import numpy as np
import matplotlib.pyplot as plt

N = 8

# Analytic DFT
k = np.arange(N)
W_N = np.exp(-2j*np.pi*k/N)
plt.stem(k, -W_N)
plt.xlabel("$k$")
plt.ylabel("$W_N$")
plt.show()


n = np.arange(N)
x = (-1.0)**n
DFT_8_point = np.fft.fft(x, 8)
DFT_9_point = np.fft.fft(x, 9)

plt.stem(abs(DFT_8_point))
plt.show()

plt.stem(abs(DFT_9_point))
plt.show()