solns2_600

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

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Unformatted text preview: ECE-600 Introduction to Digital Signal Processing Winter 2011 Homework #2 Jan. 21, 2011 HOMEWORK SOLUTIONS #2 1. (a) DTFT modulation property: F DTFT { x [ n ] e j n } = summationdisplay n =- x [ n ] e j n e- jn = summationdisplay n =- x [ n ] e- j ( - ) n = X ( e j ( - ) ) via the DTFT definition (b) DTFT shift property: F DTFT { x [ n d ] } = summationdisplay n =- x [ n d ] e- jn = summationdisplay m =- x [ m ] e- j ( m + d ) via substitution m = n d (limits dont change) = e- jd summationdisplay m =- x [ m ] e- jm = e- jd X ( e j ) via the DTFT definition (c) DTFT property of real-valued X ( e j ): x * [ n ] = parenleftbigg 1 2 integraldisplay - X ( e j ) e j (- n ) d parenrightbigg * using the 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 X ( e j ) implies X * ( e j ) = X ( e j ) = x [ n ] using the IDTFT definition 2. (a) The Fourier series says that we can write any T-period signal p ( t ) as p ( t ) = summationdisplay k =- P [ k ] e jk 2 T t with P [ k ] = 1 T integraldisplay T/ 2- T/...
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solns2_600 - ECE-600 Introduction to Digital Signal...

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