hw2_600

# hw2_600 - ECE-600 Introduction to Digital Signal Processing...

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Unformatted text preview: ECE-600 Introduction to Digital Signal Processing Autumn 2011 Homework #2 Sep. 30, 2011 HOMEWORK ASSIGNMENT #2 Due Fri. Oct. 7, 2011 (in class) 1. (a) Prove the CTFT modulation property: e j Ω t x ( t ) CTFT ←→ X c ( j (Ω- Ω )) (b) Prove the CTFT shift property: x ( t- τ ) CTFT ←→ e- j Ω τ X c ( j Ω) , τ ∈ R (c) Say x [ n ] is real-valued. For X ( e jω ) = F DTFT { x [ n ] } , prove that X ( e jω ) = X * ( e- jω ). Note: For (a)-(b) above, to prove the specified CTFT property, 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. 2. (a) Use Fourier Series to prove that ∑ ∞ m =-∞ δ ( t- mT ) = 1 T ∑ ∞ k =-∞ e j 2 π T kt . (b) Prove the Dirac scaling law δ ( t/a ) = aδ ( t ) for any a > 0. ( Hint : Exploit unit-area property.) 3. In this problem, you will verify that the DTFT and IDTFT form a valid transform pair.3....
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