# 0.13 Least squares design of iir filters  (Page 7/9)

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To solve [link] Sanathanan and Koerner defined the same linear system from [link] with the same matrix and vector definitions. However the scalar terms used in the matrix and vectors reflect the presence of the weighting function $W\left(\omega \right)$ in ${\epsilon }_{s}$ as follows,

$\begin{array}{ccc}\hfill {\lambda }_{h}& =& \sum _{l=0}^{L-1}{\omega }_{l}^{h}W\left({\omega }_{l}\right)\hfill \\ \hfill {S}_{h}& =& \sum _{l=0}^{L-1}{\omega }_{l}^{h}{R}_{l}W\left({\omega }_{l}\right)\hfill \\ \hfill {T}_{h}& =& \sum _{l=0}^{L-1}{\omega }_{l}^{h}{I}_{l}W\left({\omega }_{l}\right)\hfill \\ \hfill {U}_{h}& =& \sum _{l=0}^{L-1}{\omega }_{l}^{h}\left({R}_{l}^{2}+{I}_{l}^{2}\right)W\left({\omega }_{l}\right)\hfill \end{array}$

Then, given an initial definition of $A\left(\omega \right)$ , at the $p$ -th iteration one sets

$W\left(\omega \right)=\frac{1}{{\left|{A}_{p-1},\left({\omega }_{k}\right)\right|}^{2}}$

and solves [link] using $\left\{\lambda ,S,T,U\right\}$ as defined above until a convergence criterion is reached. Clearly, solving [link] using [link] is equivalent to solving a series of weighted least squares problems where the weighting function consists of the estimated values of $A\left(\omega \right)$ from the previous iteration. This method is similar to a time-domain method proposed by Steiglitz and McBride [link] , presented later in this chapter.

## Method of sid-ahmed, chottera and jullien

The methods by Levy and Sanathanan and Koerner did arise from an analog analysis problem formulation, and cannot therefore be used directly to design digital filters. However these two methods present important ideas that can be translated to the context of filter design. In 1978 M. Sid-Ahmed, A. Chottera and G. Jullien followed on these two important works and adapted [link] the matrix and vectors used by Levy to account for the design of IIR digital filters, given samples of a desired frequency response. Consider the frequency response $H\left(\omega \right)$ defined in [link] . In parallel with Levy's development, the corresponding equation error can be written as

${\epsilon }_{e}=\sum _{k=0}^{L}{\left|\left\{\left({R}_{k}+j{I}_{k}\right),\left(1,+,\sum _{c=1}^{N},{a}_{i},{e}^{-j{\omega }_{k}c}\right),-,\left(\sum _{c=0}^{M},{b}_{i},{e}^{-j{\omega }_{k}c}\right)\right\}\right|}^{2}$

One can follow a similar differentiation step as Levy by setting

$\frac{\partial {\epsilon }_{e}}{\partial {a}_{1}}=\frac{\partial {\epsilon }_{e}}{{a}_{2}}=...=\frac{\partial {\epsilon }_{e}}{\partial {b}_{0}}=...=0$

with as defined in [link] . Doing so results in a linear system of the form

$\mathbf{C}x=y$

where the vectors $x$ and $y$ are given by

$x=\left\{\begin{array}{c}{b}_{0}\\ ⋮\\ {b}_{M}\\ {a}_{1}\\ ⋮\\ {a}_{N}\end{array}\right\}\phantom{\rule{42.67912pt}{0ex}}y=\left\{\begin{array}{c}{\phi }_{0}-{r}_{0}\\ ⋮\\ {\phi }_{M}-{r}_{M}\\ -{\beta }_{1}\\ ⋮\\ -{\beta }_{N}\end{array}\right\}$

The matrix $\mathbf{C}$ has a special structure given by

$\mathbf{C}=\left[\begin{array}{cc}\Psi \hfill & \Phi \\ {\Phi }^{T}\hfill & Υ\end{array}\right]$

where $\Psi$ and $Υ$ are symmetric Toeplitz matrices of order $M+1$ and $N$ respectively, and their first row is given by

$\begin{array}{c}\hfill \begin{array}{cc}{\Psi }_{1m}={\eta }_{m-1}\hfill & \phantom{\rule{28.45274pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{0.277778em}{0ex}}m=1,...,M+1\hfill \\ {Υ}_{1m}={\beta }_{m-1}\hfill & \phantom{\rule{28.45274pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{0.277778em}{0ex}}m=1,...,N\hfill \end{array}\end{array}$

Matrix $\Phi$ has order $M+1×N$ and has the property that elements on a given diagonal are identical (i.e. ${\Phi }_{i,j}={\Phi }_{i+1,j+1}$ ). Its entries are given by

$\begin{array}{c}\hfill \begin{array}{cc}{\Phi }_{1m}={\phi }_{m}+{r}_{m}\hfill & \phantom{\rule{28.45274pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{0.277778em}{0ex}}m=1,...,N\hfill \\ {\Phi }_{m1}={\phi }_{m-2}-{r}_{m-2}\hfill & \phantom{\rule{28.45274pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{0.277778em}{0ex}}m=2,...,M+1\hfill \end{array}\end{array}$

The parameters $\left\{\eta ,\phi ,r,\beta \right\}$ are given by

$\begin{array}{cccc}\hfill {\eta }_{i}& =\sum _{k=0}^{L}\text{cos}i{\omega }_{k}\hfill & & \phantom{\rule{2.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}0\phantom{\rule{-0.166667em}{0ex}}\le \phantom{\rule{-0.166667em}{0ex}}i\phantom{\rule{-0.166667em}{0ex}}\le \phantom{\rule{-0.166667em}{0ex}}M\hfill \\ \hfill {\beta }_{i}& =\sum _{k=0}^{L}{|D\left({\omega }_{k}\right)|}^{2}\text{cos}i{\omega }_{k}\hfill & & \phantom{\rule{2.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}0\phantom{\rule{-0.166667em}{0ex}}\le \phantom{\rule{-0.166667em}{0ex}}i\phantom{\rule{-0.166667em}{0ex}}\le \phantom{\rule{-0.166667em}{0ex}}N-1\hfill \\ \hfill {\phi }_{i}& =\sum _{k=0}^{L}{R}_{k}\text{cos}i{\omega }_{k}\hfill & & \phantom{\rule{2.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}0\phantom{\rule{-0.166667em}{0ex}}\le \phantom{\rule{-0.166667em}{0ex}}i\phantom{\rule{-0.166667em}{0ex}}\le \phantom{\rule{-0.166667em}{0ex}}max\left(N,M-1\right)\hfill \\ \hfill {r}_{i}& =\sum _{k=0}^{L}{I}_{k}\text{sin}i{\omega }_{k}\hfill & & \phantom{\rule{2.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}0\phantom{\rule{-0.166667em}{0ex}}\le \phantom{\rule{-0.166667em}{0ex}}i\phantom{\rule{-0.166667em}{0ex}}\le \phantom{\rule{-0.166667em}{0ex}}max\left(N,M-1\right)\hfill \end{array}$

The rest of the algorithm works the same way as Levy's. For a solution error approach, one must weight each of the parameters mentioned above with the factor from [link] as in the SK method.

There are two important details worth mentioning at this point: on one hand the methods discussed up to this point (Levy, SK and Sid-Ahmed et al.) do not put any limitation on the spacing of the frequency samples; one can sample as fine or as coarse as desired in the frequency domain. On the other hand there is no way to decouple the solution of both numerator and denominator vectors. In other words, from [link] and [link] one can see that the linear systems that solve for vector $x$ solve for all the variables in it. This is more of an issue for the iterative methods (SK&Sid-Ahmed), since at each iteration one solves for all the variables, but for the purposes of updating one needs only to keep the denominator variables (they get used in the weighting function); the numerator variables are never used within an iteration (in contrast to Burrus' Prony-based method presented in [link] ). This approach decouples the numerator and denominator computation into two separate linear systems. One only needs to compute the denominator variables until convergence is reached, and only then it becomes necessary to compute the numerator variables. Therefore most of the iterations solve a smaller linear system than the methods involved up to this point.

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