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Two problem formulations

As mentioned in [link] , one can address problem [link] in two ways depending on how one views the role of the transition band in a CLS problem. The original problem posed by Adams in [link] can be written as follows,

min h D ( ω ) - H ( ω ; h ) 2 subject to | D ( ω ) - H ( ω ; h ) | τ ω [ 0 , ω p b ] [ ω s b , π ]

where 0 < ω p b < ω s b < π . From a traditional standpoint this formulation feels familiar. It assigns fixed frequencies to the transition band edges as a number of filter design techniques do. As it turns out, however, one might not want to do this in CLS design.

An alternate formulation to [link] could implicitly introduce a transition frequency ω t b (where ω p b < ω t b < ω s b ); the user only specifies ω t b . Consider

min h D ( ω ) - H ( ω ; h ) 2 ω [ 0 , π ] subject to | D ( ω ) - H ( ω ; h ) | τ ω [ 0 , ω p b ] [ ω s b , π ]

The algorithm at each iteration generates an induced transition band in order to satisfy the constraints in [link] . Therefore { ω p b , ω s b } vary at each iteration.

Two graphs showing two formulations for Constrained Least Squares Problems.
Two formulations for Constrained Least Squares problems.

It is critical to point out the differences between [link] and [link] . [link] .a explains Adams' CLS formulation, where the desired filter response is only specified at the fixed pass and stop bands. At any iteration, Adams' method attempts to minimize the least squares error ( ε 2 ) at both bands while trying to satisfy the constraint τ . Note that one could think of the constraint requirements in terms of the Chebishev error ε by writing [link] as follows,

min h D ( ω ) - H ( ω ; h ) 2 subject to D ( ω ) - H ( ω ; h ) τ ω [ 0 , ω p b ] [ ω s b , π ]

In contrast, [link] .b illustrates our proposed problem [link] . The idea is to minimize the least squared error ε 2 across all frequencies while ensuring that constraints are met in an intelligent manner. At this point one can think of the interval ( ω p b , ω s b ) as an induced transition band, useful for the purposes of constraining the filter. [link] presents the actual algorithms that solve [link] , including the process of finding { ω p b , ω s b } .

It is important to note an interesting behavior of transition bands and extrema points in l 2 and l filters. [link] shows l 2 and l length-15 linear phase filters (designed using Matlab's firls and firpm functions); the transition band was specified at { ω p b = 0 . 4 / π , ω s b = 0 . 5 / π } . The dotted l 2 filter illustrates an important behavior of least squares filters: typically the maximum error of an l 2 filter is located at the transition band. The solid l filter shows why minimax filters are important: despite their larger error across most of the bands, the filter shows the same maximum error at all extrema points, including the transition band edge frequencies. In a CLS problem then, typically an algorithm will attempt to reduce iteratively the maximum error (usually located around the transition band) of a series of least squares filters.

A graph consisting of two waveforms one is identified by a solid line and labeled L_∞. The other is identified by a dotted line and labeled L_2. The x-axis is labeled ω/π and the y-axis is labeled |H(ω)|. Both waveforms follow the same general path starting on the left high on the y-axis and dropping drastically to 0 on the y axis and bouncing along the x axis to the end of the graph.
Comparison of l 2 and l filters.

Another important fact results from the relationship between the transition band width and the resulting error amplitude in l filters. [link] shows two l designs; the transition bands were set at { 0 . 4 / π , 0 . 5 / π } for the solid line design, and at { 0 . 4 / π , 0 . 6 / π } for the dotted line one. One can see that by widening the transition band a decrease in error ripple amplitude is induced.

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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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