nlcpy.fft.irfft

nlcpy.fft.irfft(a, n=None, axis=- 1, norm=None)[source]

Computes the inverse of the n-point DFT for a real array.

This function computes the inverse of the one-dimensional n-point discrete fourier transform of a real array 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 a real array is Hermitian-symmetric, the negative frequency terms are taken to be the complex conjugates of the corresponding positive frequency terms.

Parameters

aarray_like: The input array.
nint,optional: Length of the transformed axis of the output. For 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.
axisint,optional: Axis over which to compute the FFT. If not given, the last axis is used. If axes is larger than the last axis of a, IndexError occurs.
norm{None, “ortho”},optional: Normalization mode. By default(None), the transforms are scaled by $1/n$ . It norm is set to “ortho”, the return values will be scaled by $1/\sqrt{n}$ .

Returns

outcomplex ndarray: The truncated or zero-padded input, transformed along the axis indicated by 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.

See also

rfft: Computes the one-dimensional discrete fourier transform for a real array.
fft: Computes the one-dimensional discrete fourier transform.
irfft2: Computes the 2-dimensional inverse FFT of a real array.
irfftn: Computes the inverse of the n-dimensional FFT of a real array.

Note

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 array must be given.

Examples

>>> import nlcpy as vp
>>> vp.fft.ifft([1, -1j, -1, 1j])  
array([0.+0.j, 1.+0.j, 0.+0.j, 0.+0.j])  # may vary
>>> vp.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.