rfft

→Tensor

rfft(input: Tensor, n: int | None = None, dim: int = -1, norm: str | None = None)

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1-D FFT of a real-valued input along a single axis.

Computes the one-dimensional DFT of a real signal and returns only the non-redundant complex output bins. For a real input of length $N$ , the DFT satisfies $X[N-k] = X^*[k]$ , so only $\lfloor N/2 \rfloor + 1$ unique complex values need to be stored.

The mathematical definition is identical to fft:

X[k] = \sum_{n=0}^{N-1} x[n]\, e^{-i 2\pi k n / N}, \quad k = 0, 1, \ldots, \lfloor N/2 \rfloor.

Parameters

inputTensor

Real-valued input tensor of any shape.

nint= None

Number of input samples used (along dim). The axis is zero-padded or truncated to this length before the transform. If None, the full axis length is used.

dimint= -1

Axis over which to compute the transform. Default is -1 (last axis).

normstr or None= None

Normalisation mode — "backward" (default), "forward", or "ortho". See fftn for the full description.

Returns

Tensor

Complex tensor (complex64) with the same shape as input except that axis dim has length $\lfloor n/2 \rfloor + 1$ (where $n$ is the effective input length along that axis).

Notes

Output size — for an input axis of length $N$ the output has $N//2 + 1$ complex bins:

Bin 0 is the DC component (purely real for a real input).
Bins 1 through $N//2 - 1$ are complex (positive frequencies).
Bin $N//2$ is the Nyquist component (purely real for even $N$ ).

Inverse — use irfft to reconstruct the real signal. You must pass the original length n to resolve the even/odd ambiguity when the output of rfft has an odd number of bins.

Examples

Real FFT of a length-128 signal:
>>> x = lucid.randn(128)
>>> X = lucid.fft.rfft(x)
>>> X.shape   # 128 // 2 + 1 = 65
(65,)
Round-trip:
>>> x_rec = lucid.fft.irfft(X, n=128)
>>> x_rec.shape
(128,)