Time-Frequency Representations

Time-FrequencyRepresentations

Marıa Eugenia TorresHugo L. RufinerDiego Milone

Analisis y Procesamiento Avanzado de Senales

Doctorado en Ingenierıa, FICH-UNLMaestrıa en Ingenierıa Biomedica, FI-UNER

Maestrıa en Computacion Aplicada a la Ciencia y la Ingenierıa, FICH-UNL

12 de junio de 2009

Introduction Classical TFRs Classes of TFRs Discrete calculations

Outline

Introduction

Classical Time-Frequency RepresentationsShort-Time Fourier TransformsWigner DistributionAltes Q DistributionCross-Term Problems

Classes of TFRsClasses of TFRs with Common PropertiesShift-covariant ClassAffine ClassHyperbolic and kth Power Classes

Discrete calculations of TFRs


Outline

Introduction





First... Fourier: the Fourier Transform (again!)

X(f) =

∫x(t)e−j2πftdt

x(t)←→ X(f)

x(t) =

∫X(f)ej2πftdf



X(f) =

∫x(t)e−j2πftdt

x(t)←→ X(f)

x(t) =

∫X(f)ej2πftdf



X(f) =

∫x(t)e−j2πftdt

x(t)←→ X(f)

x(t) =

∫X(f)ej2πftdf


The main lack of the Fourier analysis

Since sinusoidal basis functions are infinite durationwaveforms...

Fourier analysis implicitly assumes that each sinusoidalcomponent with nonzero weighting coefficient is always present

This is, that the spectral content of the signal under analysis isstationary












Time, frequency, Fourier and time-frequency


Instantaneous Frequency

Instantaneous Frequency is a time-varying frequency analysis...

The instantaneous frequency of a time-varying signal is definedas the instantaneous change in the phase of that signal

fx(t) =1

2πd

dtarg x(t)

For example, if x(t) = ej2π(αt)t, its instantaneous frequency is

fx(t) = 2αt





fx(t) =1

2πd

dtarg x(t)


fx(t) = 2αt





fx(t) =1

2πd

dtarg x(t)

For example, if x(t) = ej2π(αt)t, its instantaneous frequency is...?

fx(t) = 2αt





fx(t) =1

2πd

dtarg x(t)


fx(t) = 2αt



However, this time-varying spectral representations may becounterintuitive for multicomponent signals.

For example, from

y(t) = ej2πf1t + ej2πf2t

can be derived that

fy(t) =f1 + f2

2

(an interesting property...)



However, this time-varying spectral representations may becounterintuitive for multicomponent signals.For example, from


can be derived that

fy(t) =f1 + f2

2




However, this time-varying spectral representations may becounterintuitive for multicomponent signals.For example, from


can be derived that

fy(t) =f1 + f2

2




Now, what is the instantaneous frequency of

y(t) = ej2πf1t + ej2π(−f1)t . . .?

fy(t) =f1 − f1

2= 0

But note that

y(t) = cos(2πf1t) + cos(2π(−f1)t)= cos(2πf1t) + cos(2πf1t)= 2 cos(2πf1t)...

Then the real signal 2 cos(2πf1t) has an instantaneousfrequency equal to zero...?




y(t) = ej2πf1t + ej2π(−f1)t . . .?

fy(t) =f1 − f1

2= 0

But note that






y(t) = ej2πf1t + ej2π(−f1)t . . .?

fy(t) =f1 − f1

2= 0

But note that

y(t) = cos(2πf1t) + cos(2π(−f1)t)

= cos(2πf1t) + cos(2πf1t)= 2 cos(2πf1t)...





y(t) = ej2πf1t + ej2π(−f1)t . . .?

fy(t) =f1 − f1

2= 0

But note that






y(t) = ej2πf1t + ej2π(−f1)t . . .?

fy(t) =f1 − f1

2= 0

But note that




General Motivations

Nonstationary signals =⇒ time-varying spectral content.

¿How the spectral content of a signal is changing with time?

TFRs: one-dimensional signal x(t)↪→ two-dimensional function Tx(t, f)

Time-frequency plane: like a musical score...


General Motivations






General Motivations






General Motivations






Outline

Introduction





Short-Time Fourier Transforms (STFT)

Sx(t, f ; Γ) =∫x(τ)γ∗(τ − t)e−j2πfτdτ

= e−j2πtf∫X(ζ)Γ∗(ζ − f)ej2πtζdζ

Remarks...

• Sx(t, f ; Γ) is a linear function of x(t)• Typically, the analysis window, γ(t), is real and even• STFT can also be thought of as the temporal fluctuations

of the signal spectrum near the output frequency f .





Remarks...

• Sx(t, f ; Γ) is a linear function of x(t)• Typically, the analysis window, γ(t), is real and even• STFT can also be thought of as the temporal fluctuations






Remarks...• Sx(t, f ; Γ) is a linear function of x(t)

• Typically, the analysis window, γ(t), is real and even• STFT can also be thought of as the temporal fluctuations






Remarks...• Sx(t, f ; Γ) is a linear function of x(t)• Typically, the analysis window, γ(t), is real and even

• STFT can also be thought of as the temporal fluctuationsof the signal spectrum near the output frequency f .





Remarks...• Sx(t, f ; Γ) is a linear function of x(t)• Typically, the analysis window, γ(t), is real and even• STFT can also be thought of as the temporal fluctuations



STFT: the spectrogram

The STFT used for speech in the old analog analyzers wasoriginally known as “the spectrogram”

Gx(t, f ; Γ) = |Sx(t, f ; Γ)|2

=∣∣∣∣∫ x(τ)γ∗(τ − t)e−j2πfτdτ

∣∣∣∣2=

∣∣∣∣∫ X(ζ)Γ∗(ζ − f)ej2πtζdζ∣∣∣∣2

...still linear?


STFT: the spectrogram

The STFT used for speech in the old analog analyzers wasoriginally known as “the spectrogram”

Gx(t, f ; Γ) = |Sx(t, f ; Γ)|2

=∣∣∣∣∫ x(τ)γ∗(τ − t)e−j2πfτdτ

∣∣∣∣2=

∣∣∣∣∫ X(ζ)Γ∗(ζ − f)ej2πtζdζ∣∣∣∣2

...still linear?


STFT: example 1

Consider the case of x(t) = δ(t− t0)

Sx(t, f ; Γ) =∫δ(τ − t0)γ∗(τ − t)e−j2πfτdτ

= γ∗(t0 − t)e−j2πft0

Remarks...


STFT: example 1

Consider the case of x(t) = δ(t− t0)

Sx(t, f ; Γ) =∫δ(τ − t0)γ∗(τ − t)e−j2πfτdτ

= γ∗(t0 − t)e−j2πft0

Remarks...


STFT: example 2

Consider the case of Y (f) = δ(f − f0)

Sy(t, f ; Γ) = e−j2πtf∫δ(ζ − f0)Γ∗(ζ − f)ej2πtζdζ

= Γ∗(f0 − f)e−j2π(f−f0)t

Remarks...

Uncertainty Principle: Mertins 7.1.3


STFT: example 2

Consider the case of Y (f) = δ(f − f0)

Sy(t, f ; Γ) = e−j2πtf∫δ(ζ − f0)Γ∗(ζ − f)ej2πtζdζ

= Γ∗(f0 − f)e−j2π(f−f0)t

Remarks...

Uncertainty Principle: Mertins 7.1.3


STFT: main advantage

The STFT is a linear signal transformation

y(t) = αx1(t) + βx2(t)m

Sy(t, f ; Γ) = αSx1(t, f ; Γ) + βSx2(t, f ; Γ)


STFT: drawbacks

• STFT is complex-valued

• Long duration and wide bandwidth: producing a spreadingof the time-frequency support

• Tradeoff between time resolution and frequency resolution:achieving either good time resolution or good frequencyresolution but generally not both


STFT: drawbacks

• STFT is complex-valued• Long duration and wide bandwidth: producing a spreading

of the time-frequency support

• Tradeoff between time resolution and frequency resolution:achieving either good time resolution or good frequencyresolution but generally not both


STFT: drawbacks

• STFT is complex-valued• Long duration and wide bandwidth: producing a spreading

of the time-frequency support• Tradeoff between time resolution and frequency resolution:

achieving either good time resolution or good frequencyresolution but generally not both


STFT: example 3

Consider the two component signal

q(t) = δ(t− t0) + ej2πf0t

Using a Gaussian window

γ(t) =1√σe−π(t/σ)2

mΓ(f) =

√σe−π(σf)2

(i.e., the STFT becomes the Gabor transform)


STFT: example 3

Consider the two component signal

q(t) = δ(t− t0) + ej2πf0t

Using a Gaussian window

γ(t) =1√σe−π(t/σ)2

mΓ(f) =

√σe−π(σf)2

(i.e., the STFT becomes the Gabor transform)


STFT: example 3

It can be show that

Sq(t, f ; Γ) =1√σe−π(t−t0)2/σ2

e−j2πft0 +

√σe−πσ

2(f−f0)2e−j2π(f−f0)t

Remarks...


Wigner Distribution (WD)

Wx(t, f) =∫x(t+

τ

2

)x∗(t− τ

2

)e−j2πfτdτ

=∫X

(f +

ζ

2

)X∗(f − ζ

2

)ej2πtζdζ


WD: the support of x(s+ t/2)x(s− t/2)

What is the support if the WD don’t have a window?



Imagining that a signal’s support lies mainly within the interval[−a, a]...










WD: propertiesx(t) Wx(f, t)

Proofs: Boudreaux-Bartels 12.3.2 & Allen 10.4.2.2



Proofs: Boudreaux-Bartels 12.3.2 & Allen 10.4.2.2




The Ambiguity Function (AF):


AF: Autocorrelation

The autocorrelation function

rx(τ) =∫x(t+ τ)x∗(t)dt

can also be understood as the inverse Fourier transform of theenergy density spectrum

rx(τ) =1

2π

∫X(ω)X∗(ω)ejωτdω


AF: Autocorrelation

For the frequency-shifted signals we have

ρx(ζ) =∫x(t)x∗(t)ejζtdt

and

ρx(ζ) =1

2π

∫X(ω)X∗(ω + ζ)dω

in the frequency domain.


AF: Autocorrelation

For the frequency-shifted signals we have

ρx(ζ) =∫x(t)x∗(t)ejζtdt

and

ρx(ζ) =1

2π

∫X(ω)X∗(ω + ζ)dω

in the frequency domain.


The Ambiguity Function

Considering the signals

x(t− τ/2)ej(−ζ/2)t

and

x(t+ τ/2)ej(ζ/2)t

as time and frequency shifted versions of one other, centredaround x(t)...



We have the time-frequency autocorrelation function orAmbiguity Function

Ax(τ, ζ) =∫x(t+

τ

2

)x∗(t− τ

2

)e−jζtdt

and in the frequency domain (via the Parseval’s relation)

Ax(τ, ζ) =∫X

(ω − ζ

2

)X∗(ω +

ζ

2

)ejωτdω



We have the time-frequency autocorrelation function orAmbiguity Function

Ax(τ, ζ) =∫x(t+

τ

2

)x∗(t− τ

2

)e−jζtdt

and in the frequency domain (via the Parseval’s relation)

Ax(τ, ζ) =∫X

(ω − ζ

2

)X∗(ω +

ζ

2

)ejωτdω


Wigner Distribution and Ambiguity Function

By setting ζ = 0 we have

rx(τ) = Ax(τ, 0)

and the energy density spectrum

Sx(ω) =∫Ax(τ, 0)e−jωτdτ



By setting ζ = 0 we have

rx(τ) = Ax(τ, 0)

and the energy density spectrum

Sx(ω) =∫Ax(τ, 0)e−jωτdτ



For the autocorrelation function of the spectrum, with τ = 0 wehave

ρx(ζ) = Ax(0, ζ)

obtaining the temporal energy density

sx(t) =∫Ax(0, ζ)e−jζtdζ



For the autocorrelation function of the spectrum, with τ = 0 wehave

ρx(ζ) = Ax(0, ζ)

obtaining the temporal energy density

sx(t) =∫Ax(0, ζ)e−jζtdζ



The transform with respect to ζ yields the temporalautocorrelation function

φx(t, τ) =1

2π

∫Ax(τ, ζ)e−jζtdζ

= x(t+

τ

2

)x∗(t− τ

2

)

and thus the two-dimensional Fourier transform of Ax(τ, ζ) is

Wx(t, ω) =1

2π

∫ ∫Ax(τ, ζ)e−jωτe−jζtdτdζ



The transform with respect to ζ yields the temporalautocorrelation function

φx(t, τ) =1

2π

∫Ax(τ, ζ)e−jζtdζ

= x(t+

τ

2

)x∗(t− τ

2

)and thus the two-dimensional Fourier transform of Ax(τ, ζ) is

Wx(t, ω) =1

2π

∫ ∫Ax(τ, ζ)e−jωτe−jζtdτdζ


WD: the Wigner-Ville Distribution

In order to gain information on the stochastic process we definethe Wigner-Ville spectrum as the expected value of the Wignerdistribution:

Vx(t, ω) = E [Wx(t, ω)]

=∫rxx

(t+

τ

2, t− τ

2

)e−jωτdτ

where

rxx

(t+

τ

2, t− τ

2

)= E [φx(t, τ)] = E

[x(t+

τ

2

)x∗(t− τ

2

)]


WD: the Wigner-Ville Distribution

In order to gain information on the stochastic process we definethe Wigner-Ville spectrum as the expected value of the Wignerdistribution:

Vx(t, ω) = E [Wx(t, ω)]

=∫rxx

(t+

τ

2, t− τ

2

)e−jωτdτ

where

rxx

(t+

τ

2, t− τ

2

)= E [φx(t, τ)] = E

[x(t+

τ

2

)x∗(t− τ

2

)]


Altes Q Distribution

The “wideband” version of the Wigner Distribution,

Qx(t, f) = f

∫X(feu/2

)X∗(fe−u/2

)ej2πtfudu, f > 0

or the “scale-invariant” Wigner Distribution,

Qx(t, c) = f

∫x(teσ/2

)x∗(te−σ/2

)e−j2πcσdσ

by using the time-domain version of the signal x(t).




Qx(t, f) = f

∫X(feu/2

)X∗(fe−u/2

)ej2πtfudu, f > 0


Qx(t, c) = f

∫x(teσ/2

)x∗(te−σ/2

)e−j2πcσdσ





Qx(t, f) = f

∫X(feu/2

)X∗(fe−u/2

)ej2πtfudu, f > 0


Qx(t, c) = f

∫x(teσ/2

)x∗(te−σ/2

)e−j2πcσdσ




Altes Q distribution was proposed for analyzing signals thathave undergone compressions or dilations.

The Altes Q and the Wigner distributions are warped versionsof each other

QX(t, f) = WX+∗

(tf

fr, fr ln

f

fr

)WX(t, f) = QX−∗

(te−f/fr , fre

f/fr)

where the signal is first prewarped for the reference frequency fr:

X+∗(f) =√ef/frX

`fref/fr

´and X−∗(f) =

qfrfX“fr ln f

fr

”, f > 0



Altes Q distribution was proposed for analyzing signals thathave undergone compressions or dilations.

The Altes Q and the Wigner distributions are warped versionsof each other

QX(t, f) = WX+∗

(tf

fr, fr ln

f

fr

)WX(t, f) = QX−∗

(te−f/fr , fre

f/fr)

where the signal is first prewarped for the reference frequency fr:

X+∗(f) =√ef/frX

`fref/fr

´and X−∗(f) =

qfrfX“fr ln f

fr

”, f > 0


Cross-Terms on Quadratic TFR

Consider the nonlinear operation

|x(t) + y(t)|2 = |x(t)|2 + |y(t)|2 + 2R{x(t)y∗(t)}

In a more genera case, consider a multicomponent signal

y(t) =N∑i=1

xi(t)




|x(t) + y(t)|2 = |x(t)|2 + |y(t)|2 + 2R{x(t)y∗(t)}


y(t) =N∑i=1

xi(t)




|x(t) + y(t)|2 = |x(t)|2 + |y(t)|2 + 2R{x(t)y∗(t)}


y(t) =N∑i=1

xi(t)



The WD of this signal is

Wy(t, f) =N∑i=1

Wxi(t, f) + 2N−1∑i=1

N∑k=i+1

R{Wxixk(t, f)}

where

Wxixk(t, f) =∫xi

(t+

τ

2

)x∗k

(t− τ

2

)e−j2πfτdτ



The WD of this signal is

Wy(t, f) =N∑i=1

Wxi(t, f) + 2N−1∑i=1

N∑k=i+1

R{Wxixk(t, f)}

where

Wxixk(t, f) =∫xi

(t+

τ

2

)x∗k

(t− τ

2

)e−j2πfτdτ


Cross-Terms: example 1

For example, let be

xi(t) = x(t− ti)ej2πfit

using the properties of the WD we have

Wy(t, f) =

N∑i=1

Wx(t− ti, f − fi) + . . .

+ 2N−1∑i=1

N∑k=i+1

Wx

(t− ti + tk

2, f − fi + fk

2

)× . . .

× cos(

(fi − fk)t− (ti − tk)f +fi + fk

2(ti − tk)

)



For example, let be



Wy(t, f) =N∑i=1

Wx(t− ti, f − fi) + . . .

+ 2N−1∑i=1

N∑k=i+1

Wx

(t− ti + tk

2, f − fi + fk

2

)× . . .

× cos(


2(ti − tk)

)



For example, let be



Wy(t, f) =N∑i=1

Wx(t− ti, f − fi)

+ . . .

+ 2N−1∑i=1

N∑k=i+1

Wx

(t− ti + tk

2, f − fi + fk

2

)× . . .

× cos(


2(ti − tk)

)



For example, let be



Wy(t, f) =N∑i=1

Wx(t− ti, f − fi)

+ . . .

+ 2N−1∑i=1

N∑k=i+1

Wx

(t− ti + tk

2, f − fi + fk

2

)

× . . .

× cos(


2(ti − tk)

)






Outline

Introduction





Classes of TFRs

To understand the relative advantages of TFRs it is usefull togroup them into classes.

Each class is defined by the a few “ideal” TFR properties thatall members must satisfy.

Some of such classes may be:• Shift-covariant or Cohen’s class• Affine class• Hyperbolic class• Power class


Classes of TFRs





Classes of TFRs





Common Properties 1/2


Common Properties 2/2


Shift-covariant (Cohen’s) Class: definition

Cc ={Tx(t, f)

∣∣∣y(t) = x(t− t0)ej2πf0t

⇒ Ty(t, f) = Tx(t− t0, f − f0)}



Cc ={Tx(t, f)


⇒ Ty(t, f) = Tx(t− t0, f − f0)}



Cc ={Tx(t, f)


⇒ Ty(t, f) = Tx(t− t0, f − f0)}


Cc: Alternative Formulations

Any TFR within the Cohen’s class can be written in one of thefollowing equivalent “Normal Forms”

Cx(t, f ; Ψc)

=∫ ∫

φc(t− ς, τ)x(ς +

τ

2

)x∗(ς − τ

2

)e−j2πfτdςdτ

=∫ ∫

Φc(f − %, ζ)X(%+

ζ

2

)X∗(%− ζ

2

)e−j2πtζd%dζ

=∫ ∫

ψc(t− ς, f − %)Wx(ς, %)dςd%

=∫ ∫

Ψc(τ, ζ)Ax(τ, ζ)e−j2π(tζ−fτ)dτdζ

=∫ ∫

Υc(f − f1, f − f2)X(f1)X∗(f2)e−j2π(f1−f2)tdf1df2

where the kernels are interrelated by...




Cx(t, f ; Ψc) =∫ ∫


τ

2

)x∗(ς − τ

2

)e−j2πfτdςdτ

=∫ ∫

Φc(f − %, ζ)X(%+

ζ

2

)X∗(%− ζ

2

)e−j2πtζd%dζ

=∫ ∫


=∫ ∫


=∫ ∫






Cx(t, f ; Ψc) =∫ ∫


τ

2

)x∗(ς − τ

2

)e−j2πfτdςdτ

=∫ ∫

Φc(f − %, ζ)X(%+

ζ

2

)X∗(%− ζ

2

)e−j2πtζd%dζ

=∫ ∫


=∫ ∫


=∫ ∫






Cx(t, f ; Ψc) =∫ ∫


τ

2

)x∗(ς − τ

2

)e−j2πfτdςdτ

=∫ ∫

Φc(f − %, ζ)X(%+

ζ

2

)X∗(%− ζ

2

)e−j2πtζd%dζ

=∫ ∫


=∫ ∫


=∫ ∫






Cx(t, f ; Ψc) =∫ ∫


τ

2

)x∗(ς − τ

2

)e−j2πfτdςdτ

=∫ ∫

Φc(f − %, ζ)X(%+

ζ

2

)X∗(%− ζ

2

)e−j2πtζd%dζ

=∫ ∫


=∫ ∫


=∫ ∫






Cx(t, f ; Ψc) =∫ ∫


τ

2

)x∗(ς − τ

2

)e−j2πfτdςdτ

=∫ ∫

Φc(f − %, ζ)X(%+

ζ

2

)X∗(%− ζ

2

)e−j2πtζd%dζ

=∫ ∫


=∫ ∫


=∫ ∫





φc(t, τ) =∫ ∫

Φc(f, ζ)e−j2π(fτ+tζ)dfdζ

=∫

Ψc(τ, ζ)e−j2πζtdζ

F←→ Φc(f, ζ)



φc(t, τ) =∫ ∫


=∫


F←→ Φc(f, ζ)



φc(t, τ) =∫ ∫


=∫


F←→ Φc(f, ζ)



ψc(t, f) =∫ ∫

Ψc(τ, ζ)e−j2π(tζ−fτ)dfdζ

=∫

Φc(f, ζ)e−j2πζtdζ

F←→ Ψc(τ, ζ)

and

Υc(f1, f2) = Φc

(f1 + f2

2, f2 − f1

)



ψc(t, f) =∫ ∫


=∫


F←→ Ψc(τ, ζ)

and

Υc(f1, f2) = Φc

(f1 + f2

2, f2 − f1

)



ψc(t, f) =∫ ∫


=∫


F←→ Ψc(τ, ζ)

and

Υc(f1, f2) = Φc

(f1 + f2

2, f2 − f1

)



ψc(t, f) =∫ ∫


=∫


F←→ Ψc(τ, ζ)

and

Υc(f1, f2) = Φc

(f1 + f2

2, f2 − f1

)


TRFs into the Cohen’s Class (1/3)






Affine Class: definition

Ac ={Tx(t, f)

∣∣∣y(t) =√|a|x (a(t− t0))

⇒ Ty(t, f) = Tx(a(t− t0), fa

)}



Ac ={Tx(t, f)

∣∣∣y(t) =√|a|x (a(t− t0))

⇒ Ty(t, f) = Tx(a(t− t0), fa

)}



Ac ={Tx(t, f)

∣∣∣y(t) =√|a|x (a(t− t0))

⇒ Ty(t, f) = Tx(a(t− t0), fa

)}


Ac: Alternative Formulations

Ax(t, f ; Ψa)

= |f |∫ ∫

φa(f(t− ς), fτ)x(ς +

τ

2

)x∗(ς − τ

2

)e−j2πfτdςdτ

=1|f |

∫ ∫Φa

(%

f,ζ

f

)X

(%+

ζ

2

)X∗(%− ζ

2

)e−j2πtζd%dζ

=∫ ∫

ψa

(f(t− ς),− %

f

)Wx(ς, %)dςd%

=∫ ∫

Ψa

(fτ,

ζ

f

)Ax(τ, ζ)e−j2π(tζ−fτ)dτdζ

=1|f |

∫ ∫Υa

(f1f,f2f

)X(f1)X∗(f2)e−j2π(f1−f2)tdf1df2




Ax(t, f ; Ψa) = |f |∫ ∫


τ

2

)x∗(ς − τ

2

)e−j2πfτdςdτ

=1|f |

∫ ∫Φa

(%

f,ζ

f

)X

(%+

ζ

2

)X∗(%− ζ

2

)e−j2πtζd%dζ

=∫ ∫

ψa

(f(t− ς),− %

f

)Wx(ς, %)dςd%

=∫ ∫

Ψa

(fτ,

ζ

f


=1|f |

∫ ∫Υa

(f1f,f2f





Ax(t, f ; Ψa) = |f |∫ ∫


τ

2

)x∗(ς − τ

2

)e−j2πfτdςdτ

=1|f |

∫ ∫Φa

(%

f,ζ

f

)X

(%+

ζ

2

)X∗(%− ζ

2

)e−j2πtζd%dζ

=∫ ∫

ψa

(f(t− ς),− %

f

)Wx(ς, %)dςd%

=∫ ∫

Ψa

(fτ,

ζ

f


=1|f |

∫ ∫Υa

(f1f,f2f





Ax(t, f ; Ψa) = |f |∫ ∫


τ

2

)x∗(ς − τ

2

)e−j2πfτdςdτ

=1|f |

∫ ∫Φa

(%

f,ζ

f

)X

(%+

ζ

2

)X∗(%− ζ

2

)e−j2πtζd%dζ

=∫ ∫

ψa

(f(t− ς),− %

f

)Wx(ς, %)dςd%

=∫ ∫

Ψa

(fτ,

ζ

f


=1|f |

∫ ∫Υa

(f1f,f2f





Ax(t, f ; Ψa) = |f |∫ ∫


τ

2

)x∗(ς − τ

2

)e−j2πfτdςdτ

=1|f |

∫ ∫Φa

(%

f,ζ

f

)X

(%+

ζ

2

)X∗(%− ζ

2

)e−j2πtζd%dζ

=∫ ∫

ψa

(f(t− ς),− %

f

)Wx(ς, %)dςd%

=∫ ∫

Ψa

(fτ,

ζ

f


=1|f |

∫ ∫Υa

(f1f,f2f





Ax(t, f ; Ψa) = |f |∫ ∫


τ

2

)x∗(ς − τ

2

)e−j2πfτdςdτ

=1|f |

∫ ∫Φa

(%

f,ζ

f

)X

(%+

ζ

2

)X∗(%− ζ

2

)e−j2πtζd%dζ

=∫ ∫

ψa

(f(t− ς),− %

f

)Wx(ς, %)dςd%

=∫ ∫

Ψa

(fτ,

ζ

f


=1|f |

∫ ∫Υa

(f1f,f2f





φa(c, ν) F←→ Φa(b, β)

ψa(c, b)F←→ Ψa(ν, β)

and

Υa(b1, b2) = Φc

(−b1 + b2

2, b1 − b2

)





and

Υa(b1, b2) = Φc

(−b1 + b2

2, b1 − b2

)





and

Υa(b1, b2) = Φc

(−b1 + b2

2, b1 − b2

)


TRFs into the Affine Class (1/2)


TRFs into the Affine Class (2/2)


Affine-Cohen Subclass: definition

ACc ={Tx(t, f)

∣∣∣y(t) =√|a|x (a(t− t0)) ej2πf0t

⇒ Ty(t, f) = Tx(a(t− t0), f−f0

a

)}



ACc ={Tx(t, f)

∣∣∣y(t) =√|a|x (a(t− t0)) ej2πf0t

⇒ Ty(t, f) = Tx(a(t− t0), f−f0

a

)}



ACc ={Tx(t, f)

∣∣∣y(t) =√|a|x (a(t− t0)) ej2πf0t

⇒ Ty(t, f) = Tx(a(t− t0), f−f0

a

)}


TRFs into the Affine-Cohen Subclass


Hyperbolic Class: definition

Hc =

{Tx(t, f)

∣∣∣∣∣Y (f) = 1√|a|X(fa

)1√fej2πc ln(f/fr)

⇒ Ty(t, f) = Tx

(a(t− c

f

), fa

)}



Hc =

{Tx(t, f)

∣∣∣∣∣Y (f) = 1√|a|X(fa


⇒ Ty(t, f) = Tx

(a(t− c

f

), fa

)}



Hc =

{Tx(t, f)

∣∣∣∣∣Y (f) = 1√|a|X(fa


⇒ Ty(t, f) = Tx

(a(t− c

f

), fa

)}


Affine-Hyperbolic Subclass: definition

AHc =

{Tx(t, f)

∣∣∣∣∣Y (f) = 1√|a|X(fa

)ej2πc ln(f/fr)ej2πft0

⇒ Ty(t, f) = Tx

(a(t− t0 − c

f

), fa

)}



AHc =

{Tx(t, f)

∣∣∣∣∣Y (f) = 1√|a|X(fa


⇒ Ty(t, f) = Tx

(a(t− t0 − c

f

), fa

)}



AHc =

{Tx(t, f)

∣∣∣∣∣Y (f) = 1√|a|X(fa


⇒ Ty(t, f) = Tx

(a(t− t0 − c

f

), fa

)}


κth Power Class: definition

Pκc =

{Tx(t, f)

∣∣∣∣∣Y (f) = 1√|a|X(fa

)esgn(f)j2πc|(f/fr)|κ

⇒ Ty(t, f) = Tx

(a

(t− κ

fr

∣∣∣ ffr ∣∣∣κ−1), fa

)}



Pκc =

{Tx(t, f)

∣∣∣∣∣Y (f) = 1√|a|X(fa


⇒ Ty(t, f) = Tx

(a

(t− κ

fr

∣∣∣ ffr ∣∣∣κ−1), fa

)}



Pκc =

{Tx(t, f)

∣∣∣∣∣Y (f) = 1√|a|X(fa


⇒ Ty(t, f) = Tx

(a

(t− κ

fr

∣∣∣ ffr ∣∣∣κ−1), fa

)}


κth Power Class: example

Let be

ξ(f) = sgn(f)|f |κ, κ 6= 0

andY (f) = X(f)e−j2πcξ(f/fr)

For a TFR in the κth power class we have

Pκy(t, f) = Pκx

(t− c κ

fr

∣∣∣∣ ffr∣∣∣∣κ−1

, f

)

= Pκx

(t− c d

dfξ(f/fr), f

)



Let be

ξ(f) = sgn(f)|f |κ, κ 6= 0



Pκy(t, f) = Pκx

(t− c κ

fr

∣∣∣∣ ffr∣∣∣∣κ−1

, f

)

= Pκx

(t− c d

dfξ(f/fr), f

)



Let be

ξ(f) = sgn(f)|f |κ, κ 6= 0



Pκy(t, f) = Pκx

(t− c κ

fr

∣∣∣∣ ffr∣∣∣∣κ−1

, f

)

= Pκx

(t− c d

dfξ(f/fr), f

)



Let be

ξ(f) = sgn(f)|f |κ, κ 6= 0



Pκy(t, f) = Pκx

(t− c κ

fr

∣∣∣∣ ffr∣∣∣∣κ−1

, f

)

= Pκx

(t− c d

dfξ(f/fr), f

)



Let be

ξ(f) = sgn(f)|f |κ, κ 6= 0



Pκy(t, f) = Pκx

(t− c κ

fr

∣∣∣∣ ffr∣∣∣∣κ−1

, f

)

= Pκx

(t− c d

dfξ(f/fr), f

)


Classes and its Common Properties


Outline

Introduction






Sampling the signal...

...sampling the kernel...

... and defining discrete-timeand discrete-frequency distributions.

















Discrete-time Wigner Distribution

Wx(n, ejω) = 2

∑m

x(n+m)x∗(n−m)e−j2ωm



We know that for discrete signal x(n)

X(ejω) = X(ejω+k2π), k ∈ Z

However for the discrete-time WD we have the property

Wx(n, ejω) = Wx(n, ejω+kπ)

why...?



We know that for discrete signal x(n)

X(ejω) = X(ejω+k2π), k ∈ Z

However for the discrete-time WD we have the property

Wx(n, ejω) = Wx(n, ejω+kπ)

why...?


Discrete-Time Wigner Distribution

Wx(n, ejω) = 2∑m

x(n+m)x∗(n−m)e−j2ωm

Now, to avoid the aliasing in the discrete-time WD we need

fs ≥ 4fmax


General Discrete-Time TFRs

The general discrete-time time-frequency distribution ofCohen’s class is defined as

Cx(n, k) =M∑

m=−M

N∑`=−N

φ(`,m)x(`+n+m)x∗(`+n−m)e−j4πkm/L

...considering only the discrete frequencies up to 2πk/L, with Lthe DFT length.


General Discrete-Time TFRs

The general discrete-time time-frequency distribution ofCohen’s class is defined as

Cx(n, k) =M∑

m=−M

N∑`=−N

φ(`,m)x(`+n+m)x∗(`+n−m)e−j4πkm/L

...considering only the discrete frequencies up to 2πk/L, with Lthe DFT length.


The Discrete-Time Choi-Williams TFR

φCW (n,m) =

{1

|m|αm e−σn2/4m2

, m 6= 0

δ(n), m = 0

with

αm =N∑

k=−N

1|m|

e−σn2/4m2



φCW (n,m) =

{1

|m|αm e−σn2/4m2

, m 6= 0

δ(n), m = 0

with

αm =N∑

k=−N

1|m|

e−σn2/4m2



With the normalization∑

n φ(n,m) = 1 the followinginteresting properties are preserved

∑n

CWx(n, k) = |X(k)|2

and ∑k

CWx(n, k) = |x(n)|2


Bibliography

[1] A. Mertins, Signal Analysis: Wavelets, Filter Banks,Time-Frequency Transforms and Applications, John Wiley &Sons, 1999.

[2] R. Allen, D. Mills, Signal Analysis: Time, Frequency, Scale andStructure, IEEE Press, John Wiley & Sons, 2004.

[3] G. Boudreaux-Bartels, Mixed Time-Frequency SignalTransformations, Chapter 12 in: Poularikas A. (Editor), TheTransforms and Applications Handbook, CRC Press, 2000.

[4] J. Jeong and W. J. Williams, “Alias-free generalizeddiscrete-time time-frequency distributions,” IEEE Trans. SignalProcessing, vol. 40, pp. 2757-2765, Nov. 1992.

[5] L. Cohen, “Time-frequency distributions – A review,” Proc.IEEE, vol. 77, pp. 941-981, July 1989.

Documents

Time-Frequency Representations