This problem develops the simplex signaling set.¹³Consider M orthogonal signals, s_i(t), i = 0, 1, 2,...., M - 1 each with energy E_s. Compute the average of the signals 

and define a new signal set

s^'_i (t) = s_i (t) - a (t), i = 0, 1, 2,......, M - 1

(a) Show that the energy of each signal in the new set is 

(b) Show that the correlation coefficient between each signal and another is 

(c) Given that the probability of symbol error for an 

write down an expression for the symbol-error probability of the simplex signal set where, from (F.9), Q (-x) = 1 - Q (x).

(d) Simplify the expression found in part (c) using the union bound result for the probability of error for an orthogonal signaling set given by (10.67). Plot the symbol-error probability for M = 2, 4, 8, 16 and compare with that for coherent M-ary FSK.

M-1 a (t) 2 M i=0 (t) S; E = E, (1 -