MORSEREGION Generalized Morse wavelet time-frequency concentration region. MORSEREGION computes two different type of time-frequency concentration regions for the generalized Morse wavelets. This function is part of 'element analysis' described in Lilly (2017), "Element analysis: a wavelet-based method for analyzing time-localized events in noisy time series", submitted. Available at www.jmlilly.net. __________________________________________________________________ Localization regions [T,F]=MORSEREGION(A,GAMMA,BETA,FO) returns the localization region of the generalized Morse wavelets specified by GAMMA and BETA centered on *radian* frequency FO. The nonnegative number A sets the region area. T and F are then a parametric curve such that PLOT(T,F) shows the shape of the time-frequency localization region. The localization regions are based on a reconstruction of the signal from a limited inversion region, see Daubechies and Paul (1988) and Olhede and Walden (2002, 2003a) for details. Note that the definition of area used here is one-half of that used in Olhede and Walden (2002, 2003a), as we only count positive frequencies. __________________________________________________________________ Regions of influence [T,F]=MORSEREGION(LAMBDA,GAMMA,BETA,MU,FRHO) with five input arguments returns the contour at which the energy-normalized wavelet transform of another wavelet falls to a fraction LAMBDA of its peak value. The wavelet we are taking the transform of has parameters GAMMA and MU, and is characterized by a scale frequency FRHO. We then take the wavelet transform of this wavelet with a (GAMMA,BETA) wavelet. Note that this region assumes a 1/S and not a 1/SQRT(S) normalization. T and F are then a parametric curve such approximating the region where this transform takes on a fraction LAMBDA of its peak value. This approximation is formed analytically through a Taylor series expansion. Note that FRHO is the frequency of the analyzed wavelet, not the location of the maximum within the transform, which is given by FO=FRHO*MORSEFREQ(GAMMA,BETA)/MORSEFREQ(GAMMA,MU)* ((MU+1)./BETA)^(1/GAMMA). For futher details, see Lilly (2017). __________________________________________________________________ 'morseregion --f' generates a sample figure. Usage: [t,f]=morseregion(A,ga,be,fo); [t,f]=morseregion(lambda,ga,be,mu,frho); __________________________________________________________________ This is part of JLAB --- type 'help jlab' for more information (C) 2006--2017 J.M. Lilly --- type 'help jlab_license' for details