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Previous sections have discussed the importance of complex least-square and Chebishev error criteria in the context of filter design. In many applications any of these two approaches would provide adequate results. However, a case could be made where one might want to minimize the error energy in a range of frequencies while keeping control of the maximum error in a different band. This idea results particularly interesting when one considers the use of different norms in different frequency bands. In principle one would be interested in solving
where represent the pass and stopband frequencies respectively. In principle one would want . Therefore problem [link] can be written as
One major obstacle in [link] is the presence of the roots around the summation terms. These roots prevent us from writing [link] in a simple vector form. Instead, one can consider the use of a similar metric function as follows
This expression is similar to [link] but does not include the root terms. An advantage of using the IRLS approach on [link] is that one can formulate this problem in the frequency domain and properly separate residual terms from different bands into different vectors. In this manner, the modified measure given by [link] can be made into a frequency-dependent function of as follows,
Therefore this
frequency-varying
It is fundamental to note that the proposed method does not indeed solve a linear combination of norms. In fact, it can be shown that the expression [link] is not a norm but a metric. While from a theoretical perspective this fact might make [link] a less interesting distance, as it turns out one can use [link] to solve the far more interesting CLS problem, as discussed below in [link] .
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