On Quality Criteria for On Quality Criteria for Time-Varying FiltersTime-Varying Filters

By: Lior Assouline

Supervisor: Dr. Moshe Porat

Agenda

Motivation (The need for Quality Criteria).The Time-Frequency (TF) filtering problem.Short theoretical introduction to TF Filter analysis.Existing solutions.Proposed solutions.Validation and Simulations.

Motivation

In recent years there have been considerable strides made in developing a combined TF description of signals by the use of joint time-frequency distributions. This development offered the possibility of generalizing the concepts and methods of classical filtering theory to the combined TF domain.

The Problem

In many applications (e.g. Seismic analysis, Evoked potentials, E.C.G/E.E.G. analysis etc.) it is desirable to filter a signal such that its components inside given time-frequency regions are specifically weighted.

Fre

qu

ency

T im e

1

0

0.5

The TF Filtering Model M(t,f)

The Problem (cont’d)

Time-Varying filtering presents unique difficulties that have not been fully overcome:

The large variety of filtering methods is due in part to the multiple Time-Frequency representation methods, each with its own merits and drawbacks.

There is still no accepted way to determine which filtering method to use for a specific filtering task, nor is there a way to find the optimal scheme suitable for a certain filtering task.

Time-Frequency Representations and Operator Symbols

STFT

Wigner Distribution (WD)

We can generalize both representations as an inner product of the signal and a symmetric/asymmetric TF shifted version of the signal itself (WD) or a normalized window (STFT)

detxftSTFT fjtx 2*,

detxtxftWD fj

tx2*

22,

Time-Frequency Representations and Operator Symbols (cont’d)

The input-output relation of any linear, time-varying (LTV) filter H is:

where is the impulse response of H.

Weyl Symbol: Generalization of Fourier using the Heisenberg Group Theory for LTV systems.

detthftL fjH

2

2,

2,

,,

dxthtytHx

,th

Time-Frequency Representations and Operator Symbols (cont’d)

Spreading Function: The 2D Fourier of Weyl Symbol.

The Weyl Symbol can be interpreted as a TF transfer function of the operator H with certain limitations due to the uncertainty principle.

The Spreading Function can be interpreted as the amount of potential time and frequency shifts caused by the linear operator.

ftLFFS HftH ,, 1

LTV Filter Design MethodsEvery linear filter can be represented as .

Current LTV Filter design approaches are based primarily on three main methods:

1. Composition of timepass and bandpass filters.2. STFT: Restriction of the reproducing formulas

based on coherent state expansions.3. Weyl: pseudodifferential operators with

symbols of compact support.

,th

STFT based methods

Calculate the STFT of a signal :

Multiply the STFT by the mask :

Synthesize the output signal from the masked STFT :

The resultant impulse response is

STFT ANALYSISusing window l(t)

STFTSYNTHESIS

using window g(t)

MultiplicativeModification

M(t,f)

x(t)... ...

y(t)

''','

'2* dtettltxftXt

ftjtl

),(,,~

ftMftXftX

' '

'2~

'''','f t

tfj dtdfettgftXty

f t

ttfj tdfdeftlttgftMtth )(2,),()',(

tx

ty ftX ,~

ftM ,

Optimal Window STFT (Kozek-92)

Matching the analysis and synthesis windows to the filtering model .

Optimal diagonalization of the operator via a Weyl-Heisenberg matched signal set.

windowtheofFunctionAmbiguityA

ftMModeltheofFunctionSpreadingSF

ASF

M

Mopt

,

,,

1,,maxarg2

ftM ,

Weyl Correspondence based methods

Based on the interpretation of the Weyl Symbol (WS) as a TF transfer function: We construct a filter H such that its WS is the Model ,

),(, ftMftLH

dfeftt

Mtthf

ttfj

2,2

),(

where,

ftM ,

TF Weighting filter

Weyl filter has a substantial amount of TF energy displacements.A constraint on H to be a positive semi-definite operator leads to a minimization problem:

PHH

HHLMH minarg

TF Weighting filter (cont’d)

Algorithm (Hlawatsch 94)Calculate as in a Weyl filter.Calculate the eigenvalues and eigenfunctions of the filter . The TF Weighting filter (TFWF) is given by tututth kkk

k

*

0

),(

tukk

', tth

', tth

Generalized LTV Filter

The Weyl Symbol of the STFT filter is

i.e., the Model smoothed by the Cross Wigner Distribution (CWD) of the windows used in the analysis and synthesis of the STFT.

This suggests a generalized filter where the STFT and the Weyl based methods are special cases.

),(**),(),( , ftWDftMftL glH

Generalized LTV Filter (cont’d)

Proposed algorithm 2D Convolve with an arbitrary (or part of a family of a) smoothing function to yield ,

Calculate from the modified as in a Weyl filter:

),(~

ftM

),( ftM

),(~

ftM

dfeftt

Mtthf

ttfj

2,2

),(

ftS ,

),(**),(),(~

ftSftMftM

', tth

Generalized LTV Filter (cont’d)

Properties:The proposed filter is related to the STFT and the

Weyl filters:The STFT filter results fromThe Weyl filter results from

is a 2D smoothing function. Using smoothing functions along the continuum

between the two extreme cases of a physical window (Heisenberg cell) and an unrealizable Dirac TF function ( ) yields a new family of filters.

This resultant family of filters has smooth transition between the extreme properties of STFT and Weyl filters, as will be shown.

),(),( , ftWDftS gl ftftS ),(

),( ftS

ft

Current LTV Filter Quality Criteria - SNR improvement (Kozek-92)

SNR improvement :SNR_in = SNR_out = SNR_improvement is based on the difference between SNR_out

and SNR_in.

Problems:It is not clear if a small SNR improvement is due to low immunity to noise or high distortion of the internal signal.A (internal) test signal must be found: Signal synthesis is an as yet unsolved problem.

)(/)( tnts)()(~/)( tststs

noisetn

signaloutputts

signalinputts

~

Current LTV Filters Quality Criteria (cont’d), Dubiner - 97

Dubiner’s Max/Mean criteria is based on the fact that an internal (to the passing area) signal should be passed undistorted, and an external signal should be blocked.

Problems:This QC is suitable only for rectangular areas TF filters

Proposed Quality Criterion

Mean Distortion Error (MDE-QC)

Eigenvalues analysis of a self-adjoint operator

Self-Adjoint operators can be decomposed into real eigenvalues and their corresponding eigenfunctions:

The unitary property of WD enables an interpretation of concentration for the eigenvalues

tututth kk

kk *,

kHuu ftWSWDkk

),(,,

Positive and negative eigenvalues

Positive eigenvalues correspond to eigenfunctions whose TF support is inside the model, defined as weighted functions:

Negative eigenvalues correspond to eigenfunctions that are passed with inverted phase, defined as corrections:

tututtH kIk

kkWeighting

*,

tututtH kIk

kkCorrection

*,

Positive and negative eigenvalues (cont’d)

Negative eigenvalues appear when trying to filter an area sharply localized or smaller than Heisenberg cell.

This is a natural manifestation of the Heisenberg Uncertainty principle.

The proposed criterion

The QC main idea: Measure distortion i.e. difference

between resulting weighting operator and the required TF model .

Penalty for corrections induced by the operator .

We can see a filtering process as projecting a signal onto the eigenfunctions linear space. It is therefore of interest to determine the TF support of these eigenfunctions.

ttHWeighting ,

ttH sCorrection ,

),( ftM

Definition (Hlawatsch-91) : The Wigner Space (WSP) of an operator H is defined as:

where are the eigenvalues and eigenfunctions of the operator H.

The WSP is an approximate TF transfer function.

,,, , ftWDftWSPk

uukH kk

kk u,

Wigner space of an operator

MDE-QC

The Mean Distortion Error QC (MDE-QC) is defined as

where is the target TF filter model.

,,,,22

ftWSPftWSPftMHQCCorrectingWeighting HHM

),( ftM

Frequency

Ma

gn

itu

de

QC-MDE (cont’d)

This criterion measures the fidelity of the resulting linear filter with respect to the TF weighting specification .

RippleTF

H

onLocalizatiTF

HM ftWSPftWSPftMHQCCorrectingWeighting

22

,,,

This criterion is a natural generalization of the LTI case.

),( ftM

Quality Criteria - Analysis and Comparison

The MDE-QC is a suitable quality criterion:It is signal independent (unlike existing quality criteria).It permits an arbitrarily shaped TF weighting filter analysis (unlike most of the existing quality criteria).It permits an arbitrary weight specification of the TF location error (unlike SNR criteria).

Quality Criteria - Analysis and Comparison (cont’d)

It is independent of the filter implementation method (like most other methods)It accounts for the inherent tradeoff in LTV filtering: TF localization vs. TF ripple.It is a generalization of the LTI filter theory.

Quality Criteria - Analysis and Comparison (cont’d)

The choice of the WD is optimal in the sense of resolution.Linear combination of weighted WD diminishes the effect of cross-terms.

However, due to practical considerations (finite number of eigenfunctions) minimal smoothing is required for enhanced analysis.

ftWDftWSkk uu

kkH ,, ,

Upper bounds on QC values

For STFT based filters:

For Weyl based filters:

ddASHMQC MSTFTM t

,1,),( ,

ddSFddSFftWSPftM MMHH,,2,,2

Optimal filter

We can search over a family of smoothing functions to obtain the lowest value for the MDE-QC. This family of filters is defined here as GTFF (Generalized TF filter):

where is the model smoothed by .

,

~,minarg 1

,S

ftSGTFF MWSMQCH

ftM S ,~

ftS ,

ftS ,

),( ftM

Optimal filter (cont’d)

Unlike previously proposed filters such as STFT (Daubechies-88) and Weyl (Hlawatch-92), which use extreme cases of a smoothing function, the solution here is superior to both by selecting a smoothing function that suits the model and the user’s choice of TF localization and ripple errors.

ftS ,

LTV filtering comparisonThe filtering model (a) and MDE-QC weighting part (b,c,d,e) for various filtering methods.

0

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1TF model

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Weyl

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STFT

TimeF

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1.5

TF Weighting filter

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Opt. Window STFT

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(a)

(b) (c)

(d) (e)

LTV filtering comparison (cont’d)

STFT Opt. Window STFT Weyl TF Weighting

0.25

0.3

0.35

0.4

0.45

0.5

0.55

MDE-QC results

NM

SE

GTFFOGTFF synthesis and eigenvalue analysis

Signal Pass validation experiments

Validation of MDE-QC with SPNS-QC (STFT representation)

Original Signal

Time

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10 20 30 40 50 60-0.5

0

0.5STFT: PASS-QC 0.00711

Time

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10 20 30 40 50 60-0.5

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0.5OWSTFT: PASS-QC 0.00713

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Weyl: PASS-QC 0.00561

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0.5TFWF: PASS-QC 0.00536

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10 20 30 40 50 60-0.5

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0.5OGTFF: PASS-QC 0.00658

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(a) (b) (c)

(d) (e) (f)

Signal Pass validation experiments

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4Original Signal

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2.5OWSTFT: PASS-QC 0.00713

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2.5Weyl: PASS-QC 0.00561

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2.5OGTFF: PASS-QC 0.00658

Time

(a) (b) (c)

(d) (e) (f)

Noise Stop validation experimentsValidation of MDE-QC with SPNS-QC (STFT representation)

Test Noise

Time

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0.5STFT: STOP-QC 0.00087

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0.5OWSTFT: STOP-QC 0.00087

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Weyl: STOP-QC 0.00113

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0.5TFWF: STOP-QC 0.00138

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0.5OGTFF: STOP-QC 0.00095

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(a) (b) (c)

(d) (e) (f)

Noise Stop validation experiments

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4Test Noise

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1Weyl: STOP-QC 0.00114

Time0 20 40 60 80

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0.8TFWF: STOP-QC 0.00138

Time0 20 40 60 80

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1OGTFF: STOP-QC 0.00094

Time

(a) (b) (c)

(d) (e) (f)

Signal Pass/Noise Stop validation experimentsValidation experiments showing the relative quality of the filters using SPNS-QC.

TFWF Weyl GTFF STFT OWSTFT

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35PASS-QC

NM

SE

STFT OWSTFT GTFF Weyl TFWF

1

1.1

1.2

1.3

1.4

1.5

1.6STOP-QC

NM

SE

GTFF STFT OWSTFT Weyl TFWF

2.3

2.35

2.4

2.45

2.5

2.55

2.6SPNS-QC total

NM

SE

Conclusions from experiments

The MDE-QC is in agreement with the SPNS-QC criterion and can be used to predict filter performances.The optimal GTFF (OGTFF) has superior properties: TF localization and ripple, compared to both STFT and Weyl based filters.

Summary

A QC based on eigenvalue analysis is proposed.The QC can predict the filter performances for a given filtering task.LTV filtering trade-off (TF localization and ripple) are accounted for.

Summary (cont’d)

MDE-QC is in accordance with the unrelated SPNS-QC criterion.MDE-QC enables synthesis of an optimal LTV filter, which was found best also according to SPNS-QC.

Additional Quality Criteria

Extended Dubiner Quality CriterionSignal Pass/Noise Stop (SPNS) QC

Extended Dubiner Quality Criterion

(alpha) Internal/external signals are created using a (arbitrary area) TF signal synthesis technique by Hlawatsch & Krattenthaler.The filters are tested using these signals to yield the mean/max criterion.

Problem:This technique relies on a TF (bilinear) signal synthesis

technique that will bias the results to Weyl based filter techniques (since its basis functions are the eigenvalues of the filter operator). This QC is therefore suitable only for STFT based methods.

SPNS-QC based Quality Criterion

Measure the passing of internal signals:Generate a TF model based on a representative signal.Check all filtering operators available using SPNS-QC with the same signal used for the synthesis of the TF model.

SPNS-QC based Quality Criterion (cont’d)

Measure the blocking of external signals:Generate random noise distributed uniformly in the TF plane.Check the total amount of noise passed in each filtering method. The best filter passes the least amount of noise.

SPNS-QC based Quality Criterion (cont’d)

This criterion will serve as a validation QC since it relies on a test signal (available only in synthesized cases).Current methods of TF signal synthesis bias the results of the QC towards the related filtering method.

Generalization of WD and STFT

dttxtxexxSMeAt

itixx 2/2/,, *2

,

Wigner Distribution

ddAeftWD xxfti

xx ,, ,)(2

,

STFT Distribution

dttxtexSMAt

itx

*2,,

ddAeftSTFT xfti

x ,, )(2

Weyl Symbol as a generalization of Fourier

A general (time-varying) linear operator is defined by

and its 2D Fourier transform

Representing a filtering operation as a weighted superposition of TF shifted versions of the signal:

ddtxMSeSFtHx iH ,)(

detthftWS fjH

2

2,

2,

txetxM ti

2)(

)()( txtxS

ddtxMSeSFdxth iH ,,

dfeft

WSthf

tfjH

2,2

),(

ddeeSFftWS iiHH 22,,

Weighting and Correcting effects in TF plane

Weyl symbol of a Weyl operator H

Time

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0 10 20 30 40 50 60 70-0.5

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1.5Ordered Eigenvalues of the Weyl operator H

0 10 20 30 40 50 60 70-0.2

-0.1

0

0.1

0.2Eigenfunction with Eigenvalue -0.24

0 10 20 30 40 50 60 70-0.3

-0.2

-0.1

0

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0.2 Eigenfunction with Eigenvalue 1.21

The WD of the Eigenfunction

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0.5The WD of the Eigenfunction

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Time Time

Time

(a) (b)

(c) (d)

(e) (f)

WS as a weighted superposition of eigenfunctions WD

ftWD

detutu

detutu

detthftWS

kk uuk

k

fjkk

kk

fj

kkkk

fjH

,

22

22

2,

2,

,

2*

2*

2

Moyal formula:

m

kmkmm

m

mkmkm

m

uuuum

mHuu

uuuu

WDWDWSWDmmkkkk

*

,,,

,,

,,

The Heisenberg Uncertainty principle for WD

Using the Weyl correspondence, the positive semi-definite operator H obeys

Heisenberg uncertainty principle for Wigner Distributions (Folland-97)

A restriction on the minimum spread of the symbol

0,,, , t f

xxH ftWDftWSxHx

2

,,

2

222 ftWSdtdfftWSbfat

t f

The Heisenberg Uncertainty principle for WD (cont’d)

The following inequality shows that the symbol cannot be too peaked locally (Janssen-89 ):

2

2,,

t ft f

dtdfftWSdtdfftWS

Filtering Experiments – 1TF Model

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STFT Filter Output

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Filtering Experiments – 2TF Model

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STFT Filter Output

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