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1. Ashutosh Dhekne CS 8803 - MCI Signal Processing III

2. What does the Fourier Matrix provide us?   Time Domain Signal   Freq Domain 0Hz 1Hz 2Hz 3Hz 4Hz 5Hz 6Hz 7 Hz  

3. What does the Fourier Matrix provide us?   Time Domain Signal   Freq Domain 0Hz 1Hz 2Hz 3Hz 4Hz 5Hz 6Hz 7 Hz   No DC Component Signal is symmetric above and below the 0 amplitude

4. What does the Fourier Matrix provide us?   Time Domain Signal   Freq Domain 0Hz 1Hz 2Hz 3Hz 4Hz 5Hz 6Hz 7 Hz   Frequency domain vector is symmetric around 4Hz

5. What does the Fourier Matrix provide us?   Time Domain Signal   Freq Domain 0Hz 1Hz 2Hz 3Hz 4Hz 5Hz 6Hz 7 Hz   Only components are 1Hz and 3Hz (non-zero values in the frequency domain)

6. What does the Fourier Matrix provide us?   Time Domain Signal   Freq Domain 0Hz 1Hz 2Hz 3Hz 4Hz 5Hz 6Hz 7 Hz   1Hz is twice as dominant as 3Hz frequency

7. Example: Graphic Equalizer

8. How is such a transformation performed?   Time Domain Signal                                  

9. Time Domain to Frequency Domain                                      

10. Time Domain to Frequency Domain                                       Another Basis

11. Basis in the Time Domain    

12. Basis in Frequency Domain                      

13. Time Domain to Frequency Domain         =

14. Time Domain to Frequency Domain   =      

15. Time Domain to Frequency Domain   =    

16. Time Domain to Frequency Domain    

17. Time Domain to Frequency Domain       Check this at home!

18. What if the signal has higher frequencies?

19. What if the signal has higher frequencies?

20. What if the signal has higher frequencies?   Time Domain Signal

21. What if the signal has higher frequencies?   Time Domain Signal   Freq Domain 0Hz 1Hz 2Hz 3Hz 4Hz 5Hz 6Hz 7 Hz  

22. What if the signal has higher frequencies?   Time Domain Signal   Freq Domain 0Hz 1Hz 2Hz 3Hz 4Hz 5Hz 6Hz 7 Hz Does this signal have a 2Hz component?

23. Does this FD signal produce the same TD signal?   Time Domain Signal   Freq Domain 0Hz 1Hz 2Hz 3Hz 4Hz 5Hz 6Hz 7 Hz Does this signal have a 2Hz component?

24. In fact, it is ambiguous! Which of these is the correct analog signal?

25. In fact, it is ambiguous!   Time Domain Signal   Freq Domain 0Hz 1Hz 2Hz 3Hz 4Hz 5Hz 6Hz 7 Hz    

26. In fact, it is ambiguous!   Time Domain Signal   Freq Domain 0Hz 1Hz 2Hz 3Hz 4Hz 5Hz 6Hz 7 Hz There can exist infinitely many analog signals that will produce the same discrete frequency domain signal and the same discrete time domain signal!

27. Aliasing

28. Aliasing Aliasing  is an effect that causes different signals to become indistinguishable (or  aliases  of one another) when sampled.

29. Aliasing Aliasing  is an effect that causes different signals to become indistinguishable (or  aliases  of one another) when sampled. It also often refers to the distortion or artifact that results when a signal reconstructed from samples is different from the original continuous signal.

30. Aliasing : Matlab examples

31. Nyquist Frequency / Folding Frequency Harry Nyquist Feb 7, 1889 - Apr 4, 1976

32. The concept of filters

33. FFT as frequency increases

34. The difference between sine and cosine   Time Domain Signal   Freq Domain 0Hz 1Hz 2Hz 3Hz 4Hz 5Hz 6Hz 7 Hz

35. The difference between sine and cosine   Time Domain Signal   Freq Domain 0Hz 1Hz 2Hz 3Hz 4Hz 5Hz 6Hz 7 Hz

36. FFT with change in phase

37. Fourier Matrix in the more general form   0Hz 1Hz 2Hz 3Hz 4Hz 5Hz 6Hz 7Hz What if we divide the circle into N parts instead of 8?

38. Fourier Matrix in the more general form   0Hz 1Hz 2Hz 3Hz 4Hz 5Hz 6Hz 7Hz What if we divide the circle into N parts instead of 8? How will the matrix change?

39.   0Hz 1Hz 2Hz 3Hz 4Hz 5Hz 6Hz 7Hz How will the matrix change? Becomes N Extends to N-1 Extends to N-1