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# torch.fft¶

Discrete Fourier transforms and related functions.

## Fast Fourier Transforms¶

torch.fft.fft(input, n=None, dim=-1, norm=None) → Tensor

Computes the one dimensional discrete Fourier transform of input.

Note

The Fourier domain representation of any real signal satisfies the Hermitian property: X[i] = conj(X[-i]). This function always returns both the positive and negative frequency terms even though, for real inputs, the negative frequencies are redundant. rfft() returns the more compact one-sided representation where only the positive frequencies are returned.

Parameters
• input (Tensor) – the input tensor

• n (int, optional) – Signal length. If given, the input will either be zero-padded or trimmed to this length before computing the FFT.

• dim (int, optional) – The dimension along which to take the one dimensional FFT.

• norm (str, optional) –

Normalization mode. For the forward transform (fft()), these correspond to:

• "forward" - normalize by 1/n

• "backward" - no normalization

• "ortho" - normalize by 1/sqrt(n) (making the FFT orthonormal)

Calling the backward transform (ifft()) with the same normalization mode will apply an overall normalization of 1/n between the two transforms. This is required to make ifft() the exact inverse.

Default is "backward" (no normalization).

Example

>>> t = torch.arange(4)
>>> t
tensor([0, 1, 2, 3])
>>> torch.fft.fft(t)
tensor([ 6.+0.j, -2.+2.j, -2.+0.j, -2.-2.j])

>>> t = tensor([0.+1.j, 2.+3.j, 4.+5.j, 6.+7.j])
>>> torch.fft.fft(t)
tensor([12.+16.j, -8.+0.j, -4.-4.j,  0.-8.j])

torch.fft.ifft(input, n=None, dim=-1, norm=None) → Tensor

Computes the one dimensional inverse discrete Fourier transform of input.

Parameters
• input (Tensor) – the input tensor

• n (int, optional) – Signal length. If given, the input will either be zero-padded or trimmed to this length before computing the IFFT.

• dim (int, optional) – The dimension along which to take the one dimensional IFFT.

• norm (str, optional) –

Normalization mode. For the backward transform (ifft()), these correspond to:

• "forward" - no normalization

• "backward" - normalize by 1/n

• "ortho" - normalize by 1/sqrt(n) (making the IFFT orthonormal)

Calling the forward transform (fft()) with the same normalization mode will apply an overall normalization of 1/n between the two transforms. This is required to make ifft() the exact inverse.

Default is "backward" (normalize by 1/n).

Example

>>> t = torch.tensor([ 6.+0.j, -2.+2.j, -2.+0.j, -2.-2.j])
>>> torch.fft.ifft(t)
tensor([0.+0.j, 1.+0.j, 2.+0.j, 3.+0.j])

torch.fft.fft2(input, s=None, dim=(-2, -1), norm=None) → Tensor

Computes the 2 dimensional discrete Fourier transform of input. Equivalent to fftn() but FFTs only the last two dimensions by default.

Note

The Fourier domain representation of any real signal satisfies the Hermitian property: X[i, j] = conj(X[-i, -j]). This function always returns all positive and negative frequency terms even though, for real inputs, half of these values are redundant. rfft2() returns the more compact one-sided representation where only the positive frequencies of the last dimension are returned.

Parameters
• input (Tensor) – the input tensor

• s (Tuple[int], optional) – Signal size in the transformed dimensions. If given, each dimension dim[i] will either be zero-padded or trimmed to the length s[i] before computing the FFT. If a length -1 is specified, no padding is done in that dimension. Default: s = [input.size(d) for d in dim]

• dim (Tuple[int], optional) – Dimensions to be transformed. Default: last two dimensions.

• norm (str, optional) –

Normalization mode. For the forward transform (fft2()), these correspond to:

• "forward" - normalize by 1/n

• "backward" - no normalization

• "ortho" - normalize by 1/sqrt(n) (making the FFT orthonormal)

Where n = prod(s) is the logical FFT size. Calling the backward transform (ifft2()) with the same normalization mode will apply an overall normalization of 1/n between the two transforms. This is required to make ifft2() the exact inverse.

Default is "backward" (no normalization).

Example

>>> x = torch.rand(10, 10, dtype=torch.complex64)
>>> fft2 = torch.fft.fft2(t)


The discrete Fourier transform is separable, so fft2() here is equivalent to two one-dimensional fft() calls:

>>> two_ffts = torch.fft.fft(torch.fft.fft(x, dim=0), dim=1)
>>> torch.allclose(fft2, two_ffts)

torch.fft.ifft2(input, s=None, dim=(-2, -1), norm=None) → Tensor

Computes the 2 dimensional inverse discrete Fourier transform of input. Equivalent to ifftn() but IFFTs only the last two dimensions by default.

Parameters
• input (Tensor) – the input tensor

• s (Tuple[int], optional) – Signal size in the transformed dimensions. If given, each dimension dim[i] will either be zero-padded or trimmed to the length s[i] before computing the IFFT. If a length -1 is specified, no padding is done in that dimension. Default: s = [input.size(d) for d in dim]

• dim (Tuple[int], optional) – Dimensions to be transformed. Default: last two dimensions.

• norm (str, optional) –

Normalization mode. For the backward transform (ifft2()), these correspond to:

• "forward" - no normalization

• "backward" - normalize by 1/n

• "ortho" - normalize by 1/sqrt(n) (making the IFFT orthonormal)

Where n = prod(s) is the logical IFFT size. Calling the forward transform (fft2()) with the same normalization mode will apply an overall normalization of 1/n between the two transforms. This is required to make ifft2() the exact inverse.

Default is "backward" (normalize by 1/n).

Example

>>> x = torch.rand(10, 10, dtype=torch.complex64)
>>> ifft2 = torch.fft.ifft2(t)


The discrete Fourier transform is separable, so ifft2() here is equivalent to two one-dimensional ifft() calls:

>>> two_iffts = torch.fft.ifft(torch.fft.ifft(x, dim=0), dim=1)
>>> torch.allclose(ifft2, two_iffts)

torch.fft.fftn(input, s=None, dim=None, norm=None) → Tensor

Computes the N dimensional discrete Fourier transform of input.

Note

The Fourier domain representation of any real signal satisfies the Hermitian property: X[i_1, ..., i_n] = conj(X[-i_1, ..., -i_n]). This function always returns all positive and negative frequency terms even though, for real inputs, half of these values are redundant. rfftn() returns the more compact one-sided representation where only the positive frequencies of the last dimension are returned.

Parameters
• input (Tensor) – the input tensor

• s (Tuple[int], optional) – Signal size in the transformed dimensions. If given, each dimension dim[i] will either be zero-padded or trimmed to the length s[i] before computing the FFT. If a length -1 is specified, no padding is done in that dimension. Default: s = [input.size(d) for d in dim]

• dim (Tuple[int], optional) – Dimensions to be transformed. Default: all dimensions, or the last len(s) dimensions if s is given.

• norm (str, optional) –

Normalization mode. For the forward transform (fftn()), these correspond to:

• "forward" - normalize by 1/n

• "backward" - no normalization

• "ortho" - normalize by 1/sqrt(n) (making the FFT orthonormal)

Where n = prod(s) is the logical FFT size. Calling the backward transform (ifftn()) with the same normalization mode will apply an overall normalization of 1/n between the two transforms. This is required to make ifftn() the exact inverse.

Default is "backward" (no normalization).

Example

>>> x = torch.rand(10, 10, dtype=torch.complex64)
>>> fftn = torch.fft.fftn(t)


The discrete Fourier transform is separable, so fftn() here is equivalent to two one-dimensional fft() calls:

>>> two_ffts = torch.fft.fft(torch.fft.fft(x, dim=0), dim=1)
>>> torch.allclose(fftn, two_ffts)

torch.fft.ifftn(input, s=None, dim=None, norm=None) → Tensor

Computes the N dimensional inverse discrete Fourier transform of input.

Parameters
• input (Tensor) – the input tensor

• s (Tuple[int], optional) – Signal size in the transformed dimensions. If given, each dimension dim[i] will either be zero-padded or trimmed to the length s[i] before computing the IFFT. If a length -1 is specified, no padding is done in that dimension. Default: s = [input.size(d) for d in dim]

• dim (Tuple[int], optional) – Dimensions to be transformed. Default: all dimensions, or the last len(s) dimensions if s is given.

• norm (str, optional) –

Normalization mode. For the backward transform (ifftn()), these correspond to:

• "forward" - no normalization

• "backward" - normalize by 1/n

• "ortho" - normalize by 1/sqrt(n) (making the IFFT orthonormal)

Where n = prod(s) is the logical IFFT size. Calling the forward transform (fftn()) with the same normalization mode will apply an overall normalization of 1/n between the two transforms. This is required to make ifftn() the exact inverse.

Default is "backward" (normalize by 1/n).

Example

>>> x = torch.rand(10, 10, dtype=torch.complex64)
>>> ifftn = torch.fft.ifftn(t)


The discrete Fourier transform is separable, so ifftn() here is equivalent to two one-dimensional ifft() calls:

>>> two_iffts = torch.fft.ifft(torch.fft.ifft(x, dim=0), dim=1)
>>> torch.allclose(ifftn, two_iffts)

torch.fft.rfft(input, n=None, dim=-1, norm=None) → Tensor

Computes the one dimensional Fourier transform of real-valued input.

The FFT of a real signal is Hermitian-symmetric, X[i] = conj(X[-i]) so the output contains only the positive frequencies below the Nyquist frequency. To compute the full output, use fft()

Parameters
• input (Tensor) – the real input tensor

• n (int, optional) – Signal length. If given, the input will either be zero-padded or trimmed to this length before computing the real FFT.

• dim (int, optional) – The dimension along which to take the one dimensional real FFT.

• norm (str, optional) –

Normalization mode. For the forward transform (rfft()), these correspond to:

• "forward" - normalize by 1/n

• "backward" - no normalization

• "ortho" - normalize by 1/sqrt(n) (making the FFT orthonormal)

Calling the backward transform (irfft()) with the same normalization mode will apply an overall normalization of 1/n between the two transforms. This is required to make irfft() the exact inverse.

Default is "backward" (no normalization).

Example

>>> t = torch.arange(4)
>>> t
tensor([0, 1, 2, 3])
>>> torch.fft.rfft(t)
tensor([ 6.+0.j, -2.+2.j, -2.+0.j])


Compare against the full output from fft():

>>> torch.fft.fft(t)
tensor([ 6.+0.j, -2.+2.j, -2.+0.j, -2.-2.j])


Notice that the symmetric element T[-1] == T.conj() is omitted. At the Nyquist frequency T[-2] == T is it’s own symmetric pair, and therefore must always be real-valued.

torch.fft.irfft(input, n=None, dim=-1, norm=None) → Tensor

Computes the inverse of rfft().

input is interpreted as a one-sided Hermitian signal in the Fourier domain, as produced by rfft(). By the Hermitian property, the output will be real-valued.

Note

Some input frequencies must be real-valued to satisfy the Hermitian property. In these cases the imaginary component will be ignored. For example, any imaginary component in the zero-frequency term cannot be represented in a real output and so will always be ignored.

Note

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, the signal is assumed to be even length and odd signals will not round-trip properly. So, it is recommended to always pass the signal length n.

Parameters
• input (Tensor) – the input tensor representing a half-Hermitian signal

• n (int, optional) – Output signal length. This determines the length of the output signal. If given, the input will either be zero-padded or trimmed to this length before computing the real IFFT. Defaults to even output: n=2*(input.size(dim) - 1).

• dim (int, optional) – The dimension along which to take the one dimensional real IFFT.

• norm (str, optional) –

Normalization mode. For the backward transform (irfft()), these correspond to:

• "forward" - no normalization

• "backward" - normalize by 1/n

• "ortho" - normalize by 1/sqrt(n) (making the real IFFT orthonormal)

Calling the forward transform (rfft()) with the same normalization mode will apply an overall normalization of 1/n between the two transforms. This is required to make irfft() the exact inverse.

Default is "backward" (normalize by 1/n).

Example

>>> t = torch.arange(5)
>>> t
tensor([0, 1, 2, 3, 4])
>>> T = torch.fft.rfft(t)
>>> T
tensor([10.0000+0.0000j, -2.5000+3.4410j, -2.5000+0.8123j])


Without specifying the output length to irfft(), the output will not round-trip properly because the input is odd-length:

>>> torch.fft.irfft(T)
tensor([0.6250, 1.4045, 3.1250, 4.8455])


So, it is recommended to always pass the signal length n:

>>> torch.fft.irfft(T, t.numel())
tensor([0.0000, 1.0000, 2.0000, 3.0000, 4.0000])

torch.fft.rfft2(input, s=None, dim=(-2, -1), norm=None) → Tensor

Computes the 2-dimensional discrete Fourier transform of real input. Equivalent to rfftn() but FFTs only the last two dimensions by default.

The FFT of a real signal is Hermitian-symmetric, X[i, j] = conj(X[-i, -j]), so the full fft2() output contains redundant information. rfft2() instead omits the negative frequencies in the last dimension.

Parameters
• input (Tensor) – the input tensor

• s (Tuple[int], optional) – Signal size in the transformed dimensions. If given, each dimension dim[i] will either be zero-padded or trimmed to the length s[i] before computing the real FFT. If a length -1 is specified, no padding is done in that dimension. Default: s = [input.size(d) for d in dim]

• dim (Tuple[int], optional) – Dimensions to be transformed. Default: last two dimensions.

• norm (str, optional) –

Normalization mode. For the forward transform (rfft2()), these correspond to:

• "forward" - normalize by 1/n

• "backward" - no normalization

• "ortho" - normalize by 1/sqrt(n) (making the real FFT orthonormal)

Where n = prod(s) is the logical FFT size. Calling the backward transform (irfft2()) with the same normalization mode will apply an overall normalization of 1/n between the two transforms. This is required to make irfft2() the exact inverse.

Default is "backward" (no normalization).

Example

>>> t = torch.rand(10, 10)
>>> rfft2 = torch.fft.rfft2(t)
>>> rfft2.size()
torch.Size([10, 6])


Compared against the full output from fft2(), we have all elements up to the Nyquist frequency.

>>> fft2 = torch.fft.fft2(t)
>>> torch.allclose(fft2[..., :6], rfft2)
True


The discrete Fourier transform is separable, so rfft2() here is equivalent to a combination of fft() and rfft():

>>> two_ffts = torch.fft.fft(torch.fft.rfft(x, dim=1), dim=0)
>>> torch.allclose(rfft2, two_ffts)

torch.fft.irfft2(input, s=None, dim=(-2, -1), norm=None) → Tensor

Computes the inverse of rfft2(). Equivalent to irfftn() but IFFTs only the last two dimensions by default.

input is interpreted as a one-sided Hermitian signal in the Fourier domain, as produced by rfft2(). By the Hermitian property, the output will be real-valued.

Note

Some input frequencies must be real-valued to satisfy the Hermitian property. In these cases the imaginary component will be ignored. For example, any imaginary component in the zero-frequency term cannot be represented in a real output and so will always be ignored.

Note

The correct interpretation of the Hermitian input depends on the length 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, the signal is assumed to be even length and odd signals will not round-trip properly. So, it is recommended to always pass the signal shape s.

Parameters
• input (Tensor) – the input tensor

• s (Tuple[int], optional) – Signal size in the transformed dimensions. If given, each dimension dim[i] will either be zero-padded or trimmed to the length s[i] before computing the real FFT. If a length -1 is specified, no padding is done in that dimension. Defaults to even output in the last dimension: s[-1] = 2*(input.size(dim[-1]) - 1).

• dim (Tuple[int], optional) – Dimensions to be transformed. The last dimension must be the half-Hermitian compressed dimension. Default: last two dimensions.

• norm (str, optional) –

Normalization mode. For the backward transform (irfft2()), these correspond to:

• "forward" - no normalization

• "backward" - normalize by 1/n

• "ortho" - normalize by 1/sqrt(n) (making the real IFFT orthonormal)

Where n = prod(s) is the logical IFFT size. Calling the forward transform (rfft2()) with the same normalization mode will apply an overall normalization of 1/n between the two transforms. This is required to make irfft2() the exact inverse.

Default is "backward" (normalize by 1/n).

Example

>>> t = torch.rand(10, 9)
>>> T = torch.fft.rfft2(t)


Without specifying the output length to irfft2(), the output will not round-trip properly because the input is odd-length in the last dimension:

>>> torch.fft.irfft2(T).size()
torch.Size([10, 10])


So, it is recommended to always pass the signal shape s.

>>> roundtrip = torch.fft.irfft2(T, t.size())
>>> roundtrip.size()
torch.Size([10, 9])
>>> torch.allclose(roundtrip, t)
True

torch.fft.rfftn(input, s=None, dim=None, norm=None) → Tensor

Computes the N-dimensional discrete Fourier transform of real input.

The FFT of a real signal is Hermitian-symmetric, X[i_1, ..., i_n] = conj(X[-i_1, ..., -i_n]) so the full fftn() output contains redundant information. rfftn() instead omits the negative frequencies in the last dimension.

Parameters
• input (Tensor) – the input tensor

• s (Tuple[int], optional) – Signal size in the transformed dimensions. If given, each dimension dim[i] will either be zero-padded or trimmed to the length s[i] before computing the real FFT. If a length -1 is specified, no padding is done in that dimension. Default: s = [input.size(d) for d in dim]

• dim (Tuple[int], optional) – Dimensions to be transformed. Default: all dimensions, or the last len(s) dimensions if s is given.

• norm (str, optional) –

Normalization mode. For the forward transform (rfftn()), these correspond to:

• "forward" - normalize by 1/n

• "backward" - no normalization

• "ortho" - normalize by 1/sqrt(n) (making the real FFT orthonormal)

Where n = prod(s) is the logical FFT size. Calling the backward transform (irfftn()) with the same normalization mode will apply an overall normalization of 1/n between the two transforms. This is required to make irfftn() the exact inverse.

Default is "backward" (no normalization).

Example

>>> t = torch.rand(10, 10)
>>> rfftn = torch.fft.rfftn(t)
>>> rfftn.size()
torch.Size([10, 6])


Compared against the full output from fftn(), we have all elements up to the Nyquist frequency.

>>> fftn = torch.fft.fftn(t)
>>> torch.allclose(fftn[..., :6], rfftn)
True


The discrete Fourier transform is separable, so rfftn() here is equivalent to a combination of fft() and rfft():

>>> two_ffts = torch.fft.fft(torch.fft.rfft(x, dim=1), dim=0)
>>> torch.allclose(rfftn, two_ffts)

torch.fft.irfftn(input, s=None, dim=None, norm=None) → Tensor

Computes the inverse of rfftn().

input is interpreted as a one-sided Hermitian signal in the Fourier domain, as produced by rfftn(). By the Hermitian property, the output will be real-valued.

Note

Some input frequencies must be real-valued to satisfy the Hermitian property. In these cases the imaginary component will be ignored. For example, any imaginary component in the zero-frequency term cannot be represented in a real output and so will always be ignored.

Note

The correct interpretation of the Hermitian input depends on the length 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, the signal is assumed to be even length and odd signals will not round-trip properly. So, it is recommended to always pass the signal shape s.

Parameters
• input (Tensor) – the input tensor

• s (Tuple[int], optional) – Signal size in the transformed dimensions. If given, each dimension dim[i] will either be zero-padded or trimmed to the length s[i] before computing the real FFT. If a length -1 is specified, no padding is done in that dimension. Defaults to even output in the last dimension: s[-1] = 2*(input.size(dim[-1]) - 1).

• dim (Tuple[int], optional) – Dimensions to be transformed. The last dimension must be the half-Hermitian compressed dimension. Default: all dimensions, or the last len(s) dimensions if s is given.

• norm (str, optional) –

Normalization mode. For the backward transform (irfftn()), these correspond to:

• "forward" - no normalization

• "backward" - normalize by 1/n

• "ortho" - normalize by 1/sqrt(n) (making the real IFFT orthonormal)

Where n = prod(s) is the logical IFFT size. Calling the forward transform (rfftn()) with the same normalization mode will apply an overall normalization of 1/n between the two transforms. This is required to make irfftn() the exact inverse.

Default is "backward" (normalize by 1/n).

Example

>>> t = torch.rand(10, 9)
>>> T = torch.fft.rfftn(t)


Without specifying the output length to irfft(), the output will not round-trip properly because the input is odd-length in the last dimension:

>>> torch.fft.irfftn(T).size()
torch.Size([10, 10])


So, it is recommended to always pass the signal shape s.

>>> roundtrip = torch.fft.irfftn(T, t.size())
>>> roundtrip.size()
torch.Size([10, 9])
>>> torch.allclose(roundtrip, t)
True

torch.fft.hfft(input, n=None, dim=-1, norm=None) → Tensor

Computes the one dimensional discrete Fourier transform of a Hermitian symmetric input signal.

Note

hfft()/ihfft() are analogous to rfft()/irfft(). The real FFT expects a real signal in the time-domain and gives a Hermitian symmetry in the frequency-domain. The Hermitian FFT is the opposite; Hermitian symmetric in the time-domain and real-valued in the frequency-domain. For this reason, special care needs to be taken with the length argument n, in the same way as with irfft().

Note

Because the signal is Hermitian in the time-domain, the result will be real in the frequency domain. Note that some input frequencies must be real-valued to satisfy the Hermitian property. In these cases the imaginary component will be ignored. For example, any imaginary component in input would result in one or more complex frequency terms which cannot be represented in a real output and so will always be ignored.

Note

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, the signal is assumed to be even length and odd signals will not round-trip properly. So, it is recommended to always pass the signal length n.

Parameters
• input (Tensor) – the input tensor representing a half-Hermitian signal

• n (int, optional) – Output signal length. This determines the length of the real output. If given, the input will either be zero-padded or trimmed to this length before computing the Hermitian FFT. Defaults to even output: n=2*(input.size(dim) - 1).

• dim (int, optional) – The dimension along which to take the one dimensional Hermitian FFT.

• norm (str, optional) –

Normalization mode. For the forward transform (hfft()), these correspond to:

• "forward" - normalize by 1/n

• "backward" - no normalization

• "ortho" - normalize by 1/sqrt(n) (making the Hermitian FFT orthonormal)

Calling the backward transform (ihfft()) with the same normalization mode will apply an overall normalization of 1/n between the two transforms. This is required to make ihfft() the exact inverse.

Default is "backward" (no normalization).

Example

Taking a real-valued frequency signal and bringing it into the time domain gives Hermitian symmetric output:

>>> t = torch.arange(5)
>>> t
tensor([0, 1, 2, 3, 4])
>>> T = torch.fft.ifft(t)
>>> T
tensor([ 2.0000+-0.0000j, -0.5000-0.6882j, -0.5000-0.1625j, -0.5000+0.1625j,
-0.5000+0.6882j])


Note that T == T[-1].conj() and T == T[-2].conj() is redundant. We can thus compute the forward transform without considering negative frequencies:

>>> torch.fft.hfft(T[:3], n=5)
tensor([0., 1., 2., 3., 4.])


Like with irfft(), the output length must be given in order to recover an even length output:

>>> torch.fft.hfft(T[:3])
tensor([0.5000, 1.1236, 2.5000, 3.8764])

torch.fft.ihfft(input, n=None, dim=-1, norm=None) → Tensor

Computes the inverse of hfft().

input must be a real-valued signal, interpreted in the Fourier domain. The IFFT of a real signal is Hermitian-symmetric, X[i] = conj(X[-i]). ihfft() represents this in the one-sided form where only the positive frequencies below the Nyquist frequency are included. To compute the full output, use ifft().

Parameters
• input (Tensor) – the real input tensor

• n (int, optional) – Signal length. If given, the input will either be zero-padded or trimmed to this length before computing the Hermitian IFFT.

• dim (int, optional) – The dimension along which to take the one dimensional Hermitian IFFT.

• norm (str, optional) –

Normalization mode. For the backward transform (ihfft()), these correspond to:

• "forward" - no normalization

• "backward" - normalize by 1/n

• "ortho" - normalize by 1/sqrt(n) (making the IFFT orthonormal)

Calling the forward transform (hfft()) with the same normalization mode will apply an overall normalization of 1/n between the two transforms. This is required to make ihfft() the exact inverse.

Default is "backward" (normalize by 1/n).

Example

>>> t = torch.arange(5)
>>> t
tensor([0, 1, 2, 3, 4])
>>> torch.fft.ihfft(t)
tensor([ 2.0000+-0.0000j, -0.5000-0.6882j, -0.5000-0.1625j])


Compare against the full output from ifft():

>>> torch.fft.ifft(t)
tensor([ 2.0000+-0.0000j, -0.5000-0.6882j, -0.5000-0.1625j, -0.5000+0.1625j,
-0.5000+0.6882j])


## Helper Functions¶

torch.fft.fftfreq(n, d=1.0, *, dtype=None, layout=torch.strided, device=None, requires_grad=False) → Tensor

Computes the discrete Fourier Transform sample frequencies for a signal of size n.

Note

By convention, fft() returns positive frequency terms first, followed by the negative frequencies in reverse order, so that f[-i] for all $0 < i \leq n/2$ in Python gives the negative frequency terms. For an FFT of length n and with inputs spaced in length unit d, the frequencies are:

f = [0, 1, ..., (n - 1) // 2, -(n // 2), ..., -1] / (d * n)


Note

For even lengths, the Nyquist frequency at f[n/2] can be thought of as either negative or positive. fftfreq() follows NumPy’s convention of taking it to be negative.

Parameters
• n (int) – the FFT length

• d (float, optional) – The sampling length scale. The spacing between individual samples of the FFT input. The default assumes unit spacing, dividing that result by the actual spacing gives the result in physical frequency units.

Keyword Arguments
• dtype (torch.dtype, optional) – the desired data type of returned tensor. Default: if None, uses a global default (see torch.set_default_tensor_type()).

• layout (torch.layout, optional) – the desired layout of returned Tensor. Default: torch.strided.

• device (torch.device, optional) – the desired device of returned tensor. Default: if None, uses the current device for the default tensor type (see torch.set_default_tensor_type()). device will be the CPU for CPU tensor types and the current CUDA device for CUDA tensor types.

• requires_grad (bool, optional) – If autograd should record operations on the returned tensor. Default: False.

Example

>>> torch.fft.fftfreq(5)
tensor([ 0.0000,  0.2000,  0.4000, -0.4000, -0.2000])


For even input, we can see the Nyquist frequency at f is given as negative:

>>> torch.fft.fftfreq(4)
tensor([ 0.0000,  0.2500, -0.5000, -0.2500])

torch.fft.rfftfreq(n, d=1.0, *, dtype=None, layout=torch.strided, device=None, requires_grad=False) → Tensor

Computes the sample frequencies for rfft() with a signal of size n.

Note

rfft() returns Hermitian one-sided output, so only the positive frequency terms are returned. For a real FFT of length n and with inputs spaced in length unit d, the frequencies are:

f = torch.arange((n + 1) // 2) / (d * n)


Note

For even lengths, the Nyquist frequency at f[n/2] can be thought of as either negative or positive. Unlike fftfreq(), rfftfreq() always returns it as positive.

Parameters
• n (int) – the real FFT length

• d (float, optional) – The sampling length scale. The spacing between individual samples of the FFT input. The default assumes unit spacing, dividing that result by the actual spacing gives the result in physical frequency units.

Keyword Arguments
• dtype (torch.dtype, optional) – the desired data type of returned tensor. Default: if None, uses a global default (see torch.set_default_tensor_type()).

• layout (torch.layout, optional) – the desired layout of returned Tensor. Default: torch.strided.

• device (torch.device, optional) – the desired device of returned tensor. Default: if None, uses the current device for the default tensor type (see torch.set_default_tensor_type()). device will be the CPU for CPU tensor types and the current CUDA device for CUDA tensor types.

• requires_grad (bool, optional) – If autograd should record operations on the returned tensor. Default: False.

Example

>>> torch.fft.rfftfreq(5)
tensor([ 0.0000,  0.2000,  0.4000])

>>> torch.fft.rfftfreq(4)
tensor([ 0.0000,  0.2500, 0.5000])


Compared to the output from fftfreq(), we see that the Nyquist frequency at f has changed sign: >>> torch.fft.fftfreq(4) tensor([ 0.0000, 0.2500, -0.5000, -0.2500])

torch.fft.fftshift(input, dim=None) → Tensor

Reorders n-dimensional FFT data, as provided by fftn(), to have negative frequency terms first.

This performs a periodic shift of n-dimensional data such that the origin (0, ..., 0) is moved to the center of the tensor. Specifically, to input.shape[dim] // 2 in each selected dimension.

Note

By convention, the FFT returns positive frequency terms first, followed by the negative frequencies in reverse order, so that f[-i] for all $0 < i \leq n/2$ in Python gives the negative frequency terms. fftshift() rearranges all frequencies into ascending order from negative to positive with the zero-frequency term in the center.

Note

For even lengths, the Nyquist frequency at f[n/2] can be thought of as either negative or positive. fftshift() always puts the Nyquist term at the 0-index. This is the same convention used by fftfreq().

Parameters
• input (Tensor) – the tensor in FFT order

• dim (int, Tuple[int], optional) – The dimensions to rearrange. Only dimensions specified here will be rearranged, any other dimensions will be left in their original order. Default: All dimensions of input.

Example

>>> f = torch.fft.fftfreq(4)
>>> f
tensor([ 0.0000,  0.2500, -0.5000, -0.2500])

>>> torch.fft.fftshift(f)
tensor([-0.5000, -0.2500,  0.0000,  0.2500])


Also notice that the Nyquist frequency term at f was moved to the beginning of the tensor.

This also works for multi-dimensional transforms:

>>> x = torch.fft.fftfreq(5, d=1/5) + 0.1 * torch.fft.fftfreq(5, d=1/5).unsqueeze(1)
>>> x
tensor([[ 0.0000,  1.0000,  2.0000, -2.0000, -1.0000],
[ 0.1000,  1.1000,  2.1000, -1.9000, -0.9000],
[ 0.2000,  1.2000,  2.2000, -1.8000, -0.8000],
[-0.2000,  0.8000,  1.8000, -2.2000, -1.2000],
[-0.1000,  0.9000,  1.9000, -2.1000, -1.1000]])

>>> torch.fft.fftshift(x)
tensor([[-2.2000, -1.2000, -0.2000,  0.8000,  1.8000],
[-2.1000, -1.1000, -0.1000,  0.9000,  1.9000],
[-2.0000, -1.0000,  0.0000,  1.0000,  2.0000],
[-1.9000, -0.9000,  0.1000,  1.1000,  2.1000],
[-1.8000, -0.8000,  0.2000,  1.2000,  2.2000]])


fftshift() can also be useful for spatial data. If our data is defined on a centered grid ([-(N//2), (N-1)//2]) then we can use the standard FFT defined on an uncentered grid ([0, N)) by first applying an ifftshift().

>>> x_centered = torch.arange(-5, 5)
>>> x_uncentered = torch.fft.ifftshift(x_centered)
>>> fft_uncentered = torch.fft.fft(x_uncentered)


Similarly, we can convert the frequency domain components to centered convention by applying fftshift().

>>> fft_centered = torch.fft.fftshift(fft_uncentered)


The inverse transform, from centered Fourier space back to centered spatial data, can be performed by applying the inverse shifts in reverse order:

>>> x_centered_2 = torch.fft.fftshift(torch.fft.ifft(torch.fft.ifftshift(fft_centered)))
>>> torch.allclose(x_centered.to(torch.complex64), x_centered_2)
True

torch.fft.ifftshift(input, dim=None) → Tensor

Inverse of fftshift().

Parameters
• input (Tensor) – the tensor in FFT order

• dim (int, Tuple[int], optional) – The dimensions to rearrange. Only dimensions specified here will be rearranged, any other dimensions will be left in their original order. Default: All dimensions of input.

Example

>>> f = torch.fft.fftfreq(5)
>>> f
tensor([ 0.0000,  0.2000,  0.4000, -0.4000, -0.2000])


A round-trip through fftshift() and ifftshift() gives the same result:

>>> shifted = torch.fftshift(f)
>>> torch.ifftshift(shifted)
tensor([ 0.0000,  0.2000,  0.4000, -0.4000, -0.2000]) ## Docs

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