# Normalized Least-mean-squares (NLMS)¶

New in version 0.1.

Changed in version 1.0.0.

The normalized least-mean-squares (NLMS) adaptive filter [1] is an extension of the popular LMS adaptive filter (Least-mean-squares (LMS)).

The NLMS filter can be created as follows

>>> import padasip as pa
>>> pa.filters.FilterNLMS(n)


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

## Algorithm Explanation¶

The NLMS is extension of LMS filter. See Least-mean-squares (LMS) for explanation of the algorithm behind.

The extension is based on normalization of learning rate. The learning rage $$\mu$$ is replaced by learning rate $$\eta(k)$$ normalized with every new sample according to input power as follows

$$\eta (k) = \frac{\mu}{\epsilon + || \textbf{x}(k) ||^2}$$,

where $$|| \textbf{x}(k) ||^2$$ is norm of input vector and $$\epsilon$$ is a small positive constant (regularization term). This constant is introduced to preserve the stability in cases where the input is close to zero.

## Stability and Optimal Performance¶

The stability of the NLMS filter si given as follows

$$0 \le \mu \le 2 + \frac{2\epsilon}{||\textbf{x}(k)||^2}$$,

or in case without regularization term $$\epsilon$$

$$\mu \in <0, 2>$$.

In other words, if you use the zero or only small key argument eps, the key argument mu should be between 0 and 2. Best convergence should be produced by mu=1. according to theory. However in practice the optimal value can be strongly case specific.

## 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.FilterNLMS(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.FilterNLMS(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¶

class padasip.filters.nlms.FilterNLMS(n, mu=0.1, eps=1.0, w='random')[source]

Bases: padasip.filters.base_filter.AdaptiveFilter

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.

• eps : regularization term (float). It is introduced to preserve stability for close-to-zero input vectors

• 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(d, x)[source]

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.