1. Lecture 18: The Discrete Fourier Transform Instructor: Dr. Ghazi Al Sukkar Dept. of Electrical Engineering The University of Jordan Email: ghazi.alsukkar@ju.edu.jo ghazi.alsukkar@ju.edu.jo Spring 20141

2. Outline  Sampling the Fourier Transform  The Discrete Fourier Transform  Properties of DFT Spring 20142

3. 3 Sampling the Fourier Transform  Consider an aperiodic sequence with a Fourier transform  Assume that a sequence is obtained by sampling the DTFT  Since the DTFT is periodic, the resulting sequence is also periodic  We can also write it in terms of the z-transform  The sampling points are shown in figure  could be the DFS of a sequence  Write the corresponding sequence

4. Spring 20144 Sampling the Fourier Transform Cont’d  The only assumption made on the sequence is that DTFT exist  Combine the equations to get  Term in the parenthesis is  So we get

5. Spring 20145 Sampling the Fourier Transform Cont’d

6. Spring 20146 Sampling the Fourier Transform Cont’d  Samples of the DTFT of an aperiodic sequence can be thought of as DFS coefficients of a periodic sequence obtained through summing periodic replicas of original sequence  If the original sequence is of finite length and we take sufficient number of samples of its DTFT the original sequence can be recovered by  It is not necessary to know the DTFT at all frequencies To recover the discrete-time sequence in time domain  Discrete Fourier Transform Representing a finite length sequence by samples of DTFT

7. Spring 20147 The Discrete Fourier Transform

8. Spring 20148 The Discrete Fourier Transform Cont’d

9. Cont.. Spring 20149

10. Example Spring 201410

11. Spring 201411 Example

12. Spring 201412 Example Cont’d

13. Spring 201413 Properties of DFT