Advanced Digital Signal Processing

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DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF
JOENSUU JOENSUU, FINLAND

• Advanced Digital Signal Processing
• Lecture 16
• Wavelet Transform
• Alexander Kolesnikov
• (16.11.2005)

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Short-term Fourier Transform
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Two test signals again
a)
?1 10? ?2 20? ?3 40? ?4100?
x(t)cos(?1 t)cos(?2t)cos(?3)cos(?4t)
b)
x1(t)cos(?1t) x2(t)cos(?2t) x3(t)cos(?3t) x4(t)
cos(?4t)
x1(t) x2(t) x3(t) x4(t)
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Spectrums of the test signals
Signals are different, spectrums are similar
Why?
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What is wrong with the Fourier Transform?
Two basis functions sin(?t) and ?(t)
Support region In time In frequency
sin(?t) ? 0
?(t) 0 ? The basis function sin(?t) is
not localized in time, The ?(t) (sample) is not
localized in frequency.
Fourier Transform is good for stationary
signals, but real signals are non-stationary
ones. Another problem signals with
discontinuties.
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Short-term Fourier Transform (STFT)
signal x(t)
window h(t-?)
signal in window
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Short-term Fourier Transform (STFT)
Introduce basis functions which are compact
in time and frequency domains. In other words,
let us divide the input signal into
time sub-intervals, and perform DFT for every
sub-interval
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STFT Time-frequency plane
Equidistant frequencies
??
?t
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Problems with STFT
Uncertainty Principle
We cannot localize events in time and frequency
simultaneously!
Improved time resolution?Degraded frequency
resolution Improved frequency resolution
?Degraded time resolution
Problem the same ?? and ?t throught the entire
plane!
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Wavelet Transform
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The main idea
Keep the relative accuracy in the frequency
domain
Use wider time window for lower frequencies!
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Time-frequency plane
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Scale Illustration
Coast line

• Coarse scale
• flying in a jet at 5 km

• Medium scale
• bird flight at 100 m

• Fine scale
• beach walk

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Wavelet transform wavelet mother function
Two properties of mother wavelet function
mother wavelet
baby wavelets
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Wavelet transform wavelet mother function
How to obtain a set of wavelet functions?
Translation (?) and dilation (scaling, s)
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Scaling (stretching or compressing)
s1
s0.5
s0.25
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Translation (shift)
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Translation and stretching
s2-7
s2-6
s2-6
s2-6
s2-5
s2-4
s2-3
s2-3
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Localization in time
Baby wavelets Mother wavelet
...
...
k0,1,...,
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Examples of mother wavelets
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Haar wavelets
Scaling function
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Discrete wavelet transform
j1
j
k
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Hierarchical calculation of DWT
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Haar wavelet transform
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Haar DWT Example
Input data Xx1,x2,x3,, x16 Haar wavelet
transform (a,b)?(s,d) where 1) scaling
function s(ab)/2 (smooth, LPF) 2) Haar
wavelet d(a-b) (details, HPF)
X10,13, 11,14, 12,15, 12,14, 12,13, 11,13,
10,11 11.5,12.5, 13.5,13, 12.5,12, 10.5
-3, -3, -3, -2, -1,-2,-1 12, 13.25,
12.25, 10.5 -2,0.5,-0.5 -3, -3, -3, -2,
-1,-2,-1 12.625, 11.375 -1.25, 1.75
-2,0.5,-0.5 -3, -3, -3, -2, -1,-2,-1
121.25 -1.25, 1.75 -2,0.5,-0.5 -3, -3,
-3, -2, -1,-2,-1
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Inverse Haar DWT Example
Inverse Haar wavelet transform (s,d)? (a,b)
1) asd/2 2) bs?d/2
X10,13, 11,14, 12,15, 12,14, 12,13, 11,13,
10,11? 11.5,12.5, 13.5,13, 12.5,12, 10.5
-3, -3, -3, -2, -1,-2,-1 12, 13.25,
12.25, 10.5 -2,0.5,-0.5 -3, -3, -3, -2,
-1,-2,-1 12.625, 11.375 -1.25, 1.75
-2,0.5,-0.5 -3, -3, -3, -2, -1,-2,-1
121.25 -1.25, 1.75 -2,0.5,-0.5 -3, -3,
-3, -2, -1,-2,-1
Y 121.25 -1.25, 1.75 -2,0.5,-0.5 -3,
-3, -3, -2, -1,-2,-1 12.625,11.375
-1.25, 1.75 -2,0.5,-0.5 -3, -3, -3, -2,
-1,-2,-1 12, 13.25, 12.25, 10.5
-2,0.5,-0.5 -3, -3, -3, -2, -1,-2,-1
11.5,12.5, 13.5,13, 12.5,12, 10.5 -3, -3,
-3, -2, -1,-2,-1 10,13, 11,14, 12,15,
12,14, 12,13, 11,13, 10,11
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2-D Wavelet transform
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2-D wavelet transform
Transform Coeff. 4123, -12.4, -96.7, 4.5,
Original 128, 129, 125, 64, 65,
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Wavelet transform as Subband Transform
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Wavelet Transform and Filter Banks
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Wavelet Transform and Filter Banks
h0(n) is scaling function, low pass filter
(LPF) h1(n) is wavelet function, high pass filter
(HPF)
is subsampling (decimation)
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5/3 filter for lossless encoding (JPEG2000)
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9/7 filter for lossy encoding (JPEG2000)
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5-tab low pass filter (LPF)
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Filtration
Subsampling
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3-tab high pass filter (HPF)
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Subsampling
Filtration
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Filtration with 5/3-filters
Input x
Input x
LPF s
HPF d
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Inverse wavelet transform
is up-sampling (zeroes inserting)
Synthesis filters g0(n)?(-1)nh1(n)
g1(n)?(-1)nh0(n)
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Complexity of discrete wavelet transform
Without scaling-function property
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Scaling function and Wavelets
Scaling function
Wavelet function
where
This is very important property of the
wavelets. To obtain WT coefficients for level j
we can process WT coefficients for level j1.
? DWT Complexity O(N)
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Haar Scaling function and Wavelets
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Daubechies wavelets of order 2
?(t)
?(t)
Scaling function Wavelet
function
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Multiresolution discrete wavelet analysis
Low-resolution approximation
Wavelets details
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Applications
1. Image compression (JPEG200) 2. Signal
filtering (denoising) 3. Signal analysis 4. etc.