irect-sequence spread systems for any spread factor N. 1.3 UNCODED BIT ERROR PROBABILITY FOR SPECIFIC JAMMER WAVEFORMS Consider the set of orthonormal basis elsewhere kT.<t<(k+1)T, elsewhere k=0,1,...,N. , - 1. (1.39) The transmitted direct-sequence spread BPSK signal during the data bit time interval [0,T] has the form OSIST (1.40) The jammer would like to place all of its signal energy into the same sig- nal space as the transmitted signal, otherwise its power would be wasted. It, however, can only know the signal bandwidth and not the signal phase. Thus, in general it has the form (1.41) where recall J = (I,J,, JN-1) consists of the cosine components. Note that only these cosine components enter into the bit error probability bound. The total jammer signal energy during the data bit time is (1.42) (1.43) Uncoded Bit Error Probability for Specific Jammer Waveforms The worst case jammer is one that places all its energy in the cosine coor- dinates SO that (1.44) Normally, however, the jammer can only place equal energy in the cosine and sine coordinates with the result that (1.45) Figure 1.3 shows the bit error probability k=0 for the Gaussian assumption and Chernoff Bound Gaussian Assumption 10-6, Figure 1.3. Gaussian assumption and Chernoff bound. Coherent Direct-Sequence Systems the general Chernoff bound for these two cases. Note that there is only about a one dB difference for fixed bit error rates. This suggests that the Gaussian assumption is reasonable for all N. The impact of jamming signals by characterizing two types of jammers will now be illustrated. The first model assumes a deterministic jammer wave- form that is characterized by a set of parameters O. The second model assumes the jammer is a stationary random process with statistical charac- terizations. In both cases the jammer is assumed to be transmitting contin- uously with constant power J. In the next section pulse jamming will be examined along with its generalization where the jammer power can be var- ied in time while maintaining a time-