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

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

fgjfghj

yxvyxvyxcv

sdfgsdfg

time – frequency plot = Spectogram

SHORT-Time FOURIER TRANSF.

figure 1: STFT processing in time

figure 2: spectogram

Definition:

( ) ∑∞

−∞=

−−=n

njjSTFT emnvnxmeX ωω )()(,

m . . . time shift – variable ( typically an integer multiple of some fixed integer K)

ω . . . frequency – variable πωπ <≤−

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

Why is an ideal lowpass filter ?( )ωjeH

Impulse Response h(n) = 1 for all n

( ) ( )ωπδωωa

njj enheH 2)( == ∑∞

−∞=

− πωπ <≤−

only zero - frequency passesevery other frequency is completely supressed

( )0)( ωjeXny = for all n

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)(−

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

Georg Holzmann, Christian Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks figure 5: Demonstration of how STFT works

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

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

π2−

⎟⎟⎠

⎞⎜⎜⎝

− )2(

k eHeHπωω )()(0

ωω jj eVeH −=

The uniform DFT bank is a device to compute the STFT at uniformelyspaced frequencies.

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

maximally decimated analyses bank

figure 7: Analysis bank with decimators

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

Inversion of the STFT

From traditional Fourier – viewpoint

( )meX jSTFT ,ω is the FT. from the time domain product

)()( mnvnx −

( )∫=−π

ωω ωπ

,21)()( demeXmnvnx njj

Another inversion formula is given by:

( ) ( )∫ ∑ ⎟⎠

⎞⎜⎝

⎛−=

−∞=

πωω ω

*,21)( demnvmeXnx nj

which is provided by ( )∑ =m

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

Filter Bank Interpretation of the Inverse

With as synthesis - filterReconstruction can be done by the following synthesis bank:

figure 9: synthesis – bank used to reconstruct x(n)

typically for all kMnk =

The z – Transformation of is given by( )nx

( ) ( ) ( )∑−

nk zFzXzX k

in time – domain

( ) ( ) ( )∑ ∑−

−∞=

k mkkk mnnfmxnx

( ) ( )∑ ∑−

−∞=

mnjkk mnnfemny kkω

( ) tsCoefficienSTFTmny kk −K

Reconstruction is stable, if the filters are stable!)(zFk

Perfect reconstruction will be obtained, if ( ) ( )nxnx =ˆ

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 ==

The Continuous - Time Case

Main points:

( ) ( ) ( ) ( )∫∞

∞−

Ω−−=Ω STFTdtetvtxjX tjSTFT ττ,

( ) ( ) ( ) ( )STFTinvdejXtvtx tjSTFT .,

21∫∞

∞−

Ω ΩΩ=− τπ

( ) ( ) ( ) ( )∫ ∫∞

∞−

Ω⎟⎟⎠

⎞⎜⎜⎝

⎛−Ω= STFTinvdedtvjXtx tj

STFT .,21 * τττπ

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

with:E . . . window energy ( )∫= dttvE 2

Uncertainty principle:5.0≥ft DD

Iff Gaussian – window, this inequality becomes an equality !

Filter Bank Interpretation

figure 10: continuous – STFT

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.

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

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

Ratio:

Ω− −

αβαβ

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 −= )

filter ouputs can be obtained by:

( ) ( )( )∫∞

∞−

−Ω−−− dttahtxea kjk

k ττ2

Step 2:

( ) ↓→↓Ω→↑ SampleratejHofbandwidthk k

or in time domain

↑→↑→↑ sizesteplengthwindowk

Therefore:sizestepTaegernTna kk KK ,int=τ

hence:

( )( ) ( )tanThtTnaah kkk −− −=−

Summarizing, we are computing:

( ) ( ) ( )∫∞

∞−

−−−= dttanThtxankX kk

( ) ( ) ( )∫∞

∞−

−= dttTnahtxnkX kkDWT ,

figure 12: Analysis bank of DWT

DWT...Discrete Wavelet Transform

Time Frequency Grid

.constDD ft =figure 13: time – frequency grid

General Definition of the Wavelet Transform

( ) ( )∫∞

∞−⎟⎟⎠

⎞⎜⎜⎝

⎛ −= dt

pqtftx

pqpX CWT

p,q ... real – valued continuous variables

According to former definition:

kap = Tnaq k= ( ) ( )thtf −=

( ) ( ) tscoefficienwaveletnkXandqpX DWTCWT KKK,,

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

nkDWT nTatfnkXtx ,ˆ

figure 14: synthesis bank

All synthesis filters are again generated from a fixed prototype synthesis filter f(t) ( mother wavelet )

( ) ( )tafatf kk

k−−

Substituting this in the preceding equation and assuming perfect reconstruction, we get

( ) ( ) ( )∑∑ −= −−

DWT nTtafankXtx 2,

( ) ( ) ( ) ( ) ( )[ ] functionsbasisofsetTnataanTtaattft kkkkk

nk K−=−=→= −−−− ψψψψ 22

using this, we can express each basis function in terms of the filter ( )tfk

( ) ( )Tnatft kknk −=ψ

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

and get :

( ) ( ) ( )∫∞

∞−

= dtttxnkX nkDWT*, ψ

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

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

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

Inverse Transform

( ) ( ) ( ) ( ) ( ) K,, 210 zGzHzFzHzF sss ==

figure 20: synthesis filters

For perfect reconstruction ( ) ( )nxnx =ˆ we can express

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )11 211

200 ...

−−

−−−− ++++=MM

zYzFzYzFzYzFzYzFzX MMMM

and in time domain:

( ) ( ) ( ) ( ) ( )∑ ∑ ∑−

−∞=

−−−

+ −+−=2

12 22M

kkk mnfmymnfmynx

Main References

Multirate Systems and Filter Banks

(Prentice Hall Signal Processing Series)

by P. P. Vaidyanathan

Thank you for attention !

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