Let be a periodic signal with period N. A finite-duration signal x[n] is related to thr...

Let be a periodic signal with period N. A finite-duration signal x[n] is related to through

for some integer n₀. That is, x[n] is equal to over one period and zero elsewhere

(a) If l[n] has Fourier series coefficients a_k and x[n] has Fourier transform show that

regardless of the value of n_0.

(b) Consider the following two signals:

where N is a positive integer. Let a_k denote the Fourier coefficients of let denote the Fourier transform of x[n].

(i) Determine a closed-form expression for

(ii) Using the result of part (i), determine an expression for the Fourier coefficients a_k.