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# solnsmid1_600_old - ECE-600 Introduction to Digital Signal...

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ECE-600 Introduction to Digital Signal Processing Wi11 Midterm #1 Feb. 4, 2011 MIDTERM #1 SOLUTIONS 1. The DTFT of y [ n ]= x [ n - d ] e jθn can be derived as follows: Y ( e ) = summationdisplay n = −∞ y [ n ] e jωn (1) = summationdisplay n = −∞ x [ n - d ] e jθn e jωn (2) = summationdisplay n = −∞ x [ n - d ] e j ( ω θ ) n (3) = summationdisplay m = −∞ x [ m ] e j ( ω θ )( m + d ) via m = n - d (4) = e j ( ω θ ) d summationdisplay m = −∞ x [ m ] e j ( ω θ ) m (5) = e j ( ω θ ) d X ( e j ( ω θ ) ) . (6) Another approach is to realize that x [ n - d ] e jθn is a d -sample delay of x [ n ] followed by a θ rad/sec modulation of the result. Delay alone would yield the intermediate signal e jωd X ( e ), after which θ -modulation yields e j ( ω θ ) d X ( e j ( ω θ ) ). P. Schniter, 2011 1

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2. (a) Recallingthecorrespondence 1 T =20 ω =2 π ,wehave6 3 π 5 and10 π . Then, recalling that the DTFT is 2 π -periodic, we get X ( e ) ω 20 - 3 π 5 - π - 7 π 5 3 π 5 π 7 π 5 0 2 π - 2 π
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solnsmid1_600_old - ECE-600 Introduction to Digital Signal...

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