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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 l p norms in different frequency bands. In principle one would be interested in solving

min h D ( ω p b ) - H ( ω p b ; h ) p + D ( ω s b ) - H ( ω s b ; h ) q

where { ω p b Ω p b , ω s b Ω s b } represent the pass and stopband frequencies respectively. In principle one would want Ω p b Ω s b = { } . Therefore problem [link] can be written as

min h ω p b | D ( ω p b ) - H ( ω p b ; h ) | p p + ω s b | D ( ω s b ) - H ( ω s b ; h ) | q q

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

min h ω p b | D ( ω p b ) - H ( ω p b ; h ) | p + ω s b | D ( ω s b ) - H ( ω s b ; h ) | q

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 l p modified measure given by [link] can be made into a frequency-dependent function of p ( ω ) as follows,

min h D ( ω ) - H ( ω ; h ) p ( ω ) p ( ω ) = ω | D ( ω ) - H ( ω ; h ) | p ( ω )

Therefore this frequency-varying l p problem can be solved following the modified IRLS algorithm outlined in [link] with the following modification: at the i -th iteration the weights are updated according to

w i = | D - C a i | p ( ω ) - 2

It is fundamental to note that the proposed method does not indeed solve a linear combination of l p 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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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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