import numpy as np
import matplotlib.pyplot as plt

omega_c = 0.4*np.pi
omega = np.linspace(-np.pi, np.pi, 300)

# Inverse DTFT of Ideal Lowpass: $x[n] = {\omega_c \over \pi} \sinc(\omega_c n)$

# Plot $x[n]$ for values of $n$ between -50 and 50
n = np.arange(-50,51)
x = np.sinc(omega_c*n/np.pi)*omega_c/np.pi
plt.stem(n,x)
plt.ylabel("$x[n]$")
plt.xlabel("$n$")
plt.show()

# Plot the frequency response of $X(e^{j\omega})$
f_response = np.compress(np.where(omega >= 0, True, False), omega)
X_f_response = np.where(np.abs(f_response) <= omega_c, 1, 0)
plt.plot(f_response, X_f_response)
plt.ylabel("Amplitude")
plt.xlabel("Frequency [rad/sample]")
plt.show()

# Calculate truncated DTFTs of $x[n]$
plt.plot(f_response, X_f_response, label="Actual DTFT of $x[n]$")
for K in (10, 20, 30):
    n_K = np.arange(-K, K+1)
    X_K = np.zeros(f_response.shape, np.complex128)
    for n in n_K:
        x_n = np.sinc(omega_c*n/np.pi)*omega_c/np.pi
        X_K += x_n * np.exp(-1j*f_response*n)
    plt.plot(f_response, X_K, label=f"Truncated DTFT (K = {K})")

# Plot truncated DTFTs of $x[n]$
plt.ylabel("Amplitude")
plt.xlabel("Frequency [rad/sample]")
plt.legend()
plt.show()