import numpy as np
import matplotlib.pyplot as plt

def DFT(x):
    """
    Compute the Discrete Fourier Transform (DFT) of the input signal x.
    
    Parameters:
        x (array-like): Input signal
        
    Returns:
        X (array-like): DFT of the input signal
    """
    N = len(x)
    n = np.arange(N)
    k = n.reshape((N, 1))
    e = np.exp(-2j * np.pi * k * n / N)
    X = np.dot(e, x)
    return X

def DFTI(X):
    """
    Compute the Inverse Discrete Fourier Transform (IDFT) of the input signal X.
    
    Parameters:
        X (array-like): Input signal
        
    Returns:
        x (array-like): IDFT of the input signal
    """
    N = len(X)
    n = np.arange(N)
    k = n.reshape((N, 1))
    e = np.exp(2j * np.pi * k * n / N)
    x = np.dot(e, X) / N
    return x

# Define a sine function as the input signal
t = np.linspace(0, 1, 1000)  # Time array
frequency = 5  # Frequency of the sine wave
x = np.sin(2 * np.pi * frequency * t) + np.sin(np.pi * frequency * t)

# Compute the DFT of the sine wave
X = DFT(x)

# Plot the input signal
plt.figure(figsize=(12, 4))
plt.subplot(1, 3, 1)
plt.plot(t, x)
plt.title('Input Signal')
plt.xlabel('Time')
plt.ylabel('Amplitude')

# Plot the magnitude spectrum (DFT)
freq = np.fft.fftfreq(len(x), d=t[1]-t[0])
plt.subplot(1, 3, 2)
plt.stem(freq, np.abs(X))
plt.title('Magnitude Spectrum (DFT)')
plt.xlabel('Frequency')
plt.ylabel('Magnitude')
plt.xlim(-10, 10)  # Limit the x-axis for better visualization

# Compute the IDFT of the DFT signal
x_reconstructed = DFTI(X)
# Plot the reconstructed signal
plt.subplot(1, 3, 3)
plt.plot(t, x_reconstructed.real, label='Reconstructed Signal')
plt.plot(t, x, label='Original Signal')
plt.title('Reconstructed Signal vs Original Signal')
plt.xlabel('Time')
plt.ylabel('Amplitude')
plt.legend()
plt.tight_layout()
plt.show()