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Prof. Dr.-Ing. Frank H.P. Fitzek Universita di Ferrara
Code Division Multiple Access
Frank H. P. Fitzek
04 February 2003
Lecture 2
The following text is taken partially from my phd thesis. I hope it may help my
students to understand the concept of CDMA technology.
Contents
1 CDMA Concept 4
1.1 Basic Principles of Spread Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.1 Direct Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.2 Frequency Hopping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.3 Time Hopper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.4 Hybrid Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2 Basic Principles of CDMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.1 RAKE Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.2 Power Control Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.3 Spreading Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2.4 Classiﬁcation of Enhanced CDMA System . . . . . . . . . . . . . . . . . . 18
1.2.5 Comparison of APC, VSG, and MC–CDMA . . . . . . . . . . . . . . . . . 21
1.3 Wireless CDMA Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.3.1 Bit Error Probability in CDMA . . . . . . . . . . . . . . . . . . . . . . . . 23
1.3.2 Packet Error Probability in CDMA . . . . . . . . . . . . . . . . . . . . . . 30
1.3.3 Maximum Throughput for CDMA Systems for Packet Services . . . . . . 31
mail: frank@ﬁtzek.net Lecture 2 Page 1

Prof. Dr.-Ing. Frank H.P. Fitzek Universita di Ferrara List of Figures
List of Figures
1 System model for spread spectrum transmission. . . . . . . . . . . . . . . . . . . 4
2 Classification of different spread spectrum system. . . . . . . . . . . . . . . . . . 5
3 Direct sequence spread spectrum transmission process. . . . . . . . . . . . . . . . 6
4 Direct sequence spread spectrum signal generation GDS=10. . . . . . . . . . . . . 7
5 MFSK FHSS transmitter and receiver. . . . . . . . . . . . . . . . . . . . . . . . . 8
6 Example of MFSK FHSS receiving process. . . . . . . . . . . . . . . . . . . . . . 10
7 Time axis for a THSS system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
8 FDMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
9 TDMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
10 CDMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
11 FHSS multiple access interference. . . . . . . . . . . . . . . . . . . . . . . . . . . 12
12 RAKE receiver structure with three fingers and multi–path channel. . . . . . . . 12
13 Power control loops for IS–95. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
14 Inter–cell interference for adjacent cells. . . . . . . . . . . . . . . . . . . . . . . . 14
15 Auto– and cross–correlation function for Walsh and orthogonal Gold sequences. . 16
16 Auto– and cross–correlation function for m-sequences, Gold sequences, and Kasami
sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
17 Statistical multiplex effect on the air interface of CDMA. . . . . . . . . . . . . . 20
18 Multi–Code CDMA sender. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
19 General CDMA Channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
20 Random sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
21 Random subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
22 Controlled subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
23 Multi–code CDMA sender for randomly chosen spreading codes. . . . . . . . . . 27
24 Multi–code CDMA sender with overlapping sub–spaces using orthogonal and ran-
dom spreading–sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
25 Multi–code CDMA sender for controlled overlapping. . . . . . . . . . . . . . . . . 29
26 Bit error probability p∗
biterror for different Gaussian approximations and WBE
sequences with b=0 and GSpreading=16. . . . . . . . . . . . . . . . . . . . . . . . 30
27 Packet error probability ppkterror for different Gaussian approximations and WBE
sequences with b=0, e=30, LPDU =1023 bit, and GSpreading=16. . . . . . . . . . . 31
28 Throughput T for different Gaussian approximations and WBE sequences with
b=0, e=30, LPDU =1023 bit, and GSpreading=16. . . . . . . . . . . . . . . . . . . 32
List of Tables
1 Comparison of different spread–spectrum techniques with their advantages and
disadvantages. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Spreading sequence length with related number of maximal LSFR (degree f)
spreading sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Enabling CDMA technologies with their benefits and drawbacks for time–variable
QoS support. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4 Comparison of different allocation schemes. . . . . . . . . . . . . . . . . . . . . . 30

Prof. Dr.-Ing. Frank H.P. Fitzek Universita di Ferrara
1 CDMA Concept
Code Division Multiple Access (CDMA) technology was chosen for the air interface of the
upcoming third generation wireless systems (3G). Therefore we have chosen CDMA and a detailed
survey of general CDMA benefits are provided in this section. Different CDMA systems are
introduced and classified. Because Multi–Code CDMA (MC–CDMA) systems play a major role
in this work, we discuss this system in more detail. For the understanding of CDMA we begin
with the basic principles of spread spectrum systems.
1.1 Basic Principles of Spread Spectrum
Spread–spectrum techniques gained their popularity by the needs of military communications.
In contrast to narrow-band communication, spread–spectrum techniques were proved to be more
resistant against hostile jammers. If a communication system is considered a spread–spectrum
system it has to satisfy the following criteria: (1) The bandwidth of the spread signal has to
be greater than the information bandwidth. Since this criteria is satisfied also by frequency
modulation, pulse code modulation, and delta modulation, there is a second condition: (2) The
spread signal is composed of the information signal and the spreading sequence. The spreading
sequence has to be independent from the information. In Figure 1 a spread spectrum transmitter
and receiver are depicted. At the sender side the information signal i(t), with data rate Ri and
bandwidth Bi, is spread by a spreading sequence c(t). The spreading sequence has the code
symbol rate Rc, also called the chip rate. The ratio of spreading bandwidth Bs and information
bandwidth Bi is denoted as the processing gain GSpreading = Bs
Bi
of a spread–spectrum system.
Figure 1: System model for spread spectrum transmission.
The processing gain does not combat white noise as it is with frequency modulation and
pulse code modulation, because the spread signal is independent of the information signal. But

Prof. Dr.-Ing. Frank H.P. Fitzek Universita di Ferrara1.1 Basic Principles of Spread Spectrum
spread–spectrum signals offer the following applications for wireless communication system: (1)
Spread spectrum modulation is capable to deal with multi–path interference. Furthermore,
in Section 1.2.1 we will see that enhanced techniques have been developed to suppress multi-
path interference even further. Dependent on the spread spectrum scheme and modulation
methods the multi–path interference rejection gain differs. (2) The receiver of a spread spectrum
system is able to distinguish between different transmitted signals using the spreading sequence.
For multiple access capability the spreading sequence design is very important. The spreading
sequence is the identification for a transmitter–receiver pair. (3) Other advantages such as low
probability of interception, privacy, and anti–jam capability are more relevant for military needs.
Different spreading techniques are possible. Figure 2 gives a classification of spread spectrum
techniques. The diagram shows techniques such as Direct Sequence (DS), Frequency Hopping
(FH), and Time Hopping (TH) with their sub–classifications. All techniques can be combined
to combat their disadvantages and combine their benefits. These systems are referred to hybrid
spread–spectrum systems. These systems consist of combinations of two or more pure spread–
spectrum systems. There exist four possible hybrid systems, namely DS/FH, DS/TH, FH/TH,
and DS/FH/TH (see Figure 2).
Figure 2: Classification of different spread spectrum system.
In the following we give a short introduction to the different spreading techniques and discuss
their benefits.
1.1.1 Direct Sequence
In a Direct Sequence Spread Spectrum (DSSS) transmitter the information signal is directly
modulated by a spreading sequence. The spreading sequence consists of a number of spreading
chips with time duration τchip. The information signal consists of a number of information bits
with time duration τbit. Spreading is achieved if multiple chips represent one bit. If τbit is a
multiple of τchip the processing gain GDS can be easily calculated by:
GDS =
τbit
τchip
(1)
Figure 3 shows a possible DSSS transmission process. It can be seen how the original information
signal i(t) is spread before the transmission over the wireless link and how it will be despread at
the receiver side. Furthermore the influence of the presence of jammer is also mentioned. If the
data rate Ri is very small in comparison to the chip rate Rc, than the spread signal s(t) = i(t)·c(t)
will have approximately the bandwidth of the spreading signal (Bc Bs). At the receiver side
the received signal r(t) will be multiplied once again with the same spreading sequence c(t),
which results in a despreading of the original signal i(t), if the autocorrelation ϕE
cc(τ) is nearly

Figure 3: Direct sequence spread spectrum transmission process.
zero for all τ = 0. After the de-spreading the signal will be filtered with bandwidth Wi to remove
high frequencies.
The generation of DSSS signals can be achieved by a simple multiplication of information and
spreading sequence. Figure 4 shows the generation of a DSSS signal for GDS=10. A key feature
of DSSS is that multiple access capability can be achieved without synchronization between
different transmitters. Multi-path interference is combated if delayed signals differ by only one
chip duration. In this case all delay signals are treated as interference. On the other side the
transmitter receiver pair has to be fully chip–synchronized. As explained in Section 1.2.2 the
near–far effect has to be taken under consideration, because DSSS systems are using the full
bandwidth and therefore a transmitter closer to the receiver will constantly interfere and destroy
signals from transmitters that are far away.
1.1.2 Frequency Hopping
Frequency Hopping Spread Spectrum (FHSS) systems change the carrier frequency of the
modulated information signal periodically. During a time interval τhop the frequency is constant.
Afterwards both the sender and receiver hop to another frequency. The whole bandwidth Bs
is divided into frequency slices of Bs∗ . The set of available frequencies is called a hop–set

Figure 4: Direct sequence spread spectrum signal generation GDS=10.
with N frequency slices. FHSS systems diﬀer totally from DSSS systems in terms of frequency
occupancy. While DSSS systems occupy the whole bandwidth, FHSS systems use only one
frequency slice at any point in time. FHSS systems are divided into slow and fast frequency
hoppers as depicted in Figure 2. If one information bit is transmitted over several frequency slices
(τhop ≤ τbit) the frequency hopper is referred to be fast. Otherwise if several bits are transmitted
over one frequency slice it is referred to be a slow hopper. The relation of hop duration τhop to
information bit duration τbit depends upon the number of hops per information bits. If τbit is an
integer multiple of τhop, then we denote τbit = k · τhop. This leads to the processing gain GFH
(see [7]).
GFH =
Bs
Bi
= N · k · Bs∗ · τhop (2)
The advantage of FHSS systems in contrast to DSSS systems is the less strict synchronization
requirement. FHSS allows synchronization errors of the size of τhop, DSSS only in size of τchip
[25].
In Figure 5 a Multi Frequency Shift Keying (MFSK) FHSS transmitter and receiver is
depicted. The incoming data passes the MFSK modulator and then the hopping frequency
(given by a Pseudo–Noise (PN ) generator) is modulated. The receiving process is given in
Figure 6. The dashed red line represents the hopping frequency, while the relative frequency
(solid blue line) gives the information. The mapping of relative frequencies to bits is given on
the right side of Figure 6.
1.1.3 Time Hopper
Within a Time Hopping Spread Spectrum (THSS) system the time axis is divided into frames
of the duration τframe. As depicted in Figure 7 each frame is divided again into N slots with time
duration τslot. A single WT will only use one slot out of k possible slots within one frame. Within

Prof. Dr.-Ing. Frank H.P. Fitzek Universita di Ferrara 1.2 Basic Principles of CDMA
Figure 5: MFSK FHSS transmitter and receiver.
this slot the WT sends with a k times higher data rate in contrast to the situation where the
WT would transmit within the whole frame. Interference among simultaneous wireless terminals
can be minimized if coordination between terminals can be achieved. This also avoids the near–
far effect (will be introduced in Section 1.2.4). In the absence of coordination situation occur
in which more than one terminal will use a time slot. The receiver will not be able to detect
either of the signals correctly. Both terminals collide on the wireless link. For such cases error
correction schemes are required. Time hopper have nearly the same acquisition time to that of
discrete–sequence systems, but their implementation is much simpler than that of a frequency-
hopper.
1.1.4 Hybrid Systems
Hybrid Spread Spectrum systems consist of combinations of two or more pure SS systems. There-
fore there exist four possible hybrid systems by combining such systems, namely DS/FH, DS/TH,
FH/TH, and DS/FH/TH. The combination of different SS schemes leads to increased complex-
ity of the transmitter and receiver, but offers also a combination of their advantages. E.g. if
DSSS and FHSS schemes are combined, the hybrid SS system offers the multi-path interference
rejection of the DSSS system and the immunity of the FHSS system. In Table 1 the advantages
and disadvantages of different SS schemes in terms of multiple access capability, multi-path
interference rejection, synchronization, and hardware complexity are shortly listed.
1.2 Basic Principles of CDMA
For wireless communication systems the provision of a multiple access capability is indispens-
able and can be applied in different ways. The classical method is Frequency Division Multiple
Access (FDMA). A more recent technique is Time Division Multiple Access (TDMA). Both
techniques assign particular frequency or time slices to different wireless terminals. When all
slices are occupied in the system no additional wireless terminal can be accommodated. Mul-
tiple access capability is also provided by Code Division Multiple Access (CDMA). The most
common techniques of CDMA are frequency–hopped CDMA or direct–sequence CDMA. CDMA

Table1:Comparisonofdifferentspread–spectrumtechniqueswiththeiradvantagesanddisadvantages.
SSschemeadvantagedisadvantage
direct–sequence•bestbehaviorformulti-path–rejection[7]•near–farproblem[7,25,23,29]
•bestanti-jamrejection[7,24]•requirecoherentbandwidth[7]
•bestinterferencerejection[7,24]•longacquisitiontime[7,24]
•nosynchronizationamongterminals[25,23]•synchronizationofcodesignal
•simpleimplementation[25,23]•withinfractionofchiptime[25,23]
•mostdifficulttodetect[7,24]
frequency–hopper•greatamountofspreading[7,24]•complexhardware[7,25,23,24]
•noneedforcoherentbandwidth[7,25,24]•errorcorrectionisneeded[7,24]
•shortacquisitiontime[7,25,23,24]
•inherentsecurity[29]
•lessaffectedbythenear–fareffect[7,25,23,24]
time–hopper•highbandwidthefficiency[7,24]•errorcorrectionisneeded[7,25,23,24]
•lesscomplexhardware[7,25,23,24]•longacquisitiontime[7,25,23,24]
•lessaffectedbythenear–fareffect[7,25,23,24]

Figure 6: Example of MFSK FHSS receiving process.
allows multiple users to simultaneously use a common channel for transmission of information
[26]. A CDMA transmitter will code its information signal with a code sequence or spreading
sequence. Afterwards, the transmitter sends the coded signal to the receiver. Using the same
code sequence as the transmitter, the receiver is able to decode the received signal. Also, in case
the receiver receives more than one signal, it will be able to decode the information from the
desired transmitter, if the code sequences satisfy cross-correlation and auto-correlation require-
ments (see Section 1.2.3). The bandwidth of the coded signal is much larger than the information
bandwidth. One may say the information signal was spread. The coding process is therefore also
called a spread spectrum modulation, while the coded signal is called a spread-spectrum signal.
The spreading of the information signal gives the CDMA its multiple access capability. In Fig-
ure 8, 9, and 10 it is shown that subscribers can be separated in the frequency, time, or code
domain.
Figure 7: Time axis for a THSS system.

Figure 8: FDMA Figure 9: TDMA Figure 10: CDMA
Different multiple access techniques.
A key–feature of CDMA systems using Pseudo–Noise (PN ) sequences is that they can tol-
erate overload (these are situations where the actual Bit Error Probability (BEP) is very high
for a short time), if all wireless terminals in one cell can tolerate a certain degradation in their
performance. Thus CDMA systems using PN sequences do not have any sharply defined system
capacity, like TDMA or FDMA systems. But for CDMA systems using PN sequences the Bit
Error Probability (BEP) increases with the number of active terminals. To illuminate this prop-
erty we take a frequency hopping system as an example. In Figure 11 the transmission process of
two wireless terminals is given. Each terminal following the principal of frequency hopping hops
to another frequency slice at the beginning of a new slot. Under certain conditions depending on
the hopping strategy both terminals might use the same frequency slice. It is obvious that in the
same way as the number of wireless terminals within one cell increase, the number of collisions
increase as well. Note, even DS and TH spread spectrum will tolerate overload situations.
The following two sections provide a short introduction in further CDMA key features. Sec-
tion 1.2.1 introduces the RAKE receiver. Section 1.2.2 gives a more detailed view on the power
control entity. As already mentioned above the design of spreading sequences plays a major role
for a CDMA system. Therefore we introduce two different families of spreading sequences in
Section 1.2.3. Later, in Section 1.2.4 we classify the different CDMA systems.
1.2.1 RAKE Receiver
In Section ?? we have shown that multi–path degrades the system performance. CDMA receivers
resolve multi–path interference if signals arrive more than one chip apart from each other. The
direct LoS–signal is the strongest one. For this strongest signal the other multi–path signals
are regarded as interference and therefore they can be suppressed with the processing gain. A
better receiver performance is achieved by RAKE receivers. The example depicted in Figure 12
illustrates the basic principle of RAKE receiver. After spreading and modulation of the infor-
mation signal on the sender side the signal suffers from the multi–path channel. Different delay
entities and attenuation factors in Figure 12 model obstacles at different distances and with
different reflection properties. The multi–path signal (coming from different propagation paths)
is demodulated and passed to the RAKE receiver. The ideal RAKE receiver contains a receiver
finger for each multi–path component. In reality the number of fingers is limited by the receiver‘s
hardware complexity or costs. In this case only the strongest signals with a delay of less than
τmax are taken into account. In each finger the signals are despread and time aligned with one of
the multi–path channels. After the despreading process the signals are weighted and combined.

Figure 11: FHSS multiple access interference.
The weight of a signal corresponds in the best case to its attenuation factor. Roughly speaking,
the RAKE receiver is an inverse of the multi–path channel. If the conditions on the multi–path
channel change, the parameters of the RAKE receiver have to be adapted. In [3] the perfor-
mance gain achieved with RAKE receivers in DS–CDMA systems is shown. Recently, RAKE
transmitters (also called Pre–RAKE, while the RAKE receiver is also called Post–RAKE) have
been proposed. If the the multi–path channel is known at the transmitter, a RAKE transmitter
produces weighted signals such that there will be only one signal at the receiver. Barreto etal.
[3] have shown that a substantial performance improvement can be achieved by applying Pre–
and Post–RAKEs. But the trade off between channel improvements and hardware complexity
has to be taken into account.
Figure 12: RAKE receiver structure with three fingers and multi–path channel.
1.2.2 Power Control Mechanism
In a DS–CDMA system all transmitters use the same bandwidth at the same time to send
their information to the receiver. If we consider multiple transmitters sending to one receiver,
the signals are received with different power levels, because wireless terminals have different
distances to the base station (see Section ??). The signal strength can differ in the range of

100dB [30]. Due to the attenuation effect transmitters closer to the receiver have higher power
levels than transmitters that are far away. This effect is called the near–far effect. The near–far
effect plays an essential important role if multi–user interference is considered. For free space
propagation, the received power falls off as the square of the distance from the wireless terminal
to the base station. In addition to the near–far effect the signal strength differs dramatically due
to the changing propagation conditions (moving obstacles, traffic characteristics, hand over). To
overcome the changing signal strength power control entities are implemented in the transmitters.
These entities are called Transmitter Power Control (TPC). The TPC adjusts the transmission
power Ptrx at sender–side to ensure that all signals arrive at the receiver with the same power
level Prcv.
For cellular CDMA mobile communications systems the TPC plays a major role for the up–
link. Within the down–link all signals sustain the same channel propagation and no near–far
effect has to be taken under consideration. TPC for the down–link are only implemented to adjust
a required SNIR. Furthermore, it has to be mentioned that the CDMA capacity is interference
limited. Thus, the lower the power level of one terminal, the higher is the number of supported
terminals. The TPC for the down–link is only implemented to adjust a required Signal to Noise
Interference Ratio (SNIR). In contrast to the up–link the conditions on the down–link change
more slowly. Most TPCs consist of two types of power control, namely open loop and closed
loop. Some wireless CDMA communication systems also have an outer loop power control, such
as IS–95. The open loop power control measures the SNIR of the incoming signals and adjusts
the transmission power to meet the desired SNIR. Unfortunately, the conditions for up–link and
down–link can differ dramatically and therefore the open loop power control provides only a
first estimation for the TPC. The closed loop power control measures the signal power at the
receiver and controls the transmitter‘s power. IS–95 receivers send only one bit of information
to change the transmitting power in the range of 1db/1.25ms at the transmitter side. The outer
loop power control gets information about the Packet Error Probability (PEP) and compares
it with the required PEP. In case of deviations the transmitter power is adjusted. For cellular
CDMA wireless communications systems the TPC plays a major role for the up–link.
Figure 13: Power control loops for IS–95.
In [17] it was claimed that the probability density function (pdf ) of the received power
depends on the TPC algorithm, speed of the Adaptive Power Control (APC) system, dynamical
range of the transmitter, spatial distribution of the wireless terminals, and propagation statistics.

In [18] it was shown that the pdf–function can be assumed to be log normal. The pdf was given
as
f(Prcv) =
1
√
2πσPrcv
· exp −
ln2(Prcv)
2σ2
(3)
where σ is the imperfection in the TPC entity. In [17] it was shown that the performance
degradation of a DS CDMA system with σ = 1 db was about 60%. Therefore the system
performance of a CDMA systems depends particularly on the system parameter σ.
The smaller σ the better the performance. But not only the variance of the used energy within
one cell has to be taken under consideration. Also the inter–cell interference plays a major role
in CDMA systems. In Figure 14 the interference produced/received by the center cell for/from
neighboring cells is given. In [10] it is claimed that each cell in the first tier (see Figure 14)
contributes/receives about 6% of the interference. Greater tiers do not have a great impact
(second tier about 0.2% of the interference). Independent from the exact amount of interference
this small example shows that the overall energy used within the cell has to be limited. Power
control for voice oriented continuous traffic has dominated the work of most of the researchers
for decades. But the next generation wireless networks are designed to support packet oriented
data traffic in addition to the old–fashioned voice oriented traffic. Packet oriented traffic brings
up new problems for the TPC which are quite different to voice traffic.
Figure 14: Inter–cell interference for adjacent cells.
In [2] it was mentioned that packetized data without delay constraints can be sent in depen-
dency of the channel state. If the wireless terminal has to invest too much power to transmit
the packet successfully, it could also wait and send this packet later when channel conditions are
quite good. This situation is referred to as back off. In the back off case the delay of the packet
will increase – a situation which is not allowed for voice oriented services (only small buffering
can be tolerated). In a nutshell data communication introduce a new degree of freedom; the
trade–off between power consumption and delay constraints. In consideration of this new degree
of freedom Bambos etal. [2] presented a new protocol approach for adaptive power control, which
also takes delay constraints of packets into account. We will refer to this protocol in Section 1.2.4.

1.2.3 Spreading Sequences
The proper choice of spreading sequences enables multiple access capability for spread–spectrum
based wireless communication systems. A sequences is a non-ambiguous identification for a
transmitter receiver pair. Spreading sequences can be divided into orthogonal and pseudo–noise
spreading sequences. This section gives a comprehensive overview of spreading sequences of
CDMA systems. Differences between pseudo–noise and orthogonal spreading sequences in terms
of auto– and cross–correlation functions are given. The properties of spreading sequences to
accomplish multiple access capability is investigated. Furthermore the sets of spreading sequences
are discussed for multi–code CDMA.
In a spread spectrum communication system the original information signal with bandwidth
WI is spread to a much larger transmission bandwidth WSS using spreading sequences. At
the receiver side the same replica of this spreading sequence is used to recover the original
information. In cellular radio communication multiple access capability is achieved
by taking advantage of the autocorrelation and cross–correlation properties of the
spreading sequences. Such systems are refered to as code division multiple access (Therefore
the spreading sequences are also called code sequences in the literature.). A spreading sequence
consists of C units called chips. The chips are two–valued. The autocorrelation function of a
spreading sequence reflects the similarity of this sequences with a replica of itself delayed by a
time gap τgap. For a given time gap τgap = 0 the autocorrelation value is one. In any other
case τgap = 0 the autocorrelation value should be small to minimize the interference among
copies of the original signal that are generated and delayed by multi-path propagation. The
cross–correlation value of two different spreading sequences represents the interference level for
two signals from different wireless terminals with delay τgap. This value should be as small as
possible for all τgap such that a maximum number of subscribers are allowed in the cell. As a
matter of fact, the optimization of the autocorrelation and the cross–correlation can not be done
separately. They depend on each other and each optimization of one side leads to a degradation
of the other side.
Orthogonal Sequences Orthogonal sequences offer zero cross–correlation for τgap = 0. If all
transmitters are synchronized and no multi–path is considered, the multiple access interference
can be neglected. For any other value of τgap they have large cross–correlation values. There-
fore orthogonal sequences are only applied if perfect synchronism can be guaranteed within the
system. The autocorrelation properties of orthogonal sequences are also poor if τgap = 0. This
happens if we consider multi–path interference. In such a situation equalization is applied to
recover the original signal.
Several methods to generate orthogonal sequences are presented in [26, 29]. As representatives
of the orthogonal code family Walsh and orthogonal Gold sequences are presented in this section.
Walsh sequences can be easily obtained by applying the Hadamard transformation (⊗ defines
the Kronecker product), which is given by
H0 = [1], H2 =
+1 +1
+1 −1
, Hn+1 = Hn ⊗ Hn (4)
Each row of the Hadamard matrix represents one valid sequence with length N. This leads to an
overall number of sequences NS = N. Note, the overall number for PN sequences is much larger.
In general the number of achievable codes NS is limited for orthogonal spreading sequences.
Another representative of the family of orthogonal spreading sequences are the orthogonal Gold

sequences. Figure 15 (right side) shows that orthogonal Gold sequences have reasonable cross–
correlation and autocorrelation values. These values are much better than for sequences of the
Walsh family.
Walsh sequences with length 16 Orthogonal Gold sequences with length 32
Figure 15: Auto– and cross–correlation function for Walsh and orthogonal Gold sequences.
Pseudo Noise Sequences PN sequences are binary sequences, which exhibit random properties
similar to noise. Within the class of PN sequences, the most popular representatives are Maximal
Length sequences, Gold sequences, and Kasami [26] sequences. All sequences can be generated
using a Linear Feedback Shift Register (LSFR), which is build by f feedback–taps [12]. Se-
quences generated with a LSFR having the maximum possible period length for an f–stage shift
register are called Maximal Length or simply m–sequences. The length of an m–sequence can be
proven to be 2f − 1. The number of possible codes depends on the number of possible sets (also
called primitive irreducible generators) of feedback–taps. Golomb [12] showed that the overall
number of sequences generated by a LSFR of degree f equals
NS(f) =
2f − 1
f
k
i=1
Pi − 1
Pi
, (5)
where Pi equals the prime decomposition of 2f − 1. Table 2 provides the sequence length of a
given degree f and the resulting number of achievable code sequences (see [31] for higher values
of f). m–sequences have three important properties: (1) the balance property (the number of
Table 2: Spreading sequence length with related number of maximal LSFR (degree f) spreading
sequences.
f 2 4 6 8 10 12 14 16 18 20
Pi 3 15 63 255 1023 4095 16383 65535 262143 1048575
NS 1 2 6 16 60 144 756 2048 8064 24000
ones and the number of zeros diﬀer by at most one), (2) the run length property (half the runs of
ones and zeros have length 1, and 1/2k length k for k < n) and (3) the shift–and–add property

(combining two shifted replicas of an m–sequence yields another valid m–sequence). The periodic
autocorrelation function of an m–sequence is only two–valued:
ΦCC,mseq :=
1 k = l · N
− 1
N k = l · N
(6)
m-sequences have good autocorrelation properties, while the cross–correlation properties are
very poor compared to Gold sequences. Gold sequences can be achieved by combining two
preferred m–sequences. Different Gold sequences are achieved by combining one m–sequence
with a delayed replica of another m-sequence. This gives an overall number of Gold sequences
of 2f + 1 (considering two m–sequences and the possibilities of combination 2f − 1). The cross–
correlation function ΦCC,gold is three–valued {−t(f); −1; t(f) − 2} with
t(f) :=
(1 + 2
f+1
2 ) · 2−f if f odd
(1 + 2
f+2
2 ) · 2−f if f even
(7)
Gold [11] claimed that the cross–correlation between these sequences is uniform and bounded.
Combinations of Gold sequences give Kasami sequences. Kasami sequences have optimal cross–
correlation values achieving the Welsh lower bound [19]. Therefore Kasami sequences are a
promising candidate for spreading in W–CDMA. The scaled cross–correlation function ΦCC,kasami
is also three–valued {−1; −2f/2 − 1; 2f/2 − 1}. The auto– and the cross–correlation function
m–sequences with length 15 Gold sequences with length 31 Kasami sequences with length 63
Figure 16: Auto– and cross–correlation function for m-sequences, Gold sequences, and Kasami
sequences.
for m-sequences, Gold sequences and Kasami sequences are depicted in Figure 16. For these
specific examples it is shown that cross-correlation for m–sequences equals (-1/16; 1), while
Gold and Kasami sequences, which are three–valued, give (-9/35; -1/35; 7/35) and (-9/63; -
1/63; 7/63) respectively. Moreover, the Figure gives a first quantative estimate for auto– and
cross–correlation.
Whenever synchronism can not be achieved PN sequences should be applied. These se-
quences are easy to generate and offer reasonable cross–correlation and autocorrelation values.
For synchronous systems orthogonal codes should be used. In cellular wireless communication
orthogonal codes should be used for the down–link and PN sequences should be used for the
up–link. A special case are multi–code CDMA systems, which are explained in the following
section. In this work, we concentrate on the up–link transmissions and therefore use the PN
sequences. In Section ?? we give a short introduction of our work for down–link transmissions
with orthogonal sequences. Interested readers may refer to [9, 8] for further information.

1.2.4 Classification of Enhanced CDMA System
Adaptive Power Control (APC) in CDMA systems is one way to obtain heterogenous QoS sup-
port. In contrast to recent power control mechanisms, which were designed to assign equal power
to the wireless terminals, adaptive power control assigns power related to the QoS requirements.
Another subgroup of the enhanced CDMA systems are Multi–Rate CDMA systems. These
are the only systems that are allowed to change their rates instantaneously and offer variable
bit rates. The specific schemes of this subgroup are Fixed Spreading Gain (FSG), Variable
Spreading Gain (VSG), and Multi–Code CDMA (MC–CDMA). In contrast to Single–Code
CDMA systems Multi–Code systems support multiple times the basic bit rate. A more detailed
description is given in Section 1.2.4.
Adaptive Power Control As outlined above the power control is designed to overcome the
near–far effect and tries to control the received power from the WTs. Another goal of the power
control is to increase the system capacity. Under the assumption that all wireless terminals
require the same QoS, the power for each of the terminals is controlled such that sender’s signals
are received at the base station with the same power. This is the power control algorithm for
voice oriented services.
To support time–variable QoS the power level of each terminal was identified in [35, 34, 21, 36]
as a controllable parameter. The proposed approach dynamically adapts the QoS requirements
with power control schemes. As an example we assume k wireless terminals. Suppose that
terminal i = 1 has the highest QoS requirements QoS1 with power level P1. All other terminals
adjust their power level Pi according to the TPC‘s commands. The power level for WT i is given
by
Pi = P1 ·
QoSi
QoS1
. (8)
During the transmission process, the power is adjusted depending on the channel state to
achieve a required SNIR, e.g. if the channel becomes worse (higher interference) the power level is
increased to stabilize the SNIR. The advantage of such a system is that it can be easily integrated
into existing CDMA systems. Adaptive power control is a strong candidate for time–variable
QoS support. In contrast to the VSG and MC approaches adaptive power control support does
not include variable bandwidth.
Fixed Spreading Gain In the Fixed Spreading Gain (FSG) approach [16], the spreading gain
of each bit stream is maintained constant. The chip duration is also maintained constant. The
users vary the transmission time to adapt the variable bit rate requirements. The low rate users
transmit for a shorter time as opposed to the higher rate users. It can be seen that the lower
rate users suffer from more multi–access interference from the higher rate users, as the higher
rate users transmit for longer time. However, the higher rate users suffer from multi–access
interference for a shorter time. Hence, the performance for higher rate users is better than for
lower rate users. To compensate for this degradation in the performance, the lower rate users
transmit with more power to increase their signal to interference ratio.
Variable spreading gain The Variable Spreading Gain (VSG) approach [15] offers the possibil-
ity to achieve flexible data rates. Under the assumption that the bandwidth for transmission is
fixed, the spreading gain is reduced as the information bit rate increases, and vice versa. Chang-
ing the spreading gain has an impact on the energy per bit ratio. If the transmission power is

fixed an increased data rate leads to a smaller energy per bit ratio (leading to a higher bit error
probability). Therefore, the VSG approach is coupled with a variable power control entity to
satisfy the same SNIR for wireless terminals with different bit rates. In case the transmission
power is adapted to the transmission rate, high data rate terminals will influence low data rate
terminals. Note, that different terminals with the same SNIR requirement can be accommodated
by allocating power proportional to the terminals‘ data rates.
Pi = P1 ·
Ri
R1
(9)
Whenever the transmitter changes the spreading gain, the receiver has to be informed about
this change. This results in an increased signaling overhead. Since the power level varies accord-
ing to the chosen data rate the TPC will also add some more signaling overhead. In the case
of short spreading sequences, VSG–CDMA systems suffer from multi–path interference, because
of Inter Symbol Interference (ISI ). If we consider a constant delay spread for the wireless link,
the same number of chips are influenced, but the number of influenced symbols depends only
on the spreading gain. In [23] it is claimed that with VSG the high power terminals degrade
the performance of low power terminals. In addition, high power terminals are using smaller
spreading codes and therefore the spreading sequences have a significant impact on the system
performance. Thus it might be useful to implement higher bit rates with Multi–Code CDMA
systems [23].
Multi–Code Code Division Multiple Access In this section the capability of WTs to transmit
and receive on multiple CDMA channels by the means of MC–CDMA is discussed. We discuss
different families of spreading sequences with their need for signaling for MC–CDMA. We then
compare the MC–CDMA approach with the enhanced power control and the variable spreading
gain approach. In Figure 17 the statistical multiplex effect of the air interface is depicted. For
illustration purposes we choose three WTs which transmit packets over the wireless link to
the BS. For this example we assume that WTs are able to transmit also on multiple channels.
This example shows how the three different streams of each WT with highly variable load are
multiplexed on the air interface.
MC–CDMA [6] can support a basic bit rate as well as integer multiples of the basic bit
rate. In MC–CDMA a high data rate is split into smaller data rates. Each small data rate is
then spread by a different code sequence over the entire coherent bandwidth. All spread signals
are modulated and transmitted over the wireless link. Figure 18 depicts a Multi-Code CDMA
transmitter. It shows that a high data rate is split into smaller data rates. Afterwards each small
data rate is spread by a code sequence over the whole coherent bandwidth. All spread signals
are modulated and transmitted over the wireless link.
The choice of spreading codes is very important to decrease self–interference. Since orthogonal
sequences are used the number of overall sequences within the cell is limited. In case one terminal
wants to transmit on more channels it has to request these channels from a centralized code
depository. Thus delay–sensitive services are hard to support. Orthogonal code sequences do
benefit from the fact that self interference does not exist as long as the system is synchronous.
Synchronism exists only within the down–link and between channels of one wireless terminal in
the up–link. In asynchronous systems orthogonal codes lead to poor system performance.
The other choice of codes are PN code sequences. The number of these codes is large, however,
the interference among the channels is high. Lin and Gitlin [6] proposed to use pseudo random

Figure 17: Statistical multiplex effect on the air interface of CDMA.
Figure 18: Multi–Code CDMA sender.
sequences to distinguish between the asynchronous up–link transmissions of the different wire-
less terminals. Additionally, they proposed to use orthogonal sequences to distinguish between
synchronous up–link transmissions of a given wireless terminal. This approach gives a large total
number of codes in conjunction with low self–interference between the parallel channels of a given
wireless terminal. Further improvements are achieved when a central entity assigns the terminal
specific pseudo random sequences [32]. Note that this does not require additional overhead for
bit rate changes because the total number of codes is large and the codes are assigned to the
terminals during link establishment. The hardware complexity, however, is higher as RAKE re-
ceivers are required for each channel to suppress multi–path interference. It is noted in [23] that
the MC–CDMA approach is slightly more promising for multi–rate CDMA than the Variable
Spreading Gain approach. It is argued that the former has a smaller signaling overhead and
lower multi–path interference.

1.2.5 Comparison of APC, VSG, and MC–CDMA
A brief overview of benefits and drawbacks of the three approaches for supporting time–variable
QoS is given in Table 3. Note that combinations are possible to combine their benefits. In [28]
the performance in terms of SNIR and BEP of VSG and the MC–CDMA systems is compared.
The authors in [28] summarize that whether a wireless terminal uses one of theses approaches,
it has an identical effect on both the SNIR and the BEP of the other wireless terminals in
the system. Furthermore the authors remark that multi–code systems have the disadvantage of
instantaneous amplitude variations (for VSG they exhibit a constant instantaneous amplitude)
and additional hardware complexity. In [33] methods are introduced which reduce the hardware
complexity of MC–CDMA systems significantly. Nevertheless in [23] it is claimed, that multi–
rate can be implemented with MC–CDMA in a slightly better way than with VSG, because
of higher signaling overhead and lower multi–path interference rejection of the later. Adaptive
power control approaches have no multi–rate capability. For all approaches a high performance
power control entity has to be implemented. Henceforth we use the MC–CDMA approach.

Table3:EnablingCDMAtechnologieswiththeirbeneﬁtsanddrawbacksfortime–variableQoSsupport.
EnhancedVariableMulti–Code
PowerControlSpreadingGainCDMA
multi–pathnofurtherimpacthighrate→highISInofurtherimpact
interference
multipleaccess→powerlevel→bitrate→numberofchannels
interference
hardwareTPCalreadyexistfurtheroscillatorRAKEreceiver
complexityforeachchannel
highperformancepowercontrol
signalingoverheadTPCmessageTPCmessagenotnecessarily
changeinSGonlyforWBEsequences(Section1.3.1)
enablehighernoyesyes
datarates
granularity—highdegreedependsmultipleofCDMA
onspreadingcodeschannels
appliedin—UMTSFDD[1]UMTSTDD[1]

Prof. Dr.-Ing. Frank H.P. Fitzek Universita di Ferrara 1.3 Wireless CDMA Channels
1.3 Wireless CDMA Channels
In the following we investigate the characteristics of the CDMA uplink wireless channel. We
concentrate on the usage of Pseudo–Noise (PN ) sequences. In particular the CDMA based bit
error probability for non orthogonal spreading sequences depends on the overall number of active
channels in one cell. Therefore we give a detailed overview of the major bit error probability
models and the impact on throughput and capacity.
1.3.1 Bit Error Probability in CDMA
Direct Sequence (DS) CDMA communications systems achieve their multiple access capability
by assigning each WT a unique spreading sequence (see Section 1.2.3). Because of different
distances between WT and BS the transmitted signals arrive at the BS with random delays
τ, carrier phases φ and power levels P. The exact error probability depends on the particular
spreading sequences used by the WT, random amplitude, delay and carrier phase. The calcula-
tion of the exact Bit Error Probability (BEP) for a CDMA system with Pseudo–Noise (PN )
sequences is difficult to evaluate. Therefore a variety of different error models based on Gaussian
approximations can be found in [29]. The design of the spreading sequences is not taken under
consideration with the Gaussian approximations.
Figure 19: General CDMA Channel.
As given in Figure 19 we suppose that J WTs simultaneously access the wireless channel.
We further assume that each bit is spread by GSpreading chips. The transmitted signal s(k) is
assumed to be a sequence of two independent, identically distributed random variables (chips),
where the probability of each variable is given by 1/2. Therefore we use the classical expression
for uncoded BPSK modulation considering additive white Gaussian noise with power spectral
density I0 and energy per bit Eb (given in Equation 10) to calculate the bit error probability of a
DS CDMA system. Assuming that all WTs arrive chip synchronously but phase asynchronously

at the receiver side the bit error probability pBPSK
biterror results in
pBPSK
biterror = Q
√
2 · SNIR = Q
2 · Eb
I0
. (10)
where Q(x) is related to the complementary error function and is defined as
Q(x) =
1
√
2π
∞
x
e−t2/2
dt. (11)
The energy per bit Eb can be calculated by dividing the received signal power PS by the data
rate Ri.
Eb =
PS
Ri
. (12)
To calculate the power spectral density I0 we assume perfect power control for all WTs. Perfect
power control implies that all signals arrive at the receiver with the same power level. Moreover
we are not considering multi–path interference in our model. Thus the noise power spectral
density I0 is influenced by three different terms as depicted in Figure 19. The additive white
Gaussian noise power spectral density N0 is characterized by the communication system, while
the jammer density Y0 depends only on the jammer‘s hardware and his jamming technique
(broadband or band limited). For simplicity we assume only broadband jamming. The last
factor are other active WTs which are using the channel simultaneously. The sum over the
energy of all other WTs will be spread over the whole spreading bandwidth WSS. The inter-chip
interference was not taken into account within our assumptions. Because of the perfect power
control assumption each signal will arrive with the same power level, which results in a very
simply equation for the noise power spectral density I0 (see Equation 13)
I0 =
I
WSS
=
PS · (k − 1)
WSS
+ N0 + Y0, (13)
where I is the overall noise power by all other WTs and k denotes the number of active channels
within the cell. According to a Single–Code CDMA (SC–CDMA) system we assume one channel
per WT and if αi represents the activity of WT i, k is generally given by k = J
j=1 αj. Using
Equation 10, 12 and 13 we obtain:
pbiterror = Q


2 · PS
R
PS·(k−1)
WSS
+ N0 + Y0

 . (14)
GSpreading =
TB
TC
=
1
TC
1
TB
=
WSS
Ri
. (15)
The spreading gain has been defined as the number of chips per bit for a DS CDMA system.
Thus it can be calculated by the ratio of TB and TC. This is equivalent to the ratio of WSS and
Ri.
pbiterror = Q


2 · GSpreading
k − 1 + N0·WSS
PS
+ Y0·WSS
PS

 . (16)

The complementary error function Q function can be transformed in the erfc function
erfc(α) = 2 · Q
√
2 · α . (17)
We model both the background noise and the jamming (in a friendly environment the jammers
can also be neglected) as b wireless terminals that transmit all the time, resulting in a decreased
SNIR. This simplification leads to the simplified Equation 19, which can be found in [22]
pbiterror(k) = Q


2 · GSpreading
k − 1 + b

 . (18)
Hereafter, we use the following notation for p+
biterror to distinguish different CDMA systems: the
superscript + indicates if the system assumptions are based on synchronous or asynchronous
phase and chips as given in Figure 19 (e.g. PACS signifies asynchronous phases (PA) and syn-
chronous bits (CS).).
pPACS
biterror(k) = Q


2 · GSpreading
k − 1 + b

 ; k − 1 + b > 0 (19)
As mentioned before the bit error probability depends also on the delay and carrier phases.
In [29, 27] we can find further bit error probability approximations. Equation 20 gives the bit
error probability for chip and phase asynchronous CDMA systems, while Equation 21 reflects
phase and chip synchronous CDMA systems. Equation 22 reflects phase synchronous and chip
asynchronous CDMA systems.
pPACA
biterror(k) = Q


3 · GSpreading
k − 1 + b

 ; k − 1 + b > 0. (20)
pPSCS
biterror(k) = Q


GSpreading
k − 1 + b

 ; k − 1 + b > 0 (21)
pPSCA
biterror(k) = Q


1.5 · GSpreading
k − 1 + b

 ; k − 1 + b > 0 (22)
An improved Gaussian approximation has been derived by Holtzman [14] and applied in
[4]. The bit error probability for the improved Gaussian approximation is given in Equation 23
and 24. The calculations for the bit error probability are still simple enough but lead to quite
accurate results. In [29] it is claimed that the improved Gaussian approximation should be used,
if the number of WT is small or the spreading gain GSpreading is large.
pIMP
biterror(k) = 2
3 Q
3·GSpreading
k−1+b
+1
6 Q

 GSpreading
(k−1+b)N
3
+
√
3σ


+1
6 Q

 GSpreading
(k−1+b)N
3
−
√
3σ

 (23)

σ2
= (k − 1 + b)
23
360
· G2
Spreading +
1
20
+
k − 2 + b
36
(GSpreading − 1) . (24)
Until now we have focused on SC–CDMA systems. But the bit error probabilities given in
the equation above can be used even for the MC–CDMA approach. As mentioned before the
MC–CDMA performance depends significantly on the influence of parallel channels. Therefore
the motivation for the present section is to show the impact of three different spreading sequence
allocations schemes for multi–code in terms of the Signal to Noise Interference Ratio (SNIR). We
consider J WT with an asynchronous MC–CDMA channel with the Binary Phase Shift Keying
(BPSK) modulation scheme, in which each WT, say WT j, is assigned a certain number 1 ≤ Rj ≤
Rmax of spreading sequences. Spreading sequences used by user j span an N-dimensional space
Uj that is a subspace of a M-dimensional space M spanned by all k = J
i=1 Ri ≤ J ·Rmax = Smax
spreading sequences. Figure 20, 21, and 22 show three different allocation schemes with their
M-dimensional spreading sequence space. In the following section we provide a description of
the different spreading design rules for MC–CDMA.
Figure 20: Random sequences Figure 21: Random subspaces
Figure 22: Controlled sub-
spaces
Different approaches for allocation of spreading sequences.
Random Sequences The first approach called random sequences or PN sequences chooses the
spreading sequences randomly i.e. instead of carefully designing a set of spreading sequences,
the sequences are chosen randomly from the set of all possible sequences. Sequences are changed
from bit to bit1 so the vectors associated with each users jump in the space M which is illustrated
in the Figure 20. We assume the the spreading sequences are chosen randomly by a centralized
entity (e.g. base station) and signaled at the beginning of each communication session to the
WTs. This procedure takes care that sender and receiver use the same spreading sequence and
that no spreading sequence is used twice (we refer to Section ?? for details). Figure 23 gives one
possible implementation of a MC–CDMA sender. Multiple channels d1 up to dR are spreading
their information with different PN sequences. The PN sequences are created by the Generator
entity. There exists only one Oscillator entity for all channels. After the spreading process all
signals are multiplexed to signal s and send over the wireless link.
1
Such as in IS–95 between sender and receiver sequences are generated out of one long and well known PN
sequence.

Figure 23: Multi–code CDMA sender for randomly chosen spreading codes.
The advantage of random sequences is that there is no need for planning, design and/or
adaptive allocation of spreading sequences. This feature avoids overwhelming signaling. In case
the random sequences approach is used all active channels have an inﬂuence on each other. As
an example if all WT are active and the WTs are using all their channels, k (as the parameter to
calculate the BEP in Equation 19, 21, 22, 20, and 23) is given by J · Rmax. In case Rg channels
are active at WT g the number of active channels seen by WT g within the single cell for each
channel is given by
k =
J
j=1
Rj. (25)
This k value can be used to calculate the BEP.
Random Subspaces The next approach is called random subspaces. The method of allocating
spreading sequences presented here reduces the interference in some manner by exploiting the
information about the synchronous transmission channels. The idea is to make each user span a
subspace that is spanned by orthogonal sequences used on the synchronous channels. Thus, the
interference term V s
j,Rj
is equal to zero. In case random subspaces are used the interference of
parallel used channels of the same WT can be neglected and therefore k is given by
k =
J
j=1;i=g
Ri + 1. (26)
Note that spreading sequences still have to change from symbol to symbol randomly due
to the existence of other asynchronous users causing the subspaces to jump in the space M
which is illustrated in the Figure 21. For illustration purpose we assume that each WT has
the same number of maximal parallel channels Rmax (all subspace have the same size). While
the dimension of the subspace is determined by the orthogonal spreading–sequences, the hop

process is determined by the PN–sequence. Such random subspaces can be easily generated by
means of the Hadamard matrix (see Section 1.2.3). Hence, all we need to do is to generate
a random spreading sequence and multiply columns of the Hadamard matrix by the elements
of the spreading sequence so that the first row becomes the generated random sequence. This
approach shows how to use this additional information in a simple way in order to improve the
system performance. Since the subspaces are generated without considering the other spanned
subspaces by other users, they can overlap in the sense that they interfere. By overlapping,
we mean the interference measured in terms of SNIR such that stronger overlapping leads to a
smaller SNIR values.
Figure 24: Multi–code CDMA sender with overlapping sub–spaces using orthogonal and random
spreading–sequences.
In Figure 24 one possible sender structure is given. In contrast to the sender structure
in Figure 23 firstly the data signals d1 up to dR are spread with a small orthogonal spreading
sequence. The spreading sequences are different and are used to identify different channels of one
WT. After the multiplexing process the multiplexed signal is spread again with one unique PN
sequence. This PN sequence has to be assigned by one central entity and signaled once over the
control channel. While the PN sequence is unique and controlled by one central entity, the list
of orthogonal sequences can be the same for all WTs. Note, even for this sender structure only
one Oscillator entity is necessary for all channels. The advantage of this allocation schemes is
given by smaller interferences than for the random subspaces and the signaling takes place only
in the beginning of one communication session. The disadvantage is the need for a centralized
entity. This makes this approach impractical for certain scenarios such as given in ??.
Controlled Overlapping Strong overlapping of subspaces discussed in the previous section can
result in a significant performance degradation. We introduce a new approach which is called
controlled overlapping. The interference cannot be completely avoided (Figure 22) since it would
require sequences with perfect aperiodic correlation functions and such sequences do not exist.

In case the controlled overlapping approach is used and full synchronism in terms of chip and
phase can be achieved Welch Bound Equality (WBE) [20] sequences are used. The bit error
probability is given in Equation 27.
pWBE
biterror(k) =



0 : k + b ≤ GSpreading
Q
GSpreading
k−GSpreading+b : k + b > GSpreading
(27)
where k is calculated as in Equation 25.
Figure 25: Multi–code CDMA sender for controlled overlapping.
As given later in Figure 26 the WBE sequences produce smaller interferences than any other
spreading allocation scheme. The problem is that every spreading sequence in the system has
to be changed if only one channel is added or released to achieve the best system performance.
Every time a spreading sequence has to be changed signaling over the wireless link is necessary.
In case the sequences are not changed the system becomes sub–optimal. There might be a
trade–off between the signaling period and the signaling overhead. But nevertheless on wireless
links, which are highly error–prone, the signaling can be lost or be delayed and this results in
a high performance degradation. In Figure 25 one possible sender structure for the controlled
overlapping scheme is given. The only difference to Figure 23 is the control channel to the
Generator entity.
Comparison of different spreading allocation schemes In Table 4 the differences of the spread-
ing allocation schemes in terms of interference, signaling, and need for a centralized entity is
given. The symbols +, -, ++, and -- refer to the amount of interference, signaling or need for a
centralized entity.
Illustration of the BEP for different approximations Figure 26 depicts the error probability
of the four standard Gaussian approximations, the improved Gaussian approximation, and the

Table 4: Comparison of different allocation schemes.
Interference Signaling Need for centralized entity
Random Sequences ++ −− NO
Random Subspaces + − YES
Controlled Overlapping −− ++ YES
WBE sequences. The four standard Gaussian approximations show nearly the same character-
istic. Note, the asynchronous system offers the best error behavior. A dramatic performance
gain can be achieved if WBE sequences are applied. Note, that for such systems a large signal-
ing overhead and hard system requirements in terms of full synchronism has to be taken under
consideration.
Figure 26: Bit error probability p∗
biterror for different Gaussian approximations and WBE se-
quences with b=0 and GSpreading=16.
1.3.2 Packet Error Probability in CDMA
Considering the bit error probabilities in Equation 19, 21, 22, 20, 23, and 27 we give the Packet
Error Probability (PEP) for a packet data unit of the length LPDU [bit] in Equation 28 under
the assumption that the bit errors occur without correlations.
ppkterror(LPDU , k) = 1 − (1 − p∗
biterror(k))LP DU
(28)
If we assume a coding scheme that allows us to correct e bit errors the packet error probability
becomes smaller and is given in Equation 29[13, 24, 5].
ppkterror(e, LPDU , k) = 1 −
e
i=0
LPDU
i
(1 − p∗
biterror(k))LP DU −i
(p∗
biterror(k))i
(29)

In Figure 27 we depict the packet error probability ppkterror for different Gaussian approximations.
Figure 27: Packet error probability ppkterror for different Gaussian approximations and WBE
sequences with b=0, e=30, LPDU =1023 bit, and GSpreading=16.
1.3.3 Maximum Throughput for CDMA Systems for Packet Services
It was already mentioned that CDMA systems are able to tolerate overload, if all wireless ter-
minals in one cell will tolerate a certain degradation in their performance. The main goal is now
to investigate the capacity defined as the number of active terminals within a single cell CDMA
system achieving the maximal throughput. Therefore we look at Equation 30 (see [13]), which
gives the throughput T for a single CDMA cell in dependency of the packet length L, the size of
redundancy R, the number of correctable bits e, the number of active wireless terminals k, and
the type of Gaussian approximation.
T(L, e, k) = (L − R) · k · (1 − p∗
pkterror(k)) (30)
For different Gaussian approximations the throughput of one CDMA cell is depicted in Figure 28.
The behavior of all throughput curves is nearly the same. For a small number of active channels
the throughput increases linearly. At a specific point (different for all Gaussian approximations)
the throughput decreases rapidly. Thus for each Gaussian approximations an optimal number
kopt = max(T(k)) of active channels can be found, where the throughput of the cell is maximal.
For a given Gaussian approximation Figure 28 shows the optimal number of supportable active
channels. The Gaussian approximation PACA has a number of 13 active channels resulting in
an overall throughput of 9kbit per slot.
We will show in course of this work that it might lead to a better performance in terms of
delay constraints and loss probability, if other than the optimal number kopt of CDMA channels
is distributed among the WTs. From this it follows that not the optimal throughput for a
single cell CDMA system will be achieved, but WTs can be supported with their heterogenous
and time–variable QoS requirements. Note, in case we distribute the optimal number kopt of

Prof. Dr.-Ing. Frank H.P. Fitzek Universita di Ferrara References
Figure 28: Throughput T for diﬀerent Gaussian approximations and WBE sequences with b=0,
e=30, LPDU =1023 bit, and GSpreading=16.
CDMA channels, the PEP is approximately 10%. This large PEP might be insuﬃcient for some
applications.
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