C09BAF : NAG Library, Mark 24

3 Description

C09BAF 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 routine are defined as follows.

The Morlet wavelet (real part) with nondimensional wave number $κ$ is
$ψ (x) = \frac{1}{π^{1 / 4}} (\cos (κ x) - e^{- κ^{2} / 2}) e^{- x^{2} / 2},$
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, $1 / \sqrt[4]{2}$ .

5 Parameters

1: WAVNAM – CHARACTER(*)Input

On entry: the name of the mother wavelet. See the C09 Chapter Introduction for details.

$WAVNAM ='MORLET'$: Morlet wavelet.
$WAVNAM ='DGAUSS'$: Derivative of a Gaussian wavelet.
$WAVNAM ='MEXHAT'$: Mexican hat wavelet.

Constraint:

WAVNAM ='MORLET'

'DGAUSS'

'MEXHAT'

2: WPARAM – INTEGERInput

On entry: the nondimensional wave number for the Morlet wavelet or the order of the derivative for the Gaussian wavelet. It is not referenced when

WAVNAM ='MEXHAT'

Constraints:

if $WAVNAM ='MORLET'$ , $5 \leq WPARAM \leq 20$ ;
if $WAVNAM ='DGAUSS'$ , $1 \leq WPARAM \leq 8$ .

3: N – INTEGERInput

On entry: the size,

n

, of the input dataset

x

Constraint:

N \geq 2

4: X(N) – REAL (KIND=nag_wp) arrayInput

On entry: X contains the input dataset

X (j) = x_{j}

, for

j = 1, 2, \dots, n

5: NSCAL – INTEGERInput

On entry: the dimension of the array SCALES and the first dimension of the array C as declared in the (sub)program from which C09BAF is called. The number of scales to be computed.

Constraint:

NSCAL \geq 1

6: SCALES(NSCAL) – INTEGER arrayInput

On entry: the scales at which the transform is to be computed.

Constraint:

SCALES (i) \geq 1

, for

i = 1, 2, \dots, NSCAL

7: C(NSCAL,N) – REAL (KIND=nag_wp) arrayOutput

On exit: the transform coefficients at the requested scales, where

C (i, j)

is the transform coefficient

C_{i, j}

at scale

i

and position

j

8: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

IFAIL = 1

On entry,

WAVNAM \neq'MORLET'

'DGAUSS'

'MEXHAT'

IFAIL = 2

On entry,	$WAVNAM ='MORLET'$ , and $WPARAM < 5$ or $WPARAM > 20$ .
or	$WAVNAM ='DGAUSS'$ , and $WPARAM < 1$ or $WPARAM > 8$ .

IFAIL = 3

On entry,

N < 2

IFAIL = 5

On entry,

NSCAL < 1

IFAIL = - 999

Internal memory allocation failed.

NAG Library Routine Document

C09BAF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine DocumentC09BAF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

C09BAF