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

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Jackson's method

The following is a recent approach (from 2008) by Leland Jackson [link] based in the frequency domain. Consider vectors $a\in {\mathbb{R}}^{N}$ and $b\in {\mathbb{R}}^{M}$ such that

$H\left(\omega \right)=\frac{B\left(\omega \right)}{A\left(\omega \right)}$

where $H\left(\omega \right),B\left(\omega \right),A\left(\omega \right)$ are the Fourier transforms of $h,b$ and $a$ respectively. For a discrete frequency set one can describe Fourier transform vectors $B={\mathbf{W}}_{b}b$ and $A={\mathbf{W}}_{a}a$ (where ${\mathbf{W}}_{b},{\mathbf{W}}_{a}$ correspond to the discrete Fourier kernels for $b,a$ respectively). Define

${H}_{a}\left({\omega }_{k}\right)=\frac{1}{A\left({\omega }_{k}\right)}$

In vector notation, let ${\mathbf{D}}_{a}=\text{diag}\left({H}_{a}\right)=\text{diag}\left(1/A\right)$ . Then

$H\left(\omega \right)=\frac{B\left(\omega \right)}{A\left(\omega \right)}={H}_{a}\left(\omega \right)B\left(\omega \right)⇒H={\mathbf{D}}_{a}B$

Let ${H}_{d}\left(\omega \right)$ be the desired complex frequency response. Define ${\mathbf{D}}_{d}=\text{diag}\left({H}_{d}\right)$ . Then one wants to solve

$\text{min}\phantom{\rule{0.277778em}{0ex}}{E}^{*}E={\parallel E\parallel }_{2}^{2}$

where $E=H-{H}_{d}$ . From [link] one can write $H={H}_{d}+E$ as

$H={\mathbf{D}}_{a}B={\mathbf{D}}_{a}{\mathbf{W}}_{b}b$

Therefore

${H}_{d}=H-E={\mathbf{D}}_{a}{\mathbf{W}}_{b}b-E$

Solving [link] for $b$ one gets

$b=\left({\mathbf{D}}_{a}{\mathbf{W}}_{b}\right)\setminus {H}_{d}$

Also,

${H}_{d}={\mathbf{D}}_{d}\stackrel{^}{I}={\mathbf{D}}_{d}{\mathbf{D}}_{a}A={\mathbf{D}}_{a}{\mathbf{D}}_{d}A={\mathbf{D}}_{a}{\mathbf{D}}_{d}{\mathbf{W}}_{a}a$

where $\stackrel{^}{I}$ is a unit column vector. Therefore

$H-E={H}_{d}={\mathbf{D}}_{a}{\mathbf{D}}_{d}{\mathbf{W}}_{a}a$

${\mathbf{D}}_{a}{\mathbf{W}}_{b}b-E={\mathbf{D}}_{a}{\mathbf{D}}_{d}{\mathbf{W}}_{a}a$

or

${\mathbf{D}}_{a}{\mathbf{D}}_{d}{\mathbf{W}}_{a}a+E={\mathbf{D}}_{a}{\mathbf{W}}_{b}b$

which in a least squares sense results in

$a=\left({\mathbf{D}}_{a}{\mathbf{D}}_{d}{\mathbf{W}}_{a}\right)\setminus \left({\mathbf{D}}_{a}{\mathbf{W}}_{b}b\right)$

$a=\left({\mathbf{D}}_{a}{\mathbf{D}}_{d}{\mathbf{W}}_{a}\right)\setminus \left({\mathbf{D}}_{a}{\mathbf{W}}_{b}\left[\left({\mathbf{D}}_{a}{\mathbf{W}}_{b}\right)\setminus {H}_{d}\right]\right)$

As a summary, at the $i$ -th iteration one can write [link] and [link] as follows,

$\begin{array}{cc}\hfill {b}_{i}& =\left(\text{diag}\left(1/{A}_{i-1}\right){\mathbf{W}}_{b}\right)\setminus {H}_{d}\hfill \\ \hfill {a}_{i}& =\left(\text{diag}\left(1/{A}_{i-1}\right)\text{diag}\left({H}_{d}\right){\mathbf{W}}_{a}\right)\setminus \left(\text{diag}\left(1/{A}_{i-1}\right){\mathbf{W}}_{b}{b}_{i}\right)\hfill \end{array}$

Soewito's quasilinearization method

Consider the equation error residual function

$\begin{array}{c}\hfill \begin{array}{cc}\hfill e\left({\omega }_{k}\right)& =B\left({\omega }_{k}\right)-D\left({\omega }_{k}\right)·A\left({\omega }_{k}\right)\hfill \\ & =\sum _{n=0}^{M}{b}_{n}{e}^{-j{\omega }_{k}n}-D\left({\omega }_{k}\right)·\left(1,+,\sum _{n=1}^{N},{a}_{n},{e}^{-j{\omega }_{k}n}\right)\hfill \\ & ={b}_{0}+{b}_{1}{e}^{-j{\omega }_{k}}+\cdots +{b}_{M}{e}^{-j{\omega }_{k}M}\cdots \hfill \\ & \phantom{\rule{56.9055pt}{0ex}}-{D}_{k}-{D}_{k}{a}_{1}{e}^{-j{\omega }_{k}}-\cdots -{D}_{k}{a}_{N}{e}^{-j{\omega }_{k}N}\hfill \\ & =\left({b}_{0},+,\cdots ,{b}_{M},{e}^{-j{\omega }_{k}M}\right)-{D}_{k}\left({a}_{1},{e}^{-j{\omega }_{k}},+,\cdots ,{a}_{N},{e}^{-j{\omega }_{k}N}\right)-{D}_{k}\hfill \end{array}\end{array}$

with ${D}_{k}=D\left({\omega }_{k}\right)$ . The last equation indicates that one can represent the equation error in matrix form as follows,

$e=\mathbf{F}h-D$

where

$\mathbf{F}=\left[\begin{array}{ccccccc}1& {e}^{-j{\omega }_{0}}& \cdots & {e}^{-j{\omega }_{0}M}& -{D}_{0}{e}^{-j{\omega }_{0}}& \cdots & -{D}_{0}{e}^{-j{\omega }_{0}N}\\ ⋮& ⋮& & ⋮& ⋮& & ⋮\\ 1& {e}^{-j{\omega }_{L-1}}& \cdots & {e}^{-j{\omega }_{L-1}M}& -{D}_{L-1}{e}^{-j{\omega }_{L-1}}& \cdots & -{D}_{L-1}{e}^{-j{\omega }_{L-1}N}\end{array}\right]$

and

$h=\left[\begin{array}{c}{b}_{0}\\ {b}_{1}\\ ⋮\\ {b}_{M}\\ {a}_{1}\\ ⋮\\ {a}_{N}\end{array}\right]\phantom{\rule{1.em}{0ex}}\text{and}\phantom{\rule{1.em}{0ex}}D=\left[\begin{array}{c}{D}_{0}\\ ⋮\\ {D}_{L-1}\end{array}\right]$

Consider now the solution error residual function

$\begin{array}{cc}\hfill s\left({\omega }_{k}\right)& =H\left({\omega }_{k}\right)-D\left({\omega }_{k}\right)=\frac{B\left({\omega }_{k}\right)}{A\left({\omega }_{k}\right)}-D\left({\omega }_{k}\right)\hfill \\ & =\frac{1}{A\left({\omega }_{k}\right)}\left[B,\left({\omega }_{k}\right),-,D,\left({\omega }_{k}\right),·,A,\left({\omega }_{k}\right)\right]\hfill \\ & =W\left({\omega }_{k}\right)e\left({\omega }_{k}\right)\hfill \end{array}$

Therefore one can write the solution error in matrix form as follows

$s=\mathbf{W}\left(\mathbf{F}h-D\right)$

where $\mathbf{W}$ is a diagonal matrix with $\frac{1}{A\left(\omega \right)}$ in its diagonal. From [link] the least-squared solution error ${\epsilon }_{s}={s}^{*}s$ can be minimized by

$h={\left({\mathbf{F}}^{*}{\mathbf{W}}^{2}\mathbf{F}\right)}^{-1}{\mathbf{F}}^{*}{\mathbf{W}}^{2}D$

From [link] an iteration Soewito refers to this expression as the Steiglitz-McBride Mode-1 in frequency domain. could be defined as follows

${h}_{i+1}={\left({\mathbf{F}}^{*}{\mathbf{W}}_{i}^{2}\mathbf{F}\right)}^{-1}{\mathbf{F}}^{*}{\mathbf{W}}_{i}^{2}D$

by setting the weights $\mathbf{W}$ in [link] equal to ${A}_{k}\left(\omega \right)$ , the Fourier transform of the current solution for $a$ .

A more formal approach to minimizing ${\epsilon }_{s}$ consists in using a gradient method (these approaches are often referred to as Newton-like methods). First one needs to compute the Jacobian matrix $\mathbf{J}$ of $s$ , where the $pq$ -th term of $\mathbf{J}$ is given by ${\mathbf{J}}_{pq}=\frac{\partial {s}_{p}}{\partial {h}_{q}}$ with $s$ as defined in [link] . Note that the $p$ -th element of $s$ is given by

${s}_{p}={H}_{p}-{D}_{p}=\frac{{B}_{p}}{{A}_{p}}-{D}_{p}$

For simplicity one can consider these reduced form expressions for the independent components of $h$ ,

$\begin{array}{ccc}\hfill \frac{\partial {s}_{p}}{\partial {b}_{q}}& =& \frac{1}{{A}_{p}}\frac{\partial }{\partial {b}_{q}}\sum _{n=0}^{M}{b}_{n}{e}^{-j{\omega }_{p}n}={W}_{p}{e}^{-j{\omega }_{p}q}\hfill \\ \hfill \frac{\partial {s}_{p}}{\partial {a}_{q}}& =& {B}_{p}\frac{\partial }{\partial {a}_{q}}\frac{1}{{A}_{p}}=\frac{-{B}_{p}}{{A}_{p}^{2}}\frac{\partial }{\partial {a}_{q}}\left(1,+,\sum _{n=1}^{N},{a}_{n},{e}^{-j{\omega }_{p}n}\right)=\frac{-1}{{A}_{p}}·\frac{{B}_{p}}{{A}_{p}}·{e}^{-j{\omega }_{p}q}\hfill \\ & =& -{W}_{p}{H}_{p}{e}^{-j{\omega }_{p}q}\hfill \end{array}$

Therefore on can express the Jacobian $\mathbf{J}$ as follows,

$\mathbf{J}=\mathbf{W}\mathbf{G}$

where

$\mathbf{G}=\left[\begin{array}{ccccccc}1& {e}^{-j{\omega }_{0}}& \cdots & {e}^{-j{\omega }_{0}M}& -{H}_{0}{e}^{-j{\omega }_{0}}& \cdots & -{H}_{0}{e}^{-j{\omega }_{0}N}\\ ⋮& ⋮& & ⋮& ⋮& & ⋮\\ 1& {e}^{-j{\omega }_{L-1}}& \cdots & {e}^{-j{\omega }_{L-1}M}& -{H}_{L-1}{e}^{-j{\omega }_{L-1}}& \cdots & -{H}_{L-1}{e}^{-j{\omega }_{L-1}N}\end{array}\right]$

Consider the solution error least-squares problem given by

$\underset{h}{\text{min}}f\left(h\right)={s}^{T}s$

where $s$ is the solution error residual vector as defined in [link] and depends on $h$ . It can be shown [link] that the gradient of the squared error $f\left(h\right)$ (namely $\nabla f$ ) is given by

$\nabla f={\mathbf{J}}^{*}s$

A necessary condition for a vector $h$ to be a local minimizer of $f\left(h\right)$ is that the gradient $\nabla f$ be zero at such vector. With this in mind and combining [link] and [link] in [link] one gets

$\nabla f={\mathbf{G}}^{*}{\mathbf{W}}^{2}\left(\mathbf{F}h-D\right)=\mathbf{0}$

$h={\left({\mathbf{G}}^{*}{\mathbf{W}}^{2}\mathbf{F}\right)}^{-1}{\mathbf{G}}^{*}{\mathbf{W}}^{2}D$

An iteration can be defined as follows Soewito refers to this expression as the Steiglitz-McBride Mode-2 in frequency domain. Compare to the Mode-1 expression and the use of ${G}_{i}$ instead of $F$ .

${h}_{i+1}={\left({\mathbf{G}}_{i}^{*}{\mathbf{W}}_{i}^{2}\mathbf{F}\right)}^{-1}{\mathbf{G}}_{i}^{*}{\mathbf{W}}_{i}^{2}D$

where matrices $\mathbf{W}$ and $\mathbf{G}$ reflect their dependency on current values of $a$ and $b$ .

Atmadji Soewito [link] expanded the method of quasilinearization of Bellman and Kalaba [link] to the design of IIR filters. To understand his method consider the first order of Taylor's expansion near ${H}_{i}\left(z\right)$ , given by

$\begin{array}{ccc}\hfill {H}_{i+1}\left(z\right)& =& {H}_{i}\left(z\right)+\frac{\left[{B}_{i+1}\left(z\right)-{B}_{i}\left(z\right)\right]{A}_{i}\left(z\right)-\left[{A}_{i+1}\left(z\right)-{A}_{i}\left(z\right)\right]{B}_{i}\left(z\right)}{{A}_{i}^{2}\left(z\right)}\hfill \\ & =& {H}_{i}\left(z\right)+\frac{{B}_{i+1}\left(z\right)-{B}_{i}\left(z\right)}{{A}_{i}\left(z\right)}-\frac{{B}_{i}\left(z\right)\left[{A}_{i+1}\left(z\right)-{A}_{i}\left(z\right)\right]}{{A}_{i}^{2}\left(z\right)}\hfill \end{array}$

Using the last result in the solution error residual function $s\left(\omega \right)$ and applying simplification leads to

$\begin{array}{ccc}\hfill s\left(\omega \right)& =& \frac{{B}_{i+1}\left(\omega \right)}{{A}_{i}\left(\omega \right)}-\frac{{H}_{i}\left(\omega \right){A}_{i+1}\left(\omega \right)}{{A}_{i}\left(\omega \right)}+\frac{{B}_{i}\left(\omega \right)}{{A}_{i}\left(\omega \right)}-D\left(\omega \right)\hfill \\ & =& \frac{1}{{A}_{i}\left(\omega \right)}\left[{B}_{i+1},\left(\omega \right),-,{H}_{i},\left(\omega \right),{A}_{i+1},\left(\omega \right),+,{B}_{i},\left(\omega \right),-,{A}_{i},\left(\omega \right),D,\left(\omega \right)\right]\hfill \end{array}$

Equation [link] can be expressed (dropping the use of $\omega$ for simplicity) as

$s=W\left(\left(\left[{B}_{i+1},-,{H}_{i},\left({A}_{i+1}-1\right)\right],-,{H}_{i}\right),+,\left(\left[{B}_{i},-,D,\left({A}_{i}-1\right)\right],-,D\right)\right)$

One can recognize the two terms in brackets as $\mathbf{G}{h}_{i+1}$ and $\mathbf{F}{h}_{i}$ respectively. Therefore [link] can be represented in matrix notation as follows,

$s=\mathbf{W}\left[\mathbf{G}{h}_{i+1}-\left(D+{H}_{i}-\mathbf{F}{h}_{i}\right)\right]$

with $H={\left[{H}_{0},{H}_{1},\cdots ,{H}_{L-1}\right]}^{T}$ . Therefore one can minimize ${s}^{T}s$ from [link] with

${h}_{i+1}={\left({\mathbf{G}}_{i}^{*}{\mathbf{W}}_{i}^{2}{\mathbf{G}}_{i}\right)}^{-1}{\mathbf{G}}_{i}^{*}{\mathbf{W}}_{i}^{2}\left(D+{H}_{i}-\mathbf{F}{h}_{i}\right)$

since all the terms inside the parenthesis in [link] are constant at the $\left(i+1\right)$ -th iteration. In a sense, [link] is similar to [link] , where the desired function is updated from iteration to iteration as in [link] .

It is important to note that any of the three algorithms can be modified to solve a weighted ${l}_{2}$ IIR approximation using a weighting function $W\left(\omega \right)$ by defining

$V\left(\omega \right)=\frac{W\left(\omega \right)}{A\left(\omega \right)}$

Taking [link] into account, the following is a summary of the three different updates discussed so far:

$\begin{array}{cc}\hfill \text{SMB}\phantom{\rule{4.pt}{0ex}}\text{Frequency}\phantom{\rule{4.pt}{0ex}}\text{Mode-1:}& \phantom{\rule{8.5359pt}{0ex}}{h}_{i+1}={\left({\mathbf{F}}^{*}{\mathbf{V}}_{i}^{2}\mathbf{F}\right)}^{-1}{\mathbf{F}}^{*}{\mathbf{V}}_{i}^{2}D\hfill \\ \hfill \text{SMB}\phantom{\rule{4.pt}{0ex}}\text{Frequency}\phantom{\rule{4.pt}{0ex}}\text{Mode-2:}& \phantom{\rule{8.5359pt}{0ex}}{h}_{i+1}={\left({\mathbf{G}}_{i}^{*}{\mathbf{V}}_{i}^{2}\mathbf{F}\right)}^{-1}{\mathbf{G}}_{i}^{*}{\mathbf{V}}_{i}^{2}D\hfill \\ \hfill \text{Soewito's}\phantom{\rule{4.pt}{0ex}}\text{quasilinearization:}& \phantom{\rule{8.5359pt}{0ex}}{h}_{i+1}={\left({\mathbf{G}}_{i}^{*}{\mathbf{V}}_{i}^{2}{\mathbf{G}}_{i}\right)}^{-1}{\mathbf{G}}_{i}^{*}{\mathbf{V}}_{i}^{2}\left(D+{H}_{i}-\mathbf{F}{h}_{i}\right)\hfill \end{array}$

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