dft(n, scale=None)

Create the matrix that computes the discrete Fourier transform of a sequence . The nth primitive root of unity used to generate the matrix is exp(-2*pi*i/n), where i = sqrt(-1).

Notes

When :None:None:`scale` is None, multiplying a vector by the matrix returned by dft is mathematically equivalent to (but much less efficient than) the calculation performed by scipy.fft.fft .

Parameters

n : int

Size the matrix to create.

scale : str, optional

Must be None, 'sqrtn', or 'n'. If :None:None:`scale` is 'sqrtn', the matrix is divided by :None:None:`sqrt(n)`. If :None:None:`scale` is 'n', the matrix is divided by n. If :None:None:`scale` is None (the default), the matrix is not normalized, and the return value is simply the Vandermonde matrix of the roots of unity.

Returns

m : (n, n) ndarray

The DFT matrix.

Discrete Fourier transform matrix.

Examples

>>> from scipy.linalg import dft
... np.set_printoptions(precision=2, suppress=True)  # for compact output
... m = dft(5)
... m
array([[ 1.  +0.j  ,  1.  +0.j  ,  1.  +0.j  ,  1.  +0.j  ,  1.  +0.j  ],
       [ 1.  +0.j  ,  0.31-0.95j, -0.81-0.59j, -0.81+0.59j,  0.31+0.95j],
       [ 1.  +0.j  , -0.81-0.59j,  0.31+0.95j,  0.31-0.95j, -0.81+0.59j],
       [ 1.  +0.j  , -0.81+0.59j,  0.31-0.95j,  0.31+0.95j, -0.81-0.59j],
       [ 1.  +0.j  ,  0.31+0.95j, -0.81+0.59j, -0.81-0.59j,  0.31-0.95j]])

>>> x = np.array([1, 2, 3, 0, 3])
... m @ x  # Compute the DFT of x
array([ 9.  +0.j  ,  0.12-0.81j, -2.12+3.44j, -2.12-3.44j,  0.12+0.81j])

Verify that m @ x is the same as fft(x) .

>>> from scipy.fft import fft
... fft(x)     # Same result as m @ x
array([ 9.  +0.j  ,  0.12-0.81j, -2.12+3.44j, -2.12-3.44j,  0.12+0.81j])

See :

Back References

The following pages refer to to this document either explicitly or contain code examples using this.

scipy.linalg._special_matrices.dft

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GitHub : /scipy/linalg/_special_matrices.py#972
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