scipy.linalg.dft(n, scale=None)[source]

Discrete Fourier transform matrix.

Create the matrix that computes the discrete Fourier transform of a sequence [R81]. The n-th primitive root of unity used to generate the matrix is exp(-2*pi*i/n), where i = sqrt(-1).


n : int

Size the matrix to create.

scale : str, optional

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


m : (n, n) ndarray

The DFT matrix.


When 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.fftpack.fft.

New in version 0.14.0.


[R81](1, 2) “DFT matrix”,


>>> from scipy.linalg import dft
>>> np.set_printoptions(precision=5, suppress=True)
>>> x = np.array([1, 2, 3, 0, 3, 2, 1, 0])
>>> m = dft(8)
>>>   # Compute the DFT of x
array([ 12.+0.j,  -2.-2.j,   0.-4.j,  -2.+2.j,   4.+0.j,  -2.-2.j,
        -0.+4.j,  -2.+2.j])

Verify that is the same as fft(x).

>>> from scipy.fftpack import fft
>>> fft(x)     # Same result as
array([ 12.+0.j,  -2.-2.j,   0.-4.j,  -2.+2.j,   4.+0.j,  -2.-2.j,
         0.+4.j,  -2.+2.j])

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