Time Frequency Analysis

1
Time Frequency Analysis

• We want to see how the frequencies of a signal
  change with time.
• Typical example of a Time/Frequency
  representation

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

4
Spectrogram in practice

• We need to deal with the effects of the window
  main lobe and sidelobes

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
7
Time/Frequency Uncertainty Principle

• either you have good resolution in time or in
  frequency, not both.

good localization in time
either
...
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
9
Chirp in Matlab
Fs10000 Ts1/Fs t(0999)Ts ychirp(t, 100,
t(1000), 4000) plot(t(1300), y(1300))
10
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(200112000), hamming(256), 250,
256,Fs,'yaxis')
18
Spectrogram of Sealion no window
spectrogram(y(200112000), rectwin(256), 250,
256,Fs,'yaxis')
19
Spectrogram of Sealion blackman, longer
spectrogram(y(200112000), blackman(512), 500,
512,Fs,'yaxis')
20
Music
spectrogram(y(200112000), blackman(512), 500,
512,Fs,'yaxis')
21
Spectrogram
spectrogram(y(1200122000), hamming(256), 250,
256,'yaxis')
22
Spectrogram zoom
spectrogram(y(1200122000), 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 N1024
Note the slide in the video has a typo,
showing the inequality reversed.
24
Spectrogram longer window
spectrogram(y(1200122000), hamming(1024), 1000,
1024,'yaxis')
25
Spectrogram longer window (zoom)
spectrogram(y(1200122000), hamming(1024), 1000,
1024,'yaxis')
26
Spectrogram recognize some notes
spectrogram(y(1200122000), hamming(1024), 1000,
1024,'yaxis')
Closest Notes
E 330Hz, 660Hz
D 294Hz, 588Hz
27
Check with the Music Score
E
E
B
D
D
A