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# l_p error as a function of frequency

Module by: Ricardo Vargas. E-mail the author

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 lplp 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 min h D ( ω p b ) - H ( ω p b ; h ) p + D ( ω s b ) - H ( ω s b ; h ) q
(1)

where {ωpbΩpb,ωsbΩsb}{ωpbΩpb,ωsbΩsb} represent the pass and stopband frequencies respectively. In principle one would want ΩpbΩsb={}ΩpbΩsb={}. Therefore problem Equation 1 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 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
(2)

One major obstacle in Equation 2 is the presence of the roots around the summation terms. These roots prevent us from writing Equation 2 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 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
(3)

This expression is similar to Equation 2 but does not include the root terms. An advantage of using the IRLS approach on Equation 3 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 lplp modified measure given by Equation 3 can be made into a frequency-dependent function of p(ω)p(ω) as follows,

min h D ( ω ) - H ( ω ; h ) p ( ω ) p ( ω ) = ω | D ( ω ) - H ( ω ; h ) | p ( ω ) min h D ( ω ) - H ( ω ; h ) p ( ω ) p ( ω ) = ω | D ( ω ) - H ( ω ; h ) | p ( ω )
(4)

Therefore this frequency-varying


lplp problem can be solved following the modified IRLS algorithm outlined in (Reference) with the following modification: at the ii-th iteration the weights are updated according to

w i = | D - C a i | p ( ω ) - 2 w i = | D - C a i | p ( ω ) - 2
(5)

It is fundamental to note that the proposed method does not indeed solve a linear combination of lplp norms. In fact, it can be shown that the expression Equation 3 is not a norm but a metric. While from a theoretical perspective this fact might make Equation 3 a less interesting distance, as it turns out one can use Equation 3 to solve the far more interesting CLS problem, as discussed below in (Reference).

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