# Affine Projection (AP)¶

New in version 0.4.

Changed in version 1.0.0.

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.

## 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 extbf{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

$$\textbf{e}_{AP}(k) = \textbf{d}_{AP}(k) - \textbf{y}_{AP}(k)$$.

$$\textbf{w}_{AP}(k+1) = \textbf{w}_{AP}(k+1) + \mu \textbf{X}_{AP}(k) (\textbf{X}_{AP}^{T}(k) \textbf{X}_{AP}(k) + \epsilon \textbf{I})^{-1} \textbf{e}_{AP}(k)$$.

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

# 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, eps=0.001, 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 + 1*x - 1.5*x
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
# 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()


## Code Explanation¶

class padasip.filters.ap.FilterAP(n, order=5, mu=0.1, eps=0.001, 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:

• order : projection order (integer) - how many input vectors are in one input matrix

• 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 : initial offset covariance (float)

• 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.