• Details, details…

• Intro to Discrete Wavelet Transform

The Story of WaveletsTheory and Engineering Applications

Discrete Wavelet Transform

CWT computed by computers is really not CWT, it is a discretized version of the CWT.

The resolution of the time-frequency grid can be controlled (within Heisenberg’s inequality), can be controlled by time and scale step sizes.

Often this results in a very redundant representation How to discretize the continuous time-frequency plane, so that the

representation is non-redundant?Sample the time-frequency plane on a dyadic

(octave) grid

Znknt kkkn , 22)(

txx dt

s

ttx

sssCWT

1),(),(

Discrete Wavelet Transform

Dyadic sampling of the time –frequency plane results in a very efficient algorithm for computing DWT:Subband coding using multiresolution analysisDyadic sampling and multiresolution is

achieved through a series of filtering and up/down sampling operations

HHx[n]x[n] y[n]y[n]

N

k

N

k

knxkh

knhkx

nxnhnhnxny

1

1

][][

][][

][*][][*][][

Recall: Orthogonal Projection

X3E R3

X2E R2

X1E R1

e2 E W2

e1 E W1

1213

11

223

2

eeXX

eXX

eXX

V3

V2

Coarse approximation of X3 at level 2

Vector Spaces

VN-1is the next coarser vector space, it is a subspace of VN.

WN-1 is orthogonal to VN-1, and

In general

and

11 NNN WVV

1121 VWWWV NNN

1121 XeeeX NN

Details vs. Approximations

Discrete Wavelet TransformImplementation

G

H

2

2 G

H

2

2

2

2

G

H

+

2

2

G

H

+

x[n]x[n]

Decomposition Reconstruction

~

~ ~

~

n

high kngnxky ]2[][][~

n

low knhnxky ]2[][][~

k

high kngky ]2[][

k

high kngky ]2[][

2-level DWT decomposition. The decomposition can be continues as 2-level DWT decomposition. The decomposition can be continues as long as there are enough samples for down-sampling.long as there are enough samples for down-sampling.

G

H

Half band high pass filter

Half band low pass filter

2

2

Down-sampling

Up-sampling

DWT - Demystified

Length: 512B: 0 ~

g[n] h[n]

g[n] h[n]

g[n] h[n]

2

d1: Level 1 DWTCoeff.

Length: 256B: 0 ~ /2 Hz

Length: 256B: /2 ~ Hz

Length: 128B: 0 ~ /4 HzLength: 128

B: /4 ~ /2 Hz

d2: Level 2 DWTCoeff.

d3: Level 3 DWTCoeff.

…a3….

Length: 64B: 0 ~ /8 HzLength: 64

B: /8 ~ /4 Hz

2

2 2

22

|H(jw)|

w/2-/2

|G(jw)|

w- /2-/2a2

a1

Level 3 approximation

Coefficients

Quadrature Mirror Filters

It can be shown that

that is, h[] and g[] filters are related to each other:

in fact, that is, h[] and g[] are mirrors of each other, with every other coefficient negated. Such filters are called quadrature mirror filters. For example, Daubechies wavelets with 4 vanishing moments…..

1)()(1)()(

2~2~22 jGjHjGjH

][)1(]1[ ngnLh n

DB-4 Wavelets

h = -0.0106 0.0329 0.0308 -0.1870 -0.0280 0.6309 0.7148 0.2304

g = -0.0106 -0.0329 0.0308 0.1870 -0.0280 -0.6309 0.7148 -0.2304

h = 0.2304 0.7148 0.6309 -0.0280 -0.1870 0.0308 0.0329 -0.0106

g = -0.0106 -0.0329 0.0308 0.1870 -0.0280 -0.6309 0.7148 -0.2304

~

~

][][ ],[][

][)1(]1[~~ngngnhnh

ngnLh n

L: filter length (8, in this case)

Implementation of DWT on MATLAB

Load signal

Choose waveletand numberof levels

Hit Analyze button

Level 1 coeff.Highest freq.

Approx. coef. at level 5

s=a5+d5+…+d1

(Wavedemo_signal1)