原创 Adaptive filtering algorithms and implementation-1

1, Introduciton to adaptive filtering

Adaptive-Filter Structure:

* FIR: The most widely used FIR file structure is the tapped delay line,
the output y(k) is a linear combination of the filter coefficients, that yields
a quadratic mean-square error(MSE=E[|e(k)^2|]) fuction with a unique optimal
solution.the other alternative adaptive FIR realization are also used in order
to obtain improvement on computational complexity, speed of convergence and
finite wordlength properties.

* IIR: The most widely used structure is the canonic direct form
realization. It's simple to implement and analysis. But have pole-stability and
slow convergence problem, so the other realization, such as cascade, lattice and
parallel structure are considered.

 

Adaptive algorithms:

the basic objective of the AF is to set its parameters, in such a way that
its output tries to minimize a meaning ful objective function involving the
reference signal.

usually, the objective function F is a function of the input, the reference
and output signals: F[x(k),d(k),y(k)]

A consistent definition of the objective functoin must satisfy:

* Non-negativity: F[x(k),d(k),y(k)]>=0;

* Optimality:F[x(k),d(k),y(k)]=0

1)Minimization algorithm:

In practice the function F can be approximated around a given parameter
point theta(k) by a truncated Taylor series.

So many optimization methods can be used:

* Gradient methods. easy.

* Newton's method(Page5), fast to convergence.

* Quasi-Newton method, better balance.

* steepest-descent method.

2)Definition of the objective funcion:F[e(k)]

there are many way to:

* Mean-Square Error(MSE): F[e(k)]=E[|e(k)|^2];

* Least Squares(LS);

* Weighted Least Squares(WLS);

* Instantaneous Squared Value(ISV): F[e(k)]=|e(k)|^2;