where Sm(f) is the PSD of the modulation mT(t). We will now evaluate Sm(f) for two basic SS modulation designs. 1. DS/BPSK modulation: The waveform corresponding to a DS signal antipodally modulated by binary data is (1.45) in which the data value dn/Nej has the opportunity to change every N chip times (x denotes the integer part of x), and p(t) is a pulse shape which is non-zero only in the interval (0, Tc. The calculation of Sm(f) is simplified by letting the limit parameter T grow in multiples of the symbol time NCT Hence, (1.46) By converting the sum index n in (1.45) to kN + m where k ranges from - K to K-1 - and m ranges between 0 and N - 1, it is possible to show (1.47) where P(f) is the Fourier transform of the chip pulse p(t). Inserting this transform into (1.46) and ensemble averaging over the data sequence {dk) which we assume to be composed of independent binary random variables, each equally likely to be +1 or - 1, gives (1.48) A Spread-Spectrum Overview We consider two possible assumptions regarding the nature of the direct sequence {Cn}. a. Random DS Modulation: If {cn} is a sequence of independent, identically distributed random variables, each equally likely to be +1 or -1, then the ensemble average in (1.48) simplifies to N and (1.49) This PSD is sketched in Figure 1.4 for several possible pulse shapes. Note that 1/T is a rough measure of the widths of these spectra. b. Periodic DS Modulation: When {Cn} is generated by a finite state device acting as a pseudorandom number generator, then {cn} must be periodic with some period N, i.e., Cn+N = Cn for all N. (1.50) Hence, nothing random remains in (1.48) and the expected value operator can be dropped. Furthermore, the value of the sum on m in (1.48) must also be a periodic function of K, with period P corre- square pulse cosine pulse split pulse lftdl Normalized Frequency Figure 1.4. The power spectral density of random DS/BPSK modulation for three different chip pulse shapes. The Advantages of Spectrum Spreading sponding to the smallest value of k