c09ba:: Wavelet Transforms (NAG Toolbox)

nag_wav_1d_cont (c09ba) computes the real part of the one-dimensional, continuous wavelet transform

C_{s, k} = \int_{ℝ} x (t) \frac{1}{\sqrt{s}} ψ^{*} (\frac{t - k}{s}) d t,

of a signal

x (t)

at scale

s

and position

k

, where the signal is sampled discretely at

n

equidistant points

x_{i}

, for

i = 1, 2, \dots, n

ψ

is the wavelet function, which can be chosen to be the Morlet wavelet, the derivatives of a Gaussian or the Mexican hat wavelet (

*

denotes the complex conjugate). The integrals of the scaled, shifted wavelet function are approximated and the convolution is then computed.

The mother wavelets supplied for use with this function are defined as follows.

The Morlet wavelet (real part) with nondimensional wave number

κ

ψ (x) = \frac{1}{π^{1 / 4}} (\cos (κ x) - e^{- κ^{2} / 2}) e^{- x^{2} / 2},

where the correction term,

e^{- κ^{2} / 2}

(required to satisfy the admissibility condition) is included.

The derivatives of a Gaussian are obtained from

{\hat{ψ}}^{(m)} (x) = \frac{d^{m} (e^{- x^{2}})}{d x^{m}},

taking

m = 1, \dots, 8

. These are the Hermite polynomials multiplied by the Gaussian. The sign is then adjusted to give

{\hat{ψ}}^{(m)} (0) > 0

when

m

is even while the sign of the succeeding odd derivative,

{\hat{ψ}}^{(m + 1)}

, is made consistent with the preceding even numbered derivative. They are normalized by the

L^{2}

-norm,

p_{m} = {(\int_{- \infty}^{\infty} {[{\hat{ψ}}^{(m)} (x)]}^{2} d x)}^{1 / 2}

The resulting normalized derivatives can be written in terms of the Hermite polynomials,

H_{m} (x)

, as

ψ^{(m)} (x) = \frac{α H_{m} (x) e^{- x^{2}}}{p_{m}},

where

α = \{\begin{matrix} 1, & when ​ m = 0, 3 mod 4; \\ - 1, & when ​ m = 1, 2 mod 4 . \end{matrix}

Thus, the derivatives of a Gaussian provided here are,

ψ^{(1)} (x) = - {(\frac{2}{π})}^{1 / 4} 2 x e^{- x^{2}},

ψ^{(2)} (x) = - {(\frac{2}{π})}^{1 / 4} \frac{1}{\sqrt{3}} (4 x^{2} - 2) e^{- x^{2}},

ψ^{(3)} (x) = {(\frac{2}{π})}^{1 / 4} \frac{1}{\sqrt{15}} (8 x^{3} - 12 x) e^{- x^{2}},

ψ^{(4)} (x) = {(\frac{2}{π})}^{1 / 4} \frac{1}{\sqrt{105}} (16 x^{4} - 48 x^{2} + 12) e^{- x^{2}},

ψ^{(5)} (x) = - {(\frac{2}{π})}^{1 / 4} \frac{1}{3 \sqrt{105}} (32 x^{5} - 160 x^{3} + 120 x) e^{- x^{2}},

ψ^{(6)} (x) = - {(\frac{2}{π})}^{1 / 4} \frac{1}{3 \sqrt{1155}} (64 x^{6} - 480 x^{4} + 720 x^{2} - 120) e^{- x^{2}},

ψ^{(7)} (x) = {(\frac{2}{π})}^{1 / 4} \frac{1}{3 \sqrt{15015}} (128 x^{7} - 1344 x^{5} + 3360 x^{3} - 1680 x) e^{- x^{2}},

ψ^{(8)} (x) = {(\frac{2}{π})}^{1 / 4} \frac{1}{45 \sqrt{1001}} (256 x^{8} - 3584 x^{6} + 13440 x^{4} - 13440 x^{2} + 1680) e^{- x^{2}} .

The second derivative of a Gaussian is known as the Mexican hat wavelet and is supplied as an additional function in the form

ψ (x) = \frac{2}{(\sqrt{3} π^{1 / 4})} (1 - x^{2}) e^{- x^{2} / 2} .

The remaining normalized derivatives of a Gaussian can be expressed as multiples of the exponential

e^{- t^{2} / 2}

by applying the substitution

x = t / \sqrt{2}

followed by multiplication with the scaling factor,

1 / \sqrt[4]{2}

References

Daubechies I (1992) Ten Lectures on Wavelets SIAM, Philadelphia

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Errors or warnings detected by the function:

Accuracy

The accuracy of nag_wav_1d_cont (c09ba) is determined by the fact that the convolution must be computed as a discrete approximation to the continuous form. The input signal,

x

, is taken to be piecewise constant using the supplied discrete values.

Further Comments

Workspace is internally allocated by nag_wav_1d_cont (c09ba). The total size of these arrays is

2^{13} + (n + n_{k} - 1)

double elements and

n_{k}

integer elements, where

n_{k} = k \times \max (scales (i))

and

k = 17

when

wavnam ='MORLET'

'DGAUSS'

and

k = 11

when

wavnam ='MEXHAT'

Example

This example computes the continuous wavelet transform of a dataset containing a single nonzero value representing an impulse. The Morlet wavelet is used with wave number

κ = 5

and scales

1

2

3

4

c09ba example results


Number of Scales : 4
Wavelet coefficients c :
Scale:    1           2            3            4
    -1.7651e-05   1.5012e-04   5.2331e-02   1.4454e-01
    -1.3643e-03  -5.8141e-02   1.7057e-01  -8.4364e-02
     4.6511e-03   1.8442e-01  -1.4891e-01  -2.8870e-01
     8.9294e-02  -2.6380e-01  -2.6822e-01  -9.4993e-02
    -9.2563e-02   1.3289e-01   2.5680e-01   2.8293e-01
    -9.2563e-02   1.3289e-01   2.5680e-01   2.8293e-01
     8.9294e-02  -2.6380e-01  -2.6822e-01  -9.4993e-02
     4.6511e-03   1.8442e-01  -1.4891e-01  -2.8870e-01
    -1.3643e-03  -5.8141e-02   1.7057e-01  -8.4364e-02
    -1.7651e-05   1.5012e-04   5.2331e-02   1.4454e-01

NAG Toolbox: nag_wav_1d_cont (c09ba)

▸▿ Contents

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Accuracy

Further Comments

Example