 as phase modulation or amplitude modulation. Ordinary BPSK signals have a (sin2x)/x2 shaped power spectrum with one- sided first null bandwidth equal to 1/Tb. Direct sequence spreading of this BPSK signal is done with a pseudo- random (PN) binary sequence {Ck} whose elements have values 1 and are generated by the PN sequence generator N times faster than the data rate. Thus, the time T of a PN binary symbol referred to as a "chip" is (3.18) The PN chip rate might be several megabits per second while the data rate might be a few bits per second. The direct sequence spread-spectrum signal has the form x(t) = nTb +kT<1<nTb+(k+1)T k=0,1,2,...N- 1 n = integer. (3.19) This signal is similar to ordinary BPSK except that the apparent data rate is N times faster, resulting in a signal spectrum N times wider. Here the pro- cessing gain is simply PG - Wa - N, (3.20) where Wss is again the direct sequence spread signal bandwidth. Defining the data function d(t) = (3.21) for all integers n, and PN function c(t)kkT,t<(k+1)T (3.22) for all integers k, the direct sequence spread BPSK signal can be expressed x(t)=V2S sin[wotc(t)d(t)/2] = c(t)d(t)V2S cos (3.23) Basic Concepts and System Models Figure 3.4 illustrates d(t), c(t), and c(t)d(t) for N = 6. Figure 3.5(a) displays the normal form of the DS/BPSK modulator and Figure 3.5(b) shows a more convenient model for analysis. Note that in this latter form, x(t) = c(t)s(t), (3.24) where s(t) = d(t) 2S cos wot (3.25) c(t)d(t) Figure 3.4. DS/BPSK waveforms. Uncoded Direct-Sequence Spread Binary Phase-Shift-Keying d (1) MODULATOR GENERATOR (a) Normal Form S (1) MODULATOR GENERATOR (b) Convenient Representation Figure 3.5. DS/BPSK modulation. is the ordinary BPSK signal. Also, since c2(t) = 1 for all t, (3.26) it follows that c(t)x(t) = s(t). (3.27) Assuming the intended receiver has a PN sequence generator that is syn- chronized to the one in the transmitter, c(t) is available to both the trans- mitter and receiver. 3.4.1 Constant Power Broadband Noise Jammer Suppose a jammer is trans