Least-mean-fourth (LMF)¶
New in version 1.1.0.
Changed in version 1.2.0.
The least-mean-fourth (LMF) adaptive filter implemented according to the paper.
The LMF filter can be created as follows
>>> import padasip as pa
>>> pa.filters.FilterLMF(n)
where n is the size (number of taps) of the filter.
Content of this page:
See also
Algorithm Explanation¶
The LMF 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 LMF weights adaptation could be described as follows
\(\textbf{w}(k+1) = \textbf{w}(k) + \Delta \textbf{w}(k)\),
where \(\Delta \textbf{w}(k)\) is
\(\Delta \textbf{w}(k) = \frac{1}{2} \mu \frac{\partial e^4(k)} { \partial \textbf{w}(k)}\ = \mu \cdot e(k)^{3} \cdot \textbf{x}(k)\),
where \(\mu\) is the learning rate (step size) and \(e(k)\) is error defined as
\(e(k) = d(k) - y(k)\).
Minimal Working Examples¶
If you have measured data you may filter it as follows
# 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.FilterLMF(n=4, mu=0.01, 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()
