fft(x, n=None, axis=-1, overwrite_x=False)¶
Return discrete Fourier transform of real or complex sequence.
The returned complex array contains
y(0), y(1),..., y(n-1)where
y(j) = (x * exp(-2*pi*sqrt(-1)*j*np.arange(n)/n)).sum().
x : array_like
Array to Fourier transform.
n : int, optional
Length of the 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].
axis : int, optional
Axis along which the fft’s are computed; the default is over the last axis (i.e.,
overwrite_x : bool, optional
If True, the contents of x can be destroyed; the default is False.
z : complex ndarray
with the elements:
[y(0),y(1),..,y(n/2),y(1-n/2),...,y(-1)] if n is even [y(0),y(1),..,y((n-1)/2),y(-(n-1)/2),...,y(-1)] if n is odd
y(j) = sum[k=0..n-1] x[k] * exp(-sqrt(-1)*j*k* 2*pi/n), j = 0..n-1
y(-j) = y(n-j).conjugate().
The packing of the result is “standard”: If
A = fft(a, n), then
Acontains the zero-frequency term,
A[1:n/2]contains the positive-frequency terms, and
A[n/2:]contains the negative-frequency terms, in order of decreasingly negative frequency. So for an 8-point transform, the frequencies of the result are [0, 1, 2, 3, -4, -3, -2, -1]. To rearrange the fft output so that the zero-frequency component is centered, like [-4, -3, -2, -1, 0, 1, 2, 3], use
For n even,
A[n/2]contains the sum of the positive and negative-frequency terms. For n even and x real,
A[n/2]will always be real.
Both single and double precision routines are implemented. Half precision inputs will be converted to single precision. Non floating-point inputs will be converted to double precision. Long-double precision inputs are not supported.
This function is most efficient when n is a power of two, and least efficient when n is prime.
If the data type of x is real, a “real FFT” algorithm is automatically used, which roughly halves the computation time. To increase efficiency a little further, use
rfft, which does the same calculation, but only outputs half of the symmetrical spectrum. If the data is both real and symmetrical, the
dctcan again double the efficiency, by generating half of the spectrum from half of the signal.
>>> from scipy.fftpack import fft, ifft >>> x = np.arange(5) >>> np.allclose(fft(ifft(x)), x, atol=1e-15) # within numerical accuracy. True