Code Division Multiple Access

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Code Division Multiple Access

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Title: Code Division Multiple Access

1
Chapter 6

Code Division Multiple Access

We can achieve code division multiple access
(CDMA) based on spread spectrum techniques.
In particular, we can assign different spreading
codes to different users in a DSSS system so that
the users can share the communication channel.
For a DS-SS based CDMA (DSCDMA) system, multiple
access interference (MAI) is the major factor
limiting the performance and hence the capacity
of the system.
Therefore, analyses of the effect of MAI on the
system performance as well as ways to suppress
MAI have been the major focus of CDMA research.

Roughly speaking, there are two different
approaches to the problem.
The first approach is based on the concept of
single-user detection.
In this approach, we identify one of the users in
the system as the desired user and treat all
signals from the other users as interference.
The receiver (for the desired user) detects only
the desired user signal.
The second approach is called multiuser
detection, in which all signals from all users
are detected jointly and simultaneously by the
receiver.

A common receiver based on the single-user
detection approach is the matched filter receiver
matched to the desired users spreading signal.
We note that the matched filter receiver is not
optimal (in the sense of maximizing the
likelihood function) in the presence of MAI.
Optimal receivers of such a kind are discussed in
1.
However, due to their complexity, we will omit
the optimal receivers here and focus on the
simple matched filter receiver.
As a starting point, we will refine the crude
analysis of the symbol error performance.

In general, it is difficult to conduct an exact
symbol error performance analysis of a DS-CDMA
system based on the matched filter receiver even
in an AWGN channel.
Usually we have to resort to bounds and
approximations.
We will discuss several common techniques to
calculate the approximate symbol error
probability of a DS-CDMA system.
These approximate analyses are important because
they provide us simple ways to obtain the symbol
error probability, which is crucial in
determining the capacity of the system.
Towards the end of the chapter, we will also
present some techniques based on the matched
filter receiver to suppress MAI so that
performance of the system can be improved.

6
6.1 Asynchronous DS-CDMA model

In many cases, the transmissions from different
users in a DS-CDMA system are not synchronized.
One of such examples is given by the uplink
(reverse link) of IS-95 2.
In order to include these asynchronous cases, we
consider a general asynchronous model of the
DS-CDMA system here.
We assume that there are K actively transmitting
users in the system.
We associate the kth user with a data signal
bk(t) and a spreading signal ak(t), where

is the (transmitted) power of the kth
user signal.
is the sequence of data symbols for the
kth user.
is the spreading sequence assigned to
the kth user.
For simplicity, we assume that BPSK modulation is
employed, i.e., bi(k) are iid binary random
variables taking values from the set 1, -1
with equal probabilities.
We also assume that the spreading sequence
is periodic with period N, where T NTc,
and the chip waveform is time-limited to
0 Tc) and is normalized such that
Extensions to other types of modulations and long
sequences are straightforward.

The received signal at the receiver matched to
the spreading signal of the mth user, for 1?m ?
K, is
n(t) AWGN with power spectral density N0
the amplitude response
the phase response
the delay (with respect to some time
reference) of the channel from the transmitter of
the kth user (we will call it the kth
transmitter) to the receiver of the mth user (we
will call it the mth receiver).
Also, where
the term is the phase difference due to
the delay

We employ to model the asynchronous
nature of the system.
We assume that synchronization with the mth
users signal has been achieved at the mth
receiver.
Hence, we can assume
without loss of generality.
For k ? m, we model as
independent uniform random variables on the
intervals 0 T) and 0 2p), respectively.
Moreover, we also make the assumption that all
the random variables associated with different
users are independent.

Now, let us look at the mth matched filter
receiver which is designed to detect the mth
users signal.
Without loss of generality, we consider the
detection of the symbol b0(m).
Let be the received power
of the kth users signal at the mth receiver.

The decision statistic for the 0th symbol is
given by
ik,m is the interference component due to the kth
signal.
?is the component due to the AWGN that is a
zero-mean Gaussian random variable with variance
N0T.

On the other hand, is given by
where is decomposed into

are the
aperiodic, even, and odd cross correlation
functions between the sequences
respectively.

14
6.2 Error analysis of matched filter receiver

As mentioned before, it is important to determine
the error performance of a DS-CDMA system using
the matched filter receiver with the presence of
MAI.
Obviously, it would be most desirable if we could
obtain the exact average symbol error
probability.
Unfortunately, this task is exceedingly complex
for most practical scenarios in which many users
are actively transmitting.
Therefore, bounds and approximations are
typically employed.
It is the goal of this section to give an
introduction to some common bounding and
approximation techniques.
We will focus on one of the receiver, namely the
mth receiver.
To simplify notation, we write,

15
6.2.1 Error bounds

Let us assume that the set of spreading sequences
for the K users are given.
For convenience, we define a set of system
parameters
Then the conditional symbol error probability
given Sm and b0(m) 1 and that given Sm and
b0(m) -1 are, respectively,

Where
is the MAI component.
Hence the average symbol error probability is

The second equality in (6.10) is due to the fact
that the data symbols b0(k) and b-1(k) for k ? m
are symmetrically distributed about 0, i.e., the
distribution of Im is symmetric about zero.
In general, the complexity of calculating of the
expectation in (6.10) is exceedingly high even
when there are only a moderate number of users.
In many cases, we have to resort to bounds which
can be calculated with a practical level of
computational complexity.

A simple bound based on (6.10) is that
where the maximization is over all possible
choices of values of the system parameters in the
set Sm.
For example, with the rectangular chip waveform,
i.e.,
and BPSK spreading, for k ? m
Where

Hence
Therefore
where is the symbol energy of the
mth user.
Although the bound in (6.11) or (6.15) can be
calculated easily given the set of sequences
used, this bound is often not tight and hence its
usefulness is limited.

Another way to bound the average symbol error
probability is to first determine and bound the
distribution function of the MAI contribution Im
and then obtain bounds on the average symbol
error probability by taking the expectation in
(6.10) using the bounds on the distribution
function of Im.
Using this approach, we can obtain very tight
upper and lower bounds on the average symbol
error probability with a complexity which
increases linearly with K 3.
In general, it is difficult to determine the
distribution function of Im.
Only the distribution functions for some simple
cases such as BPSK spreading and QPSK spreading,
have been worked out.
Yet another way to bound the average symbol error
probability is to make use of the idea of
moment-space bounds 4.

21
6.2.2 Gaussian approximations

Instead of obtaining bounds on the average symbol
error probability, we can assume the MAI
contribution Im as a Gaussian random variable and
obtain an approximation to the average symbol
error probability based on this assumption.
There are mainly two variations to this
Gaussian-approximation approach.
Standard Gaussian approximation
Improved Gaussian approximation

22
Standard Gaussian approximation

The first method is to assume Im as a zero-mean
Gaussian random variable.
This method is usually known as the standard
Gaussian approximation (SGA) 5 and is
applicable to situations in which there are a
large number of users with similar received
powers in the system.
Its validity is justified by the central limit
theorem based on the fact that Im is a summation
of independent random variables Reik,m.

When the number of users K is large, the
distribution of Im approaches Gaussian.
With SGA, the approximate average symbol error
probability is given by
Therefore, all we need is to calculate the mean
and variance of Im

Hence, the problem reduces to the evaluation of
the variance of Reik,m.
To do this, we use the following simple identity
Then, for k ? m,
Let us write

From (6.5), we have
The second and third equalities in (6.23) are due
to the independence of the random variables
involved.
Substituting (6.21) and (6.22) into (6.23) and
making use of the fact that are
independent,

Let us define

For rectangular chip waveform, i.e.,
as in BPSK or QPSK spreading,
Hence, (6.26) reduces to
Combining (6.17), (6.18), and (6.29),

Given the set of sequences, the standard Gaussian
approximation PSGA can be calculated as easily as
the simple bound in (6.15).
Sometimes, it is more convenient to have an
approximation to the average symbol error
probability which does not depend on the set of
sequences employed.
One reasonable way to obtain such an
approximation is to assume that all the sequence
elements are zero-mean iid random variables with
and replace the terms
in (6.30) by their
expectations.

29

30

Replacing the term
in (6.30) by the expectation in (6.32), we
have the approximation,

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32

Figure 6.2 shows the plots of the approximate
symbol error probability given by the SGA in
(6.33) for the case in which all the users have
equal received powers and N 31.
Note that for K 1, the symbol error probability
is exact.
When the received powers of all users are the
same and the signal-to-noise ratio
as indicated by the plots in Figure 6.2, we
can further approximate (6.33) as

Comparing to the approximation of the average
symbol error probability given by (2.49), The
standard Gaussian approximation in (6.34) gives a
more optimistic estimate on the system capacity.
For example, in order to achieve an average
symbol error probability of 10-3, K?N/3
approximately based on (6.34).
Hence, a DS-CDMA system with a processing gain N
can accommodate N/3 users (compared to N/5 users
predicted by (2.50)).

34
Improved Gaussian approximation

In general, the SGA is reasonably accurate if the
number of users is large and the received powers
of the users are similar.
However, when either the number of active users
is not large or there are a few users with
received powers much higher than those of the
others, the SGA does not give an accurate
approximation to the symbol error probability and
another form of approximation is needed.
We would still like to approximate the MAI
component Im in the decision statistic as a
Gaussian random variable.
However, we can no longer to do so by the
reasoning employed before since Im contains of
only a few (significant) terms Reik,m now.

This difficulty can be circumvented by noting
that each ik,m is a summation of a large number
of terms involving the sequence elements (refer
to (6.5) and (6.31)) when the processing gain N
is large.
We can make use of this property to obtain the
desired Gaussian approximation for the MAI by
modeling the sequence elements as iid zero-mean
random variables with
To aid our discussion and to conform to the
notation in the literature, let us define
to be the set of all the phases and delays of
the other users and normalize the decision
statistic zm in (6.4) by the factor

Then, the real part of the normalized decision
statistic becomes
is a zero-mean (real) Gaussian random variable
with variance
It can be shown 6, 7, 8 that the normalized MAI
terms are conditional
independent given and the desired users
spreading sequence.
Based on this, one can further show 6, 7, 8
that conditioning on the set the
normalized MAI component in (6.36)
approaches a zero-mean Gaussian random variable
with variance Vm as N approaches infinity.

The conditional variance of the limiting Gaussian
random variable is given by
where corresponds to the term due to the
kth user and is a simple function (depending on
the spreading technique) of and the chip waveform
7, 8.
For example, with BPSK spreading,

While with QPSK spreading,

We note that are iid random variables given
our model of the delays and phases.
The discussion above implies that we can
accurately approximate the MAI component in
the normalized decision statistic as a
zero-mean Gaussian random variable with variance
Vm when the processing gain N is large.
Hence, the approximate conditional symbol error
probability given is

Averaging this over the delays and phases, we
obtain an accurate approximation to the
(unconditional) symbol error probability

The approximation in (6.41) is known as the
improved Gaussian approximation (IGA).
We note that the IGA is accurate, regardless of
the number of active users in the system, as long
as the processing gain is large.
On the other hand, the SGA is accurate,
regardless of the processing gain, when there are
a large number of active users with equal
received powers.
We notice that the conditional error probability
depends on the delays and phases only through Vm.
Hence, an efficient way to calculate PIGA in
(6.41) is to first obtain the probability density
function pVm(v) of the random variable Vm and
then evaluate the expectation by the integral

The density function pVm(v) of Vm, in turns, can
be easily obtained as the (K-1)-fold convolution
of the density functions of the independent
random variables
Compared to the SGA in (6.31), the computational
complexity of (6.42) is still significant higher.
We can reduce the computational complexity of the
IGA by further approximating the expectation in
(6.41) based on a Taylor series approximation 7,
9 of the conditional symbol error probability as
below

where are the mean and
variance of the random variable Vm, respectively.
From (6.37),
since are iid.
For example, with BPSK spreading,
while with QPSK spreading,

Putting these into (6.43), we obtain an
approximation of the symbol error probability
which is as simple computationally as the SGA in
(6.30).
It is shown in 7, 9 that the approximated IGA
in (6.43) is almost as accurate as the original
IGA in (6.41) in many situations.

In summary, we point out that the IGA generally
gives a more accurate approximation to the symbol
error probability than the SGA does when the
spreading gain N is reasonably large as in most
practical DS-CDMA systems.
To illustrate this point, let us consider the
symbol error probabilities of two DS-CDMA systems
with BPSK spreading and QPSK spreading,
respectively.
From (6.33), the SGA predicts that the symbol
error probabilities of the two systems are the
same.
On the other hand, the IGA (6.43) states that the
system with QPSK spreading has a smaller symbol
error probability than the system with BPSK
spreading.
The latter is in fact true for randomly selected
sequences.

46
6.3 Near-far problem

Based on the SGA in (6.30), we see that the
signal-to-noise ratio (SNR) of a user employing
the matched filter receiver in a DS-CDMA system
with K active users is degraded by the factor
as compared to the case in which only the user
is active.
When the received powers of all users are the
same and the set of spreading sequences are
properly chosen, the degradation in SNR is
relatively small if there are a moderate number
of users.
However, when the received powers of some of the
interferers are much larger than that of the
desired user, the performance degradation is
large.

In the context of wireless communications, this
situation occurs when some of the interferers are
located close to the base station while the
desired user is far away.
This problem is known as the near-far problem in
CDMA systems.
A common measure of the robustness of a receiver
against the near-far problem is the near-far
resistance, defined in 10, of the receiver.
For now, we argue the intuitive idea of the
near-far resistance measure.

To understand the concept of near-far resistance,
let us imagine that only one user, say the mth
user, were active in the DS-CDMA system
considered previously.
In this case, the optimal symbol error
probability (using the matched filter receiver)
would be which decreases
exponentially with rate approximately equal to
the SNR when the SNR is large.
We employ this as a benchmark to compare
performance of different receivers in the
multiuser scenario.
Going back to the realistic situation of multiple
active users, the performance of any receiver
will be poorer than that of the optimal receiver
of the single-user scenario just described
because of the existence of MAI.
In fact, the larger the received powers of the
interferers, the poorer is the performance.

We look at the exponential rate of decrease of
the symbol error probability given by a certain
receiver as the SNR increases to some very large
values.
The ratio of exponential decrease rate to
tells us how efficient the receiver is
compared to the optimal receiver of the
single-user scenario.
If the received MAI power increases, this ratio
will get smaller.
The ratio of the exponential decrease rate to
in the limiting case of extremely large
MAI power is the near-far resistance of the
receiver.
A receiver with near-far resistance close to 1 is
almost as efficient in any near-far situation as
the optimal receiver of the single-user scenario
(the best that could be done).
A near-far resistance of 0 indicates that the
receiver will break down in a near-far situation.

For the matched filter receiver, we can employ
(6.31) or (6.43) to conclude that the symbol
error probability levels off when the SNR
increase (for example, see Figure 6.2).
Hence the exponential decrease rate is 0 and the
near-far resistance is 0.

When the matched filter receiver does not give an
adequate level of performance, there are
basically two ways to tackle the near-far
problem.
One of the ways is to control the transmitted
powers of all the users so that the received
powers are the same. This method is known as
power control, and is employed in all current
practical CDMA systems, such as IS-95.
Another way to tackle the near-far problem is to
notice that the matched filter receiver is not
optimal in the presence of MAI and try to develop
better receivers that are near-far resistant.

52
6.4 Multiple access interference suppression

In this section, we discuss receivers based on
the single-user detection approach.
Receivers based on the multiuser detection
approach will be introduced in Chapter 7.
The optimal signal-user receiver described in 1
is near-far resistant, but it is too complex for
practical implementation.
Its development is rather involved and interested
readers are referred to 1.
Here, we focus on suboptimal receivers that are
less complex than the optimal signal-user
receiver.
The basic working principle of these receivers is
to exploit some structures of the MAI that are
different from the desired signal and to utilize
the difference to remove or suppress the MAI
component from the received signal.

The structures of MAI we can utilize depend the
design of the spreading sequences and the
availability of resources such as multiple
receive antennas.
A main dichotomy on MAI suppression receivers can
be obtained by distinguishing between
short-sequence-based and long-sequence-based
DS-CDMA systems.
In a short-sequence-based system, a simple
structural differentiation between the desired
signal and the MAI is the difference in the
sequences employed.
Since the sequences repeat every symbol period,
this structural differentiation is invariant from
symbol to symbol.
As a result, we can easily extract and utilize
this structural difference by using , for
example, some standard adaptive signal processing
techniques, making the receiver implementation
simple and desirable.

It turns out that most of these
short-sequence-based single-user MAI suppression
techniques can be interpreted as special cases of
their multiuser counterparts.
To avoid repetition, we leave their development
to Chapter 7 where we will develop the general
multiuser techniques and specialize them to
single-user receivers.

For DS-CDMA systems employing long sequences,
although the sequences are still different, the
structural differentiation mentioned above is not
very useful since the structural difference
varies from symbol to symbol, making it hard to
extract and utilize.
Therefore, we have to employ some other forms of
structural differentiation between the MAI and
the desired signal that are invariant from symbol
to symbol.
To illustrate this idea, we will present a
simplified development of the MAI suppression
receiver suggested in 11.

The MAI suppression receiver in 11 is designed
for asynchronous long-sequence-based DS-CDMA
systems.
It is based on the matched filter receiver.
Its working principle is to exploit the
structural difference between the desired signal
and the MAI caused by the fact that the delays of
the signals are different.
The problem of symbol-by-symbol varying nature of
the spreading sequences is solved by obtaining
the averaged (over all possible choices of
sequences) structural difference instead of the
instantaneous one.
This is possible since segments (over a symbol
duration) of the long sequences look like random.
Hence the statistical averaged structural
difference can be approximated by the
easy-to-obtain time-averaged structural
difference.

The abstract statement above can made precise by
looking at the output signal from the matched
filter.
First, the signal model we consider here is
exactly the same as the one described in Section
6.1except that the period of the spreading
sequences is now much large than the spreading
gain N.
To simplify our discussion, we model the sequence
elements al(k) as zero-mean iid random variables
with
In the context of long-sequence-based systems,
this model basically means that the sequences are
aperiodic and are picked randomly from the set of
all possible sequences.
Of course, this is an (good) approximation to the
actual set of sequences used in practice.
Let us focus on the detection of b0(m) , the 0th
symbol of the mth user.

In this case, the impulse response of the matched
filter is given by
Note that we have chosen to employ a non-causal
filter here to simplify our notation later.
Of course, we have to use a causal filter in
practice and the amount of delay in the causal
filter, for example T, has to be added to all the
results we are going to present.
Let us denote the output signal of the matched
filter as
is the component due to the desired user.
is the MAI plus
thermal-noise component with the subscripts I and
W standing for MAI and thermal noise,
respectively.

Conditioning on the set of delays and phases
defined in (6.35), it is easy to show that the
autocorrelation function of the matched filter
output is given by
where

The function is the autocorrelation of the
chip waveform defined by
Also, is just a
scaled version of .
We note that since the chip waveform is
time-limited to 0 Tc), both
are time-limited to (- Tc Tc) and hence
the summation in (6.50) contains only a finite
numbers of terms for any fixed values of t and s.
For example, when the chip waveform is
rectangular
then is the triangular waveform
stretching from -Tc to Tc.

Now, the structural difference between the
desired signal and the MAI can be readily
observed by considering the difference between
the autocorrelation functions,
of the desired signal
component and the MAI component nI(t),
respectively.
The autocorrelation functions are made up of
products of different delayed versions of the
functions
This difference can be easily visualized by
considering the intuition provided in Figure 6.3,
which shows the delayed versions of the function
that make up the desired signal and MAI
autocorrelation functions for the two-user case
(K 2) with rectangular chip waveform.

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63

The blue triangle corresponds to the due
to the desired signal.
The four red triangles correspond to the delayed
versions of due to the interferer.
The reason that there are four triangles
corresponding to the interferer is that for
as shown in the figure
each interferer contributes exactly four non-zero
terms in the second summation in (6.50) for the
observation interval of ?? in which
the desired signal autocorrelation function is
non-zero.
For there is only one triangle
corresponds to the interferer. This triangle
coincides with the blue triangle corresponding to
the desired user.
In this case, there is no structure difference
between the desired signal and the MAI, and hence
the MAI cannot be suppressed.

The discussion above reveals the structural
difference between the desired signal and the MAI
induced by the different delays of the signals.
We can utilize this structure to suppress the MAI
component in the received signal by observing the
matched filter output.
Since the structure does not depend on the
spreading sequences, it is invariant from symbol
to symbol and hence simple adaptive algorithms
can be employed.
A simple way 11 to suppress MAI based on this
structure is to sample the matched filter output
about the peak at each symbol (see Figure 6.3)
and then weigh the samples to form a decision
statistic for that symbol as shown in Figure 6.4.

We note that the matched filter receiver is a
special case of this method with only one sample
taken.
The weights are chosen so that the mean-squared
error (MSE) between the decision statistic and
the actual symbol is minimized.
The suppression of the MAI is performed
implicitly in the process of minimizing the MSE.

More precisely, suppose that we sample the
matched filter output 2M 1 times at
where Ts lt Tc, for the detection of the 0th
symbol of the mth user.
For example, Figure 6.3 shows that case in which
Ts Tc/2 and M 2.
For convenience, we arrange the samples into the
vector and work using vector and matrix
notation.

An estimate of the transmitted symbol
is obtained as the weighted sum of the samples
taken at the output of the matched filter, i.e.,
where is the sample at
time is the weight for that
sample.
The weight vector is chosen to minimize the
MSE defined by
where the expectation is conditioned over the
set .

It can be easily shown 11 that the optimal
choice of weight vector is the solution of the
following set of equations
where
are samples of the
autocorrelation functions of the matched filter
output and the signal respectively.
As the existence of the thermal noise component
in the signal guarantees the invertibility of the
correlation matrix the optimal weight
vector is given by

Writing the samples of the MAI-plus-thermal-noise
autocorrelation function as the
matrix
we can employ the matrix inversion lemma to show
that
Where
It can also be shown 11 that this optimal
choice of the weight vector maximizes the SNR at
the decision statistic
The maximum SNR value achieved is exactly given
by (6.58).

Suppose that we can employ the improved Gaussian
approximation here to approximate the MAI
component in the decision statistic as a Gaussian
random variable.
Then the conditional (conditioned on )
symbol error probability is given by
For example, Figure 6.5 gives the plot of the
symbol error probability (based on the Gaussian
approximation) obtained by the receiver with a
sampling scheme as shown in Figure 6.3 in the
case where there are two active users in the
system, the fractional delay of the interferer is
Tc/2, and the received power of the interferer is
20dB above that of the desired user.

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72

Also plotted in the figure are the symbol error
probability (IGA) obtained the matched filter
receiver and the symbol error probability of the
single-user scenario.
We see that the presence of the strong MAI causes
the error rate of the matched filter levels off
as the SNR increases.
With the MAI suppression receiver, the symbol
error probability decreases exponentially as the
SNR increases, showing that the MAI is suppressed
by the receiver.
In fact, with the particular set of system
parameters, the performance of the MAI
suppression receiver is only 3dB worse than the
single-user scenario.

We evaluate the robustness of the receiver
against the near-far problem based on a measure
similar to the near-far resistance suggested in
Section 6.3.
First, we note that the SNR obtained by the
receiver in any situation cannot be larger than
the SNR obtained when the mth user is the only
active user, i.e.,
Following the idea of the near-far resistance in
Section 6.3, we define the near-far efficiency
(NFE) as
We note that 0 lt NFE lt 1 and it is a function of
the delays and phases in the set

For example, in the two-user case (K 2), it can
be shown 11 that the NFE of the receiver
discussed here is upper bounded by (k ? m)
where this bound can be achieved in the limit by
sampling finer and finer.
For example, the NFE8 for the rectangular chip
waveform is shown in Figure 6.6 as a function of
(k ? m).
We see that since the NFE is positive when the
system is not chip-synchronous (i.e.,
), the receiver is robust against the near-far
problem when the two users are asynchronous.
In general, the robustness of this receiver
degrades when more and more strong interferers
are in the system.

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76

As a closing note, we emphasize that as the
structural difference between the MAI and the
desired signal is invariant from symbol to
symbol, we can employ standard adaptive
algorithms to obtain the optimal weight vector
described in (6.56).
For example, when a training data sequence is
available, we can employ the LMS algorithm to
obtain the weight vector.
In fact, it is shown in 11 that a blind
adaptive algorithm, which does not require the
availability of a training sequence, can be
developed to obtain the weight vector.

77
6.5 References

1 H. V. Poor and S. Verdu, Single-user
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Code Division Multiple Access

... users in a DSSS system so that the users can share the communication channel. ... this approach, we identify one of the users in the system as the desired user ... – PowerPoint PPT presentation