. It is a smaller population than exists in
- 57. any of the present States in the Union. From this consideration, it might have been expected that Congress would take no step whose effect would be a diminution of that population. The division, as made in the resolution, is manifestly unjust, as far as it relates to the people north of the dividing line. By it they are about to be severed from their connection with the other portion of the Territory. Mr. B. wished to know to whom they are to be attached? If attached to the Indiana Territory, the inhabitants, to arrive at the seat of Government, will be obliged to go across the new State, a distance of two or three hundred miles. Besides, after having advanced them to the second grade of territorial government, you will consign them back again to the first, and thereby give them a system of government extremely odious, and which we ought to get rid of as soon as possible. Thus, after having held out to them the flattering prospect of being elevated to the high rank of a State, you degrade them, contrary to their expectations, to the humblest condition in the Union. Mr. B., therefore, thought it would be most just and politic to include this population of five or six thousand in the bounds of the new State, subject to the reserved right of Congress to alter the boundary hereafter. Mr. Giles said he was not tenacious of his opinions; but it was necessary to justify the contents of the report by stating some considerations that might not be generally known to the members of the House. Mr. G. said he supposed the section of the Territory, not embraced in the new State, would be attached to the Indiana Territory; nor would any great hardship result from this disposition; and such as did result would arise from their local situation and not from any circumstances over which the National Legislature had a controlling power. He believed that people, to reach the seat of Government, had as far to go now as they will then have. His object was to reserve in future to Congress the right of determining the boundary of the States in the Territory. If this section should once be admitted,
- 58. he believed it would be very difficult, however proper, to detach it from the State to which it had become attached. The report contemplates the forming a constitution. Should the people on the northwardly side of the line be admitted as a part of the State, they will participate in the formation of the constitution—a constitution which will not be ultimately for themselves, but after a short time exclusively for others. This participation would be unjust. The question then is, whether you will suffer those to form a constitution who are not to be permanently affected by it; and whether, if you once constitute a State, you will be able hereafter to alter its boundaries? For if this section be now admitted, gentlemen, by looking at the map, will see that the boundary now fixed cannot be permanent. As to the remarks made by the gentleman from Delaware, Mr. G. said he was extremely glad that gentleman was for giving to the Territory the right of a State. If, however, he had attended to the report, he would have found that his calculation of numbers was incorrect. The population of five thousand had been deducted by the committee, and after that deduction forty-five thousand remained. Though the numbers in the Territory proposed to be formed into a State amounted, a year ago, to no more than forty thousand, yet it might be stated upon strong ground, that, before the new government can get into operation, there will be a sufficient population to demand admission as a matter of right. By attaching the inhabitants on the north of the line to the Indiana Territory, they will remain in the same grade of government they now are, and not be degraded, as stated by the gentleman from Delaware, to a lower state. This disposition appeared to Mr. G. the best that could be made. But if, when gentlemen came to the details of the bill, it should be thought best to introduce into the new State the population north of the line, he said he might have no objection. Mr. Fearing stated the great inconveniences that would be felt by the inhabitants north of the line, if attached to the Indiana Territory. He considered the remarks of the gentleman from Virginia, (Mr. Giles,)
- 59. respecting the participation of this description of citizens in forming a constitution for others, as entitled to little weight. Such a measure was by no means uncommon. It had been done in the case of Kentucky, and other States. Mr. F. conceived that the people of the Territory had all equal rights under the ordinance; they had been virtually promised that they should not be attached to any other Western Territory, and Congress had only reserved to themselves the right of admitting them into the Union as States. More they could not do, without their consent. Mr. Bayard moved to strike out of the resolution the words that fix the boundary, for the purpose of introducing words that should prescribe that the new State be circumscribed by the original boundaries of the Eastern State, referring to Congress the right of making one or more States in said State at any future time. Mr. Giles said that the State, as formed in the report, was one of the most compact and convenient in the Union. The amendment would materially change its character. Besides, it would in fact impair the right of Congress to accommodate the boundaries to future circumstances. It was well known, and sensibly felt, that there were many inconvenient boundaries to several of the States now in the Union; yet so great was the difficulty attending their alteration, that they could not be changed. Mr. Bayard was not so sensible of the difficulty of altering the boundaries as the gentleman from Virginia, who had stated that Congress would not have power to alter them when once fixed. This difficulty might exist as to the States now in the Union, because Congress had not the constitutional power to alter them without the consent of the adjacent States. But if this power be referred to Congress, which will be a disinterested tribunal, there will be no difficulty in varying the boundaries as circumstances shall dictate. Mr. B. asked, if, while gentlemen are attending to the interests and wishes of one part of the people, they are disposed to disregard the interests and wishes of another part? If they were not, they ought to
- 60. admit the section, proposed by the resolution to be cut off, to a participation in State rights. Mr. Bacon objected to the amendment. He said that Congress were vested by the constitution with certain powers which they cannot increase, or diminish, or delegate. By the constitution likewise, the several States are vested with certain powers which they cannot increase, diminish, or divest themselves of. By the third section of the fourth article of the constitution, new States may be admitted by the Congress into the Union. This act proposes to make this Territory a State with State powers under the constitution. How, then, can these people, once a State, divest themselves of these powers. This is a question that does not interest simply the State proposed to be formed, but every State in the Union. All are equally interested in preserving the powers vested in them by the constitution. Mr. Bayard said he did not see any occasion for striking out the proviso. The gentleman from Massachusetts (Mr. Bacon) goes on the principle that Congress has only a right to admit, without any reservation. Mr. B. said he had always believed the greater included the smaller. If you are vested with the greater power of admitting, you have certainly the minor powers included in the greater power. From the nature of the ordinance, it constitutes the fundamental principle on which the States are admitted—they are not admitted under the constitution. They are to be admitted exclusively under the provision of the ordinance. You may, therefore, say that you will not now exercise the whole power committed to you, but reserve the right of exercising it hereafter. Mr. Smilie did not consider the principle laid down by the gentleman from Delaware as constitutional. We must be governed by the constitution. If the Territory be admitted as a State into the Union, when admitted it must be bound down by the constitution, which says the boundaries of States shall not be altered but with the express permission of the State.
- 61. Mr. Giles—The gentleman from Connecticut, (Mr. Griswold,) affects lately to have discovered a great deal of disguise in the proceedings of this House. What disguise? What were the committee to do? This country is placed in a certain peculiar situation. We have waters running to the East—then to the West; and the committee thought it was desirable to connect these by good roads. With the committee, State principles or interests had no influence—they were governed entirely by general principles and the common interest. The gentleman has also insinuated that the Secretary of the Treasury holds lands that will be benefited by these roads. It may be so. Mr. G. had not inquired; but he supposed he did not hold all the lands. Congress may lay out these roads as they please. He could foresee how Congress would lay them out, and it is a million to one that they will not touch his lands. The United States are about making a new contract. These propositions are made as additional securities for the national property. The Secretary of the Treasury having estimated the annual product of these lands at four hundred thousand dollars, Mr. G. said, as chairman of the committee, he had applied to him to know his opinion of the manner in which this sum could be best secured, and he gave his opinion that this provision would be most likely to effect that object. This is all the mystery and disguise attending the resolution. Mr. Smilie said when gentlemen charge particular States with injustice, they ought to be prepared to prove what they advance. If there had been any co-operation between the delegations of Virginia and Pennsylvania on this occasion, he had never heard of it. The fact was, that no peculiar good could result to Pennsylvania from this measure. The great object was to keep up that intercourse which will attach the people of the Territory to you. When the Territory shall become a State, she will have a right to tax your lands. This benefit, together with the salt-springs, as I understand, is proposed as a substitution far the relinquishment of those rights.
- 62. Mr. Fearing said he considered a part of the rights of the Territory given up by this resolution; and though the Territory would be highly benefited by the projected roads, and the cession of the salt-springs, yet he conceived they would be much more benefited by laying out the roads within the Territory. Mr. Griswold said he was glad the honorable gentleman from Virginia had assured the House there was no disguise in this business. If the object be to make an advantageous contract with the Territory to secure our Western lands, let us offer them five per cent. of the proceeds of those lands, to be paid into their treasury. If they shall be disposed to make roads through Pennsylvania and Virginia, he should have no objection. He was as sensible as the gentleman from Virginia, that whatever improves a part of the Union improves the whole; though this was undoubtedly the case, he was not of opinion that a sum of money should be taken from the public treasury, and specially applied to local purposes. Under this resolution, according to the calculation of the Secretary of the Treasury, forty thousand dollars was the smallest sum that would be annually applied to the laying out of those roads. Mr. G. said he thought the sum too large to be withdrawn from the national treasury, and directed to local objects. The allusion of the gentlemen to light-houses raised on the Connecticut shore does not apply. There was but one light-house in Connecticut, ordered to be built by this House, for which the enormous sum of twenty-five hundred dollars had been appropriated. Yet this solitary measure had been rejected by the Senate. This is the great boon given to Connecticut! For these reasons Mr. G. hoped the article would be stricken out, and that, if it was necessary to make terms with the new State, they might receive five per cent. on the receipts of the land, to be paid into their own treasury, disposable by themselves as they saw fit. Messrs. R. Williams, Jackson, and Holland, said a few words in favor of retaining the article; when the question was taken on striking it
- 63. out, and lost—yeas 17. Mr. Fearing, wishing that half the proceeds of the Western lands should be laid out on roads within the Territory, made a motion to that effect; lost—yeas 25. The report of the select committee, without further amendment, was then agreed to, and a bill ordered in conformity thereto. Wednesday, April 7. An engrossed bill for the relief of Thomas K. Jones was read the third time, and passed. The Speaker laid before the House a letter from the Secretary of State, accompanying his report on the memorial of Fulwar Skipwith, referred to him by order of the House on the nineteenth of January last; which were read, and ordered to be committed to a Committee of the whole House on Friday next. Mr. John C. Smith, from the Committee of Claims, to whom was recommitted, on the fifteenth ultimo, their report on the memorial of Paul Coulon, a French citizen, made a supplementary report thereon; which was read, and ordered to be referred to a Committee of the whole House to-day. On motion it was Resolved, That a committee be appointed to examine and report the state of the office of the Clerk of this House. Ordered, That Mr. Clay, Mr. Huger, and Mr. Southard, be appointed a committee pursuant to the said resolution. Mr. Mitchill, from the committee to whom were referred, on the fifth instant, the amendments proposed by the Senate to the bill, entitled An act for revising and amending the acts concerning naturalization, reported that the committee had had the said amendments under consideration, and directed him to report to the House their agreement to the same.
- 64. North-western Territory. The House resolved itself into a Committee of the Whole on the bill to enable the people of the eastern division of the Territory north- west of the river Ohio to form a constitution and State Government, and for the admission of such State into the Union, on an equal footing with the original States, and for other purposes. Mr. Fearing moved to amend the bill so as to embrace the population of the eastern division as bounded by the articles of the ordinance, the effect of which motion would be to include about thirty thousand inhabitants of that division, that are excluded by the provisions of the bill, and respecting whom it is provided in the bill, that they may hereafter be added by Congress to the new State, or disposed of otherwise, as provided by the fifth article of the compact. This motion gave rise to a debate of considerable length, in which Messrs. Fearing, Bayard, Griswold, Goddard, Henderson, and Randolph, supported; and Messrs. Giles, Bacon, and R. Williams, opposed the amendment. Those who supported the amendment contended that the exclusion of that portion of territory occupied by about three thousand inhabitants was both unconstitutional and inexpedient. On the ground of constitutionality, they contended, that under the articles of the compact, which were to be considered as the constitution of the territory, Congress had only the right of forming the eastern division into one, two, or three States; and that under this power, no right existed to form one part of the division into a State, and leave the remaining section in a Territorial condition; that the rights of the whole of the inhabitants of the eastern division were equal, and if one part was, so also must the remaining part be, admitted to the privilege of a State. On the ground of expediency, it was contended that the situation of the excluded inhabitants would be peculiarly hard; that, if attached to the Indiana Territory, they would be placed two or three hundred miles from it; that they would be furthermore degraded from the
- 65. second to the first branch of Territorial government, and that they would be deprived, by the reduction of their numbers, from the prospect of being admitted for a great number of years, to State rights. On the contrary, the opponents of the amendment contended that the provisions of the bill were both constitutional and expedient; that under the compact the right was given to Congress of admitting the eastern division into the Union, in the form of one, two, or three States; that this right involved a discretion to admit a part of that division at one time, and the remaining part at a subsequent period; that if the whole division were once admitted into the Union, Congress would be prohibited from dividing hereafter, when it was acknowledged such division would be expedient, the said division into two or more States, without the consent of the State now formed. That, as to considerations of expediency, the hardships likely to be felt by the excluded inhabitants were such as arose, not from the provisions of the bill, but from their local situation; and that it was not true that they would be degraded by annexation to the Indiana Territory; to a lower grade of Territorial character than they at present enjoyed—the grade being the same. Mr. Randolph supported the amendment on peculiar ground, declaring that if the amendment should not prevail, he would still vote for the admission. He declared himself in favor of the amendment, principally from a desire to avoid the introduction of too many small States into the Union. The question was then taken on Mr. Fearing's amendment, and lost— yeas 34, nays 38. Mr. Fearing moved so to amend the bill as to leave to the new State the right of naming itself. Agreed to. After some discussion of the details of the bill, the committee rose and repeated the bill, with amendments.
- 66. Ordered, That the said bill, with the amendments, do lie on the table. Thursday, April 8. Mr. John Taliaferro, Jun., from the committee to whom was referred, on the fifth instant, the petition of sundry citizens of Georgetown, in the District of Columbia, with instruction to report thereon by bill or otherwise, presented a bill to incorporate the Directors of the Columbian Library Company; which was read twice, and committed to a Committee of the whole House on Monday next. Mr. Dennis, from the committee to whom was referred, on the fifth of February last, a motion, in the form of two resolutions of the House, respecting the adjustment of the existing disputes between the Commissioners of the City of Washington, and other persons who may conceive themselves injured by the several alterations made in the plan of the said city; also, relative to a plan of the said City of Washington, conformably, as nearly as may be, to the original design thereof, with certain exceptions, made a report thereon; which was read, and ordered to be referred to a Committee of the whole House on Monday next. Mr. John Taliaferro, Jun., from the committee appointed, presented a bill to incorporate the inhabitants of the city of Washington, in the District of Columbia; which was read twice and committed to a Committee of the whole House on Monday next. The Speaker laid before the House a letter from the Secretary of the Treasury, enclosing a statement prepared by the Register, of the application of the appropriations made by Congress for clerk-hire, in the several offices of the Treasury Department, specifying the names of the persons, and the salaries allowed to each, for the three last years, in pursuance of a resolution of this House, of the twenty-fifth ultimo; which were read, and ordered to lie on the table.
- 67. The Speaker laid before the House a letter from the Secretary of the Treasury, accompanying two statements, marked A and B, relative to expenses incurred by the United States in the exercise of jurisdiction over the territory of Columbia, since the assumption of jurisdiction by Congress, prepared in pursuance of a resolution of this House of the first instant; which were read, and ordered to be referred to the committee appointed, on the eighth of December last, to inquire whether any, and, if any, what alterations or amendments may be necessary in the existing government and laws of the District of Columbia. The House proceeded to consider the report of the select committee to whom were referred, on the fifth instant, the amendments of the Senate to the bill, entitled An act for revising and amending the acts concerning naturalization, which lay on the table: Whereupon, Resolved, That this House doth agree to the said amendments, with amendments, to the section proposed to be substituted by the Senate in lieu of the first and second sections of the original bill. Mr. Nicholson, from the committee appointed on the second instant, presented a bill to abolish the Board of Commissioners in the city of Washington, and to make provision for the repayment of loans made by the State of Maryland for the use of the city; which was read twice and committed to a Committee of the whole House on Monday next. Mr. Nicholson, from the committee appointed, presented a bill to provide more effectually for the due application of public money, and for the accountability of persons intrusted therewith; which was read twice and committed to a Committee of the whole House on Monday next. The House resolved itself into a Committee of the Whole on the supplementary report of the Committee on Claims, of the seventh instant, to whom was recommitted, on the fifteenth ultimo, their report on the memorial of Paul Coulon, a French citizen; and after some time spent therein, the committee rose and reported a
- 68. resolution, which was twice read, and agreed to by the House, as follows: Resolved, That there be paid to Paul Coulon, as agent for the captors of the ship Betty Cathcart and brig Aaron, prizes to the French privateer La Bellone, out of any moneys in the Treasury, not otherwise appropriated, the sum of six thousand two hundred and forty-one dollars and forty-four cents, being the amount retained by the Treasury Department, from the sales of the ship Betty Cathcart, and for duties on the cargo of the brig Aaron. Ordered, That a bill or bills be brought in, pursuant to the said resolution; and that the Committee on Claims do prepare and bring in the same. North-western Territory. The House proceeded to consider the amendments reported yesterday from the Committee of the Whole to the bill to enable the people of the Eastern division of the Territory north-west of the river Ohio to form a constitution and State Government, and for the admission of such State into the Union on an equal footing with the original States, and for other purposes, which lay on the table; and the same being severally twice read, were, on the question put thereupon, agreed to by the House. A motion was then made, further to amend the said bill, at the Clerk's table, by striking out, in the sixth, seventh, eighth, ninth, and tenth lines of the second section thereof, the following words: and on the north, by an east and west line, drawn through the southerly extreme of Lake Michigan, running east, after intersecting the due north line aforesaid, from the mouth of the Great Miami, until it shall intersect Lake Erie or—and inserting in lieu thereof, the word to. It passed in the negative—yeas 27, nays 44, as follows:
- 69. Yeas.—James A. Bayard, Thomas Boude, Manasseh Cutler, John Davenport, Thomas T. Davis, John Dennis, Ebenezer Elmer, Abiel Foster, Calvin Goddard, Roger Griswold, William Helms, Joseph Hemphill, Archibald Henderson, William H. Hill, Benjamin Huger, Thomas Lowndes, Lewis R. Morris, James Mott, Thomas Plater, Nathan Read, John Cotton Smith, John Stanley, John Stratton, Samuel Tenney, Thomas Tillinghast, Lemuel Williams, and Henry Woods. Nays.—Willis Alston, John Archer, John Bacon, Theodorus Bailey, Phanuel Bishop, Richard Brent, Robert Brown, William Butler, Samuel J. Cabell, Thomas Claiborne, Matthew Clay, John Clopton, John Condit, Richard Cutts, John Dawson, William Dickson, Lucas Elmendorph, William Eustis, John Fowler, William B. Giles, John A. Hanna, Daniel Heister, William Hoge, James Holland, David Holmes, George Jackson, Charles Johnson, Samuel L. Mitchill, Thomas Moore, Anthony New, Thomas Newton, jr., Joseph H. Nicholson, John Smilie, Israel Smith, John Smith, (of Virginia,) Samuel Smith, Richard Stanford, Joseph Stanton, jr., John Taliaferro, jr., Philip R. Thompson, Abram Trigg, John Trigg, Isaac Van Horne, and Robert Williams. Mr. John C. Smith moved further to amend the bill, by striking out the third section thereof, in the words following, to wit: And be it further enacted, That all male citizens of the United States, who shall have arrived at full age, and resided within the said Territory at least one year previous to the day of election, and shall have paid a territorial or county tax, and all persons having, in other respects, the legal qualifications to vote for Representatives in the General Assembly of the Territory, be, and they are hereby, authorized to choose Representatives to form a Convention, who shall be apportioned amongst the several counties within the Eastern division aforesaid, in a ratio of one Representative to every —— inhabitants of each county, according to the enumeration taken under the authority of the United States, as near as may be, that is to say: from the county of Trumbull, —— Representatives; from the county of Jefferson, —— Representatives, —— of the —— to be
- 70. elected within what is now known by the county of Belmont, taken from Jefferson and Washington Counties; from the county of Washington, —— Representatives; from the county of Ross, —— Representatives, —— of the —— to be elected in what is now known by Fairfield County, taken from Ross and Washington Counties; from the county of Adams, —— Representatives; from the county of Hamilton, —— Representatives, —— of the —— to be elected in what is now known by Clermont County, taken entirely from Hamilton County: and the elections for the Representatives aforesaid, shall take place on the second Tuesday of October next, the time fixed by a law of the Territory, entitled An act to ascertain the number of free male inhabitants of the age of twenty-one, in the Territory of the United States north-west of the river Ohio, and to regulate the elections of Representatives for the same, for electing Representatives to the General Assembly, and shall be held and conducted in the same manner as is provided by the aforesaid act, except that the qualifications of electors shall be as herein specified. The motion to strike out was supported by Messrs. John C. Smith, Goddard, Fearing, and Henderson, and opposed by Messrs. Giles, Mitchill, R. Williams, Elmer, and Holland, on the ground that the right of the United States to admit necessarily involved the power of prescribing a convention. The yeas and nays were taken, and it passed in the negative—yeas 26, nays 48, as follows: Yeas.—Thomas Boude, Manasseh Cutler, Samuel W. Dana, John Davenport, Abiel Foster, Calvin Goddard, Roger Griswold, Seth Hastings, Joseph Hemphill, Archibald Henderson, Benjamin Huger, Thomas Lowndes, Thomas Morris, Thomas Plater, Nathan Read, William Shepard, John Cotton Smith, John Stratton, Samuel Tenney, Thomas Tillinghast, George B. Upham, Killian K. Van Rensselaer, Peleg Wadsworth, Lemuel Williams, and Henry Woods. Nays.—Willis Alston, John Archer, John Bacon, Phanuel Bishop, Richard Brent, William Butler, Samuel J. Cabell, Thomas Claiborne, John Clopton, John Condit, Thomas T. Davis, John Dawson, William
- 71. Dickson, Lucas Elmendorph, Ebenezer Elmer, John Fowler, William B. Giles, Edwin Gray, John A. Hanna, Daniel Heister, William Helms, William Hoge, James Holland, David Holmes, George Jackson, Charles Johnson, Samuel L. Mitchill, Thomas Moore, James Mott, Anthony New, Thomas Newton, jr., Joseph H. Nicholson, John Smilie, Israel Smith, John Smith, (of Virginia,) Josiah Smith, Samuel Smith, Henry Southard, Richard Stanford, Joseph Stanton, jr., John Stewart, John Taliaferro, jr., David Thomas, Philip R. Thompson, Abram Trigg, John Trigg, Isaac Van Horne, and Robert Williams. Mr. Fearing said he was of opinion that some provision ought to be made for the inhabitants excluded from the new State, and the continuance of suits from the old to the new Government; for these purposes he moved the recommitment of the bill. Lost. Mr. Dana proposed so to amend the fourth section, as that a majority of the whole number of delegates elected in the Convention, instead of a majority of those present, should first determine whether it be or be not expedient to form a constitution, c. The yeas and nays were called, and the motion carried—yeas 38, nays 33, as follows: Yeas.—Thomas Boude, William Brent, John Condit, Manasseh Cutler, Samuel W. Dana, John Davenport, Thomas T. Davis, Lucas Elmendorph, Ebenezer Elmer, William Eustis, Abiel Foster, John Fowler, Calvin Goddard, Edwin Gray, Roger Griswold, John A. Hanna, Joseph Hemphill, Archibald Henderson, William Hoge, Benjamin Huger, Lewis R. Morris, Thomas Morris, James Mott, Thomas Plater, Nathan Read, William Shepard, John Cotton Smith, Henry Southard, Richard Stanford, Joseph Stanton, jr., John Stewart, John Stratton, Samuel Tenney, Thomas Tillinghast, John Trigg, George B. Upham, Peleg Wadsworth, and Lemuel Williams. Nays.—Willis Alston, John Archer, John Bacon, Robert Brown, William Butler, Samuel J. Cabell, Thomas Claiborne, Matthew Clay, John Clopton, Richard Cutts, John Dawson, William Dickson, William B. Giles, William Helms, James Holland, David Holmes, George Jackson,
- 72. Charles Johnson, Samuel L. Mitchill, Thomas Moore, Anthony New, Thomas Newton, jr., Joseph H. Nicholson, John Smilie, Israel Smith, John Smith, (of Virginia,) Samuel Smith, John Taliaferro, jr., David Thomas, Philip R. Thompson, Abram Trigg, Isaac Van Horne, and Robert Williams. The bill was then ordered to be engrossed for a third reading to- morrow. Friday, April 9. A message from the Senate informed the House that the Senate have passed a bill, entitled An act to amend the Judicial System of the United States; to which they desire the concurrence of this House. [The chief alterations made from the old system consist in the holding the Supreme Court only once a year by four justices, and the establishment of six circuits, within each district of which circuit courts are to be holden twice a year, composed of one justice of the Supreme Court and the judge of the district, in which said court is held.] The bill was read twice, and referred to a select committee. Ohio State Government. An engrossed bill to enable the people of the Eastern division of the Territory north-west of the river Ohio to form a constitution and State Government, and for the admission of such State into the Union on an equal footing with the original States, and for other purposes, was read the third time, and the blanks therein filled up: And, on the question that the same do pass, it was resolved in the affirmative—yeas 47, nays 29, as follows: Yeas.—Willis Alston, John Archer, John Bacon, Theodorus Bailey, Phanuel Bishop, Richard Brent, Robert Brown, William Butler, Samuel J. Cabell, Thomas Claiborne, Matthew Clay, John Clopton, John Condit, Thomas T. Davis, John Dawson, William Dickson, Lucas
- 73. Elmendorph, Ebenezer Elmer, William Eustis, John Fowler, William B. Giles, William Hoge, James Holland, David Holmes, George Jackson, Samuel L. Mitchill, Thomas Moore, James Mott, Anthony New, Thomas Newton, jr., Joseph H. Nicholson, John Smilie, Israel Smith, John Smith, (of New York,) Josiah Smith, Samuel Smith, Richard Stanford, Joseph Stanton, jr., John Stewart, John Taliaferro, jr., David Thomas, Philip R. Thompson, Abram Trigg, John Trigg, John P. Van Ness, Isaac Van Horne, and Robert Williams. Nays.—Thomas Boude, John Campbell, Manasseh Cutler, Samuel W. Dana, John Davenport, John Dennis, Abiel Foster, Calvin Goddard, Roger Griswold, William Barry Grove, Seth Hastings, Joseph Hemphill, Archibald Henderson, Benjamin Huger, Thomas Lowndes, Lewis R. Morris, Thomas Morris, Thomas Plater, Nathan Read, William Shepard, John Cotton Smith, John Stanley, John Stratton, Samuel Tenney, Thomas Tillinghast, George B. Upham, Killian K. Van Rensselaer, Lemuel Williams, and Henry Woods. Monday, April 12. An engrossed bill for the relief of Theodosius Fowler, was read the third time, and passed. The House went into Committee of the Whole on the bill for the relief of Paul Coulon, which was reported without amendment, and ordered to be engrossed and read the third time to-day. Mr. S. Smith, from the committee appointed, presented a bill for the relief of Lewis Tousard; which was read twice and committed to the Committee of the Whole for to-morrow. Mr. Clay, from the committee appointed on the seventh instant, to examine and report on the state of the office of the Clerk of this House, made a report: which was read, and ordered to lie on the table. The House resolved itself into a Committee of the Whole on the bill to provide for the establishment of certain districts, and therein to
- 74. amend an act, entitled An act to regulate the collection of duties on imports and tonnage, and for other purposes; and, after some time spent therein, the committee rose and reported several amendments thereto; which were severally twice read, and agreed to by the House. Ordered, That the said bill, with the amendments, be engrossed, and read the third time to-morrow. The House resolved itself into a Committee of the Whole on the report of the Secretary of State, of the seventh instant, to whom was referred, on the nineteenth of January last, the memorial of Fulwar Skipwith; and after some time spent therein, the committee rose and reported two resolutions thereupon; which were severally twice read and agreed to by the House, as follows: Resolved, That provision ought to be made by law, for the payment of four thousand five hundred and fifty dollars, unto Fulwar Skipwith, (which sum was advanced by him to the United States,) with an interest of —— per centum, from the first of November, one thousand seven hundred and ninety-five. Resolved, That provision ought to be made by law, for compensating the said Fulwar Skipwith, for his services from the first of November, one thousand seven hundred and ninety-six, to the first of May, one thousand seven hundred and ninety-nine, at the rate of —— dollars, per annum. Ordered, That a bill or bills be brought in pursuant to the said resolutions; and that Mr. Dawson, Mr. Van Cortlandt, and Mr. Stanton, do prepare and bring in the same. The House then went into Committee of the Whole on the report of the committee of the twenty-second of January, on the petition of Sarah Fletcher and Jane Ingraham, referred to them on the tenth of December last, and, after some time spent therein, the committee rose and reported several resolutions thereupon; which were severally twice read, and agreed to by the House, as follows:
- 75. Resolved, That it is expedient to grant to the widows and children, as the case may be, of the officers, seamen, and marines, who were lost at sea, on board the ship Insurgent and brigantine Pickering, lately in the service of the United States, four months' pay of their respective husbands or fathers. Resolved, That it is expedient to provide by law for the payment of five years' half pay to the widows and children, as the case may be, of such officers in the naval service of the United States as shall be slain in battle, or die, when in the actual line of their duty. Resolved, That the widows and children of those officers who were lost at sea in the ship Insurgent and brigantine Pickering, shall be entitled to this provision. Ordered, That a bill or bills be brought in pursuant to the said resolutions; and that Mr. Eustis, Mr. Goddard, and Mr. Stanton, do prepare and bring in the same. An engrossed bill for the relief of Paul Coulon was read the third time and passed. Mr. S. Smith, from the committee appointed the ninth instant, on the part of this House, jointly, with the committee appointed on the part of the Senate, to consider and report what business is necessary to be done by Congress in their present session, and when it may be expedient to close the same, made a report thereon; which was read, and ordered to lie on the table. The House went into Committee of the Whole on the bill for the relief of sick and disabled seamen. Mr. Eustis moved to strike out the first section which forms the moneys devoted to the above object into a general fund, to be applied according to the discretion of the President, instead of suffering it to remain, as heretofore, applied to the particular ports, (or those in the vicinity,) from which the moneys are derived. This motion was supported by Messrs. Eustis, Mitchill, and Dana, and opposed by Messrs. S. Smith, Milledge, Davis, Macon, and Huger.
- 76. The question was then taken on striking out the first section, and lost; when the committee rose, and reported the bill with amendments. Monday, April 19. Navy Pensions. An engrossed bill for the relief of widows and orphans of certain persons who have died, or may hereafter die, in the naval service of the United States, was read the third time; and, on the question that the same do pass, it was resolved in the affirmative—yeas 34, nays 29. Compensation of Collectors. The House went into Committee of the Whole on the bill to amend the act fixing the compensation of officers employed in the collection of duties on imposts and tonnage. This bill allows certain compensations to collectors of ports, provided the clear annual receipt does not exceed $5,000. A motion was made to strike out $5,000, for the purpose of introducing $4,000. It was contended that this latter sum was sufficient compensation to any collector; that it greatly exceeded most of the compensations allowed to the Federal officers; and that as money was appreciating, it became necessary to reduce the salaries of officers generally. In reply it was observed that very few collectors would receive so large a sum as $5,000—none other than those of New York, Philadelphia, Baltimore, and perhaps Charleston; that the responsibility attached to these officers was greater than that attached to any other, as in some instances two million of dollars passed through their hands; that the temptation to violate duty was proportionably great; and that, from these considerations, it became
- 77. the Government to afford them a liberal compensation; and that the sum was considerably below that heretofore allowed. The question was taken on striking out $5,000, and lost—yeas 26. Mr. Stanley moved to strike out that part of the bill which deducted from the compensations made to the collectors of Newbern and Edenton, the sum of $250, heretofore allowed beyond their fees. For this motion he assigned several reasons: among which were the inadequacy of the compensations, viz: about $1,600 to the duties performed, which were, notwithstanding the small amount of duties, very burdensome, owing to the smallness of the cargoes imported, and theirs being greatly inferior to the compensations allowed to the collectors of Wilmington and Petersburg. Mr. S. Smith informed the committee that the principle on which the several compensations had been graduated was, that when the gross emoluments exceed $2,000, the salary heretofore allowed by law, in addition to the emoluments, should be withdrawn. This was the fact in relation to the ports of Newbern and Edenton; and as the duties in each of these ports did not exceed $45,000, the compensation seemed adequate; he was, however, far from being tenacious, and would have little objection to a vote of the House which should increase it. Motion lost—yeas 25. The committee rose, and reported the bill without amendment. Mr. Southard renewed the motion to strike out $5,000, for the purpose of inserting $4,000, (the same motion made in committee,) and assigned substantially the same reasons above stated. Messrs. Stanley, Bacon, and Smilie, delivered a few observations for, and Mr. Huger against the motion, which was taken by yeas and nays, on the call of Mr. Southard, and lost—yeas 31, nays 40. Thursday, April 22. French Spoliations.
- 78. Mr. Giles, from the committee appointed on the fifth of February last, to whom were referred the memorials and petitions of sundry citizens of the United States, and resident merchants therein, praying relief in the case of depredations committed on their vessels and cargoes, while in pursuit of lawful commerce, by the cruisers of the French Republic, during the late European war, made a report thereon; which was read, and ordered to lie on the table. Friday, April 23. Judiciary System. The question was then put on the passage of the bill. Mr. Bayard called for the yeas and nays, which were taken, and stood —yeas 46, nays 30, as follows: Yeas.—Willis Alston, John Archer, John Bacon, Theodorus Bailey, Phanuel Bishop, Walter Bowie, Richard Brent, Robert Brown, William Butler, Thomas Claiborne, Matthew Clay, John Clopton, John Condit, Richard Cutts, John Dawson, William Dickson, Lucas Elmendorph, John Fowler, William B. Giles, Edwin Gray, John A. Hanna, Daniel Heister, William Helms, James Holland, David Holmes, Michael Leib, John Milledge, Anthony New, Joseph H. Nicholson, John Smilie, Israel Smith, John Smith, (of New York,) John Smith, (of Virginia,) Samuel Smith, Henry Southard, Richard Stanford, Joseph Stanton, jr., John Stewart, John Taliaferro, jr., Philip R. Thompson, Abram Trigg, John Trigg, Philip Van Cortlandt, John P. Van Ness, Isaac Van Horne, and Robert Williams. Nays.—James A. Bayard, Thomas Boude, John Campbell, Manasseh Cutler, Samuel W. Dana, John Davenport, Thomas T. Davis, John Dennis, Ebenezer Elmer, Abiel Foster, Calvin Goddard, Roger Griswold, Seth Hastings, Archibald Henderson, Thomas Lowndes, Lewis R. Morris, Thomas Morris, James Mott, Thomas Plater, Nathan Read, John Stanley, John Stratton, Benjamin Tallmadge, Samuel
- 79. Tenney, Thomas Tillinghast, George P. Upham, Peleg Wadsworth, Lemuel Williams, and Henry Woods. Tuesday, April 27. Naval Sites. UNAUTHORIZED PURCHASES. Mr. Mitchill, from the committee appointed on so much of the President's Message as relates to naval sites, c., made a further report. The report concludes as follows: The committee find that, prior to the fourth of March, 1801, the sum of one hundred and ninety-nine thousand and thirty dollars, and ninety-two cents, has been expended in purchasing navy yards and making improvements upon them, without any law authorizing the purchase, or any appropriation of money, either for purchase or improvements. Wednesday, April 28. Sedition Act. PETITION OF THOMAS COOPER. A petition of Thomas Cooper, of the county of Northumberland, in the State of Pennsylvania, was presented to the House and read, setting forth that, in the month of April, eighteen hundred, he was tried and condemned at Philadelphia, before Samuel Chase and Richard Peters, judges of the circuit court of the United States there sitting, for having written and published a libel upon the political character and conduct of John Adams, the then President of the United States; and was thereupon adjudged to pay a fine of four hundred dollars, and to suffer an imprisonment of six months; which
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