# NAME

PDL::FFTW3 - PDL interface to the Fastest Fourier Transform in the West v3

# DESCRIPTION

This is a PDL binding to version 3 of the FFTW library. Supported are complex <-> complex and real <-> complex FFTs.

# SYNOPSIS

```
use PDL;
use PDL::FFTW3;
use PDL::Graphics::Gnuplot;
use PDL::Complex;
# Basic functionality
my $x = sin( sequence(100) * 2.0 ) + 2.0 * cos( sequence(100) / 3.0 );
my $F = rfft1( $x );
gplot( with => 'lines', inner($F,$F));
=====>
8000 ++------------+-------------+------------+-------------+------------++
+ + + + + +
| |
| * |
7000 ++ * ++
| * |
| * |
| * |
| * |
6000 ++ * ++
| * |
| * |
| * |
5000 ++ * ++
| * |
| * |
| ** |
4000 ++ ** ++
| ** |
| * * |
| * * |
| * * |
3000 ++ * * ++
| * * |
| * * |
| * * * |
2000 ++ * * * ++
| * * * |
| * * ** |
| * * ** |
| * * ** |
1000 ++ * * * * ++
| * * * * |
| ** * * * |
+ * * + + + * * + +
0 ****-------*********************************--************************
0 10 20 30 40 50
# Correlation of two real signals
# two signals offset by 30 units
my $x = sequence(100);
my $y1 = exp( 0.2*($x - 20.5) ** (-2.0) );
my $y2 = exp( 0.2*($x - 50.5) ** (-2.0) );
# compute the correlation
my $F12 = rfft1( cat($y1,$y2) );
my $corr = irfft1( Cmul( $F12(:,:,(1)),
Cconj $F12(:,:,(0)) ) );
# and find the peak
say maximum_ind($corr);
=====> 30
```

# DESCRIPTION

## Supported operations

This module computes the Discrete Fourier Transform. In its most basic form, this transform converts a vector of complex numbers in the time domain into another vector of complex numbers in the frequency domain. These complex <-> complex transforms are supported with `fftN`

functions for a rank-`N`

transform. The opposite effect (transform data in the frequency domain back to the time domain) can be achieved with the `ifftN`

functions.

A common use case is to transform purely-real data. This data has 0 for its complex component, and FFTW can take advantage of this to compute the FFT faster and using less memory. Since a Fourier Transform of a real signal has an even real part and an odd imaginary part, only 1/2 of the spectrum is needed. These forward real -> complex transforms are supported with the `rfftN`

functions. The backward version of this transform is complex -> real and is supported with the `irfftN`

functions.

## Basic usage details

Arbitrary `N`

-dimensional transforms are supported. All functions exported by this module have the `N`

in their name, so for instance a complex <-> complex 3D forward transform is computed with the `fft3`

function. The rank *must always* be specified in this way; there is no function called simply `fft`

.

In-place operation is supported for complex <-> complex functions, but not the real ones (real function don't have mathing dimensionality of the input and output). An in-place transform of `$x`

can be computed with

` fft1( $x->inplace );`

All the functions in this module support PDL threading. For instance, if we have 4 different image piddles `$a`

, `$b`

, `$c`

, `$d`

and we want to compute their 2D FFTs at the same time, we can say

` my $ABCD_transformed = rfft2( PDL::cat( $a, $b, $c, $d) );`

This takes advantage of PDL's automatic parallelization, if appropriate (See PDL::ParallelCPU).

## Data formats

FFTW supports single and double-precision floating point numbers directly. If possible, the PDL input will be used as-is. If not, a type conversion will be made to use the lowest-common type. So as an example, the following will perform a single-precision floating point transform (and return data of that type).

` fft1( $x->byte )`

This module expects complex numbers to be stored as a (real,imag) pair in the first dimension of a piddle. Thus in a complex piddle `$x`

, it is expected that `$x->dim(0) == 2`

(this module verifies this before proceeding).

Generally, the sizes of the input and the output must match. This is completely true for the complex <-> complex transforms: the output will have the same size and the input, and an error will result if this isn't possible for some reason.

This is a little bit more involved for the real <-> complex transforms. If I'm transforming a real 3D vector of dimensions `K,L,M`

, I will get an output of dimensions `2,int(K/2)+1,L,M`

. The leading 2 is there because the output is complex; the `K/2`

is there because the input was real. The first dimension is always the one that gets the `K/2`

. This is described in detail in section 2.4 of the FFTW manual.

Note that given a real input, the dimensionality of the complex transformed output is unambiguous. However, this is *not* true for the backward transform. For instance, a 1D inverse transform of a vector of 10 complex numbers can produce real output of either 18 or 19 elements (because `int(18/2)+1 == 10`

and `int(19/2)+1 == 10`

).

*Without any extra information this module assumes the even-sized input*.

Thus `irfft1( sequence(2,10) )->dim(0) == 18`

is true. If we want the odd-sized output, we have to explicitly pass this into the function like this:

` irfft1( sequence(2,10), zeros(19) )`

Here I create a new output piddle with the `zeros`

function; `irfft1`

then fills in this piddle with the result of the computation. This module validates all of its input, so only 18 and 19 are valid here. An error will be thrown if you try to pass in `zeros(20)`

.

This all means that the following will produce surprising results if `$x->dim(0)`

isn't even

` irfft1( rfft1( $x ) )`

## FFT normalization

Following the widest-used convention for discrete Fourier transforms, this module normalizes the inverse transform (but not the forward transform) by dividing by the number of elements in the data set, so that

` ifft1( fft1( $x ) )`

is a slow approximate no-op, if `$x`

is well-behaved.

This is different from the behavior of the underlying FFTW3 library itself, but more consistent with other FFT packages for popular analysis languages including PDL.

# FUNCTIONS

## fftN (fft1, fft2, fft3, ..., fftn)

The basic complex <-> complex FFT. You can pass in the rank as a parameter with the `fftn`

form, or append the rank to the function name for ranks up to 9. These functions all take one input piddle and one output piddle. The dimensions of the input and the output are identical. The output parameter is optional and, if present, must be the last argument. If the output piddle is passed in, the user *must* make sure the dimensions match.

The 0 dim of the input PDL must have size 2 and run over (real,imaginary) components. The transform is carried out over dims 1 through N.

The fftn form takes a minimum of two arguments: the PDL to transform, and the number of dimensions to transform as a separate argument.

The following are equivalent:

```
$X = fftn( $x, 1 );
$X = fft1( $x );
fft1( $x, my $X = $x->zeros );
```

## ifftN (ifft1, ifft2, ifft3, ..., fftn)

The basic, properly normalized, complex <-> complex backward FFT. Everything is exactly like in the `fftN`

functions, except the inverse transform is computed and normalized, so that (for example)

` ifft1( fft1 ( $x ) )`

is a good approximation of `$x`

itself.

## rfftN (rfft1, rfft2, rfft3, ..., rfftn)

The real -> complex FFT. You can pass in the rank with the `rfftn`

form, or append the rank to the function name for ranks up to 9. These functions all take one input piddle and one output piddle. The dimensions of the input and the output are not identical, but are related as described in "Data formats". The output can be passed in as the last argument, if desired. If the output piddle is passed in, the user *must* make sure the dimensions match.

In the `rfftn`

form, the rank is the second argument.

The following are equivalent:

```
$X = rfftn( $x, 1 );
$X = rfft1( $x );
rfft1( $x, my $X = $x->zeroes );
```

## irfftN (irfft1, irfft2, irfft3, ..., irfftn)

The complex -> real inverse FFT. You can pass in the rank with the `irfftn`

form, or append the rank to the function name for ranks up to 9. Argument passing and interpretation is as described in `rfftN`

above. Please read "Data formats" for details about dimension interpretation. There's an ambiguity about the output dimensionality, which is described in that section.

# AUTHOR

Dima Kogan, `<dima@secretsauce.net>`

; contributions from Craig DeForest, `<craig@deforest.org>`

.

# HISTORY

## VERSION 0.02

- Use Alien::FFTW3 for build (if present)
- add parametric rank functions
- normalize inverses
- shorter abbreviation for realfft functions

## VERSION 0.01

Early working version in github

# LICENSE AND COPYRIGHT

Copyright 2013 Dima Kogan and Craig DeForest.

This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License.