0.13 Least squares design of iir filters  (Page 6/9)

The algorithm described above is an interpolation method rather than an approximation one. If $L>N+M+1$ and ${\widehat{\mathbf{H}}}_2$ is full column rank then [link] is an overdetermined linear system for which no exact solution exists; therefore an approximation must be found. From [link] we can define the solution error function ${\mathcal{E}}_s\left({\omega }_k\right)$ as

Using this notation, the design objective is to solve the nonlinear problem

Consider the system in equation [link] . If ${\mathbf{H}}_2$ is overdetermined, one can define an approximation problem by introducing an error vector $e$ ,

where

Again, it is possible to uncouple [link] as follows,

One can minimize the least-squared error norm $\parallel e_2{\parallel }_2$ of the overdetermined system [link] by solving the normal equations [link]

so that

and use this result in [link]

[link] represents the following time-domain operation,

(where $◯\mathit{\text{L}}$ denotes circular convolution ) and can be interpreted in the frequency domain as follows,

Equation [link] is a weighted version of [link] , as follows

Therefore the algorithm presented above will find the filter coefficient vectors $a$ and $b$ that minimize the equation error ${\mathcal{E}}_e$ in [link] in the least-squares sense. Unfortunately, this error is not what one may want to optimize, since it is a weighted version of the solution error ${\mathcal{E}}_s$ .

Iterative prefiltering linearization methods

[link] introduced the equation error formulation and several algorithms that minimize it. In a general sense however one is more interested in minimizing the solution error problem from [link] . This section presents several algorithms that attempt to minimize the solution error formulation from [link] by prefiltering the desired response $D\left(\omega \right)$ in [link] with $A\left(\omega \right)$ . Then a new set of coefficients $\{a_n,b_n\}$ are found with an equation error formulation and the prefiltering step is repeated, hence defining an iterative procedure.

Sanathanan-koerner ( SK ) method

The method by Levy presented in [link] suggests a relatively easy-to-implement approach to the problem of rational approximation. While interesting in itself, the equation error ${\epsilon }_e$ does not really represent what in principle one would like to minimize. A natural extension to Levy's method is the one proposed [link] by C. K. Sanathanan and J. Koerner in 1963. The algorithm iteratively prefilters the equation error formulation of Levy with an estimate of $A\left(\omega \right)$ . The SK method considers the solution error function ${\mathcal{E}}_s$ defined by

Then the solution error problem can be written as

where

Note that given $A\left(\omega \right)$ , one can obtain an estimate for $B\left(\omega \right)$ by minimizing ${\mathcal{E}}_e$ as Levy did. This approach provides an estimate, though, because one would need to know the optimal value of $A\left(\omega \right)$ to truly optimize for $B\left(\omega \right)$ . The idea behind this method is that by solving iteratively for $A\left(\omega \right)$ and $B\left(\omega \right)$ the algorithm would eventually converge to the solution of the desired solution error problem defined by [link] . Since $A\left(\omega \right)$ is not known from the beginning, it must be initialized with a reasonable value (such as $A\left({\omega }_k\right)=1$ ).