Torrieri D. Principles of Spread-Spectrum Communication Systems
Подождите немного. Документ загружается.
308
CHAPTER 6.
CODE-DIVISION MULTIPLE ACCESS
From the independence of and and the fact that they are random
binary sequences, we obtain a simplification for and
Since equals +1 or –1 with equal probability,
and thus
A similar calculation gives
Therefore,
which satisfies the definition of statistical independence of and
The same relation is trivial to establish for or
The lemma indicates that when and are given, the terms in (6-5) are
statistically independent. Since the conditional variance is
The independence of the K spreading sequences, the independence of successive
terms in each random binary sequence, and (6-50) imply that the conditional
variance of is independent of and, therefore,
Since the terms of in (6-5) are independent, zero-mean random variables
that are uniformly bounded and as the central limit the-
orem implies that converges in distribution to a Gaussian random
variable with mean 0 and variance 1. Thus, when and are given, the condi-
tional distribution of is approximately Gaussian when G is large. Since the
noise component in (2-84) has a Gaussian distribution and is independent
of has an approximate Gaussian distribution with mean given
by (6-47), and
6.2.
SYSTEMS WITH RANDOM SPREADING SEQUENCES
309
A straightforward derivation using the Gaussian distribution of the decision
statistic V indicates that the conditional symbol error probability given and
is
where is the energy per symbol in and the equivalent-noise power
spectral density is defined as
For a rectangular chip waveform, this equation simplifies to
Numerical evaluations [6] give strong evidence that the error in (6-57) due
to the Gaussian approximation is negligible if For an asynchronous
network, it is assumed that the time delays are independent and uniformly
distributed over and that the phase angles are
uniformly distributed over Therefore, the symbol error probability is
where the fact that takes all its possible values over has been used
to shorten the integration intervals. The absence of sequence parameters ensures
that the amount of computation required for (6-60) is much less than the amount
required to compute when the spreading sequence is short. Nevertheless,
the computational requirements are large enough that it is highly desirable to
find an accurate approximation that entails less computation. The conditional
symbol error probability given is defined as
A closed-form approximation to greatly simplifies the computation of
which reduces to
To approximate we first obtain upper and lower bounds on it.
For either rectangular or sinusoidal chip waveforms, elementary calculus
establishes that
310
CHAPTER 6. CODE-DIVISION MULTIPLE ACCESS
Using this upper bound successively in (6-58), (6-57), and (6-61), and perform-
ing the trivial integrations that result, we obtain
where
To apply Jensen’s inequality (2-144), the successive integrals in (6-60) are
interpreted as the evaluation of expected values. Consider the random variable
Since is uniformly distributed over straightforward calculations using
(6-52) and (6-54) give
where
The function (6-57) has the form given by (2-145). Equations (6-58), (6-63),
and yield a sufficient condition for convexity:
Application of Jensen’s inequality successively to each component of in (6-61)
yields
where
If is negligible, then (6-71) and (6-65) give Thus, a good
approximation is provided by
6.2.
SYSTEMS WITH RANDOM SPREADING SEQUENCES
311
Figure 6.5: Symbol error probability of direct-sequence system with PSK in
presence of single multiple-access interference signal and
where
If is negligible, then Therefore, in terms of the
value of needed to ensure a given the error in using approximation (6-
72) instead of (6-61) is bounded by in decibels, which equals 0.88
dB for rectangular chip waveforms and 1.16 dB for sinusoidal chip waveforms.
In practice, the error is expected to be only a few tenths of a decibel because
and coincides with neither the upper nor the lower bound.
As an example, suppose that rectangular chip waveforms are used,
and K = 2. Figure 6.5 illustrates four different evaluations of
as a function of the despread signal-to-interference ratio,
which is the signal-to-interference ratio after taking into account the beneficial
results from the despreading in the receiver. The accurate approximation is
computed from (6-57) and (6-60), the upper bound from (6-64) and (6-62), the
lower bound from (6-70) and (6-62), and the simple approximation from (6-72)
and (6-62). The figure shows that the accurate approximation moves from the
lower bound toward the simple approximation as the symbol error probability
decreases. For the simple approximation is less than 0.3 dB in error
relative to the accurate approximation.
Figure 6.6 compares the symbol error probabilities for K = 2 to K = 4,
312
CHAPTER 6.
CODE-DIVISION MULTIPLE ACCESS
Figure 6.6: Symbol error probability of direct-sequence system with PSK in
presence of K – 1 equal-power multiple-access interference signals and
rectangular chip waveforms and The simple approximation is
used for and the abscissa shows GS/I where
I
is the interference power of
each equal-power interfering signal. The figure shows that increases with K,
but the shift in is mitigated somewhat because the interference signals tend
to partially cancel each other.
The preceding bounding methods can be extended to the bounds on
by observing that and setting during the successive
applications of Jensen’s inequality, which is applicable if (6-69) is satisfied.
After evaluating (6-65), we obtain
where
A simple approximation is provided by
6.2.
SYSTEMS WITH RANDOM SPREADING SEQUENCES
313
Figure 6.7: Symbol error probability of direct-sequence system with PSK in
presence of 3 equal-power multiple-access interference signals and
If is specified, then the error in the required caused by using (6-76)
instead of (6-60) is bounded by 10 in decibels. Thus, the error is
bounded by 2.39 dB for rectangular chip waveforms and 2.66 dB for sinusoidal
ones.
The lower bound in (6-74) gives the same result as that often called the
standard Gaussian approximation, in which in (6-5) is assumed to be ap-
proximately Gaussian, each in (6-50) is assumed to be uniformly distributed
over and each is assumed to be uniformly distributed over
This approximation, gives an optimistic result for that can be as much as
4.77 dB in error for rectangular chip waveforms according to (6-74). The sub-
stantial improvement in accuracy provided by (6-72) or (6-57) is due to the
application of the Gaussian approximation only after conditioning on given
values of and The accurate approximation given by (6-57) is a version of
what is often called the improved Gaussian approximation.
Figure 6.7 illustrates the symbol error probability for 3 interferers, each
with power
I
, rectangular chip waveforms, and as a function of
GS/I. The graphs show the standard Gaussian approximation of (6-74), the
simple approximation of (6-76), and the upper and lower bounds given by (6-64),
(6-70), and (6-62). The large error in the standard Gaussian approximation is
evident. The simple approximation is reasonably accurate if
For synchronous networks, (6-57) and (6-58) can be simplified because the
314
CHAPTER 6. CODE-DIVISION MULTIPLE ACCESS
are all zero. For either rectangular or sinusoidal chip waveforms, we obtain
where
A comparison with (6-64) and (6-65) indicates that for a synchronous net-
work equals or exceeds for a similar asynchronous network when random
spreading sequences are used. This phenomenon is due to the increased band-
width of a despread asynchronous interference signal, which allows increased
filtering in the receiver.
The accurate approximation of (6-57) follows from the standard central limit
theorem, which is justified by the lemma. This lemma depends on the restriction
of the chip waveform to the interval If the chip waveform extends beyond
this interval but is time-limited, as is necessary for implementation with digi-
tal hardware, then an extension of the central limit theorem for
sequences can be used to derive an improved Gaussian approximation [7]. Alter-
natives to the analysis in this section and the next one abound in the literature,
but they are not as amenable to comparisons among systems.
Quadriphase Direct-Sequence Systems
Consider a network of quadriphase direct-sequence systems, each of which uses
dual QPSK and random spreading sequences. Each direct-sequence signal is
given by (2-123) with The multiple-access interference is
where and both have the form of (6-3) and incorporate the data
modulation. The decision variables are given by (2-124) and (2-126) with
A straightforward calculation using (6-6) indicates that
The statistical independence of the two sequences, the preceding lemma, and
analogous results for in (2-127) yield the variances of the interference terms
of the decision variables:
6.2.
SYSTEMS WITH RANDOM SPREADING SEQUENCES
315
The noise variances and the means are given by (2-130) and (2-129). Since all
variances and means are independent of the Gaussian approximation yields
a that is independent of
where
Since a similar analysis for direct-sequence systems with balanced QPSK yields
(6-83) again, both quadriphase systems perform equally well against multiple-
access interference.
Application of the previous bounding and approximation methods to (10-79)
yields
where the total interference power is defined by (6-75). A sufficient condition
for the validity of the lower bound is
A simple approximation that limits the error in the required for a specified
to 10 is
This approximation introduces errors bounded by 0.88 dB and 1.16 dB for
rectangular and sinusoidal chip waveforms, respectively. In (6-84) and (6-86),
only the total interference power is relevant, not how it is distributed among
the individual interference signals.
Figure 6.8 illustrates for a quadriphase direct-sequence system in the
presence of 3 interferers, each with power
I
, rectangular chip waveforms, and
The graphs represent the accurate approximation of (6-82), the
simple approximation of (6-86), and the bounds of (6-84) as functions of GS/I.
A comparison of Figures 6.8 and 6.7 indicates the advantage of a quadriphase
system.
For synchronous networks with either rectangular or sinusoidal chip wave-
forms, we set the equal to zero in (6-82) and obtain
316
CHAPTER 6. CODE-DIVISION MULTIPLE ACCESS
Figure 6.8: Symbol error probability of quadriphase direct-sequence system in
presence of 3 equal-power multiple-access interference signals and
Since this equation coincides with the upper bound in (6-84), we conclude that
asynchronous networks accommodate more multiple-access interference than
similar synchronous networks using quadriphase direct-sequence signals with
random spreading sequences. To compare asynchronous quadriphase direct-
sequence systems with asynchronous systems using binary PSK, we find a lower
bound on for direct-sequence systems with PSK. Substituting (6-57) into (6-
60) and applying Jensen’s inequality successively to the integrations over
K – 1, we find that a lower bound on is given by the right-hand side
of (6-82) if (6-85) is satisfied. This result implies that asynchronous quadriphase
direct-sequence systems are more resistant to multiple-access interference than
asynchronous direct-sequence systems with binary PSK.
The equations for allow the evaluation of the information-bit error prob-
ability for error-correcting codes with hard-decision decoders. To facilitate
the analysis of soft-decision decoding, two assumptions are necessary. Assume
that K is large enough that the multiple-access interference after despreading is
approximately Guassian rather than conditionally Gaussian. Since the equiv-
alent noise is a zero-mean process, the equivalent-noise power spectral density
can be obtained by averaging over the distributions of and
For asynchronous communications, (6-83) and (6-87) yield
This equation is also valid for synchronous communications if we set
6.3.
WIDEBAND DIRECT-SEQUENCE SYSTEMS
317
Thus, for a binary convolutional code with rate constraint length K, and
minimum free distance is upper-bounded by (1-112) with
The network capacity is the number of equal-power users in a network of
identical systems that can be accommodated while achieving a specified
For equal-power users, Let denote the value of
necessary for a specific error-control code to achieve the specified Equation
(6-88) implies that the network capacity is
where is the integer part of is the processing gain,
and the requirement is necessary to ensure that the specified can be
achieved for some value of K. Since in general, the factor reflects
the increased gain due to the random distributions of interference phases and
delays. If they are not random but then and the number of
users accommodated is reduced. Thus, synchronous CDMA systems require
orthogonal spreading sequences.
As an example, consider a network with systems that resemble those used
for the synchronous downlinks of an IS-95 CDMA network. We assume the
absence of fading and calculate the network capacity for power-controlled users
within a single cell. The data modulation is balanced QPSK. G = 64, and
The error-control code is a rate-1/2 binary convolutional code with
constraint length 9. If or better is desired, the performance curve
of Figure 1.8 for the convolutional code indicates that and
thus is required. Equation (6-90) then indicates that the network
capacity is K = 51 if dB and K = 57 if
6.3
Wideband Direct-Sequence Systems
A direct-sequence system is called wideband if it uses a spectral band with a
bandwidth that exceeds the coherence bandwidth of a frequency-selective fading
channel. The two most commonly proposed types of wideband direct-sequence
systems are single-carrier and multicarrier systems. A single-carrier system
uses a single carrier frequency to transmit signals. A multicarrier system parti-
tions the available spectral band among multiple direct-sequence signals, each
of which has a distinct carrier frequency. The main attractions of the multi-
carrier system are its potential ability to operate over disjoint, noncontiguous
spectral regions and its ability to avoid transmissions in spectral regions with
strong interference or where the multicarrier signal might interfere with other
signals. These features have counterparts in frequency-hopping systems.
A single-carrier system provides diversity by using a rake receiver that com-
bines several multipath signals. A multicarrier system provides diversity by