`scipy.special._basic:diric`

source: build-install/usr/lib/python3.14/site-packages/scipy/special/_basic.py:93

Signature

def diric(x, n)

Summary

Periodic sinc function, also called the Dirichlet kernel.

Extended Summary

The Dirichlet kernel is defined as

diric(x, n) = 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: Value of periodic sinc function.

Examples

import numpy as np
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))

✗

<Axes: >
[<matplotlib.lines.Line2D object at 0x7f3f68636660>]
Text(0.5, 1.0, 'diric, n=2')
<Axes: >
[<matplotlib.lines.Line2D object at 0x7f3f68636270>]
Text(0.5, 1.0, 'diric, n=3')
<Axes: >
[<matplotlib.lines.Line2D object at 0x7f3f686346e0>]
Text(0.5, 1.0, 'diric, n=4')
<Axes: >
[<matplotlib.lines.Line2D object at 0x7f3f69163770>]
Text(0.5, 1.0, 'diric, n=9')

plt.show()

✓

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.   , 2.414, 1.   , 0.414, 1.   , 0.414, 1.   , 2.414])

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.   ,  2.414,  1.   , -0.414, -1.   , -0.414,  1.   ,  2.414])

Aliases

scipy.special.diric