# MORSESPACE is the jWavelet module of jLab.

``` 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).
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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.
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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.
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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.
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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.
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