pylops.signalprocessing.FFT¶

pylops.signalprocessing.
FFT
(dims, dir=0, nfft=None, sampling=1.0, real=False, fftshift=False, engine='numpy', dtype='complex128', **kwargs_fftw)[source]¶ One dimensional FastFourier Transform.
Apply FastFourier Transform (FFT) along a specific direction
dir
of a multidimensional array of sizedim
.Note that the FFT operator is an overload to either the numpy
numpy.fft.fft
(ornumpy.fft.rfft
for real models) in forward mode and to the numpynumpy.fft.ifft
(ornumpy.fft.irfft
for real models) in adjoint mode, or to thepyfftw.FFTW
class.In both cases, scaling is properly taken into account to guarantee that the operator is passing the dottest.
Note
For a real valued input signal, it is possible to store the values of the Fourier transform at positive frequencies only as values at negative frequencies are simply their complex conjugates. However as the operation of removing the negative part of the frequency axis in forward mode and adding the complex conjugates in adjoint mode is nonlinear, the Linear Operator FTT with
real=True
is not expected to pass the dottest. It is thus only advised to use this flag when a forward and adjoint FFT is used in the same chained operator (e.g.,FFT.H*Op*FFT
) such as inpylops.waveeqprocessing.mdd.MDC
.Parameters:  dims :
tuple
Number of samples for each dimension
 dir :
int
, optional Direction along which FFT is applied.
 nfft :
int
, optional Number of samples in Fourier Transform (same as input if
nfft=None
) sampling :
float
, optional Sampling step
dt
. real :
bool
, optional Model to which fft is applied has real numbers (
True
) or not (False
). Used to enforce that the output of adjoint of a real model is real. fftshift :
bool
, optional Apply fftshift/ifftshift (
True
) or not (False
) engine :
str
, optional Engine used for fft computation (
numpy
orfftw
) dtype :
str
, optional Type of elements in input array.
 **kwargs_fftw
Arbitrary keyword arguments for
pyfftw.FTTW
Raises:  ValueError
If
dims
is not provided and ifdir
is bigger thanlen(dims)
 NotImplementedError
If
engine
is neithernumpy
nornumba
Notes
The FFT operator applies the forward Fourier transform to a signal \(d(t)\) in forward mode:
\[D(f) = \mathscr{F} (d) = \int d(t) e^{j2\pi ft} dt\]Similarly, the inverse Fourier transform is applied to the Fourier spectrum \(D(f)\) in adjoint mode:
\[d(t) = \mathscr{F}^{1} (D) = \int D(f) e^{j2\pi ft} df\]Both operators are effectively discretized and solved by a fast iterative algorithm known as Fast Fourier Transform.
Note that the FFT operator is a special operator in that the adjoint is also the inverse of the forward mode. Moreover, in case of real signal in time domain, the Fourier transform in Hermitian.
Attributes:  dims :