 5.16 (Welch's Bound [29]). Let (c4),...,CL), v=1,...,M,rep- resent L-tuples of complex numbers, and define the dinner product (5.236) Assume that each L-tuple has unit length, i.e., R = 1 for all v, and define Rmax = max RRA. (5.237) (5.238) Proof. Beginning with the definition of Rmax and the fact that the vectors Pseudonoise Generators are unit length, it is easily verified that M(M - 1)Rmax v=1A=1 M M L L i=1j=1 (5.239) Here equality (1) follows from changing the order of summation, inequal- ity (2) is achieved by neglecting terms with i #j, (3) is Cauchy's inequality for sums of squares, or equivalently is a result of the convexity of the qua- dratic function, and (4) simply uses the fact that the L-tuples are unit length. Solving the derived inequality (5.239) for Rmax gives the final result. Welch's inner-product bound can be applied to a variety of signal design situations. Here we apply it to the maximum absolute value of correlation for a set of roots-of-unity sequences. Let (5.240) for 0 < T , N and , the N-1/2 being required to make the N-tuple unit length. Here one may think of as ranging over JN values corresponding to the distinct pairs (j, T); hence, the required parameter substitution in Welch's bound is L = N and M = JN. The quantity Rmax can be shown to correspond to Pmax/N, giving the following result. COROLLARY 5.3. The maximum absolute value Pmax of the out-of-phase peri- odic autocorrelation and of the periodic cross-correlation for a set of J roots-of-unity sequences with period N is lower bounded by (5.241) When the number J of sequences is even moderately large, then (5.241) indi- cates the Pmax must be at least on the order of VN. When the set of sequences {a,}}}, 1 jJ, of period N, are composed of 1 strings, a correspondence with a code C containing JN binary code Direct-Sequence Multiple-Access Designs words bj,m, 1<j</,0<m<N, of length N over GF(2), can be established through the relation n(1)bmm (5.242) Note that the n code words bj,m, ) m m N, are cyclically equivalent for each val