1. Lecture 21: Fourier Analysis Using 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  Relationships between CTFT, DTFT, & DFT  Fourier Analysis of Signals Using DFT  Effect of Windowing on Sinusoidal Signals  Window Functions  The Effect of Spectral Sampling Spring 20142

3. Relationships between CTFT, DTFT, & DFT Spring 20143

4. Cont.. Spring 20144

5. Cont.. Spring 20145 Note: the aliasing in the time-domain occurred because the signal is time unlimited and the aliasing in the frequency-domain occurred since the signal is frequency unlimited. Dilemma: for most of the signals if it is time limited then it will be frequency unlimited and vice versa.

6. Spring 2014 6 Fourier Analysis of Signals Using DFT  One major application of the DFT: analyze signals  Let’s analyze frequency content of a continuous-time signal  Steps to analyze signal with DFT Remove high-frequencies to prevent aliasing after sampling Sample signal to convert to discrete-time Window to limit the duration of the signal (a consequence of the finite length requirement of the DFT) Take DFT of the resulting signal

7. Cont.. Spring 20147

8. 8 Example Continuous- time signal Anti-aliasing filter Signal after filter

9. Spring 2014 9 Example Cont’d Sampled discrete- time signal Frequency response of window Windowed and sampled Fourier transform

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11. Spring 2014 11 Effect of Windowing on Sinusoidal Signals  The effects of anti-aliasing filtering and sampling is known  We will analyze the effect of windowing  Choose a simple signal to analyze this effect: sinusoids  Assume ideal sampling and no aliasing we get  And after windowing we have  Calculate the DTFT of v[n] by writing out the cosines as

12. Spring 2014 12 Example

13. Spring 2014 13 Example Cont’d  Magnitude of the DTFT of the sampled signal  We expect to see dirac function at input frequencies  Due to windowing we see, instead, the response of the window  Note that both tones will affect each other due to the smearing This is called leakage: pretty small in this example

14. Spring 2014 14 Example Cont’d  If the input tones are close to each other   On the left: the tones are so close that they have considerable affect on each others magnitude  On the right: the tones are too close to even separate in this case They cannot be resolved using this particular window

15. Effects of time-windowing Spring 201415

16. Window Functions Spring 201416

17. Window choices Spring 201417

18. Cont.. Spring 201418

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20. Spring 2014 20 The Effect of Spectral Sampling

21. Spring 2014 21 Example Cont’d  Note the peak of the DTFTs are in between samples of the DFT  DFT doesn’t necessary reflect real magnitude of spectral peaks

22. Spring 2014 22 Example Cont’d  Let’s consider another sequence after windowing we have

23. Spring 2014 23 Example Cont’d  In this case N samples cover exactly 4 and 8 periods of the tones  The samples correspond to the peak of the lobes  The magnitude of the peaks are accurate  Note that we don’t see the side lobes in this case, solve the problem with zero-padding.

24. Spring 2014 24 Example: DFT Analysis with Kaiser Window  The windowed signal is given as  Where w K [n] is a Kaiser window with β=5.48 for a relative side lobe amplitude of -40 dB  The windowed signal

25. Spring 2014 25 Example Cont’d

26.  The effect of DFT length for Kaiser window of length L = 32. (b) Magnitude of DFT for N = 64. (c) Magnitude of DFT for N = 128. Spring 201426

27. Summary:  Zero-padding will increase the number of points (increase the resolution) but it will not improve the ability to resolve close frequencies.  Resolving close frequencies depends on the window length and shape, as the length increase, the ability to resolve close frequencies increase. Spring 201427