Transcript 投影片 1 - CUST

Chapter 6
Code Division Multiple Access
1
• 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.
2
•
Roughly speaking, there are two different approaches to the
problem.
1. 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.
2. The second approach is called multiuser detection, in which
all signals from all users are detected jointly and
simultaneously by the receiver.
3
• A common receiver based on the single-user detection approach
is the matched filter receiver matched to the desired user’s
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.
4
• 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.
5
6.1 Asynchronous DS-CDMA model
• In many cases, the transmissions from different users in a DSCDMA system are not synchronized.
• One of such examples is given by the uplink (reverse link) of IS95 [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
6
–
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 timelimited to [0; Tc) and is normalized such that
• Extensions to other types of modulations and long sequences are
straightforward.
7
• 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
8
• We employ
to model the asynchronous nature of the system.
• We assume that synchronization with the mth user’s 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; 2π), respectively.
• Moreover, we also make the assumption that all the random
variables associated with different users are independent.
9
• Now, let us look at the mth matched filter receiver which is
designed to detect the mth user’s signal.
• Without loss of generality, we consider the detection of the
symbol b0(m).
• Let
be the received power of the kth user’s signal
at the mth receiver.
10
• The decision statistic for the 0th symbol is given by
– ik,m is the interference component due to the kth signal.
– ηis the component due to the AWGN that is a zero-mean
Gaussian random variable with variance N0T.
11
• On the other hand,
where
is given by
is decomposed into
12
•
are the aperiodic, even, and odd cross
correlation functions between the sequences
respectively.
13
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,
14
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,
15
Where
is the MAI component.
• Hence the average symbol error probability is
where the expectation is taken over all the random variables in the set Sm.
16
• 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.
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• 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
18
• Hence
• Therefore
where  m  PmT 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.
19
• 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].
20
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
21
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
Re[ik,m].
22
• 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
23
• Hence, the problem reduces to the evaluation of the variance of
Re[ik,m].
• To do this, we use the following simple identity
• Then, for k ≠ m,
• Let us write
24
• 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,
25
• Let us define
26
• For rectangular chip waveform, i.e.,
QPSK spreading,
as in BPSK or
• Hence, (6.26) reduces to
• Combining (6.17), (6.18), and (6.29),
27
• 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.
28
29
• Replacing the term
in (6.30) by the
expectation in (6.32), we have the approximation,
30
31
• 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
32
• 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)).
33
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 Re[ik,m]
now.
34
• 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
35
• 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 user’s 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.
36
• 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,
37
• While with QPSK spreading,
38
• 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
39
• Averaging this over the delays and phases, we obtain an accurate
approximation to the (unconditional) symbol error probability
40
• 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
41
• 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:
42
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,
43
• 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.
44
• 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.
45
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 DSCDMA 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.
46
• 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.
47
• To understand the concept of near-far resistance, let us imagine
that only one user, say the mth user, were active in the DSCDMA 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.
48
• 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.
49
• 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.
50
• When the matched filter receiver does not give an adequate level
of performance, there are basically two ways to tackle the nearfar 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.
51
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.
52
• 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 longsequence-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.
53
• 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.
54
• 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].
55
• 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.
56
• 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.
57
• 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.
58
• 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
59
• The function
defined by
is the autocorrelation of the chip waveform
• 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.
60
• 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.
61
62
• 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.
63
• 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.
64
• 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.
65
• More precisely, suppose that we sample the matched filter output
2M + 1 times at
where Ts < 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.
66
• 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
.
67
• 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
68
• 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).
69
• 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.
70
71
• 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.
72
• 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 < NFE < 1 and it is a function of the delays and
phases in the set
73
• 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 NFE∞ 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.
74
75
• 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.
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6.5 References
[1] H. V. Poor and S. Verdu, “Single-user detectors for multiuser channels,” IEEE
Trans. Commun., vol. 36, no. 1, pp. 50–60, Jan. 1988.
[2] TIA/EIA/IS-95 Interim Standard, Mobile Station-Base Station Compatibility
Standard for Dual Mode Wideband Spread Spectrum Cellular System,
Telecommunications Industry Association, Washington, D.C., Jul. 1993.
[3] J. S. Lehnert, “An efficient technique for evaluating direct-sequence spreadspectrum multipleaccess communications,” IEEE Trans. Commun., vol. 37, no.
8, pp. 851–858, Aug. 1989.
[4] K. Yao, “Error probability of asynchronous spread spectrum multiple access
communication systems,” IEEE Trans. Commun., vol. 25, no. 8, pp. 803–809,
Aug. 1977.
[5] M. B. Pursley, “Performance evaluation for phase-coded spread-spectrum
multiple-access communication— Part I: System analysis,” IEEE Trans.
Commun., vol. 25, no. 8, pp. 795–799, Aug. 1977.
[6] R. K. Morrow and J. S. Lehnert, “Bit-to-bit error dependence in slotted
DS/SSMA packet systems with random signature sequences,” IEEE Trans.
Commun., vol. 37, no. 10, pp. 1052–1061, Oct. 1989.
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[7] T. M. Lok and J. S. Lehnert, “Error probabilities for generalized quadriphase
DS/SSMA communication systems with random signature sequences,” IEEE
Trans. Commun., vol. 44, no. 7, pp. 876–885, Jul. 1996.
[8] T.M. Lok and J. S. Lehnert, “An asymptotic analysis of DS/SSMA
communication systems with random signature sequences,” IEEE Trans.
Inform. Theory, vol. 42, no. 1, pp. 129–136, Jan. 1996.
[9] J. M. Holtzman, “A simple, accuratemethod to calculate spread-spectrum
multiple-access error probabilities,” IEEE Trans. Commun., vol. 40, no. 3, pp.
461–464, Mar. 1992.
[10] S. Verdu, Multiuser Detection, Cambridge University Press, 1998.
[11] T. F. Wong, T. M. Lok, and J. S. Lehnert, “Asynchronous Multiple Access
Interference Suppression and Chip Waveform Selection with Aperiodic
Random Sequences,” IEEE Trans. Commun., vol. 47, no. 1, pp. 103–114, Jan.
1999.
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