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Unformatted text preview: 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 jn can be derived as follows: Y ( e j ) = summationdisplay n = y [ n ] e jn (1) = summationdisplay n = x [ n- d ] e jn e jn (2) = summationdisplay n = x [ n- d ] e j ( ) n (3) = summationdisplay m = x [ m ] e j ( )( m + d ) via m = n- d (4) = e j ( ) d summationdisplay m = x [ m ] e j ( ) m (5) = e j ( ) d X ( e j ( ) ) . (6) Another approach is to realize that x [ n- d ] e jn is a d-sample delay of x [ n ] followed by a rad/sec modulation of the result. Delay alone would yield the intermediate signal e jd X ( e j ), after which -modulation yields e j ( ) d X ( e j ( ) )....
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