solns2_600

# solns2_600 - ECE-600 Introduction to Digital Signal...

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Unformatted text preview: ECE-600 Introduction to Digital Signal Processing Autumn 2011 Homework #2 Oct. 7, 2011 HOMEWORK SOLUTIONS #2 1. (a) CTFT modulation property: F CTFT { x ( t ) e j Ω t } = integraldisplay ∞-∞ x ( t ) e j Ω t e- j Ω t dt = integraldisplay ∞-∞ x ( t ) e- j (Ω- Ω ) t dt = X c ( j (Ω- Ω )) via the CTFT definition (b) CTFT shift property: F CTFT { x ( t- τ ) } = integraldisplay ∞-∞ x ( t- τ ) e- j Ω t dt = integraldisplay ∞-∞ x ( q ) e- j Ω( q + τ ) dq via substitution q = t- τ (limits don’t change) = e- j Ω τ integraldisplay ∞-∞ x ( q ) e- j Ω q = e- j Ω τ X c ( j Ω) via the CTFT definition (c) DTFT property of real-valued x [ n ]: X * ( e- jω ) = parenleftBigg summationdisplay n x [ n ] e- j (- ω ) n parenrightBigg * using DTFT definition = summationdisplay n x * [ n ] e- jωn distributing the conjugation & ( e jωn ) * = e- jωn = summationdisplay n x [ n ] e- jωn since real-valued x [ n ] implies x * [ n ] = x [ n ] = X ( e jω ) using DTFT definition 2. (a) The Fourier series says that we can write any T-period signal p ( t ) as...
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## This note was uploaded on 10/29/2011 for the course ECE 600 taught by Professor Clymer,b during the Fall '08 term at Ohio State.

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solns2_600 - ECE-600 Introduction to Digital Signal...

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