naginterfaces.library.wav.dim1_​cont

naginterfaces.library.wav.dim1_cont(wavnam, wparam, x, scales)

dim1_cont computes the real, continuous wavelet transform in one dimension.

For full information please refer to the NAG Library document for c09ba

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/c09/c09baf.html

Parameters

wavnamstr

The name of the mother wavelet. See the C09 Introduction for details.

$wavnam=\text{‘MORLET'}$

Morlet wavelet.

$wavnam=\text{‘DGAUSS'}$

Derivative of a Gaussian wavelet.

$wavnam=\text{‘MEXHAT'}$

Mexican hat wavelet.

wparamint

The nondimensional wave number for the Morlet wavelet or the order of the derivative for the Gaussian wavelet. It is not referenced when $wavnam=\text{‘MEXHAT'}$.

xfloat, array-like, shape $\left(n\right)$

$x$ contains the input dataset $x[\text{j}-1]=x_{\text{j}}$, for $\text{j}=1,2,\dots ,n$.

scalesint, array-like, shape $\left(\text{nscal}\right)$

The scales at which the transform is to be computed.

Returns

cfloat, ndarray, shape $(\text{nscal},n)$

The transform coefficients at the requested scales, where $c[i-1,j-1]$ is the transform coefficient $C_{i,j}$ at scale $i$ and position $j$.

Raises

NagValueError

(errno $1$)

On entry, $wavnam$ not recognised: $wavnam=⟨value⟩$.

(errno $2$)

On entry, $wavnam=\text{‘MORLET'}$ and $wparam=⟨value⟩$.

Constraint: if $wavnam=\text{‘MORLET'}$, $5\le wparam\le 20$.

(errno $2$)

On entry, $wavnam=\text{‘DGAUSS'}$ and $wparam=⟨value⟩$.

Constraint: if $wavnam=\text{‘DGAUSS'}$, $1\le wparam\le 8$.

(errno $3$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 2$.

(errno $5$)

On entry, $\text{nscal}=⟨value⟩$.

Constraint: $\text{nscal}\ge 1$.

Notes

dim1_cont computes the real part of the one-dimensional, continuous wavelet transform

$$C_{s,k}={\int }_{R}x\left(t\right)\frac{1}{\sqrt{s}}{\psi }^{\ast }\left(\frac{t-k}{s}\right)dt\text{,}$$

of a signal $x\left(t\right)$ at scale $s$ and position $k$, where the signal is sampled discretely at $n$ equidistant points $x_{\text{i}}$, for $\text{i}=1,2,\dots ,n$. $\psi$ is the wavelet function, which can be chosen to be the Morlet wavelet, the derivatives of a Gaussian or the Mexican hat wavelet ($\ast$ 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.

1. The Morlet wavelet (real part) with nondimensional wave number $\kappa$ is

   $$\psi \left(x\right)=\frac{1}{{\pi }^{1/4}}(cos\left(\kappa x\right)-e^{-{\kappa }^2/2})e^{-x^2/2}\text{,}$$

   where the correction term, $e^{-{\kappa }^2/2}$ (required to satisfy the admissibility condition) is included.

2. The derivatives of a Gaussian are obtained from

   $${\hat{\psi }}^{\left(m\right)}\left(x\right)=\frac{d^m\left(e^{-x^2}\right)}{d{x}^m}\text{,}$$

   taking $m=1,\dots ,8$. These are the Hermite polynomials multiplied by the Gaussian. The sign is then adjusted to give ${\hat{\psi }}^{\left(m\right)}\left(0\right)>0$ when $m$ is even while the sign of the succeeding odd derivative, ${\hat{\psi }}^{(m+1)}$, is made consistent with the preceding even numbered derivative. They are normalized by the $L^2$-norm,

   $$p_m={\left({\int }_{-\infty }^{\infty }{\left[{\hat{\psi }}^{\left(m\right)}\left(x\right)\right]}^{2}dx\right)}^{1/2}$$

   The resulting normalized derivatives can be written in terms of the Hermite polynomials, $H_m\left(x\right)$, as

   $${\psi }^{\left(m\right)}\left(x\right)=\frac{\alpha H_m\left(x\right)e^{-x^2}}{p_m}\text{,}$$

   where

   $$\begin{array}{c}\alpha =\{\begin{array}{cc}1\text{,}& \text{when}m=0,mod(3,4)\text{;}\\ -1\text{,}& \text{when}m=1,mod(2,4)\text{.}\end{array}\end{array}$$

   Thus, the derivatives of a Gaussian provided here are,

   $${\psi }^{\left(1\right)}\left(x\right)=-{\left(\frac{2}{\pi }\right)}^{1/4}2x{e}^{-x^2}\text{,}$$
   $${\psi }^{\left(2\right)}\left(x\right)=-{\left(\frac{2}{\pi }\right)}^{1/4}\frac{1}{\sqrt{3}}(4x^2-2)e^{-x^2}\text{,}$$
   $${\psi }^{\left(3\right)}\left(x\right)={\left(\frac{2}{\pi }\right)}^{1/4}\frac{1}{\sqrt{15}}(8x^3-12x)e^{-x^2}\text{,}$$
   $${\psi }^{\left(4\right)}\left(x\right)={\left(\frac{2}{\pi }\right)}^{1/4}\frac{1}{\sqrt{105}}(16x^4-48x^2+12)e^{-x^2}\text{,}$$
   $${\psi }^{\left(5\right)}\left(x\right)=-{\left(\frac{2}{\pi }\right)}^{1/4}\frac{1}{3\sqrt{105}}(32x^5-160x^3+120x)e^{-x^2}\text{,}$$
   $${\psi }^{\left(6\right)}\left(x\right)=-{\left(\frac{2}{\pi }\right)}^{1/4}\frac{1}{3\sqrt{1155}}(64x^6-480x^4+720x^2-120)e^{-x^2}\text{,}$$
   $${\psi }^{\left(7\right)}\left(x\right)={\left(\frac{2}{\pi }\right)}^{1/4}\frac{1}{3\sqrt{15015}}(128x^7-1344x^5+3360x^3-1680x)e^{-x^2}\text{,}$$
   $${\psi }^{\left(8\right)}\left(x\right)={\left(\frac{2}{\pi }\right)}^{1/4}\frac{1}{45\sqrt{1001}}(256x^8-3584x^6+13440x^4-13440x^2+1680)e^{-x^2}\text{.}$$

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

   $$\psi \left(x\right)=\frac{2}{\left(\sqrt{3}{\pi }^{1/4}\right)}(1-x^2)e^{-x^2/2}\text{.}$$

   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, $1/\sqrt[4]{42}$.

References

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