In this problem, we derive two important properties of the continuous-time Fourier serie...

In this problem, we derive two important properties of the continuous-time Fourier series: the multiplication property and Parseval's relation. Let x(t) and y(t) both the continuous-time periodic signals having period and with Fourier series representations given by

(a) Show that the Fourier series coefficients of the signal

are given by the discrete convolution

(b) Use the result of part (a) to compute the Fourier series coefficients of the signals depicted in Figure P3.46.

(c) Suppose that y(t) in eq. (P3.46-1) equals x* (t). Express the bk in the equation in terms of ak, and use the result of part (a) to prove Parseval's relation for periodic signals—that is,