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# cfftf_frq_reorder

Reorders the data returned by cfftf.

## Prototype

function cfftf_frq_reorder (
cf  : numeric
)

return_val  :   typeof(x)

## Arguments

cf

The array as returned by cfftf.

## Return value

A double array is returned if cf is double, otherwise a float array is returned.

## Description

Reorder the array returned by cfftf so that the data span -0.5 to 0.5.

Caveat: The reordered data can not be input to cfftb.

Background from Paul Swarztrauber:

Historically, the continuous complex transform is defined on the interval -pi to pi with wave numbers -infinity to infinity. In its discrete form it is defined on the points x sub j = j2pi/N where

if N is  odd then j = -(N-1)/2,...,(N-1)/2

if N is even then j = N/2,...,N/2-1
These integer values also correspond to wave numbers k.

The confusion arises because the description of the transform is often defined with indices j=0,..,N-1 which is true but corresponds to an aliased transform and chosen simply because one does not have to separate the description into parts corresponding to even and odd integers. That is, it is chosen to simplify math presentation but confuses the heck out of individuals that actually have to use the transform.

## Examples

Example 1

Reorder the complex frequency spectrum of Example 1 presented for cfftf. The issue is that the returned frq attribute is not monotonically increasing.

1. The simplest approach would be to replace the returned frequency attribute with values in the range 0 to 1 or create a new array containing the new values. Either is readily accomplished via
cf@frq = (ispan(0,N-1,1)*1.0)/N
; ============= or ========================
f = (ispan(0,N-1,1)*1.0)/N

The cf@frq or f could be used for the plot abscissa.

2. The convention is to use a frequency scale that spans -0.5 to 0.5. To accomplish this the returned values must be reordered.
xf = cfftf_frq_reorder( cf )
print(sprintf("%9.5f", xf@frq) +"    "+sprintf("%9.3f", xf(0,:))+"    "+sprintf("%9.3f", xf(1,:)) )

frq          real         imag
(0)      -0.50000      -43.000        0.000
(1)      -0.45833       35.417       33.699
(2)      -0.41667       -4.192      -31.658
(3)      -0.37500       26.000       30.355
(4)      -0.33333       32.500       18.187
(5)      -0.29167      -32.697       49.359
(6)      -0.25000        1.000       12.000
(7)      -0.20833      -64.781       36.235
(8)      -0.16667       39.500        4.330
(9)      -0.12500       26.000       40.355
(10)     -0.08333     -161.808       82.658
(11)     -0.04167       16.061       44.823

(12)      0.00000    24265.000        0.000

(13)      0.04167       16.061      -44.823
(14)      0.08333     -161.808      -82.658
(15)      0.12500       26.000      -40.355
(16)      0.16667       39.500       -4.330
(17)      0.20833      -64.781      -36.235
(18)      0.25000        1.000      -12.000
(19)      0.29167      -32.697      -49.359
(20)      0.33333       32.500      -18.187
(21)      0.37500       26.000      -30.355
(22)      0.41667       -4.192       31.658
(23)      0.45833       35.417      -33.699