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

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

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

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

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

where $E=H-H_d$ . From [link] one can write $H=H_d+E$ as

Therefore

Solving [link] for $b$ one gets

Also,

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

From [link] we get

or

which in a least squares sense results in

From [link] one gets

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

Soewito's quasilinearization method

Consider the equation error residual function

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

where

and

Consider now the solution error residual function

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

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

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

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

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

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

where

Consider the solution error least-squares problem given by

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

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

Solving the system [link] gives

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 $F$ .

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

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

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

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,

with $H={[H_0,H_1,\cdots ,H_{L-1}]}^T$ . Therefore one can minimize $s^{T}s$ from [link] with

since all the terms inside the parenthesis in [link] are constant at the $(i+1)$ -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

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