MORSESPACE Logarithmically-spaced frequencies for generalized Morse wavelets. F=MORSESPACE(GAMMA,BETA,N) generates a frequency array for the generalized Morse wavelet transform of an N-point time series. The wavelets are specified by GAMMA and BETA. LOG(F) is uniformly spaced, following convention for wavelet analysis. F has units of *radians* per sample point, and is a column vector with the frequencies arranged in decending order. In this usage, the frequencies F are determined using default settings, described below, which should be appropriate for most applications. Additional control over the frequency array F can be obtained using the following alternate usages. For details on this calculation, see Lilly (2016). __________________________________________________________________ High- and low-frequency specification F=MORSESPACE(GAMMA,BETA,HIGH,LOW) explicitly sets the high-frequency and low-frequency cutoffs for the frequency array. The first (largest) value of F is then just smaller than HIGH and the smallest is just larger than LOW. HIGH and LOW have units of *radian* per unit sample point. __________________________________________________________________ High-frequency cutoff The highest frequency can be set to be the minimum of a specified value and a cutoff frequency based on a Nyquist overlap condition. F=MORSESPACE(GAMMA,BETA,{ETA,HIGH},LOW) sets the highest frequency to be the minimum of the specified value HIGH, and the largest frequency for which the wavelet will satisfy the threshold level ETA. Here ETA be a number between zero and one specifying the ratio of a frequency-domain wavelet at the Nyquist frequency to its peak value. Note that in this usage, {ETA,HIGH} is a cell array with two entries. The simplified usage F=MORSESPACE(GAMMA,BETA,N) corresponds to the choice ETA=0.1, so that by default, the highest-frequency wavelet will decay to at least 10% of its peak value at the Nyquist frequency. __________________________________________________________________ Low-frequency cutoff The lowest frequency can be set to a cutoff frequency based on an endpoint overlap condition. F=MORSESPACE(GAMMA,BETA,HIGH,{P,N}) sets the lowest frequency such that the lowest-frequency wavelet will reach some number P times its central window width at the ends of the time series. P is called the packing number. A choice of P=1 corresponds to roughly 95% of the time-domain wavelet energy being contained within the time series endpoints for a wavelet at the center of the domain. F=MORSESPACE(GAMMA,BETA,HIGH,{P,N,LOW}) alternately chooses the maximum of the R-level cutoff frequency, and a specified low frequency LOW. The simplified usage F=MORSESPACE(GAMMA,BETA,N) corresponds to the default value P=5. At the lowest frequency, five wavelets will then fit into the time series, with 5% energy overlap between them. __________________________________________________________________ Wavelet density F=MORSESPACE(GAMMA,BETA,HIGH,LOW,D) controls the number of points in the frequency array through the 'density' D. Higher values of D mean more overlap in the frequency domain. The default value of the density is D=4. When D=1, the peak of one wavelet is located at the half-power points of the adjacent wavelet. D=4 means that four other wavelets will occur between the peak of one wavelet and its half-power point. __________________________________________________________________ See also MORSEWAVE, WAVETRANS. Usage: f=morsespace(ga,be,N); f=morsespace(ga,be,high,low,D); f=morsespace(ga,be,{eta,high},low,D); f=morsespace(ga,be,{eta,high},{p,N},D); f=morsespace(ga,be,{eta,high},{p,N,low},D); __________________________________________________________________ This is part of JLAB --- type 'help jlab' for more information (C) 2009--2016 J.M. Lilly --- type 'help jlab_license' for details