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 ( ) w ( t ) d bracerightbigg = integraldisplay - parenleftbiggintegraldisplay - x ( ) w ( t ) d parenrightbigg e- j t dt (1) = integraldisplay - integraldisplay - x ( ) e- j w ( t ) e- j ( t- ) dtd (2) = integraldisplay - x ( ) e- j d integraldisplay - w ( ) e- j d via substitution = t (3) (limits of integral dont change.) = X c ( j ) W c ( j ) via CTFT definition . (4) (b) CTFT multiplication property: F- 1 CTFT braceleftbigg 1 2 integraldisplay - X c ( j ) W c ( j ( )) d bracerightbigg = 1 2 integraldisplay - parenleftbigg 1 2 integraldisplay - X c ( j ) W c ( j ( )) d parenrightbigg e j t d (5) = 1 2 integraldisplay - 1 2 integraldisplay - X c ( j ) e jt W c ( j ( )) e j (- ) t d d (6) = 1 2 integraldisplay - X c ( j ) e jt d 1 2 integraldisplay - W c ( j ) e jt d via substitution = . (7) (limits of integral dont change.) = x ( t ) w ( t ) via ICTFT definition . (8) (c) DTFT property of real-valued X ( e j ): x * [ n ] = parenleftbigg 1 2 integraldisplay - X ( e j ) e j (- n ) d parenrightbigg * using IDTFT definition = 1 2 integraldisplay - X * ( e j ) e jn d distributing the conjugation & ( e- jn ) * = e jn = 1 2 integraldisplay - X ( e j ) e jn d since real-valued...
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solns2_600(2) - ECE-600 Introduction to Digital Signal...

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