%WAVE_BASES 1D Wavelet functions Morlet, Paul, or DOG % % [DAUGHTER,FOURIER_FACTOR,COI,DOFMIN] = ... % wave_bases(MOTHER,K,SCALE,PARAM); % % Computes the wavelet function as a function of Fourier frequency, % used for the wavelet transform in Fourier space. % (This program is called automatically by WAVELET) % % INPUTS: % % MOTHER = a string, equal to 'MORLET' or 'PAUL' or 'DOG' % K = a vector, the Fourier frequencies at which to calculate the wavelet % SCALE = a number, the wavelet scale % PARAM = the nondimensional parameter for the wavelet function % % OUTPUTS: % % DAUGHTER = a vector, the wavelet function % FOURIER_FACTOR = the ratio of Fourier period to scale % COI = a number, the cone-of-influence size at the scale % DOFMIN = a number, degrees of freedom for each point in the wavelet power % (either 2 for Morlet and Paul, or 1 for the DOG) % %---------------------------------------------------------------------------- % Copyright (C) 1995-1998, Christopher Torrence and Gilbert P. Compo % University of Colorado, Program in Atmospheric and Oceanic Sciences. % This software may be used, copied, or redistributed as long as it is not % sold and this copyright notice is reproduced on each copy made. This % routine is provided as is without any express or implied warranties % whatsoever. %---------------------------------------------------------------------------- function [daughter,fourier_factor,coi,dofmin] = ... wave_bases(mother,k,scale,param); mother = upper(mother); n = length(k); if (strcmp(mother,'MORLET')) %----------------------------------- Morlet if (param == -1), param = 6.;, end k0 = param; expnt = -(scale.*k - k0).^2/2.*(k > 0.); norm = sqrt(scale*k(2))*(pi^(-0.25))*sqrt(n); % total energy=N [Eqn(7)] daughter = norm*exp(expnt); daughter = daughter.*(k > 0.); % Heaviside step function fourier_factor = (4*pi)/(k0 + sqrt(2 + k0^2)); % Scale-->Fourier [Sec.3h] coi = fourier_factor/sqrt(2); % Cone-of-influence [Sec.3g] dofmin = 2; % Degrees of freedom elseif (strcmp(mother,'PAUL')) %-------------------------------- Paul if (param == -1), param = 4.;, end m = param; expnt = -(scale.*k).*(k > 0.); norm = sqrt(scale*k(2))*(2^m/sqrt(m*prod(2:(2*m-1))))*sqrt(n); daughter = norm*((scale.*k).^m).*exp(expnt); daughter = daughter.*(k > 0.); % Heaviside step function fourier_factor = 4*pi/(2*m+1); coi = fourier_factor*sqrt(2); dofmin = 2; elseif (strcmp(mother,'DOG')) %-------------------------------- DOG if (param == -1), param = 2.;, end m = param; expnt = -(scale.*k).^2 ./ 2.0; norm = sqrt(scale*k(2)/gamma(m+0.5))*sqrt(n); daughter = -norm*(i^m)*((scale.*k).^m).*exp(expnt); fourier_factor = 2*pi*sqrt(2./(2*m+1)); coi = fourier_factor/sqrt(2); dofmin = 1; else error('Mother must be one of MORLET,PAUL,DOG') end return % end of code