18.12.1 Continuous Wavelet Transform

PID1643 CWT.png

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 result can be output to a range on a worksheet and, if you select the Coefficient Matrix checkbox, a matrix. The output range on the worksheet is comprised of m columns, each of which has n rows, where m is the size of the Scale vector. Each column corresponds to a scale and each row corresponds to a position. On the other hand, the result matrix, if you choose to generate one, will have n columns and m rows. The value at a cell whose row number is M_0 and column number is N_0 is the coefficient of scale M_0 and position N_0.

Now three types of wavelet are supported in this function, including Morlet, Mexican Hat, and the Derivative of Gaussian wavelets.

The Morlet wavelet is defined as:

Cwt help English files image004.gif

where k is the wave number.

The Mexican Hat wavelet is:

Cwt help English files image006.gif

And the Gaussian wavelet is the pth derivative of the Gaussian function, which is defined as:

Cwt help English files image008.gif

where p is the derivative order.

To Perform Continuous Wavelet Transform
  1. Make a workbook or a graph active.
  2. Select Analysis: Signal Processing: Wavelet: Continuous Wavelet from the Origin menu.

Topics covered in this section: