fft.slides.printing.2

# fft.slides.printing.2 - Implementation Padding Fast Fourier...

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Implementation: Padding, Fast Fourier Transform Implementation: Padding, Fast Fourier Transform CS 450: Introduction to Digital Signal and Image Processing Bryan Morse BYU Computer Science Implementation: Padding, Fast Fourier Transform Introduction Implementing Convolution using the Fourier Transform We saw earlier: f ( t ) * h ( t ) = F - 1 ( F ( f ( t )) F ( h ( t ))) 1. Compute the Fourier Transform of the input signal f ( t ) 2. Compute the Fourier Transform of the convolution kernel h ( t ) May be done by building the kernel and applying the DFT/FFT May be calculated analytically directly in the frequency domain 3. Multiply the two Fourier Transforms together (complex multiply) 4. Apply the inverse Fourier Transform (if the signal and kernel are real-valued, just use the real part of the result)

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Implementation: Padding, Fast Fourier Transform Padding Padding the Kernel To multiply F ( u ) and H ( u ) , they must be the same size This means the discrete f ( t ) and h ( t ) must be the same size If necessary, pad the kernel h ( t ) with zeroes to be the same size as the signal f ( t ) —adding zeroes to a convolution kernel doesn’t change anything Implementation: Padding, Fast Fourier Transform Padding Convolution–What happens at the image boundary?
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