hw8 - 6. Show that the DFT matrix W = 1 1 ... 1 1 e-j 2 π...

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EE210 Signal Systems (April 9, 2009) Home Work Set 8 1. The two dimensional DTFT is defined by H ( e 1 ,e 2 ) = X m = -∞ X n = -∞ x ( m,n ) e - 1 n e - 2 m Find the expression for the inverse 2D DTFT. 2. let x ( n ) X ( e ) be a DTFT pair. Find the DTFT of (a) x ( n - n 0 ) , ( b ) x 2 ( n ). 3. Parseval’s relation for discrete sequences: Let x ( n ) X ( e ). Show that X n = -∞ | x ( n ) | 2 = 1 2 π Z -∞ | X ( e ) | 2 whenever the sum on the left hand side exists. 4. Let y ( n ) = h ( n ) * x ( n ) Find Y ( e ) the DTFT of y ( n ) in terms X ( e ) and H ( e ) 5. Let x ( n ) X ( e ). Find the DTFT of (a) x ( n ) e jnω
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Unformatted text preview: 6. Show that the DFT matrix W = 1 1 ... 1 1 e-j 2 π 1 /N ... e-j 2 π ( N-1) . 1 N . . . . . . . . . . . . 1 e-j 2 π 1 . ( N-1) /N ... e-j 2 π ( N-1)( N-1) N is orthogonal, i.e. W [ W * ] T = N. 1 ... 1 ... . . . . . . . . . ... 1 W * → complex conjugate of W . Note W = W T . 1...
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