4) We use a correlation processor to detect the presence or absence of a deterministic signal, (10') = 1 i% (u(t) u(t ~ T)) , in the presence of

This is a communication theory problem.

Can you show the steps clearly? Thank you

4) We use a correlation processor to detect the presence or absence of a deterministic signal, (10‘) = 1 i% (u(t) — u(t ~— T)) , in the presence of zero-mean white noise having spectral density equal to N0 . For an arbitrary correlation, the SNR associated with the correlationr 2 r E 2[i dta(t)f(f)] [i d? ?“f(t)]processor is SNR = 0w— = 000—. We proved, in class, that the SNR isNo‘ I M20) No' i We)maximized by choosing f0) = a(t) , which yields SNR = f,—“ . We want to investigate the0 effect of performing a suboptimal correlation, by letting f (t) = e‘a‘ a: > 0 . a) For an arbitrary or > 0 , find an explicit expression for the ratio of the suboptimalSNR to the optimal SNR. Your expression should be entirely in terms of the product,[9 E a -T . b) Find the value of ,8 that maximizes the ratio derived in part a). Find the resultingratio of the suboptimal SNR to the optimal SNR (has to be less than one!). You mayeither perform the optimization numerically, or you may differentiate the SNR ratiowith respect to fl , and then solve the resulting transcendental equation using Newton’s method for finding roots. u(t) , for some