Welcome to Time-Frequency Analysis, Adaptive Filtering and

Welcome to Aalborg University No. 1

Welcome to Time-Frequency

Analysis, Adaptive Filtering and

Source Separation

Lecture 6: Filter Banks

Wavelet Packet and Parameterization

Ernest N. Kamavuako


From surface to deep learning

Storyline

Questions and Answers


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.


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


Discrete Wavelet Transform (DWT)

DFT and CFT

Why CWT and DWT?


Multiresolution Analysis

𝑉𝑗−1 𝑉𝑗

𝑉𝑗+1

𝑉𝑗+2

𝑊𝑗

𝑊𝑗+1

𝑊𝑗+2

V: approximation space

W: detail space


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


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


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.


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


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');


Analysis: from fine scale to coarser scale



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');


Wavelet Packet


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


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

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