MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.003: Signals and Systems-Fall 2003 PROblem set 3 SOLUTION Home Study Exercise (E1)O&W3.46(a)and(c) (t) and y(t) are continuous-time periodic signals with a period= To and Fourier series representations given b r(t)= akejkw'ot y(t)=>bk k=-0 (a) Show that the Fourier series coefficients of the signal x(t)=r(t)y(t)=ko_o Ckejkwot are given by the discrete convolution Ck=>to_o anbk-n(multiplication property) z()=x(t)y(t)= ∑ bm e a. bmenwoLe3muo n=一 Let k=n+ ∑a Interchange the summations order=>i(t)= ∑ anbk-njejkwot k=-0(n=-∞ 2()=x((t)=∑ k=-∞0 (c) Suppose that y(t)=x*(t). Express bk in terms of ak, and use the result of part(a) to prove Parseval's relation for periodic signals+� +� +� +� +� +� +� +� +� +� +� +� +� +� MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.003: Signals and Systems—Fall 2003 Problem Set 3 Solution Home Study Exercise (E1) O&W 3.46 (a) and (c) x(t) and y(t) are continuous-time periodic signals with a period = T0 and Fourier series representations given by ⎨ ⎨ bkejk�0t x(t) = akejk�0t y(t) = k=−� k=−� (a) Show that the Fourier series coefficients of the signal ckejk�0t z(t) = x(t)y(t) = �+� are given by the discrete convolution k=−� ck = �+� −� anbk−n (multiplication property). n= ⎨ ⎨ ⎨ ⎨ bmejm�0t = anbmejn�0t jm�0t z(t) = x(t)y(t) = anejn�0t e n=−� m=−� n=−� −� m= ⎨ ⎨ = anbmej(n+m)�0t n=−� m=−� ⎨ ⎨ = anbk−nejk�0t Let k = n + m ⇒ m = k − n � z(t) n=−� k= � −� � ⎨ ⎨ anbk−n jk�0t Interchange the summations order � z(t) = e k=−� n=−� ⎨ ⎨ � z(t) = x(t)y(t) = ckejk�0t , where ck = anbk−n. k=−� n=−� (c) Suppose that y(t) = x�(t). Express bk in terms of ak, and use the result of part(a) to prove Parseval’s relation for periodic signals. 1