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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 and 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 -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 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 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] .
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