Normalized Leastmeanfourth (NLMF)¶
New in version 1.1.0.
The leastmeanfourth (LMF) adaptive filter implemented according to the paper [1]. The NLMF is an extension of the LMF adaptive filter (Leastmeanfourth (LMF)).
The NLMF filter can be created as follows
>>> import padasip as pa
>>> pa.filters.FilterNLMF(n)
where n is the size (number of taps) of the filter.
Content of this page:
See also
Algorithm Explanation¶
The NLMF is extension of LMF filter. See Leastmeanfourth (LMF) 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.
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]  0.3*x[:,2] + 0.5*x[:,3] + v # target
# identification
f = pa.filters.FilterNLMF(n=4, mu=0.005, 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()
References¶
[1]  Azzedine Zerguine. Convergence behavior of the normalized least mean fourth algorithm. In Signals, Systems and Computers, 2000. Conference Record of the ThirtyFourth Asilomar Conference on, volume 1, 275–278. IEEE, 2000. 
Code Explanation¶

class
padasip.filters.nlmf.
FilterNLMF
(n, mu=0.1, eps=1.0, w='random')[source]¶ Bases:
padasip.filters.base_filter.AdaptiveFilter
Adaptive NLMF 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 illconditioned input data.
eps : regularization term (float). It is introduced to preserve stability for closetozero 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 (1dimensional array)

run
(d, x)[source]¶ This function filters multiple samples in a row.
Args:
 d : desired value (1 dimensional array)
 x : input matrix (2dimensional 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.