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)

Short Time Fourier Transform (STFT)

• Given a signal we take the FFT on a window sliding with time

][nx

n

][][ nwnx

][ nw

FFT

|],[| kX

n

n

k

1N

N

Spectrogram: ideally

• The evolution of the magnitude with time is called Spectrogram.

• Ideally we would like to have this:

1

time

0n

][nx

time

frequency

1 2

12

2

0n

Spectrogram: in practice

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

time

frequency

12

21

Nm

2

N

FmF S

N STNt

Spectrogram: effect of window length

• Let N be the length of the window

• Frequency Resolution with m depending on the window;

• Time resolution:

N

FmF S

STNt

time

frequency

12

N

Nm

2

N

FmF S

STNt

Windows

0 5 10 15 20 25 30 35 40 45 500

0.5

1

1.5

-3 -2 -1 0 1 2 3-80

-70

-60

-50

-40

-30

-20

-10

0

w (rad)

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-3 -2 -1 0 1 2 3-80

-70

-60

-50

-40

-30

-20

-10

0

w (rad)-3 -2 -1 0 1 2 3

-80

-70

-60

-50

-40

-30

-20

-10

0

w (rad)

0 1N

Rectangular Hamming Blackman

][nw ][nw ][nw

)(NW )(NW )(NW

N/4 N/8 N/16

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

t

F

to

F0

t

F

to

F0

to

t

x t( )

good localization in time

good localization in freq.

either

… or

...

Time/Frequency Uncertainty Principle

Example: a Chirp

ttFAtx )(2cos)(

A “Chirp” is a sinusoid with time varying frequency, with expression

If the frequency changes linearly with time, it has the form shown below:

0F

1F

t1T Time (sec)

Frequency (Hz)

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))

0 0.005 0.01 0.015 0.02 0.025 0.03-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

time (sec)

Given Data

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

time (sec)

Four repetitions of a chirp:

Spectrogram of the Chirp: no window

spectrogram(y, rectwin(256), 250, 256,Fs,'yaxis');

Spectrogram of the Chirp: hamming window

spectrogram(y, hamming(256), 250, 256,Fs,'yaxis');

Spectrogram of the Chirp: shorter window

spectrogram(y, hamming(64), 60,64,Fs,'yaxis');

Spectrogram of the Chirp: shorter window

spectrogram(y, hamming(64), 60,256,Fs,'yaxis');

Spectrogram of the Chirp: blackman window

spectrogram(y, blackman(256), 250, 256,Fs,'yaxis');

Sealion

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

time (sec)

Spectrogram of Sealion: hamming window

spectrogram(y(2001:12000), hamming(256), 250, 256,Fs,'yaxis');

Spectrogram of Sealion: no window

spectrogram(y(2001:12000), rectwin(256), 250, 256,Fs,'yaxis');

Spectrogram of Sealion: blackman, longer

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

Music

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

Spectrogram

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

Spectrogram: zoom

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

Not enough frequency resolution!

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***: 2 15SFF HzN

This yields a window length of at least N=1024

***Note: the slide in the video has a typo, showing the inequality reversed.

Spectrogram: longer window

spectrogram(y(12001:22000), hamming(1024), 1000, 1024,'yaxis');

Spectrogram: longer window (zoom)

spectrogram(y(12001:22000), hamming(1024), 1000, 1024,'yaxis');

Spectrogram: recognize some notes

spectrogram(y(12001:22000), hamming(1024), 1000, 1024,'yaxis');

E: 330Hz, 660Hz

D: 294Hz, 588Hz

Closest Notes:

Check with the Music Score

E

D

E

D

B

A