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convolution.slides.printing.2

convolution.slides.printing.2 - Convolution and Filtering...

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Convolution and Filtering: The Convolution Theorem Convolution and Filtering: The Convolution Theorem CS 450: Introduction to Digital Signal and Image Processing Bryan Morse BYU Computer Science Convolution and Filtering: The Convolution Theorem Introduction Linear Systems and Responses Time/Spatial Frequency Input f F Output g G Impulse Response h Transfer Function H Relationship g = f * h G = FH Is there a relationship?

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Convolution and Filtering: The Convolution Theorem Convolution Theorem The Convolution Theorem Let F , G , and H denote the Fourier Transforms of signals f , g , and h respectively. g = f * h g = fh implies implies G = FH G = F * H Convolution in one domain is multiplication in the other and vice versa. Convolution and Filtering: The Convolution Theorem Convolution Theorem The Convolution Theorem Thus, F ( f ( t ) * g ( t )) = F ( f ( t )) F ( g ( t )) Likewise, F ( f ( t ) g ( t )) = F ( f ( t )) * F ( g ( t ))
Convolution and Filtering: The Convolution Theorem Convolution Theorem Filtering: Frequency-Domain vs. Spatial (Convolution) Convolution and Filtering: The Convolution Theorem Convolution Theorem Linear Systems and Responses Time/Spatial Frequency Input f F Output g G Impulse Response h Transfer Function H Relationship g = f * h G = FH Relationship: the Transfer Function H ( u ) is the Fourier Transform of the impulse response h ( u )

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