How can an adaptive filter using Python be implemented with the least mean squares method?

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An adaptive filter using the least mean squares (LMS) method can be implemented using Python by following these steps:

Step 1: Import the necessary libraries

import numpy as np
import matplotlib.pyplot as plt

Step 2: Define the input signal and the desired output signal

n = 500
x = np.random.normal(0, 1, n)
d = np.sin(2*np.pi*0.05*np.arange(n)) + np.random.normal(0, 0.1, n)

Step 3: Define the initial filter coefficients and the step size

w = np.zeros(10)
mu = 0.1

Step 4: Implement the LMS algorithm using a loop

y = np.zeros(n)
e = np.zeros(n)

for i in range(10, n):
    y[i] = np.dot(w, x[i-10:i])
    e[i] = d[i] - y[i]
    w = w + mu * e[i] * x[i-10:i]

Step 5: Plot the input signal, the desired output signal, and the filtered output signal

plt.figure()
plt.subplot(311)
plt.plot(x)
plt.title('Input Signal')

plt.subplot(312)
plt.plot(d)
plt.title('Desired Output Signal')

plt.subplot(313)
plt.plot(y)
plt.title('Filtered Output Signal')
plt.show()

The full code is shown below:

import numpy as np
import matplotlib.pyplot as plt

n = 500
x = np.random.normal(0, 1, n)
d = np.sin(2*np.pi*0.05*np.arange(n)) + np.random.normal(0, 0.1, n)

w = np.zeros(10)
mu = 0.1

y = np.zeros(n)
e = np.zeros(n)

for i in range(10, n):
    y[i] = np.dot(w, x[i-10:i])
    e[i] = d[i] - y[i]
    w = w + mu * e[i] * x[i-10:i]

plt.figure()
plt.subplot(311)
plt.plot(x)
plt.title('Input Signal')

plt.subplot(312)
plt.plot(d)
plt.title('Desired Output Signal')

plt.subplot(313)
plt.plot(y)
plt.title('Filtered Output Signal')
plt.show()