scipy.fftpack.irfft(x, n=None, axis=-1, overwrite_x=False)[source]#

Return inverse discrete Fourier transform of real sequence x.

The contents of x are interpreted as the output of the rfft function.


Transformed data to invert.

nint, optional

Length of the inverse Fourier transform. If n < x.shape[axis], x is truncated. If n > x.shape[axis], x is zero-padded. The default results in n = x.shape[axis].

axisint, optional

Axis along which the ifft’s are computed; the default is over the last axis (i.e., axis=-1).

overwrite_xbool, optional

If True, the contents of x can be destroyed; the default is False.

irfftndarray of floats

The inverse discrete Fourier transform.


The returned real array contains:


where for n is even:

y(j) = 1/n (sum[k=1..n/2-1] (x[2*k-1]+sqrt(-1)*x[2*k])
                             * exp(sqrt(-1)*j*k* 2*pi/n)
            + c.c. + x[0] + (-1)**(j) x[n-1])

and for n is odd:

y(j) = 1/n (sum[k=1..(n-1)/2] (x[2*k-1]+sqrt(-1)*x[2*k])
                             * exp(sqrt(-1)*j*k* 2*pi/n)
            + c.c. + x[0])

c.c. denotes complex conjugate of preceding expression.

For details on input parameters, see rfft.

To process (conjugate-symmetric) frequency-domain data with a complex datatype, consider using the newer function scipy.fft.irfft.


>>> from scipy.fftpack import rfft, irfft
>>> a = [1.0, 2.0, 3.0, 4.0, 5.0]
>>> irfft(a)
array([ 2.6       , -3.16405192,  1.24398433, -1.14955713,  1.46962473])
>>> irfft(rfft(a))
array([1., 2., 3., 4., 5.])