2. (Total=18%) A pulse communication system is being designed that will transmitone of two messages, mi and m2, which occur with equal a priori probability.These messages will be signaled using the following waveforms, and we wouldlike to select sets of transmitted signals that will either minimize the averagetransmitted energy or will minimize the probability of error. Assume that thechannel is corrupted by an additive white Gaussian random process. Assumehere that the inner product of two real-valued function of duration T = 2 isdefined as

 Four possible signals, sA(t), sB(t), sc(t), and sD(t) are being considered, and two ofthem will be selected as the signal set that will indicate which message had beensent. The signals are sketched in Fig. 2.

【題組】(a) (4%) Compute a set of orthonormal basis functions

using the Gram-Schmidt procedure, where

is proportional to sA(r). Thesignals sA(X), sB(t), sc(t), and sD(t) can be expressed as a linear combination ofthese basis functions as______________________________________
______________________________________
______________________________________ where the basis functions are______________________________________
______________________________________
____________________________________________________________________________