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sampling.slides.printing.6

# sampling.slides.printing.6 - Sampling Sampling Introduction...

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Sampling Sampling CS 450: Introduction to Digital Signal and Image Processing Bryan Morse BYU Computer Science Sampling Introduction Sampling Continuous f(t) t Discrete f(t) t Sampling Introduction Sampling Sampling a continuous function f to produce discrete function ˆ f ˆ f [ n ] = f ( n Δ t ) is just multiplying it by a comb: ˆ f = f comb h where h = Δ t Sampling Sampling In The Time/Spatial Domain Sampling In The Time/Spatial Domain - Graphical Example Continuous f(t) t Sampling Comb t f(t) Discrete f(t) t Sampling Sampling In The Frequency Domain Sampling In The Frequency Domain Sampling is multiplication by a comb with spacing h : ˆ f = f comb h What is happening in the frequency domain? The Fourier Transform of a comb with spacing h is a comb with spacing 1 / h : ˆ F = F * comb 1 / h Convolution of a function and a comb causes a copy of the function to “stick” to each tooth of the comb, and all of them add together . Sampling Sampling In The Frequency Domain Sampling In The Frequency Domain - Graphical Example Signal u F(u) Comb u F(u) Discrete Signal u F(u)

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Sampling Sampling In The Frequency Domain Reconstruction In theory, we can reconstruct the original continuous function by removing all of the extraneous copies of its spectrum created by the sampling process: F ( u ) = ˆ F ( u ) rect 1 / h ( u ) In other words, keep everything in the frequency domain
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sampling.slides.printing.6 - Sampling Sampling Introduction...

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