Recursive Least Squares (RLS)

New in version 0.1.

Changed in version 1.0.0.

The Recursive Least Squares filter [1] can be created as follows

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

where the n is amount of filter inputs (size of input vector).

Content of this page:

See also

Adaptive Filters

Algorithm Explanation

The RLS adaptive filter may 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 update is done as folows

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

where \(\Delta \textbf{w}(k)\) is obtained as follows

\(\Delta \textbf{w}(k) = \textbf{R}(k) \textbf{x}(k) e(k)\),

where \(e(k)\) is error and it is estimated according to filter output and desired value \(d(k)\) as follows

\(e(k) = d(k) - y(k)\)

The \(\textbf{R}(k)\) is inverse of autocorrelation matrix and it is calculated as follows

\(\textbf{R}(k) = \frac{1}{\mu}( \textbf{R}(k-1) - \frac{\textbf{R}(k-1)\textbf{x}(k) \textbf{x}(k)^{T} \textbf{R}(k-1)} {\mu + \textbf{x}(k)^{T}\textbf{R}(k-1)\textbf{x}(k)} )\).

The initial value of autocorrelation matrix should be set to

\(\textbf{R}(0) = \frac{1}{\delta} \textbf{I}\),

where \(\textbf{I}\) is identity matrix and \(\delta\) is small positive constant.

Stability and Optimal Performance

Make the RLS working correctly with a real data can be tricky. The forgetting factor \(\mu\) should be in range from 0 to 1. But in a lot of cases it works only with values close to 1 (for example something like 0.99).

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.FilterRLS(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.FilterRLS(3, mu=0.5)
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()

References

[1]Ali H Sayed and Thomas Kailath. Recursive least-squares adaptive filters. The Digital Signal Processing Handbook, pages 21–1, 1998.

Code Explanation

class padasip.filters.rls.FilterRLS(n, mu=0.99, eps=0.1, w='random')[source]

Bases: padasip.filters.base_filter.AdaptiveFilter

Adaptive RLS filter.

Args:

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

Kwargs:

  • mu : forgetting factor (float). It is introduced to give exponentially less weight to older error samples. It is usually chosen between 0.98 and 1.

  • eps : initialisation value (float). It is usually chosen between 0.1 and 1.

  • 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.