 such that kN is a multiple of N. Hence, (1.51) where 1cm() denotes the least common integer multiple of its argu- ments, and (1.52) When P = N, then (1.52) can be simplified still further by taking the k-sum inside the two sums of the squared absolute value, to give, (1.53) where Rd. is the period autocorrelation function of the sequence {Cn}, i.e., (1.54) The latter sum in (1.54) is the periodic correlation function of the sequence {CmN} which is called the decimation of {cm} by N Such dec- imations preserve the periodic autocorrelation properties of the sequence when the greatest common divisor of N and N is unity. Summing like terms in (1.53) and using the symmetry of Rc(k) gives (1.55) Since {cn} is composed of 1 elements, then R&(0) = N, and this PSD is identical to that of random DS modulation provided that Rc(k) = 0 for k = 1, 2, , N - 1. This latter condition is one reasonable objective for pseudorandom number generator design. 2. FHIFSK modulation: The use of M-ary data, frequency-shift-keyed onto an FH signal at a rate of MD hops per data symbol, creates the modu- lation (1.56) A Spread-Spectrum Overview where p(t) is a hop pulse waveform which is non-zero only in the inter- val (0, Th), {fn} is the sequence of frequency hop values, {dk} is the data sequence, and {0n} is a sequence of independent, uniformly distributed, random phase variables. In calculating Sm(f) using the approach in (1.42) we will take the limit as the integer K = T/T becomes large. Hence, (1.57) the required Fourier transform is given by (1.58) where F denotes the ordinary Fourier transform. Evaluating the trans- form in (1.58), substituting in (1.57), and averaging over the random phase sequence, gives (1.59) where P(f) is the transform of the hop pulse. Assuming that the data sequence elements are independent, identically distributed random vari- ables chosen from the set D with probability function P(), gives (1.60 We now consider two models for the hopping sequence. a. Random Hopping. When {fn} is a sequence of identically dist