Discrete Fourier Transforms
Routines in this module:
fft(a, n=None, axis=-1, norm="backward") ifft(a, n=None, axis=-1, norm="backward") rfft(a, n=None, axis=-1, norm="backward") irfft(a, n=None, axis=-1, norm="backward") hfft(a, n=None, axis=-1, norm="backward") ihfft(a, n=None, axis=-1, norm="backward") fftn(a, s=None, axes=None, norm="backward") ifftn(a, s=None, axes=None, norm="backward") rfftn(a, s=None, axes=None, norm="backward") irfftn(a, s=None, axes=None, norm="backward") fft2(a, s=None, axes=(-2,-1), norm="backward") ifft2(a, s=None, axes=(-2, -1), norm="backward") rfft2(a, s=None, axes=(-2,-1), norm="backward") irfft2(a, s=None, axes=(-2, -1), norm="backward")
i = inverse transform r = transform of purely real data h = Hermite transform n = n-dimensional transform 2 = 2-dimensional transform (Note: 2D routines are just nD routines with different default behavior.)
| Variable | array_function_dispatch |
Undocumented |
| Function | _cook_nd_args |
Undocumented |
| Function | _fft_dispatcher |
Undocumented |
| Function | _fftn_dispatcher |
Undocumented |
| Function | _get_backward_norm |
Undocumented |
| Function | _get_forward_norm |
Undocumented |
| Function | _raw_fft |
Undocumented |
| Function | _raw_fftnd |
Undocumented |
| Function | _swap_direction |
Undocumented |
| Function | fft |
Compute the one-dimensional discrete Fourier Transform. |
| Function | fft2 |
Compute the 2-dimensional discrete Fourier Transform. |
| Function | fftn |
Compute the N-dimensional discrete Fourier Transform. |
| Function | hfft |
Compute the FFT of a signal that has Hermitian symmetry, i.e., a real spectrum. |
| Function | ifft |
Compute the one-dimensional inverse discrete Fourier Transform. |
| Function | ifft2 |
Compute the 2-dimensional inverse discrete Fourier Transform. |
| Function | ifftn |
Compute the N-dimensional inverse discrete Fourier Transform. |
| Function | ihfft |
Compute the inverse FFT of a signal that has Hermitian symmetry. |
| Function | irfft |
Computes the inverse of rfft. |
| Function | irfft2 |
Computes the inverse of rfft2. |
| Function | irfftn |
Computes the inverse of rfftn. |
| Function | rfft |
Compute the one-dimensional discrete Fourier Transform for real input. |
| Function | rfft2 |
Compute the 2-dimensional FFT of a real array. |
| Function | rfftn |
Compute the N-dimensional discrete Fourier Transform for real input. |
| Constant | _SWAP_DIRECTION_MAP |
Undocumented |
Compute the one-dimensional discrete Fourier Transform.
This function computes the one-dimensional n-point discrete Fourier Transform (DFT) with the efficient Fast Fourier Transform (FFT) algorithm [CT].
n is smaller than the length of the input, the input is cropped.
If it is larger, the input is padded with zeros. If n is not given,
the length of the input along the axis specified by axis is used.Normalization mode (see numpy.fft). Default is "backward".
Indicates which direction of the forward/backward pair of transforms
is scaled and with what normalization factor.
axis, or the last one if axis is not specified.axis is not a valid axis of a.numpy.fft : for definition of the DFT and conventions used.
ifft : The inverse of fft.
fft2 : The two-dimensional FFT.
fftn : The n-dimensional FFT.
rfftn : The n-dimensional FFT of real input.
fftfreq : Frequency bins for given FFT parameters.
FFT (Fast Fourier Transform) refers to a way the discrete Fourier
Transform (DFT) can be calculated efficiently, by using symmetries in the
calculated terms. The symmetry is highest when n is a power of 2, and
the transform is therefore most efficient for these sizes.
The DFT is defined, with the conventions used in this implementation, in
the documentation for the numpy.fft module.
| [CT] | Cooley, James W., and John W. Tukey, 1965, "An algorithm for the machine calculation of complex Fourier series," Math. Comput. 19: 297-301. |
>>> np.fft.fft(np.exp(2j * np.pi * np.arange(8) / 8)) array([-2.33486982e-16+1.14423775e-17j, 8.00000000e+00-1.25557246e-15j, 2.33486982e-16+2.33486982e-16j, 0.00000000e+00+1.22464680e-16j, -1.14423775e-17+2.33486982e-16j, 0.00000000e+00+5.20784380e-16j, 1.14423775e-17+1.14423775e-17j, 0.00000000e+00+1.22464680e-16j])
In this example, real input has an FFT which is Hermitian, i.e., symmetric
in the real part and anti-symmetric in the imaginary part, as described in
the numpy.fft documentation:
>>> import matplotlib.pyplot as plt >>> t = np.arange(256) >>> sp = np.fft.fft(np.sin(t)) >>> freq = np.fft.fftfreq(t.shape[-1]) >>> plt.plot(freq, sp.real, freq, sp.imag) [<matplotlib.lines.Line2D object at 0x...>, <matplotlib.lines.Line2D object at 0x...>] >>> plt.show()
Compute the 2-dimensional discrete Fourier Transform.
This function computes the n-dimensional discrete Fourier Transform over any axes in an M-dimensional array by means of the Fast Fourier Transform (FFT). By default, the transform is computed over the last two axes of the input array, i.e., a 2-dimensional FFT.
s is not given, the shape of the input along the axes specified
by axes is used.axes means the transform over
that axis is performed multiple times. A one-element sequence means
that a one-dimensional FFT is performed.Normalization mode (see numpy.fft). Default is "backward".
Indicates which direction of the forward/backward pair of transforms
is scaled and with what normalization factor.
axes, or the last two axes if axes is not given.s and axes have different length, or axes not given and
len(s) != 2.axes is larger than than the number of axes of a.ifft2 : The inverse two-dimensional FFT. fft : The one-dimensional FFT. fftn : The n-dimensional FFT. fftshift : Shifts zero-frequency terms to the center of the array.
For two-dimensional input, swaps first and third quadrants, and second and fourth quadrants.
fft2 is just fftn with a different default for axes.
The output, analogously to fft, contains the term for zero frequency in
the low-order corner of the transformed axes, the positive frequency terms
in the first half of these axes, the term for the Nyquist frequency in the
middle of the axes and the negative frequency terms in the second half of
the axes, in order of decreasingly negative frequency.
See fftn for details and a plotting example, and numpy.fft for
definitions and conventions used.
>>> a = np.mgrid[:5, :5][0] >>> np.fft.fft2(a) array([[ 50. +0.j , 0. +0.j , 0. +0.j , # may vary 0. +0.j , 0. +0.j ], [-12.5+17.20477401j, 0. +0.j , 0. +0.j , 0. +0.j , 0. +0.j ], [-12.5 +4.0614962j , 0. +0.j , 0. +0.j , 0. +0.j , 0. +0.j ], [-12.5 -4.0614962j , 0. +0.j , 0. +0.j , 0. +0.j , 0. +0.j ], [-12.5-17.20477401j, 0. +0.j , 0. +0.j , 0. +0.j , 0. +0.j ]])
Compute the N-dimensional discrete Fourier Transform.
This function computes the N-dimensional discrete Fourier Transform over any number of axes in an M-dimensional array by means of the Fast Fourier Transform (FFT).
s is not given, the shape of the input along the axes specified
by axes is used.s is also not specified.
Repeated indices in axes means that the transform over that axis is
performed multiple times.Normalization mode (see numpy.fft). Default is "backward".
Indicates which direction of the forward/backward pair of transforms
is scaled and with what normalization factor.
axes, or by a combination of s and a,
as explained in the parameters section above.s and axes have different length.axes is larger than than the number of axes of a.ifftn : The inverse of fftn, the inverse n-dimensional FFT.
fft : The one-dimensional FFT, with definitions and conventions used.
rfftn : The n-dimensional FFT of real input.
fft2 : The two-dimensional FFT.
fftshift : Shifts zero-frequency terms to centre of array
The output, analogously to fft, contains the term for zero frequency in
the low-order corner of all axes, the positive frequency terms in the
first half of all axes, the term for the Nyquist frequency in the middle
of all axes and the negative frequency terms in the second half of all
axes, in order of decreasingly negative frequency.
See numpy.fft for details, definitions and conventions used.
>>> a = np.mgrid[:3, :3, :3][0] >>> np.fft.fftn(a, axes=(1, 2)) array([[[ 0.+0.j, 0.+0.j, 0.+0.j], # may vary [ 0.+0.j, 0.+0.j, 0.+0.j], [ 0.+0.j, 0.+0.j, 0.+0.j]], [[ 9.+0.j, 0.+0.j, 0.+0.j], [ 0.+0.j, 0.+0.j, 0.+0.j], [ 0.+0.j, 0.+0.j, 0.+0.j]], [[18.+0.j, 0.+0.j, 0.+0.j], [ 0.+0.j, 0.+0.j, 0.+0.j], [ 0.+0.j, 0.+0.j, 0.+0.j]]]) >>> np.fft.fftn(a, (2, 2), axes=(0, 1)) array([[[ 2.+0.j, 2.+0.j, 2.+0.j], # may vary [ 0.+0.j, 0.+0.j, 0.+0.j]], [[-2.+0.j, -2.+0.j, -2.+0.j], [ 0.+0.j, 0.+0.j, 0.+0.j]]])
>>> import matplotlib.pyplot as plt >>> [X, Y] = np.meshgrid(2 * np.pi * np.arange(200) / 12, ... 2 * np.pi * np.arange(200) / 34) >>> S = np.sin(X) + np.cos(Y) + np.random.uniform(0, 1, X.shape) >>> FS = np.fft.fftn(S) >>> plt.imshow(np.log(np.abs(np.fft.fftshift(FS))**2)) <matplotlib.image.AxesImage object at 0x...> >>> plt.show()
Compute the FFT of a signal that has Hermitian symmetry, i.e., a real spectrum.
n output
points, n//2 + 1 input points are necessary. If the input is
longer than this, it is cropped. If it is shorter than this, it is
padded with zeros. If n is not given, it is taken to be 2*(m-1)
where m is the length of the input along the axis specified by
axis.Normalization mode (see numpy.fft). Default is "backward".
Indicates which direction of the forward/backward pair of transforms
is scaled and with what normalization factor.
axis, or the last one if axis is not specified.
The length of the transformed axis is n, or, if n is not given,
2*m - 2 where m is the length of the transformed axis of
the input. To get an odd number of output points, n must be
specified, for instance as 2*m - 1 in the typical case,axis is not a valid axis of a.rfft : Compute the one-dimensional FFT for real input.
ihfft : The inverse of hfft.
hfft/ihfft are a pair analogous to rfft/irfft, but for the
opposite case: here the signal has Hermitian symmetry in the time
domain and is real in the frequency domain. So here it's hfft for
which you must supply the length of the result if it is to be odd.
The correct interpretation of the hermitian input depends on the length of
the original data, as given by n. This is because each input shape could
correspond to either an odd or even length signal. By default, hfft
assumes an even output length which puts the last entry at the Nyquist
frequency; aliasing with its symmetric counterpart. By Hermitian symmetry,
the value is thus treated as purely real. To avoid losing information, the
shape of the full signal must be given.
>>> signal = np.array([1, 2, 3, 4, 3, 2]) >>> np.fft.fft(signal) array([15.+0.j, -4.+0.j, 0.+0.j, -1.-0.j, 0.+0.j, -4.+0.j]) # may vary >>> np.fft.hfft(signal[:4]) # Input first half of signal array([15., -4., 0., -1., 0., -4.]) >>> np.fft.hfft(signal, 6) # Input entire signal and truncate array([15., -4., 0., -1., 0., -4.])
>>> signal = np.array([[1, 1.j], [-1.j, 2]]) >>> np.conj(signal.T) - signal # check Hermitian symmetry array([[ 0.-0.j, -0.+0.j], # may vary [ 0.+0.j, 0.-0.j]]) >>> freq_spectrum = np.fft.hfft(signal) >>> freq_spectrum array([[ 1., 1.], [ 2., -2.]])
Compute the one-dimensional inverse discrete Fourier Transform.
This function computes the inverse of the one-dimensional n-point
discrete Fourier transform computed by fft. In other words,
ifft(fft(a)) == a to within numerical accuracy.
For a general description of the algorithm and definitions,
see numpy.fft.
The input should be ordered in the same way as is returned by fft,
i.e.,
For an even number of input points, A[n//2] represents the sum of
the values at the positive and negative Nyquist frequencies, as the two
are aliased together. See numpy.fft for details.
n is smaller than the length of the input, the input is cropped.
If it is larger, the input is padded with zeros. If n is not given,
the length of the input along the axis specified by axis is used.
See notes about padding issues.Normalization mode (see numpy.fft). Default is "backward".
Indicates which direction of the forward/backward pair of transforms
is scaled and with what normalization factor.
axis, or the last one if axis is not specified.axis is not a valid axis of a.numpy.fft : An introduction, with definitions and general explanations.
fft : The one-dimensional (forward) FFT, of which ifft is the inverse
ifft2 : The two-dimensional inverse FFT.
ifftn : The n-dimensional inverse FFT.
If the input parameter n is larger than the size of the input, the input
is padded by appending zeros at the end. Even though this is the common
approach, it might lead to surprising results. If a different padding is
desired, it must be performed before calling ifft.
>>> np.fft.ifft([0, 4, 0, 0]) array([ 1.+0.j, 0.+1.j, -1.+0.j, 0.-1.j]) # may vary
Create and plot a band-limited signal with random phases:
>>> import matplotlib.pyplot as plt >>> t = np.arange(400) >>> n = np.zeros((400,), dtype=complex) >>> n[40:60] = np.exp(1j*np.random.uniform(0, 2*np.pi, (20,))) >>> s = np.fft.ifft(n) >>> plt.plot(t, s.real, label='real') [<matplotlib.lines.Line2D object at ...>] >>> plt.plot(t, s.imag, '--', label='imaginary') [<matplotlib.lines.Line2D object at ...>] >>> plt.legend() <matplotlib.legend.Legend object at ...> >>> plt.show()
Compute the 2-dimensional inverse discrete Fourier Transform.
This function computes the inverse of the 2-dimensional discrete Fourier Transform over any number of axes in an M-dimensional array by means of the Fast Fourier Transform (FFT). In other words, ifft2(fft2(a)) == a to within numerical accuracy. By default, the inverse transform is computed over the last two axes of the input array.
The input, analogously to ifft, should be ordered in the same way as is
returned by fft2, i.e. it should have the term for zero frequency
in the low-order corner of the two axes, the positive frequency terms in
the first half of these axes, the term for the Nyquist frequency in the
middle of the axes and the negative frequency terms in the second half of
both axes, in order of decreasingly negative frequency.
n for ifft(x, n).
Along each axis, if the given shape is smaller than that of the input,
the input is cropped. If it is larger, the input is padded with zeros.
if s is not given, the shape of the input along the axes specified
by axes is used. See notes for issue on ifft zero padding.axes means the transform over
that axis is performed multiple times. A one-element sequence means
that a one-dimensional FFT is performed.Normalization mode (see numpy.fft). Default is "backward".
Indicates which direction of the forward/backward pair of transforms
is scaled and with what normalization factor.
axes, or the last two axes if axes is not given.s and axes have different length, or axes not given and
len(s) != 2.axes is larger than than the number of axes of a.fft2 : The forward 2-dimensional FFT, of which ifft2 is the inverse.
ifftn : The inverse of the n-dimensional FFT.
fft : The one-dimensional FFT.
ifft : The one-dimensional inverse FFT.
ifft2 is just ifftn with a different default for axes.
See ifftn for details and a plotting example, and numpy.fft for
definition and conventions used.
Zero-padding, analogously with ifft, is performed by appending zeros to
the input along the specified dimension. Although this is the common
approach, it might lead to surprising results. If another form of zero
padding is desired, it must be performed before ifft2 is called.
>>> a = 4 * np.eye(4) >>> np.fft.ifft2(a) array([[1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j], # may vary [0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j], [0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j], [0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j]])
Compute the N-dimensional inverse discrete Fourier Transform.
This function computes the inverse of the N-dimensional discrete
Fourier Transform over any number of axes in an M-dimensional array by
means of the Fast Fourier Transform (FFT). In other words,
ifftn(fftn(a)) == a to within numerical accuracy.
For a description of the definitions and conventions used, see numpy.fft.
The input, analogously to ifft, should be ordered in the same way as is
returned by fftn, i.e. it should have the term for zero frequency
in all axes in the low-order corner, the positive frequency terms in the
first half of all axes, the term for the Nyquist frequency in the middle
of all axes and the negative frequency terms in the second half of all
axes, in order of decreasingly negative frequency.
s is not given, the shape of the input along the axes specified
by axes is used. See notes for issue on ifft zero padding.s is also not specified.
Repeated indices in axes means that the inverse transform over that
axis is performed multiple times.Normalization mode (see numpy.fft). Default is "backward".
Indicates which direction of the forward/backward pair of transforms
is scaled and with what normalization factor.
axes, or by a combination of s or a,
as explained in the parameters section above.s and axes have different length.axes is larger than than the number of axes of a.fftn : The forward n-dimensional FFT, of which ifftn is the inverse.
ifft : The one-dimensional inverse FFT.
ifft2 : The two-dimensional inverse FFT.
ifftshift : Undoes fftshift, shifts zero-frequency terms to beginning
of array.
See numpy.fft for definitions and conventions used.
Zero-padding, analogously with ifft, is performed by appending zeros to
the input along the specified dimension. Although this is the common
approach, it might lead to surprising results. If another form of zero
padding is desired, it must be performed before ifftn is called.
>>> a = np.eye(4) >>> np.fft.ifftn(np.fft.fftn(a, axes=(0,)), axes=(1,)) array([[1.+0.j, 0.+0.j, 0.+0.j, 0.+0.j], # may vary [0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j], [0.+0.j, 0.+0.j, 1.+0.j, 0.+0.j], [0.+0.j, 0.+0.j, 0.+0.j, 1.+0.j]])
Create and plot an image with band-limited frequency content:
>>> import matplotlib.pyplot as plt >>> n = np.zeros((200,200), dtype=complex) >>> n[60:80, 20:40] = np.exp(1j*np.random.uniform(0, 2*np.pi, (20, 20))) >>> im = np.fft.ifftn(n).real >>> plt.imshow(im) <matplotlib.image.AxesImage object at 0x...> >>> plt.show()
Compute the inverse FFT of a signal that has Hermitian symmetry.
n is smaller than
the length of the input, the input is cropped. If it is larger,
the input is padded with zeros. If n is not given, the length of
the input along the axis specified by axis is used.Normalization mode (see numpy.fft). Default is "backward".
Indicates which direction of the forward/backward pair of transforms
is scaled and with what normalization factor.
axis, or the last one if axis is not specified.
The length of the transformed axis is n//2 + 1.hfft, irfft
hfft/ihfft are a pair analogous to rfft/irfft, but for the
opposite case: here the signal has Hermitian symmetry in the time
domain and is real in the frequency domain. So here it's hfft for
which you must supply the length of the result if it is to be odd:
>>> spectrum = np.array([ 15, -4, 0, -1, 0, -4]) >>> np.fft.ifft(spectrum) array([1.+0.j, 2.+0.j, 3.+0.j, 4.+0.j, 3.+0.j, 2.+0.j]) # may vary >>> np.fft.ihfft(spectrum) array([ 1.-0.j, 2.-0.j, 3.-0.j, 4.-0.j]) # may vary
Computes the inverse of rfft.
This function computes the inverse of the one-dimensional n-point
discrete Fourier Transform of real input computed by rfft.
In other words, irfft(rfft(a), len(a)) == a to within numerical
accuracy. (See Notes below for why len(a) is necessary here.)
The input is expected to be in the form returned by rfft, i.e. the
real zero-frequency term followed by the complex positive frequency terms
in order of increasing frequency. Since the discrete Fourier Transform of
real input is Hermitian-symmetric, the negative frequency terms are taken
to be the complex conjugates of the corresponding positive frequency terms.
n output points, n//2+1 input points are necessary. If the
input is longer than this, it is cropped. If it is shorter than this,
it is padded with zeros. If n is not given, it is taken to be
2*(m-1) where m is the length of the input along the axis
specified by axis.Normalization mode (see numpy.fft). Default is "backward".
Indicates which direction of the forward/backward pair of transforms
is scaled and with what normalization factor.
axis, or the last one if axis is not specified.
The length of the transformed axis is n, or, if n is not given,
2*(m-1) where m is the length of the transformed axis of the
input. To get an odd number of output points, n must be specified.axis is not a valid axis of a.numpy.fft : For definition of the DFT and conventions used.
rfft : The one-dimensional FFT of real input, of which irfft is inverse.
fft : The one-dimensional FFT.
irfft2 : The inverse of the two-dimensional FFT of real input.
irfftn : The inverse of the n-dimensional FFT of real input.
Returns the real valued n-point inverse discrete Fourier transform
of a, where a contains the non-negative frequency terms of a
Hermitian-symmetric sequence. n is the length of the result, not the
input.
If you specify an n such that a must be zero-padded or truncated, the
extra/removed values will be added/removed at high frequencies. One can
thus resample a series to m points via Fourier interpolation by:
a_resamp = irfft(rfft(a), m).
The correct interpretation of the hermitian input depends on the length of
the original data, as given by n. This is because each input shape could
correspond to either an odd or even length signal. By default, irfft
assumes an even output length which puts the last entry at the Nyquist
frequency; aliasing with its symmetric counterpart. By Hermitian symmetry,
the value is thus treated as purely real. To avoid losing information, the
correct length of the real input must be given.
>>> np.fft.ifft([1, -1j, -1, 1j]) array([0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j]) # may vary >>> np.fft.irfft([1, -1j, -1]) array([0., 1., 0., 0.])
Notice how the last term in the input to the ordinary ifft is the
complex conjugate of the second term, and the output has zero imaginary
part everywhere. When calling irfft, the negative frequencies are not
specified, and the output array is purely real.
Computes the inverse of rfft2.
Normalization mode (see numpy.fft). Default is "backward".
Indicates which direction of the forward/backward pair of transforms
is scaled and with what normalization factor.
irfft2 is the inverse.rfft : The one-dimensional FFT for real input. irfft : The inverse of the one-dimensional FFT of real input. irfftn : Compute the inverse of the N-dimensional FFT of real input.
>>> a = np.mgrid[:5, :5][0] >>> A = np.fft.rfft2(a) >>> np.fft.irfft2(A, s=a.shape) array([[0., 0., 0., 0., 0.], [1., 1., 1., 1., 1.], [2., 2., 2., 2., 2.], [3., 3., 3., 3., 3.], [4., 4., 4., 4., 4.]])
Computes the inverse of rfftn.
This function computes the inverse of the N-dimensional discrete
Fourier Transform for real input over any number of axes in an
M-dimensional array by means of the Fast Fourier Transform (FFT). In
other words, irfftn(rfftn(a), a.shape) == a to within numerical
accuracy. (The a.shape is necessary like len(a) is for irfft,
and for the same reason.)
The input should be ordered in the same way as is returned by rfftn,
i.e. as for irfft for the final transformation axis, and as for ifftn
along all the other axes.
s is also the
number of input points used along this axis, except for the last axis,
where s[-1]//2+1 points of the input are used.
Along any axis, if the shape indicated by s is smaller than that of
the input, the input is cropped. If it is larger, the input is padded
with zeros. If s is not given, the shape of the input along the axes
specified by axes is used. Except for the last axis which is taken to
be 2*(m-1) where m is the length of the input along that axis.len(s) axes are used, or all axes if s is also not specified.
Repeated indices in axes means that the inverse transform over that
axis is performed multiple times.Normalization mode (see numpy.fft). Default is "backward".
Indicates which direction of the forward/backward pair of transforms
is scaled and with what normalization factor.
axes, or by a combination of s or a,
as explained in the parameters section above.
The length of each transformed axis is as given by the corresponding
element of s, or the length of the input in every axis except for the
last one if s is not given. In the final transformed axis the length
of the output when s is not given is 2*(m-1) where m is the
length of the final transformed axis of the input. To get an odd
number of output points in the final axis, s must be specified.s and axes have different length.axes is larger than than the number of axes of a.ifftn is the inverse.fft : The one-dimensional FFT, with definitions and conventions used. irfft : The inverse of the one-dimensional FFT of real input. irfft2 : The inverse of the two-dimensional FFT of real input.
See fft for definitions and conventions used.
See rfft for definitions and conventions used for real input.
The correct interpretation of the hermitian input depends on the shape of
the original data, as given by s. This is because each input shape could
correspond to either an odd or even length signal. By default, irfftn
assumes an even output length which puts the last entry at the Nyquist
frequency; aliasing with its symmetric counterpart. When performing the
final complex to real transform, the last value is thus treated as purely
real. To avoid losing information, the correct shape of the real input
must be given.
>>> a = np.zeros((3, 2, 2)) >>> a[0, 0, 0] = 3 * 2 * 2 >>> np.fft.irfftn(a) array([[[1., 1.], [1., 1.]], [[1., 1.], [1., 1.]], [[1., 1.], [1., 1.]]])
Compute the one-dimensional discrete Fourier Transform for real input.
This function computes the one-dimensional n-point discrete Fourier Transform (DFT) of a real-valued array by means of an efficient algorithm called the Fast Fourier Transform (FFT).
n is smaller than the length of the input, the input is cropped.
If it is larger, the input is padded with zeros. If n is not given,
the length of the input along the axis specified by axis is used.Normalization mode (see numpy.fft). Default is "backward".
Indicates which direction of the forward/backward pair of transforms
is scaled and with what normalization factor.
axis, or the last one if axis is not specified.
If n is even, the length of the transformed axis is (n/2)+1.
If n is odd, the length is (n+1)/2.axis is not a valid axis of a.numpy.fft : For definition of the DFT and conventions used.
irfft : The inverse of rfft.
fft : The one-dimensional FFT of general (complex) input.
fftn : The n-dimensional FFT.
rfftn : The n-dimensional FFT of real input.
When the DFT is computed for purely real input, the output is Hermitian-symmetric, i.e. the negative frequency terms are just the complex conjugates of the corresponding positive-frequency terms, and the negative-frequency terms are therefore redundant. This function does not compute the negative frequency terms, and the length of the transformed axis of the output is therefore n//2 + 1.
When A = rfft(a) and fs is the sampling frequency, A[0] contains the zero-frequency term 0*fs, which is real due to Hermitian symmetry.
If n is even, A[-1] contains the term representing both positive
and negative Nyquist frequency (+fs/2 and -fs/2), and must also be purely
real. If n is odd, there is no term at fs/2; A[-1] contains
the largest positive frequency (fs/2*(n-1)/n), and is complex in the
general case.
If the input a contains an imaginary part, it is silently discarded.
>>> np.fft.fft([0, 1, 0, 0]) array([ 1.+0.j, 0.-1.j, -1.+0.j, 0.+1.j]) # may vary >>> np.fft.rfft([0, 1, 0, 0]) array([ 1.+0.j, 0.-1.j, -1.+0.j]) # may vary
Notice how the final element of the fft output is the complex conjugate
of the second element, for real input. For rfft, this symmetry is
exploited to compute only the non-negative frequency terms.
Compute the 2-dimensional FFT of a real array.
Normalization mode (see numpy.fft). Default is "backward".
Indicates which direction of the forward/backward pair of transforms
is scaled and with what normalization factor.
>>> a = np.mgrid[:5, :5][0] >>> np.fft.rfft2(a) array([[ 50. +0.j , 0. +0.j , 0. +0.j ], [-12.5+17.20477401j, 0. +0.j , 0. +0.j ], [-12.5 +4.0614962j , 0. +0.j , 0. +0.j ], [-12.5 -4.0614962j , 0. +0.j , 0. +0.j ], [-12.5-17.20477401j, 0. +0.j , 0. +0.j ]])
Compute the N-dimensional discrete Fourier Transform for real input.
This function computes the N-dimensional discrete Fourier Transform over any number of axes in an M-dimensional real array by means of the Fast Fourier Transform (FFT). By default, all axes are transformed, with the real transform performed over the last axis, while the remaining transforms are complex.
s corresponds to n for rfft(x, n), while
for the remaining axes, it corresponds to n for fft(x, n).
Along any axis, if the given shape is smaller than that of the input,
the input is cropped. If it is larger, the input is padded with zeros.
if s is not given, the shape of the input along the axes specified
by axes is used.s is also not specified.Normalization mode (see numpy.fft). Default is "backward".
Indicates which direction of the forward/backward pair of transforms
is scaled and with what normalization factor.
axes, or by a combination of s and a,
as explained in the parameters section above.
The length of the last axis transformed will be s[-1]//2+1,
while the remaining transformed axes will have lengths according to
s, or unchanged from the input.s and axes have different length.axes is larger than than the number of axes of a.rfftn, i.e. the inverse of the n-dimensional FFTfft : The one-dimensional FFT, with definitions and conventions used. rfft : The one-dimensional FFT of real input. fftn : The n-dimensional FFT. rfft2 : The two-dimensional FFT of real input.
The transform for real input is performed over the last transformation
axis, as by rfft, then the transform over the remaining axes is
performed as by fftn. The order of the output is as for rfft for the
final transformation axis, and as for fftn for the remaining
transformation axes.
See fft for details, definitions and conventions used.
>>> a = np.ones((2, 2, 2)) >>> np.fft.rfftn(a) array([[[8.+0.j, 0.+0.j], # may vary [0.+0.j, 0.+0.j]], [[0.+0.j, 0.+0.j], [0.+0.j, 0.+0.j]]])
>>> np.fft.rfftn(a, axes=(2, 0)) array([[[4.+0.j, 0.+0.j], # may vary [4.+0.j, 0.+0.j]], [[0.+0.j, 0.+0.j], [0.+0.j, 0.+0.j]]])