x[n] is a real periodic signal with period N and complex Fourier series coefficients ak

Chapter 3, Problem 3.52

(choose chapter or problem)

x[n] is a real periodic signal with period N and complex Fourier series coefficients ak. Let the Cartesian form for ak be denoted by where bk and ck are both real. (a) Show that a-k = a'k. What is the relation between bk and b-k? What is the relation between ck and c - k? (b) Suppose that N is even. Show that aN12 is real. (c) Show that x[n] can also be expressed as a trigonometric Fourier series of the form (N-l)/2 (2 k ) (2 k ) x[n] = ao + 2 ~ bk cos : n - ck sin : n if N is odd or as (N-2l 12 (2 k ) (2 k ) x[n] = (ao + aN12(-lt) + 2 ~ bk cos : n - ck sin : n if N is even. (d) Show that ifthe polar form of ak is Akejek, then the Fourier series representation for x[n] can also be written as x[n] = ao + 2 L Akcos _!!._!!_+Ok (N-1)/2 ( 2 k ) k=l N if N is odd or as if N is even. (e) Suppose that x[n] and z[n], as depicted in Figure P3.52, have the sine-cosine series representations x[n] = ao + 2 ti ~1 bk cos (27Tkn) - . (27Tkn)j 7 - - ck sm - 7 - , z[n] =do+ 2 ti ~1 dkcos (27Tkn) - . (27Tkn)j 7 - - fksm - 7 - . Sketch the signal y[n] = a0 - do+ 2 t; 3 I dkcos (27Tkn) - . (27Tkn)j 7 - +Uk - ck)sm - 7 - .