Routines (alphabetical) > Routines: W > WV_FN_PAUL

WV_FN_PAUL

The WV_FN_PAUL function constructs wavelet coefficients for the Paul wavelet function. In real space, the Paul wavelet function is proportional to the complex polynomial (1 – ix / s)^(–m–1), where s is the wavelet scale, m is a non-dimensional parameter, and x is the position.

Syntax

Result = WV_FN_PAUL( [Order] [, Scale, N] [, /DOUBLE] [, FREQUENCY=variable] [, /SPATIAL] [, WAVELET=variable])

Return Value

The returned value of this function is an anonymous structure of information about the particular wavelet.

Structure Tags for Result
Tag	Type	Definition
FAMILY	STRING	‘Paul’
ORDER_NAME	STRING	‘Parameter’
ORDER_RANGE	DBLARR(3)	[1, 20, 4] Valid orders [first, last, default]
ORDER	DOUBLE	The chosen Order
DISCRETE	INT	0 [0=continuous, 1=discrete]
ORTHOGONAL	INT	0 [0=nonorthogonal, 1=orthogonal]
SYMMETRIC	INT	1 [0=asymmetric, 1=symm.]
SUPPORT	DOUBLE	Infinity [Compact support width]
MOMENTS	INT	1 [Number of vanishing moments]
REGULARITY	DOUBLE	Infinity [Number of continuous derivatives]
E_FOLDING	DOUBLE	1/sqrt(2) [Autocorrelation e-fold distance]
FOURIER_PERIOD	DOUBLE	Ratio of Fourier wavelength to scale

Arguments

Order

A scalar that specifies the non-dimensional order for the wavelet. The default is 4.

Scale

A scalar that specifies the scale at which to construct the wavelet function.

N

An integer that specifies the number of points in the wavelet function. For Fourier space (SPATIAL=0), the frequencies are constructed following the FFT convention:

For N even: 0, 1/N, 2/N, ..., (N–2)/(2N), 1/2, –(N–2)/(2N), ..., –1/N.
For N odd: 0, 1/N, 2/N, ..., (N–1)/(2N), –(N–1)/(2N), ..., –1/N.

For real space (/SPATIAL), the spatial coordinates are –(N–1)/2...(N–1)/2.

Note: If none of the above arguments are present then the function will simply return the Result structure using the default Order.

Keywords

DOUBLE

Set this keyword to force the computation to be done in double-precision arithmetic.

FREQUENCY

Set this keyword to a named variable in which to return the frequency array used to construct the wavelet. This variable will be undefined if SPATIAL is set.

SPATIAL

Set this keyword to return the wavelet function in real space. The default is to return the wavelet function in Fourier space.

WAVELET

Set this keyword to a named variable in which to return the wavelet function.

Reference

Torrence and Compo, 1998: A Practical Guide to Wavelet Analysis. Bull. Amer. Meteor. Soc., 79, 61–78.

Examples

Plot the Paul wavelet function at scale=100:

n = 1000 ; pick a nice number of points

info = WV_FN_PAUL( 6, 100, n, /SPATIAL, $

WAVELET=wavelet)

plot, float(wavelet), THICK=2

oplot, imaginary(wavelet)

Now plot the same wavelet in Fourier space:

info = WV_FN_PAUL( 6, 100, n, $

FREQUENCY=frequency, WAVELET=wave_fourier)

plot, frequency, wave_fourier, $

xrange=[-0.2,0.2], thick=2

Version History

5.4	Introduced