import numpy as np
import matplotlib.pyplot as plt


def d_idft(coef):
    N = len(coef)
    x = np.zeros(N, dtype='complex128')
    for n in range(N):
        for k in range(N):
            x[n] += 1j * k * coef[k] * np.exp(1j * (2 * np.pi / N) * n * k)
    x *= (2 * np.pi / N ** 2)
    return x

t = np.linspace(0, 1, 128)
x = np.cos(2 * np.pi * t)

coef = np.fft.fft(x)
xd     = np.diff(x)  # x[n] - x[n - 1]
xd_est = d_idft(coef)

plt.plot(xd)
plt.plot(xd_est.real)
plt.show()
plt.plot(t, xd_est.imag)