solns2_600(2)

solns2_600(2) - ECE-600 Introduction to Digital Signal...

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Unformatted text preview: ECE-600 Introduction to Digital Signal Processing Autumn 2010 Homework #2 Oct. 8, 2010 HOMEWORK SOLUTIONS #2 1. (a) CTFT convolution property: F CTFT braceleftbiggintegraldisplay ∞-∞ x ( τ ) w ( t − τ ) dτ bracerightbigg = integraldisplay ∞-∞ parenleftbiggintegraldisplay ∞-∞ x ( τ ) w ( t − τ ) dτ parenrightbigg e- j Ω t dt (1) = integraldisplay ∞-∞ integraldisplay ∞-∞ x ( τ ) e- j Ω τ w ( t − τ ) e- j Ω( t- τ ) dtdτ (2) = integraldisplay ∞-∞ x ( τ ) e- j Ω τ dτ integraldisplay ∞-∞ w ( ρ ) e- jωρ dρ via substitution ρ = t − τ (3) (limits of integral don’t change.) = X c ( j Ω) W c ( j Ω) via CTFT definition . (4) (b) CTFT multiplication property: F- 1 CTFT braceleftbigg 1 2 π integraldisplay ∞-∞ X c ( jξ ) W c ( j (Ω − ξ )) dξ bracerightbigg = 1 2 π integraldisplay ∞-∞ parenleftbigg 1 2 π integraldisplay ∞-∞ X c ( jξ ) W c ( j (Ω − ξ )) dξ parenrightbigg e j Ω t d Ω (5) = 1 2 π integraldisplay ∞-∞ 1 2 π integraldisplay ∞-∞ X c ( jξ ) e jξt W c ( j (Ω − ξ )) e j (Ω- ξ ) t d Ω dξ (6) = 1 2 π integraldisplay ∞-∞ X c ( jξ ) e jξt dξ 1 2 π integraldisplay ∞-∞ W c ( jθ ) e jθt dθ via substitution θ = Ω − ξ. (7) (limits of integral don’t change.) = x ( t ) w ( t ) via ICTFT definition . (8) (c) DTFT property of real-valued X ( e jω ): x * [ − n ] = parenleftbigg 1 2 π integraldisplay π- π X ( e jω ) e jω (- n ) dω parenrightbigg * using IDTFT definition = 1 2 π integraldisplay π- π X * ( e jω ) e jωn dω distributing the conjugation & ( e- jωn ) * = e jωn = 1 2 π integraldisplay π- π X ( e jω ) e jωn dω since real-valued...
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solns2_600(2) - ECE-600 Introduction to Digital Signal...

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