On Quality Criteria for TimeVarying 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 timefrequency 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.
0
0.5
Frequency
The TF Filtering Model M(t,f)
1
Time
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
STFTxt  t , f    x  * t   e  j 2f d

Wigner Distribution (WD)
   *     j 2f
WDx t  t , f    x t   x  t  e
d
 2  2
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)
Time-Frequency Representations
and Operator Symbols (cont’d)
The input-output relation of any linear, time-varying
(LTV) filter H is:
Hx t   yt    ht , x d ,
where

ht , is the impulse response of H.
Weyl Symbol: Generalization of Fourier using the
Heisenberg Group Theory for LTV systems.
   j 2f
 
LH t , f    h t  , t  e
d
2
2
 
Time-Frequency Representations
and Operator Symbols (cont’d)
Spreading Function: The 2D Fourier of Weyl Symbol.
S H  ,   Ft  Ff1  LH t , f 
 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.
LTV Filter Design Methods
Every linear filter can be represented
as ht , .
Current LTV Filter design approaches are
based primarily on three main methods:
1.
2.
3.
Composition of timepass and bandpass filters.
STFT: Restriction of the reproducing formulas
based on coherent state expansions.
Weyl: pseudodifferential operators with symbols
of compact support.
STFT based methods
Calculate the STFT of a signal
xt : X l t  t , f    xt 'l * t  t 'e j 2ft 'dt '
t'
Multiply the STFT by the mask M
t, f :X t, f   X t, f  M (t, f )
~
~
Synthesize the output signal y t  from the masked STFT X t , f :
yt     X t ' , f 'g t  t 'e j 2f 't df ' dt '
~
The resultant impulse response is
f ' t'
h(t , t ' )    M (t , f ) g t  t l t , f e j 2f (t t ) df dt 
f  t
x(t)
STFT ANALYSIS
using window l(t)
...
Multiplicative
Modification
M(t,f)
...
STFT
SYNTHESIS
using window g(t)
y(t)
Optimal Window STFT
(Kozek-92)
Matching the analysis and synthesis windows
to the filtering model M t , f .
 opt  arg max SFM , A

2
,
 1
SFM  ,   Spreading Function of the Model M t , f 
A  ,   Ambiguity Function of the window 
Optimal diagonalization of the operator via a
Weyl-Heisenberg matched signal set.
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 M t , f ,

where,

t  t   j 2f t t  

h(t , t )   M 
, f e
df
 2

f
LH t , f   M (t , f )
TF Weighting filter
Weyl filter has a substantial amount of
TF energy displacements.
A constraint on H to be a positive semidefinite operator leads to a
minimization problem:
H  arg min M  LH
H
H  HP
TF Weighting filter (cont’d)
Algorithm (Hlawatsch 94)
Calculate ht ,t ' as in a Weyl filter.
Calculate the eigenvalues k and
eigenfunctions uk t  of the filter ht ,t '.
The TF Weighting filter (TFWF) is given
by
*

h(t , t )   k uk t uk t 
k  0
Generalized LTV Filter
The Weyl Symbol of the STFT filter is
LH (t , f )  M (t , f ) * *WDl , g (t , f )
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.
Generalized LTV Filter
(cont’d)
Proposed algorithm
2D Convolve M (t , f ) with an arbitrary (or part
of a family of a) smoothing function S t , f  to
~
yield M (t , f ),
~
M (t , f )  M (t , f ) * *S (t , f )
~
Calculate ht ,t 'from the modified M (t , f ) as
in a Weyl filter:
t  t

h(t , t )   M 
,
 2
f

f e j 2f t t  df

Generalized LTV Filter (cont’d)
Properties:
The proposed filter is related to the STFT and the Weyl
filters:
The STFT filter results from S (t , f )  WDl , g (t , f )
The Weyl filter results from S (t , f )   t   f 
S (t , f ) 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 (  t   f  ) 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.
Current LTV Filter Quality Criteria SNR improvement (Kozek-92)
SNR improvement :
st   input signal
~
s t   output signal
SNR_in = s(t ) / n(t )
nt   noise
SNR_out = s(t ) / ~
s (t )  s(t )
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.
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 selfadjoint operator
Self-Adjoint operators can be decomposed
into real eigenvalues and their corresponding
eigenfunctions:
ht , t    k uk t uk* t 
k
The unitary property of WD enables an
interpretation of concentration for the
eigenvalues
WDu ,u ,WS H (t , f )  k
k
k
Positive and negative eigenvalues
Positive eigenvalues correspond to
eigenfunctions whose TF support is
inside the model, defined as weighted
functions:
*

HW eightingt , t    k uk t uk t 
kI 
Negative eigenvalues correspond to
eigenfunctions that are passed with
inverted phase, defined as corrections:
H Correctiont , t  
*



u
t
u
 k k k t
kI 
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 HW eightingt , t  and
the required TF model M (t , f ).
 Penalty for corrections induced by the
operator H Corrections t , t .
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.
Wigner space of an operator
Definition (Hlawatsch-91) : The Wigner
Space (WSP) of an operator H is
defined as:
WSPH t , f    kWDuk ,uk t , f ,
k
where k , uk are the eigenvalues and
eigenfunctions of the operator H.
 The WSP is an approximate TF transfer
function.
MDE-QC
The Mean Distortion Error QC (MDE-QC)
is defined as
QC M H   M t , f   WSPHWeightingt , f   WSPH Correctingt , f  ,
2
where M (t , f ) is the target TF filter
model.
2
QC-MDE (cont’d)
This criterion measures the fidelity of
the resulting linear filter with respect to
the TF weighting specification M (t , f ).
QC M H   M t , f   WSPHWeightingt , f   WSPH Correctingt , f 


 


2
This criterion is a natural
generalization of the LTI
case.
TF Ripple
Magnitude
TF Localization
2
Frequency
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.
WS H t , f    kWDuk ,uk t , f 
k
However, due to practical considerations (finite
number of eigenfunctions) minimal smoothing is
required for enhanced analysis.
Upper bounds on QC values
For STFT based filters:


QC ( M , H M ,STFT t  )    S M  ,  1  A  ,  dd
 
For Weyl based filters:
M 2 t , f   WSPH  H t , f   2     SFM  ,  dd     SFM  ,  dd
 
 
Optimal filter
We can search over a family of smoothing
functions S t , f  to obtain the lowest value for
the MDE-QC. This family of filters is defined
here as GTFF (Generalized TF filter):
 
H GTFF  arg min QC M ,WS
S t , f
1
  ,
~
MS
~
where M S t , f  is the model M (t , f ) smoothed
by S t , f .
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 S t , f  that
suits the model and the user’s choice of
TF localization and ripple errors.
LTV filtering comparison
The filtering model (a) and MDE-QC weighting part (b,c,d,e) for
various filtering methods.
TF model
1
0.5
(a)
Frequency
0.8
0.6
0
0.4
0.2
-0.5
10
20
30
40
Time
50
Weyl
0.5
0.5
0
60
STFT
0.8
1
0.8
0
0.6
(c)
0.4
Frequency
(b)
Frequency
1.2
0.6
0
0.4
0.2
0.2
-0.5
-0.5
10
0.5
20
30
40
Time
50
60
10
TF Weighting filter
0.5
20
30
Time
40
50
60
Opt. Window STFT
1.5
1
(e)
0
0.5
Frequency
(d)
Frequency
0.8
0.6
0
0.4
0.2
-0.5
10
20
30
40
Time
50
60
0
-0.5
10
20
30
40
Time
50
60
0
LTV filtering comparison
(cont’d)
MDE-QC results
0.55
0.5
NMSE
0.45
0.4
0.35
0.3
0.25
STFT
Opt. Window STFT
Weyl
TF Weighting
GTFF
OGTFF synthesis and eigenvalue analysis
Signal Pass validation experiments
Validation of MDE-QC with SPNS-QC (STFT representation)
STFT: PASS-QC 0.00711
(b)
-0.5
10
20
30
40
Time
50
(e)
10
20
30
40
Time
50
60
Frequency
Frequency
20
30
40
Time
50
10
20
30
40
Time
50
60
OGTFF: PASS-QC 0.00658
0.5
(f)
0
-0.5
0
-0.5
60
0.5
0
-0.5
10
TFWF: PASS-QC 0.00536
Weyl: PASS-QC 0.00561
0.5
(d)
(c)
0
-0.5
60
0.5
10
20
30
40
Time
50
60
Frequency
0
Frequency
Frequency
(a)
OWSTFT: PASS-QC 0.00713
0.5
Frequency
Original Signal
0.5
0
-0.5
10
20
30
40
Time
50
60
Signal Pass validation experiments
Original Signal
(a)
OWSTFT: PASS-QC 0.00713
STFT: PASS-QC 0.00711
4
2.5
2.5
3
2
2
2
1.5
1.5
(b)
1
(c)
1
1
0
0.5
0.5
-1
0
0
-2
0
20
40
Time
60
-0.5
80
Weyl: PASS-QC 0.00561
0
20
40
Time
60
-0.5
80
TFWF: PASS-QC 0.00536
20
40
Time
60
80
OGTFF: PASS-QC 0.00658
2.5
3
2.5
2
2.5
2
2
1.5
0
1.5
1.5
1
1
(e)
(d)
(f)
1
0.5
0.5
0.5
0
-0.5
-1
0
0
-0.5
-0.5
0
20
40
Time
60
80
-1
0
20
40
Time
60
80
-1
0
20
40
Time
60
80
Noise Stop validation experiments
Validation of MDE-QC with SPNS-QC (STFT representation)
STFT: STOP-QC 0.00087
(b)
-0.5
10
20
30
40
Time
50
(e)
10
20
30
40
Time
50
60
Frequency
Frequency
20
30
40
Time
50
10
20
30
40
Time
50
60
OGTFF: STOP-QC 0.00095
0.5
(f)
0
-0.5
0
-0.5
60
0.5
0
-0.5
10
TFWF: STOP-QC 0.00138
Weyl: STOP-QC 0.00113
0.5
(d)
(c)
0
-0.5
60
0.5
10
20
30
40
Time
50
60
Frequency
0
Frequency
Frequency
(a)
OWSTFT: STOP-QC 0.00087
0.5
Frequency
Test Noise
0.5
0
-0.5
10
20
30
40
Time
50
60
Noise Stop validation experiments
Test Noise
STFT: STOP-QC 0.00087
0.6
0.8
3
0.4
0.6
0.2
0.4
2
(a)
OWSTFT: STOP-QC 0.00087
4
0
-1
-2
0
(b)
1
0
20
40
Time
60
80
0.2
(c)
-0.2
0
-0.4
-0.2
-0.6
-0.4
-0.8
Weyl: STOP-QC 0.00114
0
20
40
Time
60
-0.6
80
TFWF: STOP-QC 0.00138
0
20
40
Time
60
80
OGTFF: STOP-QC 0.00094
1
0.8
1
0.8
0.6
0.8
0.6
0.4
0.4
0.2
0.6
0.4
(d) 0.2
(e)
(f)
0
0.2
0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
0
20
40
Time
60
80
-0.8
0
-0.2
0
20
40
Time
60
80
-0.4
0
20
40
Time
60
80
Signal Pass/Noise Stop validation
experiments
Validation experiments showing the relative quality of the filters
using SPNS-QC.
PASS-QC
STOP-QC
1.35
1.6
1.3
1.5
1.4
1.2
NMSE
NMSE
1.25
1.15
1.3
1.2
1.1
1.05
1.1
1
1
TFWF
Weyl
GTFF
STFT
OWSTFT
STFT
OWSTFT GTFF
Weyl
TFWF
SPNS-QC total
2.6
2.55
NMSE
2.5
2.45
2.4
2.35
2.3
GTFF
STFT
OWSTFT
Weyl
TFWF
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 Criterion
Signal 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
Wigner Distribution


Ax , x  ,   ei M S x, x   e 2it xt   / 2x* t   / 2dt
WDx , x t , f     e
t
2i ( t f )
Ax , x  , dd
 
STFT Distribution
Ax  ,   M S  , x    e

STFTx t , f     e

 
2it
t
2i ( t f )
 t   xt dt
*
Ax  , dd

Weyl Symbol as a generalization
of Fourier
A general (time-varying) linear operator is defined by
( Hx )t     SFH  , ei S M xt dd
 
and its 2D Fourier transform
WS H t , f     SFH  , e i 2 e i 2 dd
(S x)t   x(t   )
( M x)t   e 2it xt 
 
Representing a filtering operation as a weighted
superposition of TF shifted versions of the signal:
i






h
t
,

x

d


SF

,

e
S M xt dd

 H

 
 t    j 2f t  
h(t , )   WS H 
, f e
df
2


f

 
WS H t , f    h t  , t  e  j 2f d
2
2
 
Weighting and Correcting effects
in TF plane
Weyl symbol of a Weyl operator H
Ordered Eigenvalues of the Weyl operator H
1.5
0.5
(a)
Freq
1
(b)
0
0.5
0
-0.5
10
0.2
20
30
Time
40
50
-0.5
60
Eigenfunction with Eigenvalue 1.21
0.2
0.1
0
10
20
30 Time 40
50
60
70
Eigenfunction with Eigenvalue -0.24
0.1
0
(d)
(c)
0
-0.1
-0.1
-0.2
(e)
Freq
0.5
0
10
20
30
Time
40
50
60
-0.2
70
The WD of the Eigenfunction
0.5
0
(f)
-0.5
Freq
-0.3
0
10
20
30
Time
40
50
60
70
The WD of the Eigenfunction
0
-0.5
10
20
30
Time
40
50
60
10
20
30
Time
40
50
60
WS as a weighted superposition
of eigenfunctions WD

 
WS H t , f    h t  , t  e  j 2f d
2
2
 
   
   k uk  t  uk*  t  e  j 2f d
 2  2
 k
   
  k  uk  t  uk*  t  e  j 2f d
 2  2
k

Moyal formula:
  kWDuk ,uk t , f 
k
WDuk ,uk , WS H   m WDuk ,uk ,WDum ,um
m
  m uk , um uk , um
m
  m km km
m
 m
*
The Heisenberg Uncertainty
principle for WD
Using the Weyl correspondence, the positive semidefinite operator H obeys
Hx , x    WS H t , f WDx , x t , f   0
t f
Heisenberg uncertainty principle for Wigner
Distributions (Folland-97)
  t  a
t f
2

 f  b WS t , f dtdf 
2
WS t , f  2
2
2
 A restriction on the minimum spread of the symbol
The Heisenberg Uncertainty
principle for WD (cont’d)
The following inequality shows that the
symbol cannot be too peaked locally
(Janssen-89 ):
  WS t, f 
t f
2



dtdf    WS t , f dtdf 


t f

2
Filtering Experiments – 1
TF Model
Test Signal
0.5
Frequency
Frequency
0.5
0
-0.5
0
-0.5
10
20
30
Time
40
50
60
20
STFT Filter Output
60
OPTTF Filter Output
0.5
Frequency
0.5
Frequency
40
Time
0
-0.5
0
-0.5
20
40
Time
60
20
40
Time
60
Filtering Experiments – 2
TF Model
Test Signal
0.5
Frequency
Frequency
0.5
0
-0.5
0
-0.5
10
20
30
Time
40
50
60
20
STFT Filter Output
60
OPTTF Filter Output
0.5
Frequency
0.5
Frequency
40
Time
0
-0.5
0
-0.5
20
40
Time
60
20
40
Time
60