k - spsc - Graz University of Technology
k - spsc - Graz University of Technology
k - spsc - Graz University of Technology
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SPSC – Signal Processing & Speech Communication Lab<br />
Wavelet Transform and<br />
its relation to<br />
multirate filter banks<br />
Christian Wallinger<br />
ASP Seminar 12 th June 2007<br />
<strong>Graz</strong> <strong>University</strong> <strong>of</strong> <strong>Technology</strong>, Austria<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
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SPSC – Signal Processing & Speech Communication Lab<br />
Outline<br />
Short – Time Fourier – Transformation<br />
‣ Interpretation using Bandpass Filters<br />
‣ Uniform DFT Bank<br />
‣ Decimation<br />
‣ Inverse STFT and filter - bank interpretation<br />
‣ Basis Functions and Orthonormality<br />
‣ Continuous Time STFT<br />
Wavelet – Transformation<br />
‣ Passing from STFT to Wavelets<br />
‣ General Definition <strong>of</strong> Wavelets<br />
‣ Inversion and filter - bank interpretation<br />
‣ Orthonormal Basis<br />
‣ Discrete – Time Wavelet Transf.<br />
‣ Inverse<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
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SPSC – Signal Processing & Speech Communication Lab<br />
SHORT-Time FOURIER TRANSF.<br />
sdfgsdfg<br />
yxvyxvyxcv<br />
fgjfghj<br />
figure 1: STFT processing in time<br />
time – frequency plot = Spectogram<br />
figure 2: spectogram<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
3
SPSC – Signal Processing & Speech Communication Lab<br />
Definition:<br />
X<br />
STFT<br />
(<br />
jω<br />
) ∑ ∞ e , m =<br />
n=<br />
−∞<br />
x(<br />
n)<br />
v(<br />
n − m)<br />
e<br />
− jωn<br />
m . . . time shift – variable<br />
( typically an integer multiple <strong>of</strong> some fixed integer K)<br />
ω . . . frequency – variable<br />
− π ≤ ω < π<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
4
SPSC – Signal Processing & Speech Communication Lab<br />
Interpretation using Bandpass Filters<br />
Traditional Fourier Transform as a Filter Bank<br />
figure 3: Representation <strong>of</strong> FT in terms <strong>of</strong> a linear system<br />
e<br />
0<br />
− jω<br />
n<br />
1. Modulator : performs a frequency shift<br />
(<br />
jω<br />
)<br />
H e<br />
2. LTI – System : ideal lowpass filter<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
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SPSC – Signal Processing & Speech Communication Lab<br />
(<br />
jω)<br />
H e<br />
Why is an ideal lowpass filter ?<br />
Impulse Response h(n) = 1 for all n<br />
(<br />
jω<br />
)<br />
− jωn<br />
H e h( n)<br />
e = 2πδ<br />
( ω)<br />
= ∑ ∞<br />
−∞<br />
n=<br />
a<br />
−π ≤ ω < π<br />
only zero - frequency passes<br />
every other frequency is completely supressed<br />
(<br />
jω<br />
)<br />
0<br />
e<br />
y ( n)<br />
= X<br />
for all n<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
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SPSC – Signal Processing & Speech Communication Lab<br />
STFT as a Bank <strong>of</strong> Filters<br />
Expansion <strong>of</strong> Definiton for further insight!<br />
X<br />
STFT<br />
(<br />
jω<br />
)<br />
− jωm<br />
∞ e , m = e ∑<br />
n=<br />
−∞<br />
x(<br />
n)<br />
v(<br />
n<br />
−<br />
m)<br />
e<br />
jω(<br />
m−n)<br />
with:<br />
v(<br />
n − m)<br />
e<br />
jω(<br />
m−n)<br />
=<br />
v(<br />
−(<br />
m − n))<br />
e<br />
jω(<br />
m−n)<br />
Convolution <strong>of</strong> x(n) with the impulse response <strong>of</strong> the LTI – System<br />
v(−n)<br />
e<br />
jωn<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
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SPSC – Signal Processing & Speech Communication Lab<br />
figure 4: Representation <strong>of</strong> STFT in terms <strong>of</strong> a linear system<br />
In most applications, v(n) has a lowpass transform V(e jω ).<br />
<br />
jω<br />
v( − n) V ( e<br />
− )<br />
0<br />
v(<br />
jω<br />
n<br />
− j( 0<br />
− n)<br />
e V ( e<br />
ω−ω ) )<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
8
SPSC – Signal Processing & Speech Communication Lab<br />
figure 5: Demonstration <strong>of</strong> how STFT works<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
9
SPSC – Signal Processing & Speech Communication Lab<br />
In practice, we are interested in computing the Fourier transform at a discrete<br />
set <strong>of</strong> frequencies<br />
0 ≤ ω 0<br />
< ω 1<br />
< … < ω M-1<br />
< 2π<br />
Therefore the STFT reduces to a filter bank with M bandpass filters<br />
H<br />
k<br />
( e<br />
jω<br />
) = V ( e<br />
− j( ω−ωk<br />
)<br />
)<br />
figure 6: STFT viewed as a filter bank<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
10
SPSC – Signal Processing & Speech Communication Lab<br />
Uniform DFT bank<br />
If the frequencies ω k<br />
are uniformly spaced, then the system<br />
becomes the uniform DFT bank.<br />
The M filters are related as in the following manner<br />
H<br />
k<br />
z)<br />
H<br />
0<br />
(<br />
k<br />
zW )<br />
( = 0 ≤ k ≤ M −1<br />
W<br />
=<br />
e<br />
j 2π<br />
−<br />
M<br />
<br />
H<br />
k<br />
( e<br />
jω<br />
)<br />
=<br />
H<br />
0<br />
⎛<br />
⎜<br />
e<br />
⎝<br />
2π<br />
j(<br />
ω− k )<br />
M<br />
⎞<br />
⎟<br />
⎠<br />
H 0( e<br />
) = V ( e<br />
jω − jω<br />
)<br />
The uniform DFT bank is a device to compute the STFT at uniformely<br />
spaced frequencies.<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
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SPSC – Signal Processing & Speech Communication Lab<br />
Decimation<br />
if passband width <strong>of</strong> V(e jω ) is narrow<br />
output signals y k<br />
(n) are narrowband lowpass signals<br />
this means, that yk(n) varies slowly with time<br />
According to this variying nature, one can exploit that to decimate the<br />
output.<br />
Decimation Ratio <strong>of</strong> M = moving the window v(k) by M samples at a time<br />
if filters have equal bandwidth<br />
<br />
n k<br />
= M<br />
maximally decimated analyses bank<br />
figure 7: Analysis bank with decimators<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
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SPSC – Signal Processing & Speech Communication Lab<br />
Time – Frequency Grid<br />
Uniform sampling <strong>of</strong> both, ‘time’ n and ‘frequency’ ω<br />
figure 8: time – frequency grid<br />
Time spacing M corresponds to moving the window M units ( = samples ) at a time.<br />
2π<br />
frequency spacing <strong>of</strong> adjacent filters = M<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
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SPSC – Signal Processing & Speech Communication Lab<br />
X<br />
Inversion <strong>of</strong> the STFT<br />
From traditional Fourier – viewpoint<br />
(<br />
jω<br />
e m)<br />
STFT<br />
,<br />
is the FT. from the time domain product<br />
x( n)<br />
v(<br />
n − m)<br />
<br />
2π<br />
1<br />
− = ∫ (<br />
jω<br />
)<br />
jωn<br />
x( n)<br />
v(<br />
n m)<br />
X<br />
STFT<br />
e , m e dω<br />
2π<br />
0<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
14
SPSC – Signal Processing & Speech Communication Lab<br />
Another inversion formula is given by:<br />
2π<br />
∞<br />
1 ⎛<br />
∫ ∑ (<br />
jω<br />
)<br />
* ⎞<br />
= ⎜<br />
( − )<br />
jωn<br />
x( n)<br />
X<br />
STFT<br />
e , m v n m ⎟e<br />
dω<br />
2π<br />
⎝ m=−∞<br />
⎠<br />
0<br />
2<br />
∑ v = 1<br />
which is provided by ( m)<br />
m<br />
if<br />
∑<br />
m<br />
( m)<br />
2<br />
v ≠ 1<br />
but finite divide right side <strong>of</strong> the formula by<br />
∑ m<br />
v<br />
( m)<br />
2<br />
but if window energy is infinite one cannot apply this formulation<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
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SPSC – Signal Processing & Speech Communication Lab<br />
Filter Bank Interpretation <strong>of</strong> the Inverse<br />
F k<br />
(z)<br />
With as synthesis - filter<br />
Reconstruction can be done by the following synthesis bank:<br />
figure 9: synthesis – bank used to reconstruct x(n)<br />
typically n k<br />
= M for all k<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
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SPSC – Signal Processing & Speech Communication Lab<br />
( n)<br />
The z – Transformation <strong>of</strong> is given by<br />
Xˆ<br />
M<br />
∑ − 1<br />
k = 0<br />
xˆ<br />
nk<br />
( z) = X ( z ) F ( z)<br />
k<br />
k<br />
xˆ<br />
M − 1<br />
in time – domain<br />
∞<br />
( n) = x ( m) f ( n − n m)<br />
∑∑<br />
k=<br />
0 m=−∞<br />
k<br />
k<br />
k<br />
=<br />
M − 1<br />
∞<br />
∑∑<br />
k=<br />
0 m=−∞<br />
y<br />
k<br />
jω ( n m)<br />
( n m) e k k<br />
f ( n − n m)<br />
k<br />
k<br />
k<br />
y<br />
k<br />
( n m) K STFT − Coefficien ts<br />
k<br />
Reconstruction is stable, if the filters F k<br />
(z) are stable!<br />
Perfect reconstruction will be obtained, if x ˆ(<br />
n) = x( n)<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
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SPSC – Signal Processing & Speech Communication Lab<br />
Basis Functions and Orthonormality<br />
η<br />
km<br />
Functions <strong>of</strong> interest<br />
( n) =ˆ f ( n − n m) Kbasis<br />
functions<br />
k<br />
k<br />
For these double indexed functions ( basis functions n ),<br />
the orthonormality property means that<br />
{ ( )}<br />
η km<br />
∑ ∞<br />
n=<br />
−∞<br />
( n − n m ) f ( n − n m ) = δ ( k − k ) ( m − m )<br />
*<br />
f<br />
k1 k1<br />
1 k 2 k 2 2 1 2<br />
δ<br />
1 2<br />
should be zero, except for those cases where k<br />
1<br />
= k2<br />
and m1<br />
= m2<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
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SPSC – Signal Processing & Speech Communication Lab<br />
The Continuous - Time Case<br />
Main points:<br />
X<br />
STFT<br />
∞<br />
∫<br />
−∞<br />
− jΩ<br />
t<br />
( jΩ,τ<br />
) = x( t) v( t −τ<br />
) e dt ( STFT )<br />
x<br />
∞<br />
1<br />
2π<br />
∫<br />
j t<br />
()( t v t −τ<br />
) = X ( jΩ,<br />
τ ) e<br />
Ω dΩ<br />
( inv STFT )<br />
−∞<br />
STFT<br />
.<br />
x<br />
1 ∞ ∞<br />
*<br />
STFT<br />
.<br />
jΩ<br />
t<br />
() t = ⎜ X ( jΩ,<br />
τ ) v ( t −τ<br />
) dτ<br />
⎟e<br />
dΩ<br />
( inv STFT )<br />
2π<br />
∫⎜<br />
∫<br />
−∞<br />
⎛<br />
⎝<br />
−∞<br />
⎞<br />
⎟<br />
⎠<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
19
SPSC – Signal Processing & Speech Communication Lab<br />
Choice <strong>of</strong> “Best Window”<br />
R oot<br />
M ean<br />
S quare<br />
duration <strong>of</strong> window function v(t) in<br />
time domain D t<br />
∞<br />
2 1 2 2<br />
D t ∫ t v<br />
E<br />
−∞<br />
() t<br />
= dt<br />
frequency domain D f<br />
∞<br />
2 1 2<br />
2<br />
D f ∫ Ω V ( jΩ)<br />
2πE<br />
−∞<br />
= dΩ<br />
with:<br />
2<br />
E . . . window energy E = v () t dt<br />
∫<br />
Uncertainty principle:<br />
D t D f<br />
≥ 0.5<br />
Iff Gaussian – window, this inequality becomes an equality !<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
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SPSC – Signal Processing & Speech Communication Lab<br />
Filter Bank Interpretation<br />
figure 10: continuous – STFT<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
21
SPSC – Signal Processing & Speech Communication Lab<br />
THE WAVELET TRANSFORM<br />
Disadvantage <strong>of</strong> STFT<br />
uniform time – frequency box ( D = const., D const. )<br />
t f<br />
=<br />
The accuracy <strong>of</strong> the estimate <strong>of</strong> the Fourier transform<br />
is poor at low frequencies, and improves as the frequency increases.<br />
Expected properties for a new function:<br />
‣ window width should adjust itself with ‘frequency’<br />
‣ as the window gets wider in time, also the step sizes for<br />
moving the window should become wider.<br />
These goals are nicely accomplished by the wavelet transform.<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
22
SPSC – Signal Processing & Speech Communication Lab<br />
Passing from STFT to Wavelets<br />
Step 1:<br />
Giving up the STFT modulation scheme and obtain filters<br />
h<br />
k<br />
−k<br />
() t a ( ) 2 −k<br />
= h a t a > 1Kscaling<br />
factor,<br />
k = int eger<br />
in the frequency domain:<br />
k<br />
k<br />
( jΩ) = a H ( ja Ω)<br />
2<br />
H<br />
k<br />
all reponses are obtained by frequency – scaling <strong>of</strong> a prototype response H(<br />
jΩ)<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
23
SPSC – Signal Processing & Speech Communication Lab<br />
Example:<br />
H<br />
( jΩ)<br />
Assuming is a bandpass with cut<strong>of</strong>f frequencies α and β.<br />
Also a = 2 , β = 2α and the center frequency should be the geometrical<br />
mean <strong>of</strong> the two cut<strong>of</strong>f edges<br />
Ω<br />
k<br />
=<br />
2<br />
−k<br />
αβ = α 2<br />
−k<br />
2<br />
figure 11: frequency – response obtained by scaling process<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
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SPSC – Signal Processing & Speech Communication Lab<br />
Ratio:<br />
bandwidth<br />
center − frequency<br />
Ω<br />
k<br />
=<br />
2<br />
−k<br />
2<br />
( β − α )<br />
−k<br />
αβ<br />
=<br />
1<br />
2<br />
is independent <strong>of</strong> integer k<br />
In electrical filter theory such a system is <strong>of</strong>ten said to be a ‘constant Q’ system!<br />
( Q ... Quality factor<br />
center − frequency<br />
Q = )<br />
bandwidth<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
25
SPSC – Signal Processing & Speech Communication Lab<br />
filter ouputs can be obtained by:<br />
a<br />
−k<br />
2<br />
e<br />
− jΩ<br />
k<br />
∞<br />
τ<br />
∫<br />
−∞<br />
x<br />
−k<br />
() t h a ( τ − t)<br />
( )<br />
dt<br />
Step 2:<br />
k<br />
↑<br />
→<br />
bandwidth<br />
<strong>of</strong><br />
H<br />
k<br />
( jΩ) ↓ → Samplerate ↓<br />
or in time domain<br />
k<br />
↑<br />
→<br />
window length<br />
↑<br />
→<br />
step<br />
size<br />
↑<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
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SPSC – Signal Processing & Speech Communication Lab<br />
Therefore:<br />
τ =<br />
na<br />
k<br />
T<br />
nK int eger,<br />
a<br />
k<br />
T Kstep<br />
size<br />
hence:<br />
h<br />
(<br />
−k<br />
(<br />
k<br />
) (<br />
−k<br />
a na T − t = h nT − a t)<br />
Summarizing, we are computing:<br />
<br />
X<br />
X<br />
DWT<br />
DWT<br />
,<br />
∞<br />
∫<br />
−∞<br />
−k<br />
2<br />
−k<br />
( k,<br />
n) a x() t h( nT − a t)<br />
= dt<br />
∞<br />
∫<br />
−∞<br />
k<br />
( k n) x( t) h ( na T − t)<br />
= dt<br />
k<br />
DWT...Discrete Wavelet Transform<br />
figure 12: Analysis bank <strong>of</strong> DWT<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
27
SPSC – Signal Processing & Speech Communication Lab<br />
Time Frequency Grid<br />
figure 13: time – frequency grid<br />
D t<br />
D f<br />
=<br />
const.<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
28
SPSC – Signal Processing & Speech Communication Lab<br />
General Definition <strong>of</strong> the Wavelet Transform<br />
∞<br />
1 ⎛ t − q ⎞<br />
X CWT<br />
( p,<br />
q) = ∫ x()<br />
t f ⎜ ⎟dt<br />
p −∞ ⎝ p ⎠<br />
p,q ... real – valued continuous variables<br />
According to former definition:<br />
p =<br />
k<br />
a<br />
q<br />
k<br />
= a Tn f ( t) = h( − t)<br />
X<br />
CWT<br />
( p,<br />
q) and X ( k,<br />
n) KKK<br />
wavelet coefficients<br />
DWT<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
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SPSC – Signal Processing & Speech Communication Lab<br />
Inversion <strong>of</strong> Wavelet Transform<br />
x<br />
( t) X ( k,<br />
n) ψ ( t)<br />
= ∑∑<br />
ψ k n<br />
k<br />
( t)<br />
n<br />
DWT<br />
k n<br />
where are the basis functions<br />
Filter Bank Interpretation <strong>of</strong> Inversion<br />
Reconstruction <strong>of</strong> x(t) as a designing problem <strong>of</strong> the following synthesis filter bank<br />
( k,<br />
n) K sequence<br />
( jΩ) K continuous intime<br />
X DWT<br />
F k<br />
output <strong>of</strong> synthesis filter bank :<br />
k<br />
( t) = ∑∑X<br />
( k n) f ( t − a nT )<br />
x ˆ<br />
,<br />
k n<br />
DWT<br />
k<br />
figure 14: synthesis bank<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
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SPSC – Signal Processing & Speech Communication Lab<br />
All synthesis filters are again generated from a fixed prototype synthesis filter<br />
f(t) ( mother wavelet )<br />
f<br />
k<br />
−k<br />
2 −k<br />
() t = a f ( a t)<br />
Substituting this in the preceding equation and assuming perfect reconstruction, we get<br />
with:<br />
−k<br />
() = ∑∑ ( ) ( − k<br />
t X k,<br />
n a f a t − nT )<br />
2<br />
x<br />
DWT<br />
k n<br />
−k<br />
k<br />
k<br />
−<br />
2 −<br />
−k<br />
k<br />
() t = f () t → ψ () t = a ψ ( a t − nT ) = a ψ [ a ( t − na T )] Kset<br />
<strong>of</strong> basis functions<br />
2<br />
ψ<br />
k n<br />
using this, we can express each basis function in terms <strong>of</strong> the filter f k<br />
( t)<br />
ψ<br />
k n<br />
( ) (<br />
k<br />
t = f t − na T )<br />
k<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
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SPSC – Signal Processing & Speech Communication Lab<br />
Orthonormal Basis<br />
{ ()}<br />
Of particular interest is the case where ψ k<br />
t is a set <strong>of</strong> orthonormal<br />
n<br />
functions<br />
Therefore, we expect:<br />
∞<br />
∫ψ *<br />
k<br />
−∞<br />
n<br />
() t ψ () t dt = δ ( k − l) δ ( n − m)<br />
l m<br />
using Parseval’s theorem, this becomes<br />
1<br />
2π<br />
∞<br />
∫<br />
−∞<br />
Ψ<br />
*<br />
k n<br />
( jΩ) Ψ ( jΩ) dΩ = δ ( k − l) δ ( n − m)<br />
l m<br />
and get :<br />
X<br />
DWT<br />
∞<br />
∫<br />
−∞<br />
*<br />
( k, n) x() t ψ () t<br />
= dt<br />
k n<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
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SPSC – Signal Processing & Speech Communication Lab<br />
Comparing these results, we can conclude:<br />
ψ<br />
k n<br />
And in particular for k = 0 and n = 0:<br />
( ) (<br />
k<br />
t = h<br />
* a nT − t)<br />
*<br />
ψ ( t) = ( t) = h ( − t)<br />
for the orthonormal case fk<br />
( t) = h<br />
*<br />
k ( − t)<br />
00<br />
ψ<br />
k<br />
H<br />
Discrete – Time Wavelet Transform<br />
Starting with the frequency domain relation and a scaling factor a = 2<br />
k<br />
k<br />
j<br />
( e ) ( ) ω j2<br />
ω<br />
= H e K k is a nonnegative int eger<br />
H<br />
jω<br />
( e )<br />
for highpass and k = 1, k = 2<br />
figure 15: Magnitude responses<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
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SPSC – Signal Processing & Speech Communication Lab<br />
Let G(z) be a lowpass with response<br />
Using QMF – banks<br />
figure 16: Magnitude – response <strong>of</strong> G(z)<br />
or<br />
its equivalent<br />
figure 17: 3 level binary tree-structured QMF<br />
figure 18: equivalent 4-channel system<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
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SPSC – Signal Processing & Speech Communication Lab<br />
2<br />
2 4<br />
Responses <strong>of</strong> the filters H ( z) , G( z) H ( z ),<br />
G( z) G( z ) H ( z ),......<br />
figure 19: combinations <strong>of</strong> H(z) and G(z)<br />
Defining the Discrete –Time Wavelet Transform<br />
M −1<br />
k +<br />
( n) = x( m) h ( 2 1 n − m)<br />
∑ ∞<br />
m=<br />
−∞<br />
y , 0 ≤ k ≤ M − 2<br />
k<br />
k<br />
M −<br />
( n) = x( m) h ( 2 1 n − m) ( D T WT )<br />
∑ ∞<br />
m=<br />
−∞<br />
y ,<br />
M −1<br />
iscrete<br />
ime<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
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SPSC – Signal Processing & Speech Communication Lab<br />
Inverse Transform<br />
F<br />
2<br />
( z) = H ( z) , F ( z) H ( z ) G ( ),<br />
K<br />
0 s 1<br />
=<br />
s s<br />
z<br />
figure 20: synthesis filters<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
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SPSC – Signal Processing & Speech Communication Lab<br />
For perfect reconstruction x ˆ( n) = x( n)<br />
we can express<br />
X<br />
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )<br />
2<br />
4<br />
2<br />
2<br />
z = F z Y z + F z Y z + + F z Y z F z Y z<br />
0 0<br />
1 1<br />
...<br />
M −1<br />
M − 2 M −2<br />
+<br />
M −1<br />
M −1<br />
M −1<br />
and in time domain:<br />
x<br />
M −2<br />
∞<br />
∞<br />
2k<br />
+ 1<br />
M −1<br />
( n) = y ( m) f ( n − 2 m) + y ( m) f ( n − 2 m)<br />
∑∑<br />
∑<br />
k k<br />
k=<br />
0 m=−∞<br />
m=−∞<br />
M −1<br />
M −1<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
37
SPSC – Signal Processing & Speech Communication Lab<br />
Main References<br />
Multirate Systems and Filter Banks<br />
(Prentice Hall Signal Processing Series)<br />
by P. P. Vaidyanathan<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
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SPSC – Signal Processing & Speech Communication Lab<br />
Thank you for attention !<br />
Pr<strong>of</strong>essor Georg Holzmann, Horst Cerjak, Christian 19.12.2005 Wallinger 12.06.07 Wavelet T. - Relation to Filter Banks<br />
39