e stage for our discussion of slowly fading non-uniform chan- nels in Section 2.6, which is based on the work of Avidor [25], [26]. Assume that whenever a hopped MFSK signal is transmitted the average signal energy at the receiver is E independent on the part of the spread- spectrum frequency band that the signal hopped into during its transmission. This is what we refer to as the uniform channel and is the channel assumed in Section 2.4. Divide the spread-spectrum frequency band into Q equal size sub-bands where in each sub-band the jammer places noise with constant spectral den- sity across the sub-band. Let Jq be the jammer noise power in the q-th sub- band of bandwidth W q = WSIQ. Thus, the q-th sub-band was white Gaussian noise of single-sided spectral density Nq = Jq/W = QJq/Wss q=1,2,...,Q (2.241) This is illustrated in Figure 2.116. Note that for large values of Q this can accurately approximate all noise jammer power spectral densities of inter- est. With total jammer power J, the total power constraint is J - Er (2.242) 2.5.1 Uncoded The uncoded (no diversity) FH/MFSK signal that hops with uniform prob- ability across the total spread-spectrum frequency band has a bit error prob- ability given by6 (2.243) 6We assume the FH/MFSK signal will hop into each sub-band with equal probability 1/Q. Non-Coherent Frequency-Hopped Systems Figure 2.116. Jammer noise power distribution. where Pb(x) is the Rayleigh fading uncoded non-coherent MFSK bit error probability in white Gaussian noise with average symbol-energy-to-noise ratio X. This is given by (2.208). As was done in Section 2.5, consider uncoded FH/BFSK in a Rayleigh fad- ing uniform channel where the Rayleigh fading statistics are the same in each sub-band. In this case from (2.208) (2.244) and the average bit error probability is (2.245) Now consider the worst distribution of jammer power J = which satisfies constraint (2.242). This equality constraint can be incorpo- rated into the maximization of Pb given by (2.243) by using a Lagrange mul- tiplier l > 