18.12.1.2 Algorithms (Continuous Wavelet Transform)

Continuous Wavelet Transform

This function computes the real continuous wavelet coefficient for each given scale presented in the Scale vector and each position b from 1 to n, where n is the size of the input signal.

Let x(t) be the input signal and ψ be the chosen wavelet function, the continuous wavelet coefficient of x(t) at scale a and position b is:

$C_{a,b} = \int_R {x(t)\frac{1}{{\sqrt a }}\psi^* } (\frac{{t - b}}{a})dt$

The computation is implemented with a NAG function: nag_cwt_real(). It does not compute the coefficients with the definition of CWT. Instead, the integrals of the scaled, shifted wavelet function are approximated and the convolution is then computed.

Origin Wavelet Types

Morlet
The Morlet wavelet:

$\psi (x) = \pi ^{ - 1/4} \cos (kx)e^{ - x^2 /2}$

where k is the wave number.
DGauss
The Derivative of a Gaussian wavelet, which is the pth derivative of the Gaussian function:

$\psi (x) = (\frac{2}{\pi })^{ - 1/4} e^{ - x^2 }$

where p is the derivative order.
MexHat
The Mexican Hat Wavelet:

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

Convert Scale to Pseudo Frequency

For a given wavelet, you can map a scale and convert to pseudo-frequency by ways below:

$F_a=\frac{F_c}{s\cdot \Delta }$

In this formula:

s is a scale.
$\Delta$ is the sampling period.
$F_c$ is the center frequency of a wavelet in Hz.
$F_a$ is the pseudo-frequency corresponding to the scale a, in Hz.

The $F_c$ is the frequency contributes most to the variability of the wavelet, and it can be derived from maximizing the FFT of the wavelet modulus.