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.