Example 7.2.2. Let us construct the binary Welch-bound ensemble of K ¼ 16 sequences of length N ¼ 14. For this we may make use of the matrix H16 of Example 7.2.1 and discard two arbitrary (e.g. the last two) rows. The matrix A obtained this way is exactly what is needed, and its 16 columns are Welch-bound signatures of length 14. TSC for the ensemble thus found equates accurately to its minimum value determined by (7.30):

TSC ¼ K 2 ¼ 256

N 14

The floor SIR estimated with respect to average MAI power per receiver is according to (7.33) qI2 ¼ N/(K N) ¼ 7.

If belonging to a PSK alphabet is the only restriction on the signature code sequences, then the algorithm for constructing Welch-bound sets described above works universally. For example, rows of the matrix A may always be taken as K cyclically shifted replicas of the Chu sequence of length K. As was shown in Section 6.11.2, Chu codes exist for any length and all their different cyclic replicas are orthogonal. On the other hand, when all signatures should be binary (ak, i ¼ 1) orthogonality of all N rows of the matrix A with N > 2 is possible only for K divisible by four (see Problem 7.14). This implies that for K 6¼0 mod4 binary signatures the Welch bound (7.30) is not tight, and more precise lower borders should exist. Derivations of them may be found in [64,65] (see also Problem 7.17).

7.3Approaches to designing signature ensembles for asynchronous DS CDMA

Let us extend the issue of signature design to the case of asynchronous DS CDMA, where time and phase shifts between individual user signals are random. On the assumption of employing a single-user receiver, the decision on the current symbol of the kth user is again done on the basis of correlation (7.25). Now, however, a tough alignment between the boundaries of data symbols and the chips of different users is not maintained due to arbitrary mutual time shifts of users’ signals. Suppose that the kth user data receiver is explored and l is the delay of the lth signal against the kth signal. In order to concentrate only on the issue of designing signature codes let us assume that chip boundaries of all K signatures are synchronized, i.e. mutual delays are multiples of D: l ¼ nlD, where nl is integer, 0 nl < N. Then the situation is well explained by Figure 7.15 (for k ¼ 1), stressing that in asynchronous CDMA, unlike synchronous (see Figure 7.13), data symbols of other users may change during the reception of the kth user’s current data symbol. Still the main factor making design of asynchronous signature sets harder is the necessity to distinguish every signature from all possible shifted replicas of the others, which is not necessary in synchronous CDMA.

Suppose first that no change of data symbols of all users happens during the received symbol of the kth user, i.e. bl, i 1 ¼ bl, l ¼ 2, 3, . . . , K. Then the situation is different from the synchronous one only in the mutual time-mismatch of signatures. Start with the assumption that the signature period L coincides with the processing gain N, which

Figure 7.15 Datastreams and signatures in asynchronous CDMA

is the number of chips per data symbol duration or, equivalently, of chips integrated in the correlator. If no restriction is imposed on the possible range of mutual delays, the lth signature may be presented by any of its N cyclically shifted replicas, so that there are N(K 1) different N-dimensional vectors, each of which is a potential source of MAI in the kth receiver. When a channel is subject to multipath effects every cyclic replica of the kth signal may also create interference at the kth receiver. Let us admit that up to N 1 such replicas may exist, i.e. the multipath delay spread ranges up to the period of the signature. Another reason to include the cyclic replicas in the explored vector set is a desire to have low autocorrelation sidelobes, which is important in the search problem (see Section 8.2). With such an extension we have a total of KN vectors, whose correlations should be as small as possible.

The Welch bound is again a good instrument to estimate the lower limit of the average

squared correlation 2 of those KN vectors. For that it is enough to replace K by KN in (7.31). Since KN > N for any K 2 this gives:

This inequality shows the fundamental lower limit, which an average squared correlation between all cyclic replicas of all K signatures of length N (own replicas of each one included) can never fall below. When the number of users is about ten or more this version of the Welch bound becomes especially simple:

Suppose now that the signature period in the number of L chips covers several data symbols L > N and data of no user changes during the kth current data symbol, as

before.1 Again, let the delays range up to the signature period. Since the number of chips per data symbol (integration interval) remains N, we, as previously, deal with N-dimensional vectors, but the number of vectors whose correlations are controlled is now KL instead of KN, so that the bound stems from (7.31):

which again turns into (7.35) with K 1. The last result makes it possible to demonstrate that data modulation can in no way lower the bounds obtained. Indeed, every data-modulated signature may be considered as a new sequence of some (possibly very big) period Lk. Then all modulated signatures will have a common period L, being a least common multiple of all Lk, and the average squared correlation will be bordered from below by (7.36), again meaning that (7.35) is valid for the case of many users.

The derivations just undertaken establish a criterion of asynchronous signature set design: the ensemble of many signatures may be considered appropriate if its average squared correlation is close to the bound (7.35). Let us demonstrate that ensembles of random signatures attain this bound. Let all signatures be composed independently of each other by a random independent choice of elements of each of them. The whole procedure is similar to drawing balls out of an urn. Set an M-ary PSK alphabet and treat it as an urn with different M balls (code symbols). Pick out one ball K times, each time noting the result and returning the ball to the urn. This gives the first symbols of K signatures. The next symbols of all the signatures are generated the same way. Since all M-ary symbols in this scheme are equiprobable, uniformly spaced on the plane (see Figure 2.6c) and independent of each other, we have the following expectations:

the second equation stemming from the fact that the expectation of the product of independent entities equals the product of their expectations. Let us use this in an estimation of the average squared correlation of signatures at the integration interval of N chips:

Physically (7.38) is nothing but the expectation of MAI power (k 6¼l), or of multipath interference power (k ¼ l) created by the lth signature shifted by m chips at the kth

1 Saving symbol N for the processing gain, i.e. the number of chips per data symbol, we will from now on denote a signature period by L, whenever they may be different.

It is clearly seen now that all unwanted squared correlations in the ensemble under consideration attain the lower bound (7.35), i.e. sets of random signatures are optimal when the number of users is around ten or more. It is extremely important to emphasize that the data modulation of random sequences meeting (7.37) (multiplying them by data symbols independent of them) does not destroy (7.37) (see Problem 7.20). Therefore, the presence or absence of modulation does not affect all of the derivations above, as well as the final result (7.40) and conclusion on the set optimality.

Equations (7.37) seem to give an unequivocal instruction for designing signature ensembles. In practice, however, signatures cannot be random, since the receiver should be a priori aware of the signature modulation law in order to generate the necessary correlator reference. To realize the randomness properties (7.37) by the deterministic coding rules, so-called pseudorandom sequences are necessary.

Take the deterministic PSK signature of period L and treat it as though it is one of several equiprobable realizations of a stationary ergodic random sequence fak, ig (random discrete-time process) [14,66]. The other realizations may be all cyclic shifts of the initial sequence. Then, due to the ergodicity property, each realization presents the whole random process exhaustively, and statistical averaging fak, ig over all realizations is equivalent to time averaging, i.e. evaluating the expectation ak, i and correlation moment ak, iak, i m via constant component and periodic ACF of the deterministic signature, respectively:

In the same way, treating two deterministic signatures as realizations of two jointly ergodic random sequences fak, ig and fal, ig, we have equality between the correlation moments of two random sequences and CCF of two deterministic signatures:

The comparison of (7.41) and (7.42) with (7.37) sets a criterion of pseudorandomness: to serve as signatures in asynchronous DS CDMA all the deterministic sequences of the ensemble should ideally have zero constant component, perfect periodic ACF and zero periodic CCF:

In the case of unconstrained mutual time shifts (any m in the range 0, 1, . . . , L 1 are probable) the last demands obviously contradict each other, making ensembles of this sort hypothetical for any finite L. Indeed (see also Problem 7.21), the requirements for perfect ACF and zero CCF mean nothing but zero level of correlations between all cyclic shifts of K sequences of period L, i.e. zero value of the average of unwanted squared correlations 2. As (7.34) and (7.35) show, this is impossible with K 2, and in particular with many users 2 cannot fall smaller than 1/L.