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Exam 2 Solution Spring 209 - EEE4001F Digital Signal...

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EEE4001F: Digital Signal Processing Class Test 2 30 April 2008 SOLUTIONS Name: Student number: Information ± The test is closed-book. ± This test has four questions, totalling 20 marks. ± Answer all the questions. ± You have 45 minutes.
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1. (5 marks) Find wŒn D xŒn ² yŒn with xŒn D . NUL 1/ n and yŒn D sin .±n=3/ ±n : We can think of wŒn as the output of a system with impulse response yŒn to the input signal xŒn . From the tables of Fourier transform pairs, the frequency response of the system is Y.e j! / D 8 < : 1 j ! j ³ ±=3 0 ±=3 < j ! j < ±: Since x.t/ D . NUL 1/ n D e j±n is a complex exponential of frequency ! D ± , the output is wŒn D Y.e /e j±n D 0: (Alternatively find and sketch X.e j! / and Y.e j! / , and show that the product W.e j! / D X.e j! /Y.e j! / is zero.)
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2. (5 marks) Consider the following discrete-time signal: xŒn D 8 < : sin . 4 / when n=2 is an integer 0 otherwise : Calculate and sketch the 8-point DFT (magnitude and phase) of the first 8 samples of xŒn , i.e. xŒ0 ; xŒ1 ; : : : ; xŒ7 . Show and motivate your calculations. The only nonzero samples of xŒn over the range 0 to 7 are xŒ2 D 1 and xŒ6 D NUL 1 . The DFT is therefore XŒk D 7 X n D 0 xŒn W kn 8 D 7 X
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