cmorwavf

Complex Morlet wavelet

Syntax

[psi,x] = cmorwavf(lb,ub,n)

[psi,x] = cmorwavf(lb,ub,n,fb,fc)

Description

[psi,x] = cmorwavf(lb,ub,n) returns the complex Morlet wavelet, psi, with time-decay parameter, fb, and center frequency, fc, both equal to 1. The wavelet is evaluated on an n-point regular grid, x, for the interval [lb,ub]. The general expression for the complex Morlet wavelet is

$$\psi (x)=\frac{1}{\sqrt{\pi \text{fb}}}\exp (2\pi i\text{fc}x)\exp (-x^2\text{/}\text{fb}).$$

[psi,x] = cmorwavf(lb,ub,n,fb,fc) returns the complex Morlet wavelet with time-decay parameter, fb, and center frequency, fc.

Examples

Complex Morlet Wavelet

Construct a complex-valued Morlet wavelet with a bandwidth parameter of 1.5 and a center frequency of 1. Set the effective support to $$[-8,8]$$ and the number of sample points to 1000.

N = 1000;
Lb = -8;
Ub = 8;
fb = 1.5;
fc = 1;
[psi,x] = cmorwavf(Lb,Ub,N,fb,fc);

Plot the real and imaginary parts of the wavelet.

tiledlayout(2,1)
nexttile
plot(x,real(psi))
title("Real Part")
nexttile
plot(x,imag(psi))
title("Imaginary Part")

Effect of Bandwidth Parameter on Morlet Wavelet Shape

This example shows how the complex Morlet wavelet shape in the frequency domain is affected by the value of the bandwidth parameter (Fb). Both wavelets have a center frequency of 1. One wavelet has an Fb value of 0.5 and the other wavelet has a value of 8.

f = -5:.01:5;  
Fc = 1;  
Fb1 = 0.5;  
Fb2 = 8;  
psihat1 = exp(-pi^2*Fb1*(f-Fc).^2);  
psihat2 = exp(-pi^2*Fb2*(f-Fc).^2);  
plot(f,psihat1)  
hold on
plot(f,psihat2)
hold off
legend("Fb = 0.5","Fb = 8")

The Fb bandwidth parameter for the complex Morlet wavelet is the inverse of the variance in frequency. Therefore, increasing Fb results in a narrower concentration of energy around the center frequency.

Input Arguments

lb — Left endpoint

scalar

Left endpoint of the closed interval, specified as a scalar. lb is strictly less than ub.

ub — Right endpoint

scalar

Right endpoint of the closed interval, specified as a scalar. ub is strictly greater than ub.

n — Number of regularly spaced points

positive integer

Number of regularly spaced points in the interval [lb,ub], specified as a positive integer.

fb — Time-decay parameter

positive scalar

Time-decay parameter, specified as a positive scalar. fb controls the decay in the time domain and the corresponding energy spread (bandwidth) in the frequency domain. fb is the inverse of the variance in the frequency domain.

• Increasing fb makes the wavelet energy more concentrated around the center frequency and results in slower decay of the wavelet in the time domain.

• Decreasing fb results in faster decay of the wavelet in the time domain and less energy spread in the frequency domain.

The value of fb does not affect the center frequency. When converting from scale to frequency, only the center frequency affects the frequency values. The energy spread or bandwidth parameter affects how localized the wavelet is in the frequency domain.

fc — Center frequency

positive scalar

Center frequency, specified as a positive scalar.

Output Arguments

psi — Complex Morlet wavelet

vector

Complex Morlet wavelet, returned as a complex-valued 1-by-n vector.

x — Sample points

vector

Sample points where the complex Morlet wavelet is evaluated, returned as a 1-by-n vector. The sample points are evenly distributed between lb and ub.

References

[1] Teolis, Anthony. Computational Signal Processing with Wavelets. Boston, MA: Birkhäuser Boston, 1998. https://doi.org/10.1007/978-1-4612-4142-3.

Version History

Introduced before R2006a