# MORSEWAVE is the jWavelet module of jLab.

```  MORSEWAVE  Generalized Morse wavelets of Olhede and Walden (2002).
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PSI=MORSEWAVE(N,GAMMA,BETA,FS) returns an  N x LENGTH(FS) array PSI
which contains time-domain versions of the generalized Morse wavelet
specified by GAMMA and BETA, concentrated at frequencies FS.

The vector FS specifically denote the *radian* frequencies at which
the Fourier transform of the wavelets reach their maximum amplitudes.

A set of frequencies appropriate for analyzing a given length time
series can be easily chosen using MORSESPACE.

Note that the wavelets are centered at the midpoint in time, that is,
row number ROUND(SIZE(PSI,1)/2).  FS assumes a unit sample rate.

[PSI,PSIF]=MORSEWAVE(...) optionally returns a frequency-domain version
PSIF of the wavelets.  PSIF is the same size as PSI.
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Normalization

MORSEWAVE supports two kinds of normalization for the wavelets.

MORSEWAVE(...,'bandpass') uses "bandpass normalization", meaning that
the FFT of the wavelet has a peak value of 2 for all frequencies FS.

MORSEWAVE(...,'energy') uses the unit energy normalization.  The time-
domain wavelet energy SUM(ABS(PSI).^2,1) is then always unity.

The bandpass normalization corresponds to having 1/S in the time-domain
wavelet transform defintion, where S is the scale, while the unit
energy normalization corresponds to 1/SQRT(S).

MORSEWAVE uses bandpass normalization by default.
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Multiple orthogonal wavelets

MORSEWAVE can compute multiple orthogonal versions of the generalized
Morse wavelets, characterized by the order K.

PSI=MORSEWAVE(N,K,GAMMA,BETA,FS) with a fifth numerical argument K
returns an N x LENGTH(FS) x K array PSI which contains time-domain
versions of the first K orthogonal generalized Morse wavelets.

These K different orthogonal wavelets have been employed in
multiwavelet polarization analysis, see Olhede and Walden (2003a,b).

Again either bandpass or energy normalization can be applied.  With
bandpass normalization, all wavelets are divided by a constant, setting
the peak value of the first frequency-domain wavelet equal to 2.
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Background

For further details on generalized Morse wavelets, see the following
publications.

Lilly and Olhede (2012), Generalized Morse wavelets as a superfamily
of analytic wavelets. IEEE Trans. Sig. Proc., 60 (11), 6036--6041.

Lilly and Olhede (2009),  Higher-order properties of analytic
wavelets.  IEEE Trans. Sig. Proc., 57 (1), 146--160.

Olhede and Walden (2002),  Generalized Morse Wavelets. IEEE Trans.
Sig. Proc., 50 (11), 2661--2670.
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'morsewave --t' runs a test.
'morsewave --f' generates some sample figures.

Usage: psi=morsewave(N,ga,be,fs);
[psi,psif]=morsewave(N,ga,be,fs,'bandpass');
[psi,psif]=morsewave(N,K,ga,be,fs,'energy');
[psi,psif]=morsewave(N,K,ga,be,fs,'bandpass');
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This is part of JLAB --- type 'help jlab' for more information
(C) 2004--2016 J.M. Lilly and F. Rekibi
--- type 'help jlab_license' for details
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