 of these waveforms can be achieved, regardless of the data values (d,d') and random phase values (, 'o), if one or more of the following conditions holds: d, d'. (1.37) bo # b'o. (1.38) fo - so integer, k # 0. (1.39) The signal variables, known a priori to the receiver, but not the jammer, are the pseudorandom DS chip values {Coma: d = 1, 2, =1,2,...,No), the hop frequency fo, and the time interval index b0. Hence, the receiver must A Spread-Spectrum Overview observe the signal in a 2M-dimensional space whose basis over the interval (0, Ts) consists of Re{c(d)(t) exp(j2ft)} evaluated for each of the M values of d, with the random hop phase set at 0 and at T12. The jammer, how- ever, must choose his waveform to jam a signal space of dimension 2NMMM, whose basis consists of the sines and cosines of N orthogonal DS modula- tions hopped over MF orthogonal tones and MT disjoint time intervals. Therefore, the nominal energy gain EG of this system is EG=N.Mr., (1.40) The nominal minimum bandwidth Wss required to implement the orthog- onality requirements (1 - 3) is determined by the minimum hop frequency spacing 1/T to be MFT Using this fact and substituting (1.35) into (1.40) indicates that the energy gain for this hybrid SS system in a closely packed design is TWSSIM. 1.5 THE ADVANTAGES OF SPECTRUM SPREADING We have seen the advantages of making a jammer counteract an ensemble of orthogonal communication systems. The bandwidth increase which must accompany this SS strategy has further advantages which we will outline here. 1.51 Low Probability of Intercept (LPI) Spectrum spreading complicates the signal detection problem for a sur- veillance receiver in two ways: (1) a larger frequency band must be moni- tored, and (2) the power density of the signal to be detected is lowered in the spectrum-spreading process. The signal may have further desirable attributes based in part on LPI, such as low probability of position fix (LPPF) which includes both intercept and direction finding (DFing) in its evalua- tion, or low prob