from scipy.special import iv
import numpy as np


def discrete_gaussian_kernel(n, t):
    T = np.exp(-t) * iv(n, t)
    return T


def sampled_gaussian_kernel(n, t):
    G = 1/np.sqrt(2*np.pi*t) * np.exp(-n**2/(2*t))
    return G


if __name__ == '__main__':
    import matplotlib.pyplot as plt

    plt.figure(figsize=(5, 6))
    for t in (0.5, 1, 2, 4):
        N = 6
        n = np.arange(-N, N+1)
        p = plt.plot(n, discrete_gaussian_kernel(n, t),  '.-',
                     label=f'$t={t:.1f}$')[0]
        plt.plot(n, sampled_gaussian_kernel(n, t),  ':', color=p.get_color())
    plt.grid(True)
    plt.ylim(0, 0.7)
    plt.xlim(-6, 6)
    plt.legend()
    plt.xlabel('$n$')
    plt.ylabel('$T(n,t)$')