HW6F13_sol

# HW6F13_sol - EE 113 Digital Signal Processing Solutions for...

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EE 113 Digital Signal Processing Solutions for Homework VI Instructor: Professor Ali H. Sayed TA: Zaid Towfic, Xiaochuan Zhao November 12, 2013

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Problem 13.9. Find the DTFTs of the following sequences: (a). x ( n ) = ( 1 2 ) n 1 u ( n + 1) . (b). x ( n ) = ( 1 3 ) 2 n 1 u ( n 1) . (c). x ( n ) = ( 1 4 ) n 2 u ( n 1) . In each case, ±nd expressions for the magnitude and phase of the DTFT. Solution. (a). Notice that x ( n ) = 4 · p 1 2 P n +1 u ( n + 1) Thus, we get X ( e ) = 4 e 1 1 2 e = 4 · exp( ) r 5 4 cos( ω ) · exp ± j arctan ± sin( ω ) 2 cos( ω ) ²² where the amplitude and phase are given by | X ( e ) | = 4 r 5 4 cos( ω ) , X ( e ) = ω arctan p sin( ω ) 2 cos( ω ) P (b). Notice that x ( n ) = 1 3 · p 1 9 P n 1 u ( n 1) Thus, we get X ( e ) = 1 3 · e 1 1 9 e = 1 3 · exp( ) r 82 81 2 9 · cos( ω ) · exp ± j arctan ± sin( ω ) 9 cos( ω ) ²² where the amplitude and phase are given by | X ( e ) | = 1 3 r 82 81 2 9 · cos( ω ) , X ( e ) = ω arctan p sin( ω ) 9 cos( ω ) P (c). Notice that x ( n ) = 64 · 4 n 1 u ( n 1) which is a left-sided sequence. Its z -transform is given by X ( z ) = 64 · z 1 4 · z and its ROC is given by 0 ≤ | z | < 1 4 Since on every point of the unit circle the X ( z ) diverges, its DTFT does not exist. 338
Problem 13.11. Find the DTFTs of the following sequences: (a). x ( n ) = cos ( π 3 n ) sin ( π 3 n ) .

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