hw2_600 - ECE-600 Homework #2 Introduction to Digital...

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ECE-600 Introduction to Digital Signal Processing Winter 2011 Homework #2 Jan. 14, 2011 HOMEWORK ASSIGNMENT #2 Due Fri. Jan. 21, 2011 (in class) 1. (a) Prove the DTFT modulation property: e 0 n x [ n ] DTFT ←→ X ( e j ( ω - ω 0 ) ) (b) Prove the DTFT shift property: x [ n - d ] DTFT ←→ e - jωd X ( e ) , d Z (c) Say X ( e ) = F DTFT { x [ n ] } is real-valued. Prove x [ n ] = x * [ - n ]. Note: For (a)-(b) above, to prove the speci±ed DTFT property, you can either start with the time domain expression and show that a forward transform yields the frequency domain expression, or you can start with the frequency domain expression and show that an inverse transform yields the time-domain expression. ( Hint : In each case, one of these two approaches may be much easier than the other!) 2. (a) Use Fourier Series to prove that n = -∞ δ ( t - nT ) = 1 T k = -∞ e j 2 π T kt . (b) Prove the Dirac scaling law
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