c09baf (dim1_cont) : NAG Library, Mark 28

2 Specification

Fortran Interface

Subroutine c09baf (

wavnam, wparam, n, x, nscal, scales, c, ifail)

Integer, Intent (In)	::	wparam, n, nscal, scales(nscal)
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	x(n)
Real (Kind=nag_wp), Intent (Out)	::	c(nscal,n)
Character (*), Intent (In)	::	wavnam

C Header Interface

#include <nag.h>

void	c09baf_ (const char wavnam, const Integer wparam, const Integer n, const double x[], const Integer nscal, const Integer scales[], double c[], Integer *ifail, const Charlen length_wavnam)

The routine may be called by the names c09baf or nagf_wav_dim1_cont.

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.

1.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.
2.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,

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

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

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

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

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

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

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

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

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'

wparam

– Integer Input

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 $wavnam ='DGAUSS'$ , $1 \leq wparam \leq 8$ .

n

– Integer Input

On entry: the size,

n

, of the input dataset

x

Constraint:

n \geq 2

x (n)

– Real (Kind=nag_wp) array Input

On entry: x contains the input dataset

x (j) = x_{j}

, for

j = 1, 2, \dots, n

nscal

– Integer Input

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

scales (nscal)

– Integer array Input

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

Constraint:

scales (i) \geq 1

, for

i = 1, 2, \dots, nscal

c (nscal, n)

– Real (Kind=nag_wp) array Output

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

ifail

– Integer Input/Output

On entry: ifail must be set to

0

−1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

−1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

−1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. 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 = ⟨ value ⟩

was an illegal value.

ifail = 2

On entry,

wavnam ='MORLET'

and

wparam = ⟨ value ⟩

.
Constraint: if

wavnam ='MORLET'

5 \leq wparam \leq 20

On entry,

wavnam ='DGAUSS'

and

wparam = ⟨ value ⟩

.
Constraint: if

wavnam ='DGAUSS'

1 \leq wparam \leq 8

ifail = 3

On entry,

n = ⟨ value ⟩

.
Constraint:

n \geq 2

ifail = 5

On entry,

nscal = ⟨ value ⟩

.
Constraint:

nscal \geq 1

ifail = - 99

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 7 in the Introduction to the NAG Library FL Interface for further information.

ifail = - 399

Your licence key may have expired or may not have been installed correctly.

See Section 8 in the Introduction to the NAG Library FL Interface for further information.

ifail = - 999

Dynamic memory allocation failed.

See Section 9 in the Introduction to the NAG Library FL Interface for further information.

NAG FL Interface
c09baf (dim1_cont)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interfacec09baf (dim1_​cont)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
c09baf (dim1_cont)