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soln8 - R Balan Homework#8 Solutions MATH 464 I The window...

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Unformatted text preview: R. Balan Homework #8 Solutions MATH 464 I . The window is g ( x ) = Π( x ) . a) f ( x ) = e 2 πix . The windowed Fourier transform is V g f ( w,t ) = Z ∞-∞ e- 2 πiwx f ( x ) g ( x- t ) dx = Z e- 2 πix ( w- 1) Π( x- t ) dx = = e- 2 πit ( w- 1) Z e- 2 πiy ( w- 1) Π( y ) dy = e- 2 πit ( w- 1) sinc ( w- 1) b) f ( x ) = e 2 πiax . The windowed Fourier transform is V g f ( w,t ) = Z ∞-∞ e- 2 πiwx f ( x ) g ( x- t ) dx = Z e- 2 πix ( w- a ) Π( x- t ) dx = = e- 2 πit ( w- a ) Z e- 2 πiy ( w- a ) Π( y ) dy = e- 2 πit ( w- a ) sinc ( w- a ) c) f ( x ) = e 2 πiax + e 2 πibx . The windowed Fourier transform is V g f ( w,t ) = Z ∞-∞ e- 2 πiwx f ( x ) g ( x- t ) dx = Z e- 2 πix ( w- a ) Π( x- t ) dx + Z e- 2 πix ( w- b ) Π( x- t ) dx = = e- 2 πit ( w- a ) Z e- 2 πiy ( w- a ) Π( y ) dy + e- 2 πit ( w- b ) Z e- 2 πiy ( w- b ) Π( y ) dy == e- 2 πit ( w- a ) sinc ( w- a )+ e- 2 πit ( w- b ) sinc ( w V g f ( w,t ) = Z ∞-∞ e- 2 πiwx f ( x ) g ( x- t ) dx = Z...
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