scipy.special.diric

scipy.special.diric(x, n)

Periodic sinc function, also called the Dirichlet function.

The Dirichlet function is defined as:

diric(x) = sin(x * n/2) / (n * sin(x / 2)),

where n is a positive integer.

Parameters:

x : array_like

Input data

n : int

Integer defining the periodicity.

Returns:

diric : ndarray

Examples

>>> from scipy import special
>>> import matplotlib.pyplot as plt

>>> x = np.linspace(-8*np.pi, 8*np.pi, num=201)
>>> plt.figure(figsize=(8, 8));
>>> for idx, n in enumerate([2, 3, 4, 9]):
...     plt.subplot(2, 2, idx+1)
...     plt.plot(x, special.diric(x, n))
...     plt.title('diric, n={}'.format(n))
>>> plt.show()

(Source code)

The following example demonstrates that diric gives the magnitudes (modulo the sign and scaling) of the Fourier coefficients of a rectangular pulse.

Suppress output of values that are effectively 0:

>>> np.set_printoptions(suppress=True)

Create a signal x of length m with k ones:

>>> m = 8
>>> k = 3
>>> x = np.zeros(m)
>>> x[:k] = 1

Use the FFT to compute the Fourier transform of x, and inspect the magnitudes of the coefficients:

>>> np.abs(np.fft.fft(x))
array([ 3.        ,  2.41421356,  1.        ,  0.41421356,  1.        ,
        0.41421356,  1.        ,  2.41421356])

Now find the same values (up to sign) using diric. We multiply by k to account for the different scaling conventions of numpy.fft.fft and diric:

>>> theta = np.linspace(0, 2*np.pi, m, endpoint=False)
>>> k * special.diric(theta, k)
array([ 3.        ,  2.41421356,  1.        , -0.41421356, -1.        ,
       -0.41421356,  1.        ,  2.41421356])