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**Unformatted text preview: **UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN Department of Electrical and Computer Engineering ECE 310 Digital Signal Processing Problem Set 7 Fall 2003 Due: Wednesday, October 15, 2003 Problem 7.1 Compute the discrete-time Fourier transform X d ( ) of the following signals: 1. x ( n ) = n sin( n ) u ( n ) for | | < 1 2. x ( n ) = 2 n u (- n ) 3. x ( n ) = sin 2 n n sin 4 n n 4. x ( n ) = ( 1 2 ) | n | 5. x ( n ) = n ( 1 2 ) | n | 6. x ( n ) = (- 1) n Problem 7.2 Compute the signal x ( n ) for the following spectra 1. X d ( ) = cos 2 ( ) 2. X d ( ) = e- j 1+ 1 6 e- j- 1 6 e- j 2 Problem 7.3 Consider an LSI-system with impulse response h ( n ) = sin 3 n n Compute the output signal y ( n ) for the following input signals: 1. x ( n ) = k =- ( n- 8 k ) 2. x ( n ) = ( n + 1) + ( n- 1) 3. x ( n ) = sin 4 n n 4. x ( n ) = sin 4 n n cos( 3 4 n ) Problem 7.4 Suppose X d ( ) = n 1 , | | 6 , 6 < | | 1. Sketch X d ( ) 2. Find { x ( n ) } n =- 3. If y ( n ) = e j 3 n x ( n ), sketch Y d ( ) 4. If y ( n ) = cos( 3 n ) x ( n ), sketch Y d ( ) Problem 7.5 Let x ( n ) = n 1 N , n N- 1 , else 1. Find closed-form expressions (no sums) for | X d ( ) | , and X d ( ) 2. For N = 6, plot | X d ( ) | . How will the shape of | X d ( ) | change if N increases?change if N increases?...

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