0.13 Least squares design of iir filters  (Page 8/9)

Steiglitz-mcbride iterative algorithm

In 1965 K. Steiglitz and L. McBride presented an algorithm [link] , [link] that has become quite popular in statistics and engineering applications. The Steiglitz-McBride method ( SMB ) considers the problem of deriving a transfer function for either an analog or digital system from their input and output data; in essence it is a time-domain method. Therefore it is mentioned in this work for completeness as it closely relates to the methods by Levy, SK and Sid-Ahmed, yet it is far better known and understood.

The derivation of the SMB method follows closely that of SK. In the Z-domain, the transfer function of a digital system is defined by

Furthermore

Steiglitz and McBride define the following problem,

where $X\left(z\right)={\sum }_j{x}_j{z}^{-j}$ and $D\left(z\right)={\sum }_j{d}_j{z}^{-j}$ represent the z-transforms of the input and desired signals respectively. Equation [link] is the familiar nonlinear solution error function expressed in the Z-domain. Steiglitz and McBride realized the complexity of such function and proposed the iterative solution [link] using a simpler problem defined by

This linearized error function is the familiar equation error in the Z-domain. Steiglitz and McBride proposed a two-mode iterative approach. The SMB Mode 1 iteration is similar to the SK method, in that at the $k$ -th iteration a linearized error criterion based on [link] is used,

where

Their derivation For more details the reader should refer to [link] , [link] . leads to the familiar linear system

with the following vector definitions

The vector $q_j$ is referred to as the input-output vector. Then

SMB Mode 2 is an attempt at reducing further the error once Mode 1 produces an estimate close enough to the actual solution. The idea behind Mode 2 is to consider the solution error defined by [link] and equate its partial derivatives with respect to the coefficients to zero. Steiglitz and McBride showed [link] , [link] that this could be attained by defining a new vector

Then

The main diference between Mode 1 and Mode 2 is the fact that Mode 1 uses the desired values to compute its vectors and matrices, whereas Mode 2 uses the actual output values from the filter. The rationale behind this is that at the beggining, the output function $y\left(t\right)$ is not accurate, so the desired function provides better data for computations. On the other hand, Mode 1 does not really solve the desired problem. Once Mode 1 is deemed to have reached the vicinity of the solution, one can use true partial derivatives to compute the gradient and find the actual solution; this is what Mode 2 does.

It has been claimed that under certain conditions the Steiglitz-McBride algorithm converges. However no guarantee of global convergence exists. A more thorough discussion of the Steiglitz-McBride algorithm and its relationships to other parameter estimation algorithms (such as the Iterative Quadratic Maximum Likelihood algorithm, or IQML) are found in [link] , [link] , [link] .