Affine Projection (AP)¶
New in version 0.4.
Changed in version 1.2.0.
The Affine Projection (AP) algorithm is implemented according to paper. Usage of this filter should be benefical especially when input data is highly correlated. This filter is based on LMS. The difference is, that AP uses multiple input vectors in every sample. The number of vectors is called projection order. In this implementation the historic input vectors from input matrix are used as the additional input vectors in every sample.
The AP filter can be created as follows
>>> import padasip as pa
>>> pa.filters.FilterAP(n)
where n is the size of the filter.
Content of this page:
See also
Algorithm Explanation¶
The input for AP filter is created as follows
\(\textbf{X}_{AP}(k) = (\textbf{x}(k), ..., \textbf{x}(k-L))\),
where \(\textbf{X}_{AP}\) is filter input, \(L\) is projection order, \(k\) is discrete time index and \(\textbf{x}_{k}\) is input vector. The output of filter si calculated as follows:
\(\textbf{y}_{AP}(k) = \textbf{X}^{T}_{AP}(k) \textbf{w}(k)\),
where \(\textbf{x}(k)\) is the vector of filter adaptive parameters. The vector of targets is constructed as follows
\(\textbf{d}_{AP}(k) = (d(k), ..., d(k-L))^T\),
where \(d(k)\) is target in time \(k\).
The error of the filter is estimated as
\(\textbf{e}_{AP}(k) = \textbf{d}_{AP}(k) - \textbf{y}_{AP}(k)\).
And the adaptation of adaptive parameters is calculated according to equation
\(\textbf{w}_{AP}(k+1) = \textbf{w}_{AP}(k+1) + \mu \textbf{X}_{AP}(k) (\textbf{X}_{AP}^{T}(k) \textbf{X}_{AP}(k) + \epsilon \textbf{I})^{-1} \textbf{e}_{AP}(k)\).
During the filtering we are interested just in output of filter \(y(k)\) and the error \(e(k)\). These two values are the first elements in vectors: \(\textbf{y}_{AP}(k)\) for output and \(\textbf{e}_{AP}(k)\) for error.
Minimal Working Example¶
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.FilterAP(n=4, order=5, mu=0.5, ifc=0.001, 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.FilterAP(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()
