# MSP15 The Fourier Transform (cont’) Lim, 1990. MSP16 The Fourier Series Expansion Suppose g(t) is a transient function that is zero outside the interval.

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MSP15 The Fourier Transform (cont’) Lim, 1990

MSP16 The Fourier Series Expansion Suppose g(t) is a transient function that is zero outside the interval [-T/2,T/2] (e.g., a cycle of a periodic function). We can obtain a sequence of coefficients by making s a discrete variable and integrating over the interval (with period T), so that

MSP17 The Fourier Series Expansion (cont’) where

MSP18 The Discrete Fourier Transform (DFT) If we discretize both time and frequency the Fourier transform pair of a series become

MSP19 The DFT (cont’) If {f i } is a sequence of length N (by taking samples of a continuous function at equal intervals) then its discrete Fourier transform pair is given by

MSP20 Properties of the Fourier Transform The addition theorem (addition in time/spatial domain corresponds to addition in frequency) The shift theorem (shifting a function causes to only phase shift) The convolution theorem (convolution is equivalent to multiplication in the other domain) …

MSP21 The Addition Theorem Castleman, 1996

MSP22 The Fourier Transform of a 2D Sequence x(m,n) The Fourier Transform Pair

MSP23 A 2D Fourier Transform Castleman, 1996

MSP24 Properties of 2D Fourier Transform Castleman, 1996

MSP25 The Fourier Transform (cont’) Example 1 (x  h) Lim, 1990

MSP26 The Fourier Transform (cont’) Lim, 1990

MSP27 The Fourier Transform (cont’) Lim, 1990

MSP28 The Fourier Transform (cont’) Lim, 1990

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