0.13 Least squares design of iir filters  (Page 3/9)

A common notion used in this work (as well as some of the literature related to linearization and filter design) is that there are two different error measures that authors often refer to. It is important to recognize the differences between them as one browses through literature. Typically one would be interested in minimizing the $l_2$ error given by:

This quantity is often referred to as the solution error (denoted by ${\epsilon }_s$ ); we refer to the function $\mathcal{E}\left(\omega \right)$ in [link] as the solution error function , denoted by ${\mathcal{E}}_s\left(\omega \right)$ . Also, in linearization algorithms the following measure often arises,

This measure is often referred to as the equation error ${\epsilon }_e$ ; we denote the function $\mathcal{E}\left(\omega \right)$ in [link] as the equation error function ${\mathcal{E}}_e\left(\omega \right)$ . Keeping the notation previously introduced, it can be seen that the two errors relate by one being a weighted version of the other,

Levy's method

E. C. Levy [link] considered in 1959 the following problem in the context of analog systems (electrical networks to be more precise): define For consistency with the rest of this document, notation has been modified from the author's original paper whenever deemed necessary.

Given $L$ samples of a desired complex-valued function $D\left(j{\omega }_k\right)=R\left({\omega }_k\right)+jI\left({\omega }_k\right)$ (where $R,I$ are both real funtions of $\omega$ ), Levy defines

or

Observing the linear structure (in the coefficients $A_k,B_k$ ) of equation [link] , Levy proposed minimizing the quantity $\epsilon$ . He actually realized that this measure (what we would denote as the equation error) was indeed a weighted version of the actual solution error that one might be interested in; in fact, the denominator function $A\left(\omega \right)$ became the weighting function.

Levy's proposed method for minimizing [link] begins by writing $\epsilon$ as follows,

by recognizing that [link] can be reformulated in terms of its real and imaginary parts,

and performing appropriate manipulations For further details on the algebraic manipulations involved, the reader should refer to [link] . . Note that the optimal set of coefficients $A_k$ , $B_k$ must satisfy

The conditions introduced above generate a linear system in the filter coefficients. Levy derives the system

where

and

with

Solving for the vector $x$ from [link] gives the desired coefficients (note the trivial assumption that $A_0=1$ ). It is important to remember that although Levy's algorithm leads to a linear system of equations in the coefficients, his approach is indeed an equation error method. Matlab's invfreqz function uses an adaptation of Levy's algorithm for its least-squares equation error solution.