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Welcome to Time-Frequency
Analysis, Adaptive Filtering and
Source Separation
Lecture 6: Filter Banks
Wavelet Packet and Parameterization
Ernest N. Kamavuako
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From surface to deep learning
Storyline
Questions and Answers
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Continuous Wavelet Transform (CWT)
From french: ondelette (small wave)
Finite in time
π π, π = π₯ π‘ β1
π
+β
ββΟβπ‘βπ
πdt
Different values of a and b gives a serie of wavelets that may
be addedd together to reconstruct the signal
They are all localized in both time and frequency, but not
precisely localized in either.
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Continuous Wavelet Transform (CWT)
π π, π = π₯ π‘ β1
π
+β
ββΟβπ‘βπ
πdt
π₯(π‘) = 1
πΆ π(π, π) β
+β
ββΟβ π‘ dπππ
+β
ββ
CWT
iCWT
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Discrete Wavelet Transform (DWT)
DFT and CFT
Why CWT and DWT?
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Multiresolution Analysis
ππβ1 ππ
ππ+1
ππ+2
ππ
ππ+1
ππ+2
V: approximation space
W: detail space
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2βπ2 β π 2βππ‘ β π ππ πππ‘βπππππππ πππ ππ πππ ππ
π½ π is called Scaling function
ππβ1 = ππ+π
+β
π=0
2βπ2 β Ο 2βππ‘ β π ππ πππ‘βπππππππ πππ ππ πππ ππ
Ο π is called wavelet function
Multiresolution Analysis
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Discrete Wavelet transform
π π, π = π π‘ β1
π
+β
ββ
Οβπ‘ β π
πdt
π = 2π and b = 2ππ : Dyadic wavelet transform
π½π,π = π π‘ β1
2π
+β
ββ
Οβπ‘ β 2ππ
2πdt
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Filter Banks
A filter bank is an array of band-pass filters that separates the
input signal into multiple components, each one carrying a
single frequency subband of the original signal.
We have seen that multiresolution Analysis allows us to
decompose a signal into approximations and details.
Filter Bank is a way to implement the MRA and DWT.
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Filter Banks
ππ0π = πππ π‘ β π = π(π‘)
π
ππ1π = ππ1
2ππ‘
2β π
π
ππ1π = ππ1
2Ξ¨π‘
2β π
π
We would like to find ππ and ππ, not by using π(π‘) but its
representation in π0(ππ). ππ, ππ?
π0 π1
π2
π1
π2
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Analysis: from fine scale to coarser scale
C[n] g[n]
h[n]
2
2 a1[n]
d1[n]
g[n] 2 d2[n]
h[n] 2 a2[n] Matlab functions: dwt and
wavedec
[cA, cD] = dwt(x, Lo, Hi);
= dwt(x, 'wname');
[C, L] = wavedec(x, N, Lo, Hi);
= wavedec(x, N, 'wname');
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Analysis: from fine scale to coarser scale
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Synthesis: from coarse scale to fine scale
g[n]
h[n]
2
a1[n]
d1[n]
2
+ C[n]
Matlab functions: idwt and waverec
x = idwt(cA, cD, Lo, Hi);
= idwt(cA, cD, 'wname'); x = waverec(C, L, Lo, Hi);
= waverec(C,L, 'wname');
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Wavelet Packet
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Wavelet Parameterization
WT requires the selection of the mother wavelet.
Wavelet usually designed similar to the signal.
Here The mother wavelet is parameterized.
Ο is defined by a low-pass filter h and its associated
high-pass filter g.
)22/())sin()cos(1(3
)22/())sin()cos(1(2
)22/())sin()cos(1(1
)22/())sin()cos(1(0
h
h
h
h
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Wavelet Parameterization
If Ξ± = 0, β = 0,1
2,1
2, 0 g = 0,
1
2, β1
2, 0
[h,g] = wfilters(βdb2β) Flip h and change signs of odd values
β = β0.1294, 0.2241, 0.8365, 0.4830 , g = β0.4830, 0.8365,β0.2241,β0.1294
]1[)1(][ 1 nhng n
)22/())sin()cos(1(3
)22/())sin()cos(1(2
)22/())sin()cos(1(1
)22/())sin()cos(1(0
h
h
h
h