nag_cwt_1d_real (c09bac) : NAG Library, Mark 24

3 Description

nag_cwt_1d_real (c09bac) 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}

5 Arguments

1: wavnam – Nag_WaveletInput

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 – 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 = Nag_MexHat

Constraints:

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

3: n – IntegerInput

On entry: the size,

n

, of the input dataset

x

Constraint:

n \geq 2

4: x[n] – const doubleInput

On entry: x contains the input dataset

x [j - 1] = x_{j}

, for

j = 1, 2, \dots, n

5: nscal – IntegerInput

On entry: the number of scales to be computed.

Constraint:

nscal \geq 1

6: scales[nscal] – const IntegerInput

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

] – doubleOutput

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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.

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.

NAG Library Function Document

nag_cwt_1d_real (c09bac)

+− 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 Library Function Documentnag_cwt_1d_real (c09bac)

+− 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 Library Function Document

nag_cwt_1d_real (c09bac)