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One of the common obstacles to innovation occurs when knowledge settles on a particular way of dealing with problems. While new ideas keep appearing suggesting innovative approaches to design digital filters, it is all too common in practice that l 2 and l dominate error criteria specifications. This section is devoted to exploring a different way of thinking about digital filters. It is important to note that up to this point we are not discussing an algorithm yet. The main concern being brought into play here is the specification (or description) of the design problem. Once the Constrained Least Squares (CLS) problem formulation is introduced, we will present an IRLS implementation to solve it, and will justify our approach over other existing approaches. It is the author's belief that under general conditions one should always use our IRLS implementation over other methods, especially when considering the associated management of transition regions.

The CLS problem was introduced in [link] and is repeated here for clarity,

min h D ( ω ) - H ( ω ; h ) 2 subject to | D ( ω ) - H ( ω ; h ) | τ

To the best of our knowledge this problem was first introduced in the context of filter design by John Adams [link] in 1991. The main idea consists in approximating iteratively a desired frequency response in a least squares sense except in the event that any frequency exhibits an error larger than a specified tolerance τ . At each iteration the problem is adjusted in order to reduce the error on offending frequencies (i.e. those which do not meet the constraint specifications). Ideally, convergence is reached when the altered least squares problem has a frequency response whose error does not exceed constraint specifications. As will be shown below, this goal might not be attained depending on how the problem is posed.

Adams and some collaborators have worked in this problem and several variations [link] . However his main (and original) problem was illustrated in [link] with the following important assumption: the definition of a desired frequency response must include a fixed non-zero width transition band . His method uses Lagrange multiplier theory and alternation methods to find frequencies that exceed constraints and minimize the error at such locations, with an overall least squares error criterion.

Burrus, Selesnick and Lang [link] looked at this problem from a similar perspective, but relaxed the design specifications so that only a transition frequency needs to be specified. The actual transition band does indeed exist, and it centers itself around the specified transition frequency; its width adjusts as the algorithm iterates (constraint tolerances are still specified). Their solution method is similar to Adams' approach, and explicitly uses the Karush-Kuhn-Tucker (KKT) conditions together with an alternation method to minimize the least squares error while constraining the maximum error to meet specifications.

C. S. Burrus and the author of this work have been working on the CLS problem using IRLS methods with positive results. This document is the first thorough presentation of the method, contributions, results and code for this approach, and constitutes one of the main contributions of this work. It is crucial to note that there are two separate issues in this problem: on one hand there is the matter of the actual problem formulation, mainly depending on whether a transition band is specified or not; on the other hand there is the question of how the selected problem description is actually met (what algorithm is used). Our approach follows the problem description by Burrus et al. shown in [link] with an IRLS implementation.

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Source:  OpenStax, Iterative design of l_p digital filters. OpenStax CNX. Dec 07, 2011 Download for free at http://cnx.org/content/col11383/1.1
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