sampling_example.py

import numpy as np
import matplotlib.pyplot as plt

# Ts = 0.01
# N = 400
t = np.arange(0, 4, 0.01)
x = np.cos(2 * np.pi * t)
# plt.plot(t, x)
# plt.show()

# If we take FFT, the frequency of the sinusoid is 1 Hz
# f_s / N = 100 / 400 = 0.25 Hz
# So we will get two peaks, one at k = 4 and another at k = 396

X = np.fft.fft(x)
# plt.stem(np.abs(X))
# plt.show()

# For this one:
y = np.cos(2 * 1.125 * np.pi * t)
Y = np.fft.fft(y)
# plt.stem(np.abs(Y))
# plt.show()

# Reason is this: if omega_0 is a multiple of 2pi / N, you get nice peak
print(np.abs(np.sum(x * np.exp(-1j * 2 * np.pi * t))))
print(np.abs(np.sum(y * np.exp(-1j * 2 * np.pi * t))))
print(np.abs(np.sum(y * np.exp(-1j * 2 * 2 * np.pi * t))))