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hw5_600(2) - ECE-600 Homework#5 Introduction to Digital...

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ECE-600 Introduction to Digital Signal Processing Autumn 2010 Homework #5 Oct. 29, 2010 HOMEWORK ASSIGNMENT #5 Due Fri. Nov. 5, 2010 (in class) 1. Say that we sampled a signal x ( t ) at the rate 1 T = 990 MHz to produce x [ n ]. Now, however, we wish that we had samples of x ( t ) at the rate 999 MHz, which we will refer to as y [ m ]. Describe how x [ n ] can be resampled to get y [ m ]. 2. After a recent lecture, a student asked, “What does the DTFT of a periodic signal look like?” This problem is meant to address that question. (a) Say that ˜ x [ n ] is an N -periodic signal (i.e., ˜ x [ n ] = ˜ x [ n + N ] for all n ) and say that { x [ n ] } N - 1 n =0 is a length- N signal that contains one period of ˜ x [ n ] (i.e., x [ n ] = ˜ x [ n ] for n = 0 , 1 ,...,N 1). Derive a simplified expression for ˜ X ( e ) in terms of the DFT X [ k ]. ( Hint : In stating your answer, remember that the DFT X [ k ] is valid only for k = 0 , 1 ,...,N 1. Thus, the expression X [ ( k ) N ] may come in useful.) (b) Confirm your answer with the signal x [ n ] = e j 2 π N n defined for n = 0 , 1 ,...,N 1. To do this, compute the DFT X [ k
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