20102ee113_1_hw7_2010

20102ee113_1_hw7_2010 - EE113 Digital Signal Processing...

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EE113: Digital Signal Processing Spring 2010 Prof. Mihaela van der Schaar Homework #7 The following problems are from the electronic versions of the chapters of An Undergraduate Course on Discrete-Time Signal Processing , by Prof. A.H. Sayed (posted on the class website): Problem 15.1, 15.8, 17.3 Additional Problems: Problem A. Consider the sequence x [ n ] given by x [ n ] = α n u [ n ], where | α | < 1. A periodic sequence ˜ x [ n ] is constructed from x [ n ] in the following way: ˜ x [ n ] = X r = -∞ x [ n + rN ] . (a) Determine the Fourier transform X ( e jw ) of x [ n ]. (b) Determine the discrete Fourier series ˜ X [ k ] of ˜ x [ n ]. (c) How is ˜ X [ k ] related to X ( e jw )? Problem B. Two finite-length sequences x [ n ] and x 1 [ n ] are shown in Fig. 1. The N -point DFTs of these sequences, X [ k ] and X 1 [ k ], respectively, are related by the equation X 1 [ k ] = X [ k ] e j 2 π (2 k ) N , where N is an unknown constant. Can you determine a value of N consistent with Fig. 1? Is your choice for N unique? Figure 1: Problem B.
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