# 18.12.1 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 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:

where k is the wave number.

The Mexican Hat wavelet is:

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

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: