NLMS: Normalized Least-mean-square

nlms.py

import numpy as np
import matplotlib.pyplot as plt
from scipy.signal import lfilter

# Step 1: Create a synthetic dataset
np.random.seed(42)  # For reproducibility
N = 1000  # Number of samples
x = np.random.randn(N)  # Input signal (random noise)
w_true = np.array([0.5, -0.3, 0.1])  # True filter coefficients
e_ = 0.01 * np.random.randn(N)  # Noise
d = lfilter(w_true, 1, x) + e_  # Desired signal

# Step 2: NLMS Algorithm Implementation
def nlms(x, d, mu=0.01, eps=1e-6, M=3):
    n = len(x)
    w = np.zeros(M)  # Initial filter coefficients
    e = np.zeros(n)  # Error signal
    for i in range(M, n):
        x_i = x[i-M+1:i+1]
        y = np.dot(x_i, w)
        e[i] = d[i] - y
        norm = np.dot(x_i, x_i) + eps
        w += (mu / norm) * x_i * e[i]
    return w[::-1], e

# Run NLMS
w_est, e = nlms(x, d)

# Step 3: Performance Metrics and Plots
# Mean Squared Error (MSE)
mse = np.mean(e**2)

# Plotting
plt.figure(figsize=(14, 5))
plt.subplot(1, 2, 1)
plt.plot(e, label='Error Signal')
plt.title('Error Signal Over Time')
plt.xlabel('Sample')
plt.ylabel('Error')
plt.legend()

plt.subplot(1, 2, 2)
plt.plot(w_true, label='True Coefficients', marker='o')
plt.plot(w_est, label='Estimated Coefficients', marker='x')
plt.title('Filter Coefficients')
plt.xlabel('Coefficient Index')
plt.ylabel('Value')
plt.legend()

plt.tight_layout()
plt.show()

# Interpretations
print(f"Final Estimated Coefficients: {w_est}")
print(f"Mean Squared Error: {mse}")