rfftn

→Tensor

rfftn(input: Tensor, s: int | Sequence[int] | None = None, dim: int | Sequence[int] | None = None, norm: str | None = None)

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N-dimensional FFT of a real-valued input, exploiting conjugate symmetry.

When the input is real, its N-dimensional DFT satisfies the Hermitian (conjugate-symmetric) property:

X[k_1, \ldots, k_d] = X^*[N_1 - k_1, \ldots, N_d - k_d].

Only the non-redundant half of the spectrum along the last transformed axis needs to be stored. For a last-axis length $N$ , the output contains only $\lfloor N/2 \rfloor + 1$ unique complex values along that axis. All other axes are full-length ( $N_i$ bins each).

Parameters

inputTensor

Real-valued input tensor of any shape. Passing a complex tensor is accepted but the imaginary part is silently discarded by the engine.

sint or sequence of int= None

Signal length(s) along each transformed axis. When given, the last element s[-1] determines the last-axis size before the real FFT; the output last axis will be s[-1] // 2 + 1. len(s) must equal len(dim) when both are sequences. If None, each axis is transformed at its current size.

dimint or sequence of int= None

Axis or axes over which to compute the transform. The last element of dim is the axis that gets the conjugate-symmetry compression. Defaults to all axes when None.

normstr or None= None

Normalisation mode — "backward" (default), "forward", or "ortho".

N

is the product of the full (uncompressed) lengths of all transformed axes.

Returns

Tensor

Complex tensor (complex64) with the same shape as input except:

All transformed axes except the last have their original lengths (or s[i] when specified).
The last transformed axis has length $\lfloor N_{\text{last}} / 2 \rfloor + 1$ .

Notes

Why n//2 + 1? — A real-to-complex 1-D DFT of length $N$ has the symmetry $X[N-k] = X^*[k]$ . The unique bins are therefore $k = 0, 1, \ldots, \lfloor N/2 \rfloor$ , giving $\lfloor N/2 \rfloor + 1$ complex values. For even $N$ this is $N/2 + 1$ ; for odd $N$ it is $(N+1)/2$ .

Nyquist theorem — for a real discrete signal sampled at rate $f_s$ , the highest representable frequency is the Nyquist frequency $f_s / 2$ . The bin at index $\lfloor N/2 \rfloor$ corresponds exactly to this frequency. Any signal energy above $f_s / 2$ aliases back into the $[0, f_s/2]$ range — this is the aliasing artefact that the sampling theorem warns against.

Memory saving — compared to fftn, rfftn stores roughly half the complex coefficients along the last axis, reducing peak memory and compute by ~2× for large real tensors.

Examples

1-D real FFT:
>>> x = lucid.randn(128)
>>> X = lucid.fft.rfftn(x)
>>> X.shape   # 128 // 2 + 1 = 65
(65,)
3-D real FFT over all axes:
>>> x = lucid.randn(16, 32, 64)
>>> X = lucid.fft.rfftn(x)
>>> X.shape   # last axis: 64 // 2 + 1 = 33
(16, 32, 33)
Explicit output size for zero-padding:
>>> X = lucid.fft.rfftn(lucid.randn(60), s=[64])
>>> X.shape   # 64 // 2 + 1 = 33
(33,)