EECE 340 Homework, Spring 2008

EECE 340-Signals and Systems Homework #10 Problem #1 Determine the expressions of the discrete Fourier transforms of the following sequences assumed to be of length N. a) x(n) = d(n) b) x(n) = d(n-1) °) af vane uth 0, otherwise Problem # 2 Let x() be the discrete sequence shown below. x(n) @ teen o_o o—e > 2 4 0 F 2 3 Plot x((-n)), which denotes the periodic repetition of x(—n) with period equal to 4. Problem #3 Verify that x((N-—n)), =x((-”)),, with x((-n)),q being the periodic repetition of x(—n) with period equal to N. Problem # 4 Consider two finite duration sequences x(n) and y(#) where both are zero for n<0 and with x(n)=0, n>8, y(n) =0,n2 20. The 20 point DFT’s of each of the sequences are multiplied and the inverse DFT is computed. Let r(n) denote the inverse DFT. Specify which points in r(n) correspond to points that would be obtained in a linear convolution of x(n) and y(n). Page | of 1 EECE 340 Homework 410 SohTeor )) 4 Xn) =SCn VV -! kn X (kK) =) x(n) W =I, R=0,1, 2,02, WO b) x(n) = §(h-n,) N~I Ly kn X (k) = > b~7gy) Wy = Wy ; b=s]2 h=0 ae 1/3 XK = D_ al We fl =o V—} N = > @up)'= 1. ia) N=0 _— aw | — Qn " F l-awk ~ F Ot, 2, voy N-I 3h | ID AGO) = O40) chase A enites a tact contin. Teg G04 of Ag §4+20~/ = 27 ctucl r(n) = [D7 20h) | K (n) hk =x—po