uantities N (and integer) and TI to satisfy (1.89) (n+1)T for m = N, + n (1.90) for m = N 1 +n+ 0, otherwise where Xp(T, v; T) is the ambiguity function of the pulse waveform p(t) gen- erally defined as (1.91) for pulse waveforms p(t) non-zero only in the time interval (0, T). The lim- ited values of m which produce non-zero results in (1.90) further simplify (1.88) to P)() = (1.92) Equation (1.92) is a convenient starting point for time and ensemble averaging operations. The Advantages of Spectrum Spreading We will carry out the ensemble-averaging process under the assump- tion that the spectrum-spreading sequence {cn} is composed of inde- pendent, identically distributed random variables, equally likely to be +1 or - -1,and that the independent data sequence {dk} likewise, is composed solely of +1's and -1's. The expansion of (1.92) then requires the moments 0 otherwise 0, n = m (1.93) = 0 for all N, m, n, (1.94) and the fact that kN,-1 (1.95) n=(k-1)Nc Expanding the squared sum in (1.92) as a double sum and simplifying, gives for N1 = 0, E{|UM(k)|2} for N1 = -1, otherwise. (1.96) Because of the manner in which we modelled {Cn} and {dk}, the above mean- squared value of vm(k) is not a function of k, and hence, no further averag- ing is necessary. When the chip pulse p(t) is unit amplitude and rectangular in shape, with duration T, then (1.97) otherwise. A Spread-Spectrum Overview With the aid of (1.89) and (1.97), the average ambiguity surface (1.96) can be evaluated as a function of t and v, and the results are displayed in Figure 1.6 for the case N = 100. When there is no time or frequency mismatch, i.e., when t = 0 and v = 0, then the normalized detector output (1.96) is N&T2. Alternatively, when t1 T, C then (1.96) is upper-bounded by N&T2. Hence, multipath and repeater jamming are reduced by a factor 1/N when the interfer- ence's incremental delay exceeds a chip time. This is a firm basis for stat- ing that the TOA resolution of a high energy gain DS/BPSK system is approximately one chip time. 2. FH/FSK mo