1. Time Frequency Analysis
We want to see how the frequencies of a signal change with time.
Typical example of a Time/Frequency representation:
time
log(freq)

2. Short Time Fourier Transform (STFT)
Given a signal we take the FFT on a window sliding with time

3. Spectrogram: ideally
The evolution of the magnitude with time is called Spectrogram.
Ideally we would like to have this:
time
frequency

4. Spectrogram: in practice
We need to deal with the effects of the window: main lobe and sidelobes
time
frequency

5. Spectrogram: effect of window length
Let N be the length of the window
frequency
time
Frequency Resolution with m depending on the window;
Time resolution:

6. Windows
Rectangular
Hamming
Blackman

7. Time/Frequency Uncertainty Principle
either you have good resolution in time or in frequency, not both.
either
good localization in time
...
… or
good localization in freq.

8. Example: a Chirp
A “Chirp” is a sinusoid with time varying frequency, with expression
If the frequency changes linearly with time, it has the form shown below:
Time (sec)
Frequency (Hz)

9. Chirp in Matlab Fs=10000; Ts=1/Fs; t=(0:999)*Ts;
y=chirp(t, 100, t(1000), 4000);
plot(t(1:300), y(1:300))

10. Four repetitions of a chirp:
Given Data
Four repetitions of a chirp:

11. Spectrogram of the Chirp: no window
spectrogram(y, rectwin(256), 250, 256,Fs,'yaxis');

12. Spectrogram of the Chirp: hamming window
spectrogram(y, hamming(256), 250, 256,Fs,'yaxis');

13. Spectrogram of the Chirp: shorter window
spectrogram(y, hamming(64), 60,64,Fs,'yaxis');

14. Spectrogram of the Chirp: shorter window
spectrogram(y, hamming(64), 60,256,Fs,'yaxis');

15. Spectrogram of the Chirp: blackman window
spectrogram(y, blackman(256), 250, 256,Fs,'yaxis');

16. Sealion

17. Spectrogram of Sealion: hamming window
spectrogram(y(2001:12000), hamming(256), 250, 256,Fs,'yaxis');

18. Spectrogram of Sealion: no window
spectrogram(y(2001:12000), rectwin(256), 250, 256,Fs,'yaxis');

19. Spectrogram of Sealion: blackman, longer
spectrogram(y(2001:12000), blackman(512), 500, 512,Fs,'yaxis');

20. Music
spectrogram(y(2001:12000), blackman(512), 500, 512,Fs,'yaxis');

21. Spectrogram
spectrogram(y(12001:22000), hamming(256), 250, 256,'yaxis');

22. Not enough frequency resolution!
Spectrogram: zoom
spectrogram(y(12001:22000), hamming(256), 250, 256,'yaxis');
Not enough frequency resolution!

23. Frequencies we expect to see
Since this signal contains music we expect to distinguish between musical notes. These are the frequencies associated to it (rounded to closest integer): C Db D Eb E F Gb G Ab A Bb B
262
277
294
311
330
349
370
392
415
440
466
494
Notes
Freq. (Hz)
Desired Frequency Resolution***:
This yields a window length of at least N=1024
***Note: the slide in the video has a typo, showing the inequality reversed.

24. Spectrogram: longer window
spectrogram(y(12001:22000), hamming(1024), 1000, 1024,'yaxis');

25. Spectrogram: longer window (zoom)
spectrogram(y(12001:22000), hamming(1024), 1000, 1024,'yaxis');

26. Spectrogram: recognize some notes
spectrogram(y(12001:22000), hamming(1024), 1000, 1024,'yaxis');
Closest Notes:
E: 330Hz, 660Hz
D: 294Hz, 588Hz

27. Check with the Music ScoreB D D A