# mars.tensor.fft.fft¶

mars.tensor.fft.fft(a, n=None, axis=- 1, norm=None)[source]

Compute the one-dimensional discrete Fourier Transform.

This function computes the one-dimensional n-point discrete Fourier Transform (DFT) with the efficient Fast Fourier Transform (FFT) algorithm [CT].

Parameters
• a (array_like) – Input tensor, can be complex.

• n (int, optional) – Length of the transformed axis of the output. If n is smaller than the length of the input, the input is cropped. If it is larger, the input is padded with zeros. If n is not given, the length of the input along the axis specified by axis is used.

• axis (int, optional) – Axis over which to compute the FFT. If not given, the last axis is used.

• norm ({None, "ortho"}, optional) – Normalization mode (see mt.fft). Default is None.

Returns

out – The truncated or zero-padded input, transformed along the axis indicated by axis, or the last one if axis is not specified.

Return type

complex Tensor

Raises

IndexError – if axes is larger than the last axis of a.

`mt.fft`

for definition of the DFT and conventions used.

`ifft`

The inverse of fft.

`fft2`

The two-dimensional FFT.

`fftn`

The n-dimensional FFT.

`rfftn`

The n-dimensional FFT of real input.

`fftfreq`

Frequency bins for given FFT parameters.

Notes

FFT (Fast Fourier Transform) refers to a way the discrete Fourier Transform (DFT) can be calculated efficiently, by using symmetries in the calculated terms. The symmetry is highest when n is a power of 2, and the transform is therefore most efficient for these sizes.

The DFT is defined, with the conventions used in this implementation, in the documentation for the numpy.fft module.

References

CT

Cooley, James W., and John W. Tukey, 1965, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19: 297-301.

Examples

```>>> import mars.tensor as mt
```
```>>> mt.fft.fft(mt.exp(2j * mt.pi * mt.arange(8) / 8)).execute()
array([-2.33486982e-16+1.14423775e-17j,  8.00000000e+00-6.89018570e-16j,
2.33486982e-16+2.33486982e-16j,  0.00000000e+00+0.00000000e+00j,
-1.14423775e-17+2.33486982e-16j,  0.00000000e+00+1.99159850e-16j,
1.14423775e-17+1.14423775e-17j,  0.00000000e+00+0.00000000e+00j])
```

In this example, real input has an FFT which is Hermitian, i.e., symmetric in the real part and anti-symmetric in the imaginary part, as described in the numpy.fft documentation:

```>>> import matplotlib.pyplot as plt
>>> t = mt.arange(256)
>>> sp = mt.fft.fft(mt.sin(t))
>>> freq = mt.fft.fftfreq(t.shape[-1])
>>> plt.plot(freq.execute(), sp.real.execute(), freq.execute(), sp.imag.execute())
[<matplotlib.lines.Line2D object at 0x...>, <matplotlib.lines.Line2D object at 0x...>]
>>> plt.show()
```