1
Lecture 13: Multirate processing
and wavelets fundamentals
Instructor:
Dr. Gleb V. Tcheslavski
Contact:
gleb@ee.lamar.edu
Office Hours: Room 2030
Class web site:
http://ee.lamar.edu/gleb/ds
p/index.htm
ELEN 5346/4304
DSP and Filter Design
Fall 2008
2
Filter banks
A digital filter bank (FB) is a set of
digital bandpass (usually) filters with
either a common input (analysis FB)
or a summed output (synthesis FB).
Analysis FB separates the analyzed signal xn into a set of M spectral bands.
A uniform filter bank – equal passband widths
Let a causal IIR LPF with a real impulse response h0,n have a transfer function:

H 0 (z) 

h0 , n z
n
(13.2.1)
n0
Assume that the filter has its passband and stopband frequencies around /M as
indicated in the picture, where M is an arbitrary integer.
ELEN 5346/4304
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Fall 2008
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Filter banks
Let us consider next the casual filter with
the impulse response obtained by
modulating h0,n with the exponent:
h k , n  h0 , n e
j 2  kn M
(13.3.1)
The corresponding transfer function is

H k (z) 
h
n0
k ,n
z
n


h
n0
o ,n
  j 2M k 
 ze



n
 j 2M k 
 H 0  ze



(13.3.2)
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DSP and Filter Design
Fall 2008
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Filter banks
The frequency response of the filter is
H k e
j

 j    2  k  
 H 0  e  M   ;0  k  M  1




(13.4.1)
Therefore, the frequency response is obtained by shifting the frequency response
of the LPF “to the right” by the amount 2k/M as shown.
By designing M-1 filters obtained by modulating (shifting) the low pass prototype,
we get M-1 uniformly shifted (in frequency) versions of the prototype H0.
These M filters can be arranged into a uniform filter bank.
Note: a DFT algorithm can be viewed in terms of a uniform filter bank.
ELEN 5346/4304
DSP and Filter Design
Fall 2008
5
Filter banks
Two-channel Quadrature Mirror (QM) filter bank
In many applications, a discrete-time signal xn is first split into a number of
subband signals {vk,n} by an analysis FB; the subband signals are then (somehow)
processed and finally combined by a synthesis FB to form an output yn.
If the bandwidths of the subband signals are (chosen to be) much smaller than the
bandwidth of the original signal, subband signals can be downsampled (sampled
at a lower sampling rate) before the processing. This allows more efficient
processing.
Downsampling by the factor Q
p
s
ELEN 5346/4304
N
DSP and Filter Design
Qp
’N = N/Q
Qs
Fall 2008
6
Filter banks
After processing, these signals are upsampled and then combined by the
synthesis bank into the output (higher rate) signal. The structure performing such
operations is called a quadrature-mirror (QM) filter bank.
Assuming no processing,
a two-channel QM FB is:
Note that for two
channels, Q = 2.
Analyzing this structure:
Vk ( z )  H k ( z ) X ( z )
(13.6.1)
Downsampling and upsampling can be represented as:
U k (z) 
1
V k  z 1 2   V k   z 1 2  

2
2
Vˆk ( z )  U k ( z )
Here, k = 0,1.
ELEN 5346/4304
DSP and Filter Design
Fall 2008
(13.6.2)
(13.6.3)
7
Filter banks
The last equation can be rewritten as:
1
1
Vˆk ( z )  V k ( z )  V k (  z )    H k ( z ) X ( z )  H k (  z ) X (  z ) 
2
2
(13.7.1)
Therefore, the output of the FB is:
Y ( z )  G 0 ( z )Vˆ0 ( z )  G 1 ( z )Vˆ1 ( z )

1
2
 H 0 ( z ) G 0 ( z )  H 1 ( z ) G1 ( z )  X ( z ) 
1
2
 H 0 (  z ) G 0 ( z )  H 1 (  z ) G1 ( z )  X (  z )
(13.7.2)
Which can be rewritten as:
Y ( z )  T ( z ) X ( z )  A( z ) X ( z )
T (z) 
where
1
 H 0 ( z ) G 0 ( z )  H 1 ( z ) G1 ( z ) 
2
is the distortion transfer function
1
A ( z )   H 0 (  z )G 0 ( z )  H 1 (  z ) G1 ( z ) 
2
is the aliasing term.
ELEN 5346/4304
DSP and Filter Design
Fall 2008
(13.7.3)
(13.7.4)
(13.7.5)
8
Filter banks
In general, QM FB are Linear Time-Varying systems. However, by selecting the
analysis and synthesis filters such that aliasing effects are canceled, QM FB can
be made an LTI system.
Aliasing cancellation condition
A ( z )  0  H 0 (  z ) G 0 ( z )  H 1 (  z ) G1 ( z )  0
Will hold when, for instance:
In this situation:
(13.8.1)
G 0 ( z )  H 1 (  z );
(13.8.2)
G1 ( z )   H 0 (  z )
(13.8.3)
Y (z)  T (z) X (z)
(13.8.4)
If T(z) is an allpass filter, i.e. |T(z)| = c  0
Y (z)  c X (z)
(13.8.5)
Therefore, amplitude distortions are possible. We can also show that no phase
distortions will be introduced if T(z) has a linear phase.
ELEN 5346/4304
DSP and Filter Design
Fall 2008
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Filter banks
Perfect reconstruction condition
An alias-free QM FB having no amplitude and phase distortions is called a
perfect reconstruction QM FB. In this case:
H 0 ( z ) G 0 ( z )  H 1 ( z ) G1 ( z )  2 z
which implies that:
Therefore:
and
T (z)  z
l
l
l
(13.9.1)
(13.9.2)
Y (z)  z X (z)
(13.9.3)
y n  xn l
(13.9.4)
Indicating that the reconstructed output is a delayed by l samples replica
of the input.
ELEN 5346/4304
DSP and Filter Design
Fall 2008
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Filter banks (QM FB example)
Let us consider a two-channel QM FB with the analysis filters:
1
H 0 (z) 
2
1  z a  LPF ; H
1
(z) 
1
1
2
1  z a  H PF
(13.10.1)
  1  z a  H PF
(13.10.2)
1
and the synthesis filters:
G0 ( z ) 
1
2
1  z
1
a  LPF ;G1 ( z ) 
1
2
1
Therefore:
A ( z ) H 0 (  z ) G 0 ( z )  H 1 (  z ) G 1 ( z ) 

1
2
1  z
2

1
2
1
2
1
 1  z   0
2
DSP and Filter Design
1
1
1
2
(13.10.3)
the two-channel QM FB is an alias-free system.
ELEN 5346/4304
1
 1  z  1  z   1  z    1  z 
Fall 2008
11
Filter banks (QM FB example)
On the other hand:
T (z) 
1
2
 H 0 ( z ) G 0 ( z )  H 1 ( z ) G1 ( z )  
1
4
1  z
1
 1  z
1

1
4
1  z    1  z   2 z
1
1
1
(13.11.1)
Therefore, this FB is a perfect reconstruction system.
In a time domain, the outputs of the analysis filters are:
u 0 ,n 
Note: the scaling factor 1
1
2
 x n  x n 1  ,u1, n 
1
2
 x n  x n 1 
2 preserves energy. In a matrix form:
 u 0 ,n 
 xn 

  H2 

u
x
1,
n
 n 1 


where
H2 
1 1

2 1
The reconstruction:
ELEN 5346/4304
DSP and Filter Design
(13.11.2)
1 

 1
is the Haar transform matrix
 u 0 ,n 
 xn 
t


  H2 
u
1,
n
 x n 1 


Fall 2008
(13.11.3)
(13.11.4)
(13.11.5)
12
Filter banks
From alias-free and perfect reconstruction requirements:
G0 ( z )  H 0 ( z )
Therefore:
For a real filter:
(13.12.1)
H 1 ( z )  G0 ( z )  H 0 ( z )
(13.12.2)
G1 ( z )   H 1 ( z )   H 0 (  z )
(13.12.3)
H1 e
j


 H0 e
j   

(13.12.4)
Indicating that if h0 is a LPF, than h1 is a HPF; in fact, its mirror image with
respect to the /2 – the quadrature frequency. Therefore, two analysis filters and
two synthesis filters in the QM FB are essentially determined from one prototype
transfer function.
QM FB can be build from FIR or IIR filters.
ELEN 5346/4304
DSP and Filter Design
Fall 2008
13
Filter banks
Orthogonal QM FB
G0 ( z )  z
N
1
H 0 ( z ),G 1 ( z )  z
N
1
H 1 ( z ), H 1 ( z )  z
N
1
H 0 ( z )
(13.13.1)
Implying that if H0(z) is a causal FIR filter, three other filters must be causal FIR
filters. Also, if H0(z) is a LPF, than H1(z) is a HPF.
Third-order FIR filters
forming analysis QM FB
7-order FIR filters
forming analysis QM FB
Biorthogonal QM FB – QM FB
obtained for the perfect
reconstruction condition and
represented by nonorthogonal
transfer matrices (and filters).
ELEN 5346/4304
DSP and Filter Design
Fall 2008
14
Filter banks
L-channel QM FB
is a generalization of a
two-channel QM FB.
0  k  L 1
Vk ( z )  H k ( z ) X ( z )
U k (z) 
1
L 1

L
Hk z
1 L
(13.14.1)
e
 j 2 l L
 X z
1 L
e
 j 2 l L

(13.14.2)
l0
L
Vˆk ( z )  U k ( z )
ELEN 5346/4304
DSP and Filter Design
(13.14.3)
Fall 2008
15
Filter banks
The vector of downsampled subband signals of an L-channel QM FB can be
expressed in the matrix form as:
u(z) 
1
H
L
 z  
1 L
t
x
(m )
z 
1 L
u ( z )  U 0 ( z )U 1 ( z )...U L 1 ( z ) 
Where:
The input is:
x
(m )
H 0 (z)

 H0 z e
(m )
H (z)  

 H z  e  j 2   L 1  L
 0
t
(13.15.2)
H1(z)
H1  z e
 j 2 L

DSP and Filter Design
(13.15.1)
 j 2   L 1  L
 j 2 L
( z )   X ( z ) X ( z  e
)... X ( z  e
)


The analysis FB

modulation matrix: 
ELEN 5346/4304
(m )


H1 z e
Fall 2008
 j 2 L
t
(13.15.3)
H L 1 ( z )

 j 2   L 1  L
H L 1  z  e


H L 1 z  e
 j 2 L

 j 2   L 1  L




 (13.15.4)



16
Filter banks
The output of the QM FB is
L 1
G
Y (z) 
k
( z )Vˆk ( z )
(13.16.1)
k 0
which in the matrix form is
y
where the output is:
y
(m )
(m )
(z)  G
( z )   Y ( z )Y ( z  e

The synthesis FB

modulation matrix: 
G0 ( z)
T(z) 

1
L
ELEN 5346/4304
DSP and Filter Design
 j 2 L
G1  z  e
 j 2 L

( z )u  z
L

(13.16.2)
)...Y ( z  e
 j 2   L 1  L
G1 ( z )

 G0  z  e
(m )
G (z)  

 G z  e  j 2   L 1 L
 0
The transfer matrix:
(m )

G1 z  e
G
(m )
Fall 2008
(13.16.3)
G L 1 ( z )
 j 2 L

 j 2   L 1  L
( z )  H
)

t
(m )
G L 1  z  e

( z ) 

G L 1 z  e
 j 2 L

 j 2   L 1  L




 (13.16.4)



t
(13.16.5)
17
Filter banks
From the alias cancellation condition, the set of synthesis filters can be determined:
G ( z )   H
(m )
( z ) 
1
t( z)
t ( z )   L  T ( z )...0 
Where
and the distortion transfer function is:
T ( z )  a0 ( z ) 
1
(13.17.1)
t
L 1
H

L
k
( z )G k ( z )
k 0
If T(z) has a constant magnitude, the QM FB is magnitude-preserving.
If T(z) has a linear phase, the QM FB has no phase distortion.
If T(z) is a pure delay, it is a perfect reconstruction QM FB i.e.:
T (z)  z
 n0
(13.17.4)
Three-channel FIR QM FB
ELEN 5346/4304
DSP and Filter Design
(13.17.2)
Fall 2008
(13.17.3)
18
Filter banks
Multilevel FB
A multichannel FB can be replaced by a multilevel FB called a tree-structured FB.
Analysis and synthesis filters can be related between these two structures:
H 00 ( z )  H 0 ( z ) H 0 ( z ), H 01 ( z )  H 0 ( z ) H 1 ( z )
2
2
H 10 ( z )  H 1 ( z ) H 0 ( z ), H 11 ( z )  H 1 ( z ) H 1 ( z )
2
2
G 00 ( z )  G 0 ( z ) G 0 ( z ),G 01 ( z )  G 0 ( z ) G 1 ( z )
2
2
G 10 ( z )  G 1 ( z ) G 0 ( z ),G 11 ( z )  G 1 ( z ) G 1 ( z )
2
ELEN 5346/4304
DSP and Filter Design
2
Fall 2008
(13.18.1)
19
Filter banks
FB considered so far have equal passband widths (full tree FB). Sometimes, it is
beneficial to make FB with unequal passband widths (partial tree FB).
H 000 ( z )  H 0 ( z ) H 0 ( z ) H 0 ( z ), H 001 ( z )  H 0 ( z ) H 0 ( z ) H 1 ( z )
2
4
2
4
H 01 ( z )  H 0 ( z ) H 1 ( z ), H 1 ( z )
(13.19.1)
2
G 000 ( z )  G 0 ( z ) G 0 ( z ) G 0 ( z ),G 001 ( z )  G 0 ( z ) G 0 ( z ) G 1 ( z )
2
4
2
G 01 ( z )  G 0 ( z ) G 1 ( z ),G 1 ( z )
2
Corresponding filters’ magnitude responses
ELEN 5346/4304
DSP and Filter Design
Fall 2008
4
20
Discrete Wavelet Transform (DWT)
It turns out that perfect reconstruction octave band QM FB can be represented
by a certain discrete wavelet transform (DWT).
For this binary
tree, there are
two “parent”
analysis FIR
filters:
H0(z) – LPF, and
H1(z) – HPF.
Which can also be represented by the equivalent
structure where 0  k  L and:
i
k 1
 k 2

2
2
H k ,1 ( z )    H 0 z  H 1 z
 i0

 
k 1
H k ,0 ( z ) 

i0
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DSP and Filter Design


 
H0 z
2
(13.20.1)
i
(13.20.2)
Fall 2008
21
Discrete Wavelet Transform (DWT)
Assuming that the input xn is a sequence of length M, the outputs will have the
following approximate lengths and bandwidths:
The downsampling leads to:

2

 
downsampling by 2
 
2
In the wavelet decomposition, u0,n represents the “finest details” (the highest
frequency components) of the input xn, while u4,n is its “coarsest” representation
(its lowest frequency components).
ELEN 5346/4304
DSP and Filter Design
Fall 2008
22
Discrete Wavelet Transform (DWT)
The process of generating the set of output subsequences uk,n where 0  k  L
from the finite-length input xn by the structure that can be represented as an Llevel octave-band perfect reconstruction QM FB is called the discrete wavelet
transform (DWT). The analysis sequences for the equivalent representation are:
X k ( z )  H k ,1 ( z ) X ( z ),0  k  L  1
X k ( z )  H L ,1 ( z ) X ( z ), k  L
or in the time domain:
(13.22.1)

xk ,n 

h k ,1, n  m x m ,0  k  L  1
m  
(13.22.2)

x L ,n 

h L ,0 , n  m x m , k  L
m  
The subband sequences:

u k ,n 

m  
h k ,1, 2 k 1 n  m x m ,0  k  L  1
(13.22.3)

u L ,n 

m  
ELEN 5346/4304
DSP and Filter Design
h L ,0 , 2 k n  m x m ,k  L
Fall 2008
23
Discrete Wavelet Transform (DWT)
The process of reconstruction (synthesis) of the output sequence yn – which is
a replica of the input xn – from the subband sequences un can be accomplished
by the L-level octave-band perfect reconstruction QM FB:
Which has an
equivalent structure,
where
i
k 1
 k 2

2
2
G k ,1 ( z )    G 0 z  G 1 z
 i0

 
k 1
G k ,0 ( z ) 

i0
ELEN 5346/4304
DSP and Filter Design


 
G0 z
2
(13.23.1)
i
(13.23.2)
Fall 2008
24
Discrete Wavelet Transform (DWT)
Therefore, the output will be:
 G
L
Y (z) 
G
k ,1
( z )U k 1 z
k 1
2
 
(13.24.1)
g L ,0 , n  2 L m u L , m
(13.24.2)
k
L ,0
( z )U L z
2
L
or, in the time domain:
L
yn 

 
k  1 m  
Using the notation:

g k ,1, n  2 k 1 m u k , m 
 k , m , n  g k ,1, n  2
k 1
 L , m , n  g L ,0 , n  2
the output will be:
L
yn 
L
m
m

m  
,0  k  L  1
, k  L
(13.24.3)

 
 k ,m ,n u k ,m
(13.24.4)
k  0 m  
Which is the inverse DWT. uk,m are the wavelet coefficients of xm with respect to
the basis function k,m,n
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Discrete Wavelet Transform (DWT)
Biorthogonal wavelets
The biorthogonal wavelets are generated from biorthogonal octave-band
perfect reconstruction QM FB.
The simplest
examples are Haar
wavelets:
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Fall 2008
26
Discrete Wavelet Transform (DWT)
Orthogonal wavelets
The orthogonal wavelets are generated from orthogonal octave-band perfect
reconstruction QM FB.
One example would
be the four-level
Daubechies 4/4-tap
wavelet:
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Fall 2008
27
On the Uncertainty Principle
Observations:
• A short-duration (“narrow”) waveform (signal) yields a long-duration (“wide”)
spectrum.
• A narrow spectrum yields a long-duration (“wide”) waveform.
• Waveform and spectrum cannot be made arbitrary small simultaneously.
These observations are explained by the Uncertainty Principle:
     C onst 
2
t
2
1
(13.27.1)
4
“=“ for Gaussian signals
Here t2 is a variance in time – a measure of the duration of the signal;
2 is a variance in frequency – a measure of the bandwidth of the signal (most of
the signal energy will be concentrated in 2B = 2 rad/s.
Consequence: the same signal cannot have both: fine resolution in time and fine
resolution in frequency.
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DSP and Filter Design
Fall 2008
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Non-stationary… So what?
Signals
Estimated spectra
2
1.5
Case 1:
1
Stationary:
0.5
0
20
0
-20
-1.5
0
300
400
500
600
700
800
900
0
100
1000
1
0.8
0.6
Case 2:
0.4
0.2
Non –
stationary:
0
-0.2
-0.4
S1  S2
-0.6
-0.8
200
400
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600
800
1000
1200
DSP and Filter Design
1400
1600
1800
2000
Fall 2008
1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Normalized Frequency ( rad/sample)
0.9
1
50
0
-50
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Normalized Frequency ( rad/sample)
0.9
1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Normalized Frequency ( rad/sample)
0.9
1
5
1
x 10
0
-1
-2
0
0.9
-1
Magnitude (dB)
200
0.3
0.4
0.5
0.6
0.7
0.8
Normalized Frequency ( rad/sample)
0
Phase (degrees)
100
0.2
x 10
-2
0
0.1
5
1
Phase (degrees)
-1
-1
40
S1 + S2
-0.5
-2
Magnitude (dB)
60
0
29
Non-stationary… So what?
Ideally:
from “Modern DSP” by Cristi
ELEN 5346/4304
DSP and Filter Design
Fall 2008
30
Short-Time Fourier Transform*
The Short-Time Fourier Transform
(STFT) of a windowed signal with the
window centered at t is:
X (t ,  ) 
w

x ( ) w (  t ) e
*
 j 
d  (13.30.1)
where w(t) is a low pass analysis
window function, and (typically):

2
w ( t ) dt  1
(13.30.2)
The inverse STFT is defined as:
x (t ) 
1
2

X ( ,  ')  ( t   ) e
w
j 't
d d  '
(13.30.3)
Given that:
 w ( t ) ( t ) dt  1
*
(13.30.4)
* - based on the lecture notes for “Advanced topics in Signal Processing” by Dr. Ben H. Jansen, UH, 2006
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Short-Time Fourier Transform
A natural choice for the synthesis window (t), is to make it equal to the analysis
window w(t).
1. STFT is a linear time-frequency representation;
2. STFT depends on the choice of w(t);
3. STFT is generally complex.
Defining a spectrogram as:
X (t ,  )
w
2
which is a representation of the energy distribution in the time-frequency plane, i.e.
a quadratic time-frequency representation. We can show that:
ELEN 5346/4304
2
E 


X (t ,  ) d  
w

X ( t ,  ) dt 
w
DSP and Filter Design

X ( t ,  ) dtd  
w
2
2


x ( )
2
X (  ')
2
2
x ( t ) dt   w ( t ) dt
2
w (  t ) d   x ( t )
2
2
(13.31.1)
2
W (    ') d  '  X (  )
Fall 2008
(13.31.2)
2
(13.31.3)
32
Short-Time Fourier Transform
We can notice that:
 e  j t x (t )   w * (  t ) 



x ( ) w (  t ) e
*
 j 
d
(13.32.1)
Therefore, the STFT can be viewed as a
filtering operation by a LPF with impulse
response w*(-t) as shown in (A):
Also:
*
 j t

x ( t )   w (  t ) e


e
x ( ) w (  t ) e
*
 j t
X (t ,  )
 j 
e
j t
d
(13.32.2)
Therefore, the STFT can also be viewed as a filtering operation with a BPF with
impulse response ejt w*(-t) as shown in (B):
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Discrete STFT
Define a sampled version of STFT as:

X ( nt , kF ) 
w
x ( ) w (  nT ) e
*
 jk 2  F 
d
(13.33.1)
Therefore, the DSTFT:
X n ,k 
w
x
 j
*
m
wm n e
2  km
, k  0,1, ... L  1
L
(13.33.2)
m
The IDSTFT:
x (t ) 

n
or:
xm 

n
X ( nT , kF )  ( t  nT ) e
w
jk 2  Ft
(13.33.3)
k
X n ,k  m  n e
w
j
2  kn
L
,k  0,1, ..., L  1
(13.33.4)
k
The IDSTFT holds if:
1
F
ELEN 5346/4304
DSP and Filter Design

  t 
n

 *
 nT  w  t  nT    ( k ) for  all t
F

k
Fall 2008
(13.33.5)
34
STFT: Gabor expansion (transform)
Note that (13.33.3) can be viewed as a series expansion of a 1-D signal in terms
of 2-D time-frequency functions. The Gabor expansion is defined as:
x (t ) 
G
n
x
( n , k ) g n ,k (t )
(13.34.1)
k
where g(t) is a 1-D function:
g n ,k (t )  g (t  n T )e
jk 2  F t
(13.34.2)
If the basis functions gn,k(t) are selected such that they are well localized in time
and well concentrated in frequency, then the Gabor coefficients Gx(n,k) will reflect
the signal’s time-frequency content around (nT, kF). Gabor proposed
g (t ) 
which is a Gaussian signal with
  e
 t
2
(13.34.3)
   t  1 4
2
2
The Gabor expansion is always possible if FT  1, but the Gabor coefficients are
not unique (the DSTFT is just one possible solution).
Note: FT of a Gaussian is a Gaussian.
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Time-Frequency resolution of STFT
Using the filter representation of STFT, we observe:
1. All the BPFs have the same duration impulse response, hence the time
resolution of the STFT is the same at all frequencies and is proportional to
window length.
2. All the BPFs have the same bandwidth, hence the frequency resolution is the
same at all times and is inversely proportional to window length.
1/T
T
Although, according to the uncertainty principle,  2   t2  C onst , t could be made
large for low frequency components and small for high frequency components…
leading to corresponding changes in the frequency resolution.
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Fall 2008
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Continuous Wavelet Transform (CWT)
Rather than using constant bandwidth BPFs (as STFT does), we use constant Q
BPFs, where
Q 
c
B
center frequency
(13.36.1)
bandwidth
In other words, the impulse response will decrease in duration (and the bandwidth
increase) as center frequency increases: therefore, use scaled versions of w*(-t).
For simplification, we will use (as window functions) the scaled versions of the
same prototype (t):
  

(13.36.2)
  (t ) 
 
t
c
 c 
Here (t) is the analyzing (mother) wavelet function: a bandpass function centered
around c.
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Continuous Wavelet Transform (CWT)
The CWT is defined as:
( )
Wx
(t ,  ) 

x ( )
 

*
 
  t   d 
 c


c
(13.37.1)
Using b = c/, the CWT can also be defined as a time-scale representation:
( )
Wx
(t , b ) 

x ( )
*   t 
 
 d
 b 
b
1
The energy distribution in the time-domain is evaluated by the scalogram:
W
ELEN 5346/4304
DSP and Filter Design
( )
x
2
(t , b )
Fall 2008
(13.37.2)
38
STFT vs. CWT
STFT estimates the similarity (cross-correlation) between x(t) and g,(t):
X ( ,  ) 
Where:

*
x ( t ) g  , ( t ) dt
g  , ( t )  w ( t   ) e
j t
(13.38.1)
(13.38.2)
CWT estimates the similarity (cross-correlation) between x(t) and hb,(t):
W ( , b ) 
Where:
ELEN 5346/4304
DSP and Filter Design
hb , ( t ) 

*
x ( t ) hb , ( t ) dt
(13.38.1)
  t 


b


b
(13.38.2)
1
Fall 2008
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Same time and frequency resolutions
STFT vs. CWT
STFT:
FT = const, F = const, T = const
CWT:
FT = const, F  const, T  const
from “Modern DSP” by Cristi
ELEN 5346/4304
DSP and Filter Design
Fall 2008