Introduction to Timefrequencyanalysis Ssp&Mm 2013-14

8/14/2019 Introduction to Timefrequencyanalysis Ssp&Mm 2013-14

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Introduction to

Time-Frequency Analysis

Lorenzo Galleani

Department of Electronics and Telecommunications, Politecnico di Torino

Statistical Signal Processing and Multimedia



2

Time-Frequency Analysis (TFA)

• Born around the middle of 1900• “Boom” around 1980

• Extension of classical frequency analysis



3

Classical Frequency Analysis 1/2

• Fourier: Engineer • 1700-1800

• Decomposition of a signal in a sum ofsinusoids

• Heat transfer



4

Classical Frequency Analysis 2/2

It tells what frequencies existed in the signal



5

The Need for Time-Frequency Analysis

• In the real world, signals have, in general,time-varying frequencies

– Speech

– Doppler effect

– Sunset

– Astronomical signals: QPO, etc.

– …



6

We want more

We want to know

what frequencies existed in the signal

and

when they existed



7

Quiz 1/4

C D E F

Spectrogram



8

Quiz 2/4

C E D F

Spectrogram



9

Quiz 3/4

C F

Wigner distribution



10

Quiz 4/4

C+D

Wigner distribution

E+F



11

Terminology 1/5

t

f

0 100 200 300 400 500 600 700 800 9000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

One-component signal

Time-frequency domain

[Time-frequency plane]

Component[Red→High Energy→Large Amplitude]

Time-frequency region[Blue→Low Energy→Small Amplitude]

t

f

0 100 200 300 400 500 600 700 800 9000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Multicomponent signal

Components



12

Terminology 2/5

2.5

x 105

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

200 400 600 800 1000−2

0

2

t

f

0 200 400 600 8000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Sinusoid

Short

duration

sinusoid

Linear chirp



13

Terminology 3/5

10

0.050.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

200 400 600 800 10000

0.5

1

t

f

0 200 400 600 8000

0.050.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Delta function



14

Terminology 4/5

6965.62260

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

200 400 600 800 1000

−2

0

2

t

f

0 200 400 600 8000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Noise components

White Gaussian noise



15

Terminology 5/5

832.87070

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

200 400 600 800 1000

−1

−0.5

0

0.5

t

f

0 200 400 600 8000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Linear chirp withGaussian amplitude modulation

Instantaneous

frequency

Instantaneous

bandwidth



16

Turbo-diesel engine

Choi-Williams distribution

Pilot fuelinjection

Main fuel

injection

Vibrational

modes of

the chamber

Interference

terms



17

Instantaneous Frequency

Derivative of the phase of the analytic signal (Gabor)

dt

t d t i

)()( ϕ ω =

where

)()()( t i

a et At x ϕ =

is the analytic signal associated to the real signal x(t )



18

The Analytic Signal 1/5

Defined in the frequency domain as

)()(2)( ω ω ω X u X a =

∫

∫∞+

∞−

−

+∞

∞−

+

=

=

dt et x X

d e X t x

t i

t i

ω

ω

π ω

ω ω

π

)(2

1)(

)(

2

1)(

Note: we use the Fourier transform pair given by

where f π ω 2= is the angular frequency

where )(ω u is the Heaviside step function



19


ω 0

)(ω X

)(ω a X 2



20


We derive the connection between )(t xa and )(t x

∫+∞

∞−

+= ω ω π

ω d e X t x t i

aa )(2

1)(

)()(2)( ω ω ω X u X a =

In the time domain, we have

It is

∫+∞

∞−

−= dt et x X t iω

π ω )(

2

1)(

substituting

∫ ∫+∞ +∞

∞−

+−=0

' ')'(21

212)( ω

π π ω ω d dt eet xt x t it i

a

∫+∞

+=0

)(2

12 ω ω

π

ω d e X t i



21


∫ ∫+∞ +∞

∞−

−+=0

)'( ')'(1

)( ω π

ω d dt et xt x t t i

a

We now consider the identity

z

i z d e z i +=∫

+∞

)(0

πδ ω ω

By setting

∫

+∞

∞− ⎥

⎦

⎤⎢⎣

⎡

−

+−= '

'

)'()'(1

)( dt

t t

it t t xt xa πδ

π

't t z −= it is

)(t x

∫∫ +∞

∞−

+∞

∞− −+−= '

'

)'(')'()'( dt

t t

t xidt t t t x

π δ



22


∫+∞

∞− −+= '

'

)'()()( dt

t t

t xit xt xa

π

We note that

{ } )()( t xt xa =ℜ

which is the reason for the “2” in the definition of theanalytic signal

Hilbert transform of )(t x



23

Example 1: Linear Chirp

Satisfies our intuition…

Instantaneous frequency

E l 2 S f T Li Chi !



24

Example 2: Sum of Two Linear Chirps!

Instantaneous frequency

Paradoxes of the instantaneous frequency ωi(t )

1. It can take values outside of the signal bandwidth

2. It can take negative values even if the signal is analytic

3. It is a local quantity computed from the entire signal

b a n d w i d t h

Negative values!



25

Abundance of TF distributions

There are infinite Time-Frequency Distributions• Spectrogram

• Wigner distribution (or Wigner-Ville)

• Cohen’s class

– Smoothed Pseudo-Wigner

– Choi-Williams



26

Cohen’s class

• 1966: Journal of Mathematical Physics• It collects all TF distributions

• Every TFD has a kernel

• A new TFD can be designed by imposing someconstraints of the kernel

θ τ τ θ φ τ τ π ω θ τω θτ

∫∫∫ +−−

+−= d dud eu xu xt C uiii

),()2/()2/(*4

1

),( 2



27

Spectrogram

• Most intuitive TF distribution

• h(t ) is the window

• h(t -τ ) x(τ ) is the windowed signal

• Squared magnitude of the Short Time Fourier

Transform (STFT)

2

d)()(2

1),( ∫

∞+

∞−

−−= τ τ τ π

ω ω τ ie xt ht P



28

Positivity and Marginals

1.

2. Marginals are not satisfied

2)(),( ω ω X dt t P ≠∫2

)(),( t xd t P ≠∫ ω ω

0),( ≥ω t P



29

Window length 1/3

Short



30

Window length 2/3

Long



31

Window length 3/3

Medium



32

Support: Strong and Weak



33

Wigner distribution

∫

+∞

∞−

−

+−= τ τ τ π ω

ω τ

d)2/()2/(2

1

),(* i

et xt xt W

• Nonlinear (quadratic or bilinear )• Many mathematical properties

• Prototype of many distributions



34

Positivity



35

Interference terms 1/2

A consequence of the quadratic nature

of the Wigner distribution



36

Interference terms 2/2

We can filter the interference terms

Smoothed Pseudo-Wigner



37

Excellent localization

Linear chirp



38

Support: Strong and Weak



39

Filtering of TF distributions

∫∫ −= '')',')W('-,'(),( ω ω ω ω ϕ ω d dt t t t t C

Any distribution

∫∫ ++= ω ω ϕ

π τ θ φ τω θ td et t d),(

21),( ii

The kernel is defined as),( τ θ φ



40

Constraints on the kernel

• Positivity

• Time and frequency marginals

• Interference terms

• Strong and weak support

• Instantaneous frequency

• ...



41

TFA of random processes

• Very young field

– Wigner spectrum

– Sliding Welch spectrum

Wi S



42

Wigner Spectrum

∫+∞

∞−

−+−= τ τ τ π

ω ω τ d)]2/()2/([2

1),( * iet X t X E t W

• Expected value of the Wigner distribution

• Simple connection to the autocorrelation

∫+∞

∞−

−−+= τ τ τ ω ω τ d)2/,2/(),( i

X et t Rt W

)](*)([),( 2121 t X t X E t t R X =

Sliding Welch spectrum



4343


Almost identical to the spectrogram:

rather than computing the squared FFT ofthe truncated signal (basic periodogram), we

evaluate its Welch periodogram

A li i



44

Applications

• Time-frequency representation of dynamical

systems

– Deterministic case

– Random case• Quasi-Periodic Oscillations in a binary star

The Study of Dynamical Systems



45

The Study of Dynamical Systems

Transformation of a dynamical system in thetime-frequency domain...

)()()()()(01

1

1 t f t xadt

t dxadt

t xd adt

t xd an

nn

n

n =++++−

− K

…of the Wigner distribution

Can we write an equation for W x(t,ω ) ?

∫ ∞+

∞−

−+−= τ τ τ π

ω τω d et xt xt W i

x )2/()2/(*2

1),(

h i i h f



46

Why it is worth to transform

1. Better understanding of the system behavior

2. New system identification methods

3. New system design methods



47

Numerical approach

Harmonic oscillator with a chirp driving term

We choose μ < ω0

so that

the system is underdamped

t it it

e xdt

dx

dt

xd 1

22 2/2/2

02

2

2

ω β α

ω μ

++−

=++



48

−8 −6 −4 −2 0 2 4 6 8 10

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

x 10−3

t (s)

x ( t )

−10 −8 −6 −4 −2 0 2 4 6 8 10 12

10−7

10−6

10−5

10−4

10−3

f (Hz)

| X ( f ) | 2

Time Frequency

Classical representations



49

Time-Frequency Representation

t (s)

f ( H z )

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

t

f

Sinusoidal FM chirp



50

p

t it ie x

dt

dx

dt

xd 21 sin2

02

2

2 ω α ω

ω μ +=++

0 2 4 6 8 10 12 14 16 18

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

t (s)

x ( t )

0 1 2 3 4 5 6

10−5

10−4

10

−3

10−2

10−1

100

f (Hz)

| X ( f ) | 2

Time Frequency

3 f [Hz]

i i



51

Time-Frequency Representation

t (s)

f ( H

z )

2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

Transformation to the Wigner domain



52

g

)()()(

2)( 2

02

2

t f t xdt

t dx

dt

t xd =++ ω μ

)2(2

1)(

)(2)(

4)()(

222

02

2201

22222

00

μ ω ω ω

ω ω μ ω

ω μ ω ω ω

++=

+=

+−=

a

a

a

),(),()()()()()( 012

2

23

3

34

4

4 ω ω ω ω ω ω ω t W t W a

t

a

t

a

t

a

t

a f x =⎟⎟

⎠

⎞⎜⎜

⎝

⎛ +

∂

∂+

∂

∂+

∂

∂+

∂

∂

16/1)(

2

1)(

4

3

=

=

ω

μ ω

a

a

Time system

Time-frequency system

Impulse response



5353

t

ω

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

10

20

30

40

50

60

Impulse response

L. Galleani, “Response of dynamical systems to nonstationary inputs,” IEEE

Trans. Signal Process., vol. 60, no. 11, pp. 5775–5786, November 2012.

The Transient Spectrum

http://ieeexplore.ieee.org/xpl/articleDetails.jsp?tp=&arnumber=6272388&contentType=Journals+%26+Magazines&sortType%3Ddesc_p_Publication_Year%26searchField%3DSearch_All%26queryText%3Dgalleani




5454

p

L. Galleani, “The transient spectrum of a random system,” IEEE Trans.Signal Process., vol. 58, no. 10, pp. 5106–5117, October 2010.

Sliding Welch Spectrum 1/3





55


X-Ray binary star




56


Measured signal: photon arrival time

0

0.10.12

0.15...




57

g p

X-Ray data from the binary star XTE-J1550




58



59



60

Conclusions



61

Conclusions

• Time-frequency analysis characterizes signalswhose frequency content changes with time

• The perfect TF distribution does not exist

• We must choose the TFD for every signal: not sodifficult…

• We can study systems in the time-frequencydomain

• Better understanding of the system behavior,especially in nonstationary situations

Documents

Introduction to Timefrequencyanalysis Ssp&Mm 2013-14