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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

𝑊 𝑎, 𝑏 = 𝑥 𝑡 ∙1

𝑎

+∞

−∞ψ∗𝑡−𝑏

𝑎dt

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)

𝑊 𝑎, 𝑏 = 𝑥 𝑡 ∙1

𝑎

+∞

−∞ψ∗𝑡−𝑏

𝑎dt

𝑥(𝑡) = 1

𝐶 𝑊(𝑎, 𝑏) ∙

+∞

−∞ψ∗ 𝑡 d𝑎𝑑𝑏

+∞

−∞

CWT

iCWT

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Discrete Wavelet Transform (DWT)

DFT and CFT

Why CWT and DWT?

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Multiresolution Analysis

𝑉𝑗−1 𝑉𝑗

𝑉𝑗+1

𝑉𝑗+2

𝑊𝑗

𝑊𝑗+1

𝑊𝑗+2

V: approximation space

W: detail space

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2−𝑗2 ∙ 𝜃 2−𝑗𝑡 − 𝑛 𝑎𝑠 𝑜𝑟𝑡ℎ𝑜𝑛𝑜𝑟𝑚𝑎𝑙 𝑏𝑎𝑠𝑒𝑠 𝑓𝑜𝑟 𝑉𝑗

𝜽 𝒕 is called Scaling function

𝑉𝑗−1 = 𝑊𝑗+𝑘

+∞

𝑘=0

2−𝑗2 ∙ ψ 2−𝑗𝑡 − 𝑛 𝑎𝑠 𝑜𝑟𝑡ℎ𝑜𝑛𝑜𝑟𝑚𝑎𝑙 𝑏𝑎𝑠𝑒𝑠 𝑓𝑜𝑟 𝑊𝑗

ψ 𝒕 is called wavelet function

Multiresolution Analysis

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Discrete Wavelet transform

𝑊 𝑎, 𝑏 = 𝑓 𝑡 ∙1

𝑎

+∞

−∞

ψ∗𝑡 − 𝑏

𝑎dt

𝑎 = 2𝑗 and b = 2𝑗𝑛 : Dyadic wavelet transform

𝛽𝑛,𝑗 = 𝑓 𝑡 ∙1

2𝑗

+∞

−∞

ψ∗𝑡 − 2𝑗𝑛

2𝑗dt

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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

𝑃𝑉0𝑓 = 𝑐𝑛𝜃 𝑡 − 𝑛 = 𝑓(𝑡)

𝑛

𝑃𝑉1𝑓 = 𝑎𝑘1

2𝜃𝑡

2− 𝑛

𝑘

𝑃𝑊1𝑓 = 𝑑𝑘1

2Ψ𝑡

2− 𝑛

𝑘

We would like to find 𝑎𝑘 and 𝑑𝑘, not by using 𝑓(𝑡) but its

representation in 𝑉0(𝑐𝑛). 𝑎𝑘, 𝑑𝑘?

𝑉0 𝑉1

𝑉2

𝑊1

𝑊2

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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.

ψ 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 α = 0, ℎ = 0,1

2,1

2, 0 g = 0,

1

2, −1

2, 0

[h,g] = wfilters(‘db2’) Flip h and change signs of odd values

ℎ = −0.1294, 0.2241, 0.8365, 0.4830 , g = −0.4830, 0.8365,−0.2241,−0.1294

]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