o 30 MHz). These may be known to the transmitter and receiver by using channel sounder signals that periodically measure the channel. The jammer may also be transmitting the jammer signal over the non-uniform channel, thus resulting in propagation parameters B1, B2, BQ. The jammer has control over the noise power distribution J = (J1,J2, ..,Jo) . under constraint (2.242) while the FH/MFSK anti-jam system con- trols the frequency hopping pattern defined by the probability vector p = In addition to the generalizations described above, assume there is receiver white Gaussian noise with single-sided spectral density No across the total spread-spectrum band. 2.6.1 Uncoded The uncoded (no diversity) FH/MFSK signal has conditional symbol error probability (2.291) when the FH/MFSK signal hops into the q-th sub-band. This is the usual sym- bol error probability for non-coherent Rayleigh fading channels of average signal energy aqE in additive white Gaussian noise of single-sided spectral density N° + Bq1q. The symbol error probability averaged Non-Coherent Frequency-Hopped Systems over the random hopping depends on the jammer noise power distribution J and the hopping probability vector p as follows: (2.292) which is a concave function of J. Without loss of generality let (2.293) Thus with no jammer the sub-band symbol error probabilities satisfy (2.294) Suppose the jammer is very weak SO that even if the jammer used all its power to jam the 1st sub-band, it would still have lower symbol error probability than the other sub-bands. That is, with J = J and Jq = 0, q = 2, 3, , Q P1(J)<P2(0)P(0). (2.295) Clearly in this case the optimum choice for the hopping probability vector is P1 = 1 and pq=0,q = = 2,3,...,Q. Using only the 1st sub-band would max- imize the bit error probability. Now define threshold T1 that satisfies (2.296) No + B 1 1 No (2.297) Then T1 is the jammer power in sub-band 1 that makes it exactly as bad as sub-band 2 without jamming. It is clear that if J > T1, then the optimum hop- ping probability would no