Least-mean-square (LMS)¶
New in version 0.1.
Changed in version 1.2.0.
The least-mean-square (LMS) adaptive filter is the most popular adaptive filter.
The LMS filter can be created as follows
>>> import padasip as pa
>>> pa.filters.FilterLMS(n)
where n is the size (number of taps) of the filter.
Content of this page:
See also
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-square (NLMS)).
Minimal Working Examples¶
If you have measured data you may filter it as follows
import numpy as np
import matplotlib.pylab as plt
import padasip as pa
# 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.subplot(211);plt.title("Adaptation");plt.xlabel("samples - k")
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
import padasip as pa
# 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
filt.adapt(d, x)
# log values
log_d[k] = d
log_y[k] = y
### show results
plt.figure(figsize=(15,9))
plt.subplot(211);plt.title("Adaptation");plt.xlabel("samples - k")
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()
