The conclusion we just arrived at explains why so many efforts have been dedicated to searching for ensembles whose characteristics approach those of the hypothetical ensembles mentioned above when length L grows. Quite a popular criterion of this approximation is the minimax one, orienting the ensemble design towards minimizing maximum value among all unwanted correlations. Define the correlation peak max as the greater of two entities: the maximal autocorrelation sidelobe amax among all sequences and the maximal cross-correlation peak cmax among all pairs of sequences:

Naturally, for the hypothetical perfect ensemble, max along with 2 is zero, and for any real ensemble max may serve as an adequate measure of its proximity to the perfect one.

Since the maximal value of any variable can never be smaller than its average,

2max 2, which spreads the Welch bounds (7.34) and (7.35) on the correlation peak:

where, again, the last approximation corresponds to the case K 1. With additional limitations on the PSK alphabet, the bound above may appear rather loose, especially when the number of sequences approaches L. In particular, for sufficiently large ensembles of binary f 1g sequences the Sidelnikov bound holds [67,68]:

Ensembles having max attaining the limit predicted by the lower bounds are certainly optimal in the correlation peak criterion and are sometimes called minimax. Some of them are discussed in Section 7.5.

7.4 Time-offset signatures for asynchronous CDMA

In many real situations mutual time shifts of asynchronous signatures may vary only within a restricted range. The finiteness of a channel delay spread on the one hand, and system geometry on the other are the most typical factors setting such limitations. To be specific, let us turn to the uplink of a cellular mobile radio. The local clock of an active MS is synchronized with the received BS signal and has a delay 1 versus the BS clock determined by the distance D from BS to MS as 1 ¼ D/c, where c is the speed of light. Since the signal transmitted by a specific MS reaches the BS receiver with the same delay, the total delay of the signal arriving at BS versus the BS clock is 2 ¼ 2 1 ¼ 2D/c. Let Dmax be the maximal distance, the signal from which has intensity perceptible by the BS receiver. Strong path attenuation (see Section 4.6) permits signals arriving from distances markedly exceeding the cell radius Dc to be ignored, which gives a rough estimation Dmax Dc. Then the maximal value of 2 is 2Dc/c and signals from mobiles at distances from BS ranging between zero and Dc arrive at BS within the time window

Figure 7.17 Signatures formed as time-shifted copies of the initial one

a sidelobe of 11(m), whose level was assumed to be negligible. We have thus proved that properly offset copies of the initial sequence with good periodic ACF produce the ensemble where conditions of pseudorandomness (7.43) hold within the full range of possible mutual signature shifts jmj mmax. It follows immediately that this ensemble achieves (when 11(m) is perfect) or approaches very closely the lowest level (7.40) of average unwanted effects due to MAI and multipath propagation or, equivalently, the Welch bounds (7.36) or (7.35). We again emphasize strongly the validity of this statement in the presence of DS data modulation of signatures, since conditions (7.43) are sufficient to minimize unwanted MAI/multipath effects in this case (see the remark following (7.40)).2

Example 7.4.1. Consider the system with chip duration D ¼ 1 ms, number of users K ¼ 60, channel delay spread ds ¼ 20 ms and cell radius Dc ¼ 15 km. In this case max ¼ 2Dc /c þ ds ¼ 120 ms and mmax ¼ 120. The signature ensemble may be arranged starting with the initial sequence fa1, i g whose period L K (mmax þ 1) ¼ 60 121 ¼ 7260. Since fa1, i g should have a good periodic ACF the relevant candidates may be the ternary perfect ACF sequence of length L ¼ 8011, a binary m-sequence (L ¼ 213 1 ¼ 8191) or a Legendre sequence (L ¼ 7283). The 60 signatures are then just 60 cyclic replicas of the fa1, i g offset from each other by 121 chips. Clearly, there is no upper limit on the length of the sequence and it may be advisable to take it longer with an appropriate increase in signature offset to secure some safety margin.

The uplinks of the 2G cdmaOne (IS-95) and 3G cdma2000 standards present very good examples of implementation of this version of asynchronous CDMA [69]. A binary m-sequence of an extremely long length L ¼ 242 1 extended by one symbol is used as the initial one and the user-specific signatures of all mobiles are just its relevant cyclic replicas. Pseudonoise properties of an m-sequence along with signature offsets exceeding possible variations of time of arrival of signal at the BS receiver guarantee a minimal

2 Without DS data modulation, perfection of periodic ACF of the initial sequence secures zero level of both MAI and multipath interference for any m mmax in the described signature construction.