(a) Referring to the signal set defined by (11.98), show that the minimum possible f = ...

(a) Referring to the signal set defined by (11.98), show that the minimum possible Δf = Δω / 2π such that (s_i, s_j) = 0 is Δf = 1/(2T_s).

(b) Using the result of part (a), show that for a given time-bandwidth product WT_s the maximum number of signals for M-ary FSK signaling is given by M = 2WT_s, where W is the transmission bandwidth and T_s is the signal duration. Use null-to-null bandwidth. Thus, W = M / (2T_s). (Note that this is smaller than the result justified in Chapter 10 because a wider tone spacing was used there.)

(c) For vertices-of-a-hyper-cube signaling, described in Problem 11.22, show that the number of signals grows with WT_s as M = 2^2WTs. Thus, W = log₂ M / (2T_s), which grows slower with M than does FSK.