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