Overview and A Short Tutorial¶
Before we begin, we assume that you are already familiar with the discrete Fourier transform, and why you want a faster library to perform your FFTs for you.
FFTW is a very fast FFT C library. The way it is designed to work is by planning in advance the fastest way to perform a particular transform. It does this by trying lots of different techniques and measuring the fastest way, so called planning.
One consequence of this is that the user needs to specify in advance
exactly what transform is needed, including things like the data type,
the array shapes and strides and the precision. This is quite
different to how one uses, for example, the
The purpose of this library is to provide a simple and pythonic way
to interact with FFTW, benefiting from the substantial speed-ups it
offers. In addition to the method of using FFTW as described above,
a convenient series of functions are included through
that make using
pyfftw almost equivalent to
This tutorial is split into three parts. A quick introduction to the
pyfftw.interfaces module is given, the
most simple and direct way to use
pyfftw. Secondly an
overview is given of
pyfftw.FFTW, the core
of the library. Finally, the
pyfftw.builders helper functions are
introduced, which ease the creation of
Quick and easy: the
The easiest way to begin using
pyfftw is through the
pyfftw.interfaces module. This module implements three APIs:
which are (apart from a small
caveat 1) drop in replacements for
>>> import pyfftw >>> import numpy >>> a = pyfftw.empty_aligned(128, dtype='complex128', n=16) >>> a[:] = numpy.random.randn(128) + 1j*numpy.random.randn(128) >>> b = pyfftw.interfaces.numpy_fft.fft(a) >>> c = numpy.fft.fft(a) >>> numpy.allclose(b, c) True
We initially create and fill a complex array,
a, of length 128.
pyfftw.empty_aligned() is a helper function that works like
numpy.empty() but returns the array aligned to a particular number of
bytes in memory, in this case 16. If the alignment is not specified then the
library inspects the CPU for an appropriate alignment value. Having byte aligned
arrays allows FFTW to performed vector operations, potentially speeding up the
FFT (a similar
pyfftw.byte_align() exists to align a pre-existing array as
a gives the same
output (to numerical precision) as calling
If you wanted to modify existing code that uses
numpy.fft to use
pyfftw.interfaces, this is done simply by replacing all instances of
pyfftw.interfaces.numpy_fft (similarly for
pyfftw.interfaces.scipy_fftpack), and then,
optionally, enabling the cache (see below).
The first call for a given transform size and shape and dtype and so on may be slow, this is down to FFTW needing to plan the transform for the first time. Once this has been done, subsequent equivalent transforms during the same session are much faster. It’s possible to export and save the internal knowledge (the wisdom) about how the transform is done. This is described below.
Even after the first transform of a given specification has been performed,
subsequent transforms are never as fast as using
directly, and in many cases are substantially slower. This is because of the
internal overhead of creating a new
pyfftw.FFTW object on every call.
For this reason, a cache is provided, which is recommended to be used whenever
pyfftw.interfaces is used. Turn the cache on using
pyfftw.interfaces.cache.enable(). This function turns the cache on
globally. Note that using the cache invokes the threading module.
The cache temporarily stores a copy of any interim
objects that are created. If they are not used for some period of time,
which can be set with
then they are removed from the cache (liberating any associated memory).
The default keepalive time is 0.1 seconds.
Integration with 3rd party libraries¶
SciPy versions 1.4 and above have support for installing different FFT
pyfftw.interfaces.scipy_fft support the use as a backend. Note
that the interfaces (and builders) all currently default to a single thread. The
number of threads to use can be configured by assigning a positive integer to
pyfftw.config.NUM_THREADS (see more details under :ref:configuration
<interfaces_tutorial>). The following code demonstrates using the
backend to speed up
import pyfftw import multiprocessing import scipy.signal import scipy.fft import numpy from timeit import Timer a = pyfftw.empty_aligned((128, 64), dtype='complex128') b = pyfftw.empty_aligned((128, 64), dtype='complex128') a[:] = numpy.random.randn(128, 64) + 1j*numpy.random.randn(128, 64) b[:] = numpy.random.randn(128, 64) + 1j*numpy.random.randn(128, 64) t = Timer(lambda: scipy.signal.fftconvolve(a, b)) print('Time with scipy.fft default backend: %1.3f seconds' % t.timeit(number=100)) # Configure PyFFTW to use all cores (the default is single-threaded) pyfftw.config.NUM_THREADS = multiprocessing.cpu_count() # Use the backend pyfftw.interfaces.scipy_fft with scipy.fft.set_backend(pyfftw.interfaces.scipy_fft): # Turn on the cache for optimum performance pyfftw.interfaces.cache.enable() # We cheat a bit by doing the planning first scipy.signal.fftconvolve(a, b) print('Time with pyfftw backend installed: %1.3f seconds' % t.timeit(number=100))
which outputs something like:
Time with scipy.fft default backend: 0.267 seconds Time with pyfftw backend installed: 0.162 seconds
Prior to SciPy 1.4 it was necessary to monkey patch the libraries
pyfftw.interfaces.scipy_fftpack are drop-in replacements for the
scipy.fftpack libraries respectively so it is
possible to use them as replacements at run-time through monkey patching.
# Monkey patch fftpack with pyfftw.interfaces.scipy_fftpack scipy.fftpack = pyfftw.interfaces.scipy_fftpack scipy.signal.fftconvolve(a, b)
Note that prior to SciPy 0.16, it was necessary to patch the individual
scipy.signal.signaltools. For example:
scipy.signal.signaltools.ifftn = pyfftw.interfaces.scipy_fftpack.ifftn
The core of this library is provided through the
class. FFTW is fully encapsulated within this class.
The following gives an overview of the
pyfftw.FFTW class, but
the easiest way to of dealing with it is through the
pyfftw.builders helper functions, also
discussed in this tutorial.
For users that already have some experience of FFTW, there is no
interface distinction between any of the supported data types, shapes
or transforms, and operating on arbitrarily strided arrays (which are
common when using
numpy) is fully supported with no copies
In its simplest form, a
pyfftw.FFTW object is created with
a pair of complementary
numpy arrays: an input array and an
output array. They are complementary insomuch as the data types and the
array sizes together define exactly what transform should be performed.
We refer to a valid transform as a scheme.
Internally, three precisions of FFT are supported. These correspond
to single precision floating point, double precision floating point
and long double precision floating
point, which correspond to
longdouble dtypes respectively (and the corresponding
complex types). The precision is decided by the relevant scheme,
which is specified by the dtype of the input array.
Various schemes are supported by
pyfftw.FFTW. The scheme
that is used depends on the data types of the input array and output
arrays, the shape of the arrays and the direction flag. For a full
discussion of the schemes available, see the API documentation for
We will first consider creating a simple one-dimensional transform of a one-dimensional complex array:
import pyfftw a = pyfftw.empty_aligned(128, dtype='complex128') b = pyfftw.empty_aligned(128, dtype='complex128') fft_object = pyfftw.FFTW(a, b)
In this case, we create 2 complex arrays,
b each of
length 128. As before, we use
make sure the array is aligned.
Given these 2 arrays, the only transform that makes sense is a
1D complex DFT. The direction in this case is the default, which is
forward, and so that is the transform that is planned. The
fft_object represents such a transform.
In general, the creation of the
pyfftw.FFTW object clears the
contents of the arrays, so the arrays should be filled or updated
Similarly, to plan the inverse:
c = pyfftw.empty_aligned(128, dtype='complex128') ifft_object = pyfftw.FFTW(b, c, direction='FFTW_BACKWARD')
In this case, the direction argument is given as
(to override the default of
pyfftw.FFTW also supports all of the discrete sine and cosine
transformations (also called real to real transformations) implemented by
FFTW: for example
d = pyfftw.empty_aligned(128, dtype='float64') e = pyfftw.empty_aligned(128, dtype='float64') dct_transform = pyfftw.FFTW(d, e, direction='FFTW_REDFT00')
creates an instance of
pyfftw.FFTW which can execute the
discrete cosine boundary condition with even boundary conditions on
both ends (also known as the DCT-1).
The actual FFT is performed by calling the returned objects:
import numpy # Generate some data ar, ai = numpy.random.randn(2, 128) a[:] = ar + 1j*ai fft_a = fft_object()
Note that calling the object like this performs the FFT and returns
the result in an array. This is the same array as
>>> fft_a is b True
This is particularly useful when using
Calling the FFT object followed by the inverse FFT object yields
an output that is numerically the same as the original
(within numerical accuracy).
>>> fft_a = fft_object() >>> ifft_b = ifft_object() >>> ifft_b is c True >>> numpy.allclose(a, c) True >>> a is c False
In this case, the normalisation of the DFT is performed automatically
by the inverse FFTW object (
ifft_object). This can be disabled
by setting the
It is possible to change the data on which a
pyfftw.FFTW.__call__() accepts both an
input_array and an
output_array argument to update the
arrays. The arrays should be compatible with the arrays with which
pyfftw.FFTW object was originally created. Please read the
API docs on
pyfftw.FFTW.__call__() to fully understand the
requirements for updating the array.
>>> d = pyfftw.empty_aligned(4, dtype='complex128') >>> e = pyfftw.empty_aligned(4, dtype='complex128') >>> f = pyfftw.empty_aligned(4, dtype='complex128') >>> fft_object = pyfftw.FFTW(d, e) >>> fft_object.input_array is d # get the input array from the object True >>> f[:] = [1, 2, 3, 4] # Add some data to f >>> fft_object(f) array([ 10.+0.j, -2.+2.j, -2.+0.j, -2.-2.j]) >>> fft_object.input_array is d # No longer true! False >>> fft_object.input_array is f # It has been updated with f :) True
If the new input array is of the wrong dtype or wrongly strided,
pyfftw.FFTW.__call__() method will copy the new array into the
internal array, if necessary changing it’s dtype in the process.
It should be made clear that the
is simply a helper routine around the other methods of the object.
Though it is expected that most of the time
pyfftw.FFTW.__call__() will be sufficient, all the FFTW
functionality can be accessed through other methods at a slightly
Arrays of more than one dimension are easily supported as well.
In this case, the
axes argument specifies over which axes the
transform is to be taken.
import pyfftw a = pyfftw.empty_aligned((128, 64), dtype='complex128') b = pyfftw.empty_aligned((128, 64), dtype='complex128') # Plan an fft over the last axis fft_object_a = pyfftw.FFTW(a, b) # Over the first axis fft_object_b = pyfftw.FFTW(a, b, axes=(0,)) # Over the both axes fft_object_c = pyfftw.FFTW(a, b, axes=(0,1))
For further information on all the supported transforms, including
real transforms, as well as full documentation on all the
instantiation arguments, see the
When creating a
pyfftw.FFTW object, it is possible to instruct
FFTW how much effort it should put into finding the fastest possible
method for computing the DFT. This is done by specifying a suitable
planner flag in
flags argument to
of the planner flags can take a very long time to complete which can
When the a particular transform has been created, distinguished by things like the data type, the shape, the stridings and the flags, FFTW keeps a record of the fastest way to compute such a transform in future. This is referred to as wisdom. When the program is completed, the wisdom that has been accumulated is forgotten.
It is possible to output the accumulated wisdom using the
wisdom output routines.
pyfftw.export_wisdom() exports and returns the wisdom as a tuple
of strings that can be easily written to file. To load the wisdom back
in, use the
pyfftw.import_wisdom() function which takes as its
argument that same tuple of strings that was returned from
If for some reason you wish to forget the accumulated wisdom, call
If you absolutely need the flexibility of dealing with
pyfftw.FFTW directly, an easier option than constructing valid
arrays and so on is to use the convenient
These functions take care of much of the difficulty in specifying the
exact size and dtype requirements to produce a valid scheme.
pyfftw.builders functions are a series of helper functions
that provide an interface very much like that provided by
numpy.fft, only instead of returning the result of the
pyfftw.FFTW object (or in some cases a wrapper
pyfftw.FFTW) is returned.
import pyfftw a = pyfftw.empty_aligned((128, 64), dtype='complex128') # Generate some data ar, ai = numpy.random.randn(2, 128, 64) a[:] = ar + 1j*ai fft_object = pyfftw.builders.fft(a) b = fft_object()
fft_object is an instance of
the result of the DFT.
Note that in this example, unlike creating a
object using the direct interface, we can fill the array in advance.
This is because by default all the functions in
keep a copy of the input array during creation (though this can
pyfftw.builders functions construct an output array of
the correct size and type. In the case of the regular DFTs, this
always creates an output array of the same size as the input array.
In the case of the real transform, the output array is the right
shape to satisfy the scheme requirements.
The precision of the transform is determined by the dtype of the input array. If the input array is a floating point array, then the precision of the floating point is used. If the input array is not a floating point array then a double precision transform is used. Any calls made to the resultant object with an array of the same size will then be copied into the internal array of the object, changing the dtype in the process.
numpy.fft, it is possible to specify a length (in the
one-dimensional case) or a shape (in the multi-dimensional case) that
may be different to the array that is passed in. In such a case,
a wrapper object of type
pyfftw.builders._utils._FFTWWrapper is returned. From an
interface perspective, this is identical to
difference is in the way calls to the object are handled. With
pyfftw.builders._utils._FFTWWrapper objects, an array that
is passed as an argument when calling the object is copied into the
internal array. This is done by a suitable slicing of the new
passed-in array and the internal array and is done precisely because
the shape of the transform is different to the shape of the input
a = pyfftw.empty_aligned((128, 64), dtype='complex128') fft_wrapper_object = pyfftw.builders.fftn(a, s=(32, 256)) b = fft_wrapper_object()
Inspecting these objects gives us their shapes:
>>> b.shape (32, 256) >>> fft_wrapper_object.input_array.shape (32, 256) >>> a.shape (128, 64)
It is only possible to call
fft_wrapper_object with an array
that is the same shape as
a. In this case, the first axis of
is sliced to include only the first 32 elements, and the second axis
of the internal array is sliced to include only the last 64 elements.
This way, shapes are made consistent for copying.
numpy.fft, these functions are largely
self-explanatory. We point the reader to the
for more information.
If you like the
pyfftw.builders functions, but do not need or wish to
pyfftw.FFTW-instances directly, the third party
planfftw package provides helper functions that return planned functions
similar to those in
numpy.fft, as well as FFTW-powered versions of some
Configuring FFTW planning effort and number of threads¶
The user may set the default number of threads used by the interfaces and
builders at run time by assigning to
may be set by assigning a string such as
For example, to change the effort to
'FFTW_MEASURE' and specify 4 threads:
import pyfftw pyfftw.config.NUM_THREADS = 4 pyfftw.config.PLANNER_EFFORT = 'FFTW_MEASURE'
All functions in
pyfftw.builders use the
pyfftw.config when determining the default number of threads
and planning effort.
The initial values in pyfftw.config at import time can be controlled via the environment variables as detailed in the configuration documentation.
pyfftw.interfacesdeals with repeated values in the
axesargument differently to
numpy.fft(and probably to
scipy.fftpackto, but that’s not documented clearly). Specifically,
numpy.ffttakes the transform along a given axis as many times as it appears in the
pyfftw.interfacestakes the transform only once along each axis that appears, regardless of how many times it appears. This is deemed to be such a fringe corner case that it is ignored.