Real-Valued Transforms

rfft, irfft, hfft, ihfft – real-input and Hermitian-symmetric transforms following numpy.fft conventions. Wrappers that call the complex FFT from mpmath_fft via lazy import to avoid circular dependencies.

Real-valued FFT transforms for mpmath.

mpmath_fft._real_transforms.hfft(x, n=None, axis=-1)

Hermitian FFT.

Computes the inverse FFT of a Hermitian-symmetric half-spectrum. Equivalent to n_used * irfft(conj(x), n=n, axis=axis).

Parameters

xndarray[dtype=object] of mp.mpc

Half-spectrum input.

nint, optional

Output length. Default: 2 * (x.shape[axis] - 1).

axisint

Transform axis (default -1).

Returns

yndarray[dtype=object] of mp.mpf

Real-valued output.

mpmath_fft._real_transforms.ihfft(x, n=None, axis=-1)

Inverse Hermitian FFT.

Computes the forward FFT of a real signal and returns the non-redundant half-spectrum, with 1/n scaling. Equivalent to conj(rfft(x, n=n, axis=axis)) / n_used.

Parameters

xndarray[dtype=object] of mp.mpf

Real-valued input.

nint, optional

Transform length. Default: x.shape[axis].

axisint

Transform axis (default -1).

Returns

yndarray[dtype=object] of mp.mpc

Half-spectrum output.

mpmath_fft._real_transforms.irfft(x, n=None, axis=-1)

Complex-to-real inverse FFT.

Reconstructs the full Hermitian-symmetric spectrum from the non-redundant half, computes the inverse DFT, and returns the real part.

Parameters

xndarray[dtype=object] of mp.mpc

Half-spectrum, shape = (…, M, …) where M = n_orig//2 + 1.

nint, optional

Output length along transform axis. Default: 2 * (x.shape[axis] - 1).

axisint

Transform axis (default -1).

Returns

yndarray[dtype=object] of mp.mpf

Real-valued inverse transform.

mpmath_fft._real_transforms.rfft(x, n=None, axis=-1)

Real-to-complex FFT.

Computes the forward DFT of real-valued input and returns only the non-redundant half of the spectrum (indices 0..n//2).

Parameters

xndarray[dtype=object] of mp.mpf

Real-valued input array.

nint, optional

Transform length. If n > x.shape[axis], zero-pad. If n < x.shape[axis], truncate. Default: x.shape[axis].

axisint

Transform axis (default -1).

Returns

yndarray[dtype=object] of mp.mpc

Half-spectrum, shape = (…, n//2 + 1, …).

mpmath_fft._real_transforms.rfftfreq(n, d=1.0)

Frequency bins for an n-point real FFT.

Returns the non-negative frequencies f_k = k / (n*d) for k = 0..n//2.

Parameters

nint

Number of sample points.

dfloat or mpmath scalar, optional

Sample spacing (default 1.0).

Returns

fndarray[dtype=object] of mp.mpf

Frequencies, shape (n//2 + 1,).