p the signal amplitude A at the receiver is a Rayleigh random variable with probability density function (2.191) where this fade is independent from hop to hop. Amplitude A is constant during each hop interval. This is called a uniform channel since the proba- bility density function (2.191) does not depend on which part of the spread spectrum frequency band the signal hops to at any time. Consider a CW signal of T seconds duration at frequency wo with ampli- tude A and phase together with white Gaussian noise n(t) of double-sided power spectral density No/2, which is given by y(t) Asin(wyt+d)+n(1) (2.192) Using the sine expansion y(t) = A sin cos wot + A cos sin wot + n(t) OSIT. (2.193) In detecting such a signal, there is no loss of generality3 in basing decisions on the normalized cosine and sine components of y(t) given by (2.194) 3This is because the signal has only cosine and sine components. Slow Fading Uniform Channels where nc and n's are independent zero mean Gaussian random variables with variance No/2. Note that if A is a Rayleigh random variable with probability density given by (2.191) and is independent of A and uniformly distributed over [0, ),2n], Zs=Asin o Zc=Acos o (2.195) are independent zero mean Gaussian random variables with variance o2. Thus, yc and ys are independent zero mean Gaussian random variables with variance o T/2 1 No/2. Denoting (2.196) as the average signal energy, the joint density for yc and ys is given by (2.197) If the received signal was noise alone then y(t) = n(t) (2.198) and the joint probability density of yc and y's is (2.199) Thus in a Rayleigh fading channel, at the receiver each FH/MFSK signal is a narrowband Gaussian random process with the resulting sine and cosine components being zero mean independent Gaussian random variables with variance (E + No)/2. At a frequency4 where there is no FH/MFSK signal, the corresponding cosine and sine components would be zero mean Gaussian random variables with variance No/2. Note that when the CW signal is present and A and ar