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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.

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Presentation on theme: "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."— Presentation transcript:

1 MSP15 The Fourier Transform (cont’) Lim, 1990

2 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

3 MSP17 The Fourier Series Expansion (cont’) where

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

5 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

6 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) …

7 MSP21 The Addition Theorem Castleman, 1996

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

9 MSP23 A 2D Fourier Transform Castleman, 1996

10 MSP24 Properties of 2D Fourier Transform Castleman, 1996

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

12 MSP26 The Fourier Transform (cont’) Lim, 1990

13 MSP27 The Fourier Transform (cont’) Lim, 1990

14 MSP28 The Fourier Transform (cont’) Lim, 1990


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