If all the spreading sequences in a network of asynchronous CDMA systems have a common period equal to the data-symbol duration, then by the proper selection of the sequences and their relative phases, one can obtain a system performance better than that theoretically attainable with random sequences.
However, the number of suitable sequences is too small for many applications, and long sequences that extend over many data symbols provide more system security. Furthermore, long sequences ensure that successive data symbols are covered by different sequences, thereby limiting the time duration of an un- favorable cross-correlation due to multiple-access interference. Even if short sequences are used, the random-sequence model gives fairly accurate perfor- mance predictions.
Direct-Sequence Systems with PSK
Consider the direct-sequence receiver of Figure 2.14 when the modulation is PSK and multiple-access interference is present. If the spreading sequence of the desired signal is modeled as a random binary sequence and the chip waveform confined to then the input V to the decision device is given by (2-84) and has mean value
The interference component is given by (6-5), (6-6), and (6-1). Since the data modulation in an interference signal is modeled as a random binary se- quence, it can be subsumed into given by (6-3) with no loss of generality.
Since is determined by an independent, random spreading sequence, only time delays are significant and, thus, we can assume that
in (6-1) without loss of generality.
Since is confined to and the substitution of (6-1) and
(6-3) into (6-6) yields
The partial autocorrelation for the chip waveform is defined as
Substitution into (6-48) and appropriate changes of variables in the integrals yield
For rectangular chips in the spreading waveform,
Consequently,
For sinusoidal chips in the spreading waveform,
Substituting this equation into (6-49), using a trigonometric identity, and per- forming the integrations, we obtain
Since both and contain the same random variable it does not appear at first that the terms in (6-50) are statistically independent even when
and are given. The following
lemma [6] resolves this issue.
Lemma. Suppose that and are statistically independent, random binary sequences. Let and denote arbitrary constants. Then and
are statistically independent random variables when
Proof: Let denote the joint probability that
and where From the theorem of
total probability, it follows that
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
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
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
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,
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
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 powerI, 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
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:
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
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
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