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, $1 / \sqrt[4]{2}$ .

4 References

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

5 Arguments

1: $wavnam$ – Nag_Wavelet Input

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

$wavnam = Nag_Morlet$: Morlet wavelet.
$wavnam = Nag_DGauss$: Derivative of a Gaussian wavelet.
$wavnam = Nag_MexHat$: Mexican hat wavelet.

Constraint:

wavnam = Nag_Morlet

Nag_DGauss

Nag_MexHat

2: $wparam$ – 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 = Nag_MexHat

Constraints:

if $wavnam = Nag_Morlet$ , $5 \leq wparam \leq 20$ ;
if $wavnam = Nag_DGauss$ , $1 \leq wparam \leq 8$ .

3: $n$ – Integer Input

On entry: the size,

n

, of the input dataset

x

Constraint:

n \geq 2

4: $x [n]$ – const double Input

On entry: x contains the input dataset

x [j - 1] = x_{j}

, for

j = 1, 2, \dots, n

5: $nscal$ – Integer Input

On entry: the number of scales to be computed.

Constraint:

nscal \geq 1

6: $scales [nscal]$ – const Integer Input

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

Constraint:

scales [i - 1] \geq 1

, for

i = 1, 2, \dots, nscal

7: $c [nscal \times n]$ – double Output

Note: the

(i, j)

th element of the matrix

C

is stored in

c [(j - 1) \times nscal + i - 1]

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

c [(j - 1) \times nscal + i - 1]

is the transform coefficient

C_{i, j}

at scale

i

and position

j

8: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_INT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 2$ .

On entry, $nscal = ⟨ value ⟩$ .
Constraint: $nscal \geq 1$ .

On entry, $wavnam = Nag_DGauss$ and $wparam = ⟨ value ⟩$ .
Constraint: if $wavnam = Nag_DGauss$ , $1 \leq wparam \leq 8$ .

On entry, $wavnam = Nag_Morlet$ and $wparam = ⟨ value ⟩$ .
Constraint: if $wavnam = Nag_Morlet$ , $5 \leq wparam \leq 20$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

7 Accuracy

The accuracy of c09bac 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.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

c09bac is not threaded in any implementation.

9 Further Comments

Workspace is internally allocated by c09bac. 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 - 1])

and

k = 17

when

wavnam = Nag_Morlet

Nag_DGauss

and

k = 11

when

wavnam = Nag_MexHat

10 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: FL CL CPP AD PY MB

NAG CL Interfacec09bac (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 CL Interface
c09bac (dim1_cont)