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hw4_600 - ECE-600 Introduction to Digital Signal Processing...

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ECE-600 Introduction to Digital Signal Processing Autumn 2011 Homework #4 Oct. 14, 2011 HOMEWORK ASSIGNMENT #4 Due Fri. Oct. 21, 2011 (in class) 1. Say that a symmetric real signal x ( t ) = x ( - t ) has the CTFT below. For each of the parts below, . f (MHz) X c ( j 2 πf ) 0 0 . 2 × 10 6 5 10 - 5 - 10 sketch the DTFT X ( e ) that results from sampling x ( t ) uniformly at the specified rate. In your sketches, show the range ω ( - 2 π, 2 π ). (a) 1 /T = 60 MHz (b) 1 /T = 30 MHz (c) 1 /T = 15 MHz (d) 1 /T = 10 MHz (e) 1 /T = 5 MHz Hint : Since you know the DTFT phase response for a real-valued symmetric signal, you know how the aliasing copies will add. 2. Say that a signal x ( t ) is sampled at rate 1 /T 1 Hz, producing x [ n ], and then sinc-reconstructed at rate 1 /T 2 Hz, producing y ( t ). Assuming that x ( t ) is bandlimited to 1 2 T 1 , give an expression for Y c ( j Ω) in terms of X c ( j Ω). ( Hint : Relate X c ( j Ω) to X ( e ), then relate X ( e ) to Y c ( j Ω), and finally relate X c ( j Ω) to Y c ( j Ω). The final relationship should not include a summation of spectral copies!) 3. Suppose again that a signal x
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