Complex Morlet wavelet
[PSI,X] = cmorwavf(LB,UB,N)
[PSI,X] = cmorwavf(LB,UB,N,FB,FC)
[PSI,X] = cmorwavf(LB,UB,N) returns the
complex Morlet wavelet,
PSI, with time-decay parameter,
and center frequency,
FC, both equal to 1. The
general expression for the complex Morlet wavelet is
PSI(X) = ((pi*FB)^(-0.5))*exp(2*pi*i*FC*X)*exp(-(X^2)/FB)
Xis evaluated on an N-point regular grid in the interval [
[PSI,X] = cmorwavf(LB,UB,N,FB,FC) returns
values of the complex Morlet wavelet defined by a positive time-decay
FB, and positive center frequency,
FB controls the decay in the time domain
and the corresponding energy spread (bandwidth) in the frequency domain.
the inverse of the variance in the frequency domain. Increasing
the wavelet energy more concentrated around the center frequency and
results in slower decay of the wavelet in the time domain. Decreasing
in faster decay of the wavelet in the time domain and less energy
spread in the frequency domain. The value of
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.
Construct a complex-valued Morlet wavelet with a bandwidth parameter of 1.5 and a center frequency of 1. Set the effective support to and the length of the wavelet 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.
subplot(2,1,1) plot(x,real(psi)); title('Real Part'); subplot(2,1,2) plot(x,imag(psi)); title('Imaginary Part');
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,'r') legend('Fb = 0.5','Fb = 8')
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.
Teolis, A. (1998), Computational signal processing with wavelets, Birkhauser, p. 65.