Least-mean-squares (LMS)¶

New in version 0.1.

Changed in version 1.0.0.

The least-mean-squares (LMS) adaptive filter [1] is the most popular adaptive filter.

The LMS filter can be created as follows

>>> pa.filters.FilterLMS(n)

where n is the size (number of taps) of the filter.

Algorithm Explanation¶

The LMS adaptive filter could be described as

$$y(k) = w_1 \cdot x_{1}(k) + ... + w_n \cdot x_{n}(k)$$,

or in a vector form

$$y(k) = \textbf{x}^T(k) \textbf{w}(k)$$,

where $$k$$ is discrete time index, $$(.)^T$$ denotes the transposition, $$y(k)$$ is filtered signal, $$\textbf{w}$$ is vector of filter adaptive parameters and $$\textbf{x}$$ is input vector (for a filter of size $$n$$) as follows

$$\textbf{x}(k) = [x_1(k), ..., x_n(k)]$$.

The LMS weights adaptation could be described as follows

$$\textbf{w}(k+1) = \textbf{w}(k) + \Delta \textbf{w}(k)$$,

where $$\Delta \textbf{w}(k)$$ is

$$\Delta \textbf{w}(k) = \frac{1}{2} \mu \frac{\partial e^2(k)} { \partial \textbf{w}(k)}\ = \mu \cdot e(k) \cdot \textbf{x}(k)$$,

where $$\mu$$ is the learning rate (step size) and $$e(k)$$ is error defined as

$$e(k) = d(k) - y(k)$$.

Stability and Optimal Performance¶

The general stability criteria of LMS stands as follows

$$|1 - \mu \cdot ||\textbf{x}(k)||^2 | \leq 1$$.

In practice the key argument mu should be set to really small number in most of the cases (recomended value can be something in range from 0.1 to 0.00001). If you have still problems stability or performance of the filter, then try the normalized LMS (Normalized Least-mean-squares (NLMS)).

Minimal Working Examples¶

If you have measured data you may filter it as follows

import numpy as np
import matplotlib.pylab as plt

# creation of data
N = 500
x = np.random.normal(0, 1, (N, 4)) # input matrix
v = np.random.normal(0, 0.1, N) # noise
d = 2*x[:,0] + 0.1*x[:,1] - 4*x[:,2] + 0.5*x[:,3] + v # target

# identification
f = pa.filters.FilterLMS(n=4, mu=0.1, w="random")
y, e, w = f.run(d, x)

# show results
plt.figure(figsize=(15,9))
plt.plot(d,"b", label="d - target")
plt.plot(y,"g", label="y - output");plt.legend()
plt.subplot(212);plt.title("Filter error");plt.xlabel("samples - k")
plt.plot(10*np.log10(e**2),"r", label="e - error [dB]");plt.legend()
plt.tight_layout()
plt.show()

An example how to filter data measured in real-time

import numpy as np
import matplotlib.pylab as plt

# these two function supplement your online measurment
def measure_x():
# it produces input vector of size 3
x = np.random.random(3)
return x

def measure_d(x):
# meausure system output
d = 2*x[0] + 1*x[1] - 1.5*x[2]
return d

N = 100
log_d = np.zeros(N)
log_y = np.zeros(N)
filt = pa.filters.FilterLMS(3, mu=1.)
for k in range(N):
# measure input
x = measure_x()
# predict new value
y = filt.predict(x)
# do the important stuff with prediction output
pass
# measure output
d = measure_d(x)
# update filter
# log values
log_d[k] = d
log_y[k] = y

### show results
plt.figure(figsize=(15,9))
plt.plot(log_d,"b", label="d - target")
plt.plot(log_y,"g", label="y - output");plt.legend()
plt.subplot(212);plt.title("Filter error");plt.xlabel("samples - k")
plt.plot(10*np.log10((log_d-log_y)**2),"r", label="e - error [dB]")
plt.legend(); plt.tight_layout(); plt.show()

References¶

 [1] Ali H Sayed. Fundamentals of adaptive filtering. John Wiley & Sons, 2003.

Code Explanation¶

This class represents an adaptive LMS filter.

Args:

• n : length of filter (integer) - how many input is input array (row of input matrix)

Kwargs:

• mu : learning rate (float). Also known as step size. If it is too slow, the filter may have bad performance. If it is too high, the filter will be unstable. The default value can be unstable for ill-conditioned input data.

• w : initial weights of filter. Possible values are:

• array with initial weights (1 dimensional array) of filter size
• “random” : create random weights
• “zeros” : create zero value weights

Adapt weights according one desired value and its input.

Args:

• d : desired value (float)
• x : input array (1-dimensional array)
run(d, x)[source]

This function filters multiple samples in a row.

Args:

• d : desired value (1 dimensional array)
• x : input matrix (2-dimensional array). Rows are samples, columns are input arrays.

Returns:

• y : output value (1 dimensional array). The size corresponds with the desired value.
• e : filter error for every sample (1 dimensional array). The size corresponds with the desired value.
• w : history of all weights (2 dimensional array). Every row is set of the weights for given sample.