adaptfilt.ap - FIR adaptive filter that uses direct matrix inversion

Syntax

ha = adaptfilt.ap(l,step,projectord,offset,coeffs,states,...errstates,epsstates)

Description

ha = adaptfilt.ap(l,step,projectord,offset,coeffs,states,... errstates,epsstates) constructs an affine projection FIR adaptive filter ha using direct matrix inversion.

Input Arguments

Entries in the following table describe the input arguments for adaptfilt.ap.

Input Argument	Description
`l`	Adaptive filter length (the number of coefficients or taps) and it must be a positive integer. `l` defaults to 10.
`step`	Affine projection step size. This is a scalar that should be a value between zero and one. Setting `step` equal to one provides the fastest convergence during adaptation. `step` defaults to 1.
`projectord`	Projection order of the affine projection algorithm. `projectord` defines the size of the input signal covariance matrix and defaults to two.
`offset`	Offset for the input signal covariance matrix. You should initialize the covariance matrix to a diagonal matrix whose diagonal entries are equal to the offset you specify. `offset` should be positive. `offset` defaults to one.
`coeffs`	Vector containing the initial filter coefficients. It must be a length `l` vector, the number of filter coefficients. `coeffs` defaults to length `l` vector of zeros when you do not provide the argument for input.
`states`	Vector of the adaptive filter states. states defaults to a vector of zeros which has length equal to (`l` + `projectord` - 2).
`errstates`	Vector of the adaptive filter error states. `errstates` defaults to a zero vector with length equal to (`projectord` - 1).
`epsstates`	Vector of the epsilon values of the adaptive filter. `epsstates` defaults to a vector of zeros with (`projectord` - 1) elements.

Properties

Since your adaptfilt.ap filter is an object, it has properties that define its behavior in operation. Note that many of the properties are also input arguments for creating adaptfilt.ap objects. To show you the properties that apply, this table lists and describes each property for the affine projection filter object.

Name	Range	Description
`Algorithm`	None	Defines the adaptive filter algorithm the object uses during adaptation
`FilterLength`	Any positive integer	Reports the length of the filter, the number of coefficients or taps
`ProjectionOrder`	1 to as large as needed.	Projection order of the affine projection algorithm. `ProjectionOrder` defines the size of the input signal covariance matrix and defaults to two.
`OffsetCov`	Matrix of values	Contains the offset covariance matrix
`Coefficients`	Vector of elements	Vector containing the initial filter coefficients. It must be a length `l` vector, the number of filter coefficients. `coeffs` defaults to length `l` vector of zeros when you do not provide the argument for input.
`States`	Vector of elements, data type double	Vector of the adaptive filter states. states defaults to a vector of zeros which has length equal to (`l` + `projectord` - 2).
`ErrorStates`	Vector of elements	Vector of the adaptive filter error states. `errstates` defaults to a zero vector with length equal to (`projectord` - 1).
`EpsilonStates`	Vector of elements	Vector of the epsilon values of the adaptive filter. `epsstates` defaults to a vector of zeros with (`projectord` - 1) elements.
`StepSize`	Any scalar from zero to one, inclusive	Specifies the step size taken between filter coefficient updates
`PersistentMemory`	`false` or `true`	Determine whether the filter states get restored to their starting values for each filtering operation. The starting values are the values in place when you create the filter. `PersistentMemory` returns to zero any state that the filter changes during processing. States that the filter does not change are not affected. Defaults to `true`.

Examples

Quadrature phase shift keying (QPSK) adaptive equalization using a 32-coefficient FIR filter. Run the adaptation for 1000 iterations.

D  = 16;                     % Number of samples of delay
b  = exp(j*pi/4)*[-0.7 1];   % Numerator coefficients of channel
a  = [1 -0.7];               % Denominator coefficients of channel
ntr= 1000;                   % Number of iterations
s  = sign(randn(1,ntr+D)) + j*sign(randn(1,ntr+D));% Baseband Signal
n  = 0.1*(randn(1,ntr+D) + j*randn(1,ntr+D));      % Noise signal
r  = filter(b,a,s)+n;        % Received signal
x  = r(1+D:ntr+D);    % Input signal (received signal)
d  = s(1:ntr);        % Desired signal (delayed QPSK signal)
mu = 0.1;             % Step size
po = 4;               % Projection order
offset = 0.05;        % Offset for covariance matrix
ha = adaptfilt.ap(32,mu,po,offset);
[y,e] = filter(ha,x,d);
subplot(2,2,1); plot(1:ntr,real([d;y;e])); title('In-Phase Components');
legend('Desired','Output','Error');
xlabel('Time Index'); ylabel('Signal Value');
subplot(2,2,2); plot(1:ntr,imag([d;y;e]));
title('Quadrature Components');
legend('Desired','Output','Error');
xlabel('Time Index'); ylabel('Signal Value');
subplot(2,2,3); plot(x(ntr-100:ntr),'.');
axis([-3 3 -3 3]); title('Received Signal Scatter Plot');
axis('square');  xlabel('Real[x]'); ylabel('Imag[x]'); grid on;
subplot(2,2,4); plot(y(ntr-100:ntr),'.'); axis([-3 3 -3 3]);
title('Equalized Signal Scatter Plot');
axis('square'); xlabel('Real[y]'); ylabel('Imag[y]'); grid on;

The four plots shown reveal the QPSK process at work.

References

[1] Ozeki, K. and Umeda, T., "An Adaptive Filtering Algorithm Using an Orthogonal Projection to an Affine Subspace and Its Properties," Electronics and Communications in Japan, vol.67-A, no. 5, pp. 19-27, May 1984

[2] Maruyama, Y., "A Fast Method of Projection Algorithm," Proc. 1990 IEICE Spring Conf., B-744

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