SPSC – Signal Processing & Speech Communication Lab · 2007-06-22 · SPSC – Signal Processing & Speech Communication Lab Professor Horst Cerjak, 19.12.2005 3 Georg Holzmann,
Post on 16-Jul-2020
4 Views
Preview:
Transcript
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.20051
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks
Wavelet Transform and its relation to
multirate filter banks
Christian WallingerASP Seminar 12th June 2007
Graz University of Technology, Austria
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.20052
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks
OutlineShort – Time Fourier – Transformation
Interpretation using Bandpass FiltersUniform DFT BankDecimationInverse STFT and filter - bank interpretationBasis Functions and OrthonormalityContinuous Time STFT
Wavelet – TransformationPassing from STFT to WaveletsGeneral Definition of WaveletsInversion and filter - bank interpretationOrthonormal BasisDiscrete – Time Wavelet Transf.Inverse
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.20053
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks
fgjfghj
yxvyxvyxcv
sdfgsdfg
time – frequency plot = Spectogram
SHORT-Time FOURIER TRANSF.
figure 1: STFT processing in time
figure 2: spectogram
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.20054
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks
Definition:
( ) ∑∞
−∞=
−−=n
njjSTFT emnvnxmeX ωω )()(,
m . . . time shift – variable ( typically an integer multiple of some fixed integer K)
ω . . . frequency – variable πωπ <≤−
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.20055
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks
Interpretation using Bandpass Filters
Traditional Fourier Transform as a Filter Bank
1. Modulator : performs a frequency shiftnje 0ω−
figure 3: Representation of FT in terms of a linear system
( )ωjeH2. LTI – System : ideal lowpass filter
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.20056
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks
Why is an ideal lowpass filter ?( )ωjeH
Impulse Response h(n) = 1 for all n
( ) ( )ωπδωωa
n
njj enheH 2)( == ∑∞
−∞=
− πωπ <≤−
only zero - frequency passesevery other frequency is completely supressed
( )0)( ωjeXny = for all n
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.20057
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks
STFT as a Bank of FiltersExpansion of Definiton for further insight!
( ) ∑∞
−∞=
−− −=n
nmjmjjSTFT emnvnxemeX )()()(, ωωω
with:)()( ))(()( nmjnmj enmvemnv −− −−=− ωω
Convolution of x(n) with the impulse response of the LTI – System njenv ω)(−
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.20058
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks
figure 4: Representation of STFT in terms of a linear system
In most applications, v(n) has a lowpass transform V(ejω).
)( nv − )( ωjeV −
njenv 0)( ω− )( )( 0ωω−− jeV
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.20059
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks figure 5: Demonstration of how STFT works
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.200510
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks
In practice, we are interested in computing the Fourier transform at a discreteset of frequencies
0 ≤ ω0 < ω1 < … < ωM-1 < 2π
Therefore the STFT reduces to a filter bank with M bandpass filters
)()( )( kjjk eVeH ωωω −−=
figure 6: STFT viewed as a filter bank
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.200511
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks
Uniform DFT bank
If the frequencies ωk are uniformly spaced, then the systembecomes the uniform DFT bank.
The M filters are related as in the following manner
( )kk zWHzH 0)( = 10 −≤≤ Mk M
jeW
π2−
=
⎟⎟⎠
⎞⎜⎜⎝
⎛=
− )2(
0)(k
Mjj
k eHeHπωω )()(0
ωω jj eVeH −=
The uniform DFT bank is a device to compute the STFT at uniformelyspaced frequencies.
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.200512
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks
Decimation
if passband width of V(ejω) is narrow
output signals yk(n) are narrowband lowpass signals
this means, that yk(n) varies slowly with time
According to this variying nature, one can exploit that to decimate theoutput.Decimation Ratio of M = moving the window v(k) by M samples at a time
if filters have equal bandwidth
Mnk =
maximally decimated analyses bank
figure 7: Analysis bank with decimators
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.200513
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks
Time – Frequency Grid
Uniform sampling of both, ‘time’ n and ‘frequency’ ω
figure 8: time – frequency grid
Time spacing M corresponds to moving the window M units ( = samples ) at a time.
frequency spacing of adjacent filters = Mπ2
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.200514
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks
Inversion of the STFT
From traditional Fourier – viewpoint
( )meX jSTFT ,ω is the FT. from the time domain product
)()( mnvnx −
( )∫=−π
ωω ωπ
2
0
,21)()( demeXmnvnx njj
STFT
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.200515
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks
Another inversion formula is given by:
( ) ( )∫ ∑ ⎟⎠
⎞⎜⎝
⎛−=
∞
−∞=
πωω ω
π
2
0
*,21)( demnvmeXnx nj
m
jSTFT
which is provided by ( )∑ =m
mv 12
if but finite divide right side of the formula by ( )∑ ≠m
mv 12 ( )∑mmv 2
but if window energy is infinite one cannot apply this formulation
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.200516
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks
Filter Bank Interpretation of the Inverse
With as synthesis - filterReconstruction can be done by the following synthesis bank:
)(zFk
figure 9: synthesis – bank used to reconstruct x(n)
typically for all kMnk =
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.200517
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks
The z – Transformation of is given by( )nx
( ) ( ) ( )∑−
=
=1
0
ˆM
kk
nk zFzXzX k
in time – domain
( ) ( ) ( )∑ ∑−
=
∞
−∞=
−=1
0
ˆM
k mkkk mnnfmxnx
( ) ( )∑ ∑−
=
∞
−∞=
−=1
0
)(M
k mkk
mnjkk mnnfemny kkω
( ) tsCoefficienSTFTmny kk −K
Reconstruction is stable, if the filters are stable!)(zFk
Perfect reconstruction will be obtained, if ( ) ( )nxnx =ˆ
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.200518
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks
Basis Functions and Orthonormality
Functions of interest
( ) ( ) functionsbasismnnfn kkkm K−=η
For these double indexed functions ( basis functions ), the orthonormality property means that
( ) nkmη
( ) ( ) ( ) ( )∑∞
−∞=
−−=−−n
kkkk mmkkmnnfmnnf 212122211*1 δδ
should be zero, except for those cases where 2121 mmandkk ==
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.200519
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks
The Continuous - Time Case
Main points:
( ) ( ) ( ) ( )∫∞
∞−
Ω−−=Ω STFTdtetvtxjX tjSTFT ττ,
( ) ( ) ( ) ( )STFTinvdejXtvtx tjSTFT .,
21∫∞
∞−
Ω ΩΩ=− τπ
τ
( ) ( ) ( ) ( )∫ ∫∞
∞−
Ω∞
∞−
Ω⎟⎟⎠
⎞⎜⎜⎝
⎛−Ω= STFTinvdedtvjXtx tj
STFT .,21 * τττπ
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.200520
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks
Choice of “Best Window”
Root Mean Square duration of window function v(t) in
frequency domain Dftime domain Dt
( )∫∞
∞−
= dttvtE
Dt222 1 ( )∫
∞
∞−
ΩΩΩ= djVE
D f222
21π
with:E . . . window energy ( )∫= dttvE 2
Uncertainty principle:5.0≥ft DD
Iff Gaussian – window, this inequality becomes an equality !
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.200521
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks
Filter Bank Interpretation
figure 10: continuous – STFT
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.200522
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks
THE WAVELET TRANSFORM
Disadvantage of STFTuniform time – frequency box ( ).., constDconstD ft ==
The accuracy of the estimate of the Fourier transform is poor at low frequencies, and improves as the frequency increases.
Expected properties for a new function:window width should adjust itself with ‘frequency’as the window gets wider in time, also the step sizes for moving the window should become wider.
These goals are nicely accomplished by the wavelet transform.
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.200523
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks
Passing from STFT to Wavelets
Step 1:
Giving up the STFT modulation scheme and obtain filters
( ) ( ) egerkfactorscalingatahath kk
k int,12 =>= −−K
in the frequency domain:
( ) ( )Ω=Ω kk
k jaHajH 2
all reponses are obtained by frequency – scaling of a prototype response ( )ΩjH
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.200524
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks
Example:
Assuming is a bandpass with cutoff frequencies α and β. Also and the center frequency should be the geometricalmean of the two cutoff edges
( )ΩjHαβ 2,2 ==a
222 kkk
−− ==Ω ααβ
figure 11: frequency – response obtained by scaling process
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.200525
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks
Ratio:
( )2
12
2=
−=
Ω− −
−
αβαβ
k
k
kfrequencycenterbandwidth
is independent of integer k
In electrical filter theory such a system is often said to be a ‘constant Q’ system!
( Q ... Quality factorbandwidth
frequencycenterQ −= )
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.200526
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks
filter ouputs can be obtained by:
( ) ( )( )∫∞
∞−
−Ω−−− dttahtxea kjk
k ττ2
Step 2:
( ) ↓→↓Ω→↑ SampleratejHofbandwidthk k
or in time domain
↑→↑→↑ sizesteplengthwindowk
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.200527
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks
Therefore:sizestepTaegernTna kk KK ,int=τ
hence:
( )( ) ( )tanThtTnaah kkk −− −=−
Summarizing, we are computing:
( ) ( ) ( )∫∞
∞−
−−−= dttanThtxankX kk
DWT2,
( ) ( ) ( )∫∞
∞−
−= dttTnahtxnkX kkDWT ,
figure 12: Analysis bank of DWT
DWT...Discrete Wavelet Transform
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.200528
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks
Time Frequency Grid
.constDD ft =figure 13: time – frequency grid
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.200529
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks
General Definition of the Wavelet Transform
( ) ( )∫∞
∞−⎟⎟⎠
⎞⎜⎜⎝
⎛ −= dt
pqtftx
pqpX CWT
1,
p,q ... real – valued continuous variables
According to former definition:
kap = Tnaq k= ( ) ( )thtf −=
( ) ( ) tscoefficienwaveletnkXandqpX DWTCWT KKK,,
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.200530
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks
Inversion of Wavelet Transform
( ) ( ) ( )∑∑=k n
nkDWT tnkXtx ψ,
where are the basis functions( )tnkψ
Filter Bank Interpretation of Inversion
Reconstruction of x(t) as a designing problem of the following synthesis filter bank
( ) sequencenkX DWT K,
( ) timeincontinuousjFk KΩ
output of synthesis filter bank :
( ) ( ) ( )∑∑ −=k
k
nkDWT nTatfnkXtx ,ˆ
figure 14: synthesis bank
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.200531
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks
All synthesis filters are again generated from a fixed prototype synthesis filter f(t) ( mother wavelet )
( ) ( )tafatf kk
k−−
= 2
Substituting this in the preceding equation and assuming perfect reconstruction, we get
( ) ( ) ( )∑∑ −= −−
k n
kk
DWT nTtafankXtx 2,
with:
( ) ( ) ( ) ( ) ( )[ ] functionsbasisofsetTnataanTtaattft kkkkk
nk K−=−=→= −−−− ψψψψ 22
using this, we can express each basis function in terms of the filter ( )tfk
( ) ( )Tnatft kknk −=ψ
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.200532
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks
Orthonormal Basis
Of particular interest is the case where is a set of orthonormal functions
( ) tnkψ
Therefore, we expect:
( ) ( ) ( ) ( )∫∞
∞−
−−= mnlkdttt mlnk δδψψ *
using Parseval’s theorem, this becomes
( ) ( ) ( ) ( )∫∞
∞−
−−=ΩΩΨΩΨ mnlkdjj mlnk δδπ
*
21
and get :
( ) ( ) ( )∫∞
∞−
= dtttxnkX nkDWT*, ψ
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.200533
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks
Comparing these results, we can conclude:
( ) ( )tnTaht kknk −= *ψ
And in particular for k = 0 and n = 0:
( ) ( ) ( )thtt −== *00 ψψ for the orthonormal case ( ) ( )thtf kk −= *
Discrete – Time Wavelet Transform
Starting with the frequency domain relation and a scaling factor a = 2
( ) ( ) egerenonnegativaiskeHeHkjj
k int2 Kωω =
for highpass and k = 1, k = 2( )ωjeH
figure 15: Magnitude responses
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.200534
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks
Let G(z) be a lowpass with response
figure 16: Magnitude – response of G(z)
or its equivalentUsing QMF – banks
figure 18: equivalent 4-channel systemfigure 17: 3 level binary tree-structured QMF
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.200535
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks
Responses of the filters ( ) ( ) ( ) ( ) ( ) ( ),......,, 422 zHzGzGzHzGzH
figure 19: combinations of H(z) and G(z)
Defining the Discrete –Time Wavelet Transform
( ) ( ) ( )∑∞
−∞=
+ −≤≤−=m
kkk Mkmnhmxny 20,2 1
( ) ( ) ( ) ( )∑∞
−∞=
−−− −=
mimeiscrete
MMM WTTDmnhmxny ,2 1
11
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.200536
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks
Inverse Transform
( ) ( ) ( ) ( ) ( ) K,, 210 zGzHzFzHzF sss ==
figure 20: synthesis filters
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.200537
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks
For perfect reconstruction ( ) ( )nxnx =ˆ we can express
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )11 211
222
411
200 ...
−−
−−−− ++++=MM
zYzFzYzFzYzFzYzFzX MMMM
and in time domain:
( ) ( ) ( ) ( ) ( )∑ ∑ ∑−
=
∞
−∞=
∞
−∞=
−−−
+ −+−=2
0
111
12 22M
k m m
MMM
kkk mnfmymnfmynx
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.200538
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks
Main References
Multirate Systems and Filter Banks
(Prentice Hall Signal Processing Series)
by P. P. Vaidyanathan
SPSC – Signal Processing & Speech Communication Lab
Professor Horst Cerjak, 19.12.200539
Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks
Thank you for attention !
top related