notes-Convolution Theorem

notes-Convolution Theorem - = 1 M 2 M-1 X k =0 f ( k ) M-1...

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Proof of the Convolution Theorem Written up by Josh Wills January 21, 2002 f ( x ) * h ( x ) = ←→ F ( u ) H ( u ) (1) g ( x ) = 1 M M - 1 X x =0 f ( k ) h ( x - k ) (2) Perform a Fourier Transform on each side of the equation: G ( u ) = 1 M 1 M ± M - 1 X x =0 ± M - 1 X k =0 f ( k ) h ( x - k ) ! e - j 2 πux/M ! (3) Factor 1 into two exponentials and subsitute them into the equation: = 1 M 2 M - 1 X x =0 M - 1 X k =0 f ( k ) h ( x - k ) e - j 2 πuk/M e j 2 πuk/M e - j 2 πux/M (4) Note: f(x) and h(x) are both assumed to be periodic with period M:
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Unformatted text preview: = 1 M 2 M-1 X k =0 f ( k ) M-1 X x =0 h ( x-k ) e-j 2 u ( x-k ) /M ! e-j 2 uk/M (5) Since M-1 X x =0 h ( x ) e-j 2 ux/M = H ( u ) M : G ( u ) = M M 2 M-1 X k =0 f ( k ) H ( u ) e-j 2 uk/M (6) = 1 M F ( u ) H ( u ) M (7) = F ( u ) H ( u ) (8) 1...
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