Complex Sinusoid and DFT in Python

# Create subplots
fig, axes = plt.subplots(4, 1, figsize=(10, 8))  # 4 rows, 1 column

# Plot each sine wave in its respective subplot
axes[0].plot(timeAxis, sineWave1, color='b')
axes[0].set_title(f'Sine Wave 1: {freq1} Hz')
axes[0].set_xlabel('t/s')
axes[0].set_ylabel('Amplitude = x(t)')

axes[1].plot(timeAxis, sineWave2, color='g')
axes[1].set_title(f'Sine Wave 2: {freq2} Hz')
axes[1].set_xlabel('t/s')
axes[1].set_ylabel('Amplitude = x(t)')

axes[2].plot(timeAxis, sineWave3, color='r')
axes[2].set_title(f'Sine Wave 3: {freq3} Hz')
axes[2].set_xlabel('t/s')
axes[2].set_ylabel('Amplitude = x(t)')

axes[3].plot(timeAxis, sineWaveSum, color='k')
axes[3].set_title(f'Sine Wave Sum')
axes[3].set_xlabel('t/s')
axes[3].set_ylabel('Amplitude = x(t)')


# Clean
plt.tight_layout()
plt.show()

# Create subplots
fig, axes = plt.subplots(3, 1, figsize=(10, 8))  # 4 rows, 1 column

# Plot each sine wave in its respective subplot
axes[0].plot(timeAxis, sineWaveSum, color='b')
axes[0].set_title(f'SineSum')
axes[0].set_xlabel('t/s')
axes[0].set_ylabel('Amplitude = x(t)')

axes[1].plot(timeAxis, sineWave1, color='g')
axes[1].set_title(f'Sine Wave1: {freq1} Hz')
axes[1].set_xlabel('t/s')
axes[1].set_ylabel('Amplitude = x(t)')

axes[2].plot(timeAxis, cross_corr[999:1999], color='g')
axes[2].set_title(f'Cross Correlation')
axes[2].set_xlabel('t/s')
axes[2].set_ylabel('c(t)')


# Clean
plt.tight_layout()
plt.show()

# Correlation of the 10Hz SineWave with our sum of Sinusoidals
cross_corr = np.correlate(sineWaveSum, sineWave2, mode='full')

# Create subplots
fig, axes = plt.subplots(3, 1, figsize=(10, 8))  # 3 rows, 1 column

# Plot each sine wave in its respective subplot
axes[0].plot(timeAxis, sineWaveSum, color='b')
axes[0].set_title('SineSum')
axes[0].set_xlabel('t/s')
axes[0].set_ylabel('Amplitude = x(t)')

axes[1].plot(timeAxis, sineWave2, color='g')
axes[1].set_title(f'Sine Wave1: {freq2} Hz')
axes[1].set_xlabel('t/s')
axes[1].set_ylabel('Amplitude = x(t)')

axes[2].plot(timeAxis, cross_corr[999:1999], color='g')
axes[2].set_title(f'Cross Correlation')
axes[2].set_xlabel('t/s')
axes[2].set_ylabel('c(t)')


# Clean
plt.tight_layout()
plt.show()

import numpy as np
import matplotlib.pyplot as plt

# Total samples of 32
N = 32
n = np.arange(N)

# Generate a complex Sinusoid
k0 = np.exp(-1j * 2 * np.pi * 0 * n / N)  # DC
cosine = k0.real
sine = k0.imag

# Create subplot
plt.figure(figsize=(10, 6))
plt.plot(n, cosine, 'b-o', label='Real (Cosine)')
plt.plot(n, sine, 'g-o', label='Imaginary (Sine)')
plt.title('Complex Sinusoid k=0 (DC)')
plt.xlabel('n (Sample Index)')
plt.ylabel('Amplitude')
plt.grid(True)
plt.legend()

# Generate a complex Sinusoid
k1 = np.exp(-1j * 2 * np.pi * 1 * n / N)  # First Harmonic
cosine = k1.real
sine = k1.imag

# Create subplot
plt.figure(figsize=(10, 6))
plt.plot(n, cosine, 'b-o', label='Real (Cosine)')
plt.plot(n, sine, 'g-o', label='Imaginary (Sine)')
plt.title('Complex Sinusoid k=1')
plt.xlabel('n (Sample Index)')
plt.ylabel('Amplitude')
plt.grid(True)
plt.legend()

# Generate a complex Sinusoid
k2 = np.exp(-1j * 2 * np.pi * 2 * n / N)  # Second Harmonic
cosine = k2.real
sine = k2.imag

# Create subplot
plt.figure(figsize=(10, 6))
plt.plot(n, cosine, 'b-o', label='Real (Cosine)')
plt.plot(n, sine, 'g-o', label='Imaginary (Sine)')
plt.title('Complex Sinusoid k=2')
plt.xlabel('n (Sample Index)')
plt.ylabel('Amplitude')
plt.grid(True)
plt.legend()

# Generate a complex Sinusoid
k3 = np.exp(-1j * 2 * np.pi * 3 * n / N)
cosine = k3.real
sine = k3.imag

# Create subplot
plt.figure(figsize=(10, 6))
plt.plot(n, cosine, 'b-o', label='Real (Cosine)')
plt.plot(n, sine, 'g-o', label='Imaginary (Sine)')
plt.title('Complex Sinusoid k=3')
plt.xlabel('n (Sample Index)')
plt.ylabel('Amplitude')
plt.grid(True)
plt.legend()

# Generate sample points
N = 32
n = np.arange(N)

# Generate example signal (you can replace this with your actual signal)
x = np.cos(2 * np.pi * 2 * n / N)

# Generate basis functions
kComplex = [np.exp(-1j * 2 * np.pi * k * n / N) for k in range(4)]

# Initialize arrays with complex dtype
X_sum = np.zeros(len(kComplex), dtype=complex)
X_corr = np.zeros(len(kComplex), dtype=complex)

for kInd, k in enumerate(kComplex):
    # Method 1: Using the Formula
    X_sum[kInd] = np.sum(x * k)
    
    # Method 2: Using correlation
    X_corr[kInd] = np.correlate(x, k, mode='full')[N-1]

# Create subplots
fig, axes = plt.subplots(2, 1, figsize=(12, 12))

# Plot input signal
axes[0].plot(n, x, 'b-', label='Signal')
axes[0].set_title('Input Cosine')
axes[0].set_xlabel('n')
axes[0].set_ylabel('Amplitude = x[n]')
axes[0].grid(True)
axes[0].legend()

# Plot DFT magnitudes from both methods
k_vals = np.arange(4)
axes[1].plot(k_vals, np.abs(X_sum), 'b-o', label='Sum Method')
axes[1].plot(k_vals, np.abs(X_corr), 'r--x', label='Correlation Method')
axes[1].set_title('DFT Magnitude |X[k]|')
axes[1].set_xlabel('Frequency Bin k')
axes[1].set_ylabel('|X[k]|')
axes[1].grid(True)
axes[1].legend()


plt.tight_layout()
plt.show()