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U. Glaeser

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This constrained magnitude problem can be solved by Mathematical programming, which takes different<br />

forms depending on the norm. For the least-squares norm, the solution can be found by using quadratic<br />

programming.<br />

Mathematical programming is also a tool for the nonlinear-phase Chebyshev problem [20], which can<br />

be rewritten as a constrained problem:<br />

This problem is a semi-infinite linear minimization (SILM). Linear programming can then be applied<br />

to the problem by sampling the parameters ω and θ. The algorithm is not efficient for high-order filters<br />

because dense parameter sampling is needed to design filters with high precision. In order to improve<br />

efficiency, the SILM can be rewritten in a dual form [21–24].<br />

26.5 Recent Design Methods<br />

Because conventional design methods are available for only special types of digital filters, e.g., linearphase,<br />

researchers have proposed various new methods that use complex approximation in filter design.<br />

Among those, only a few are discussed because they are elegant and useful in various applications.<br />

Complex Remez Algorithm<br />

The complex Chebyshev design problem is one of the most important approaches for designing digital<br />

filters. Unfortunately, it might need a general algorithm such as SILM, which requires a large number of<br />

frequency samples (with resulting high computation and high memory) when high precision is desired.<br />

For high order filters, linear programming is very inefficient for Chebyshev filter design. Instead, modifications<br />

of the Remez Exchange algorithm would be more desirable. Therefore, the complex Remez<br />

algorithm (CRemez) [25,26] was proposed using an exchange method search that is similar to the Remez<br />

Exchange; however, the original CRemez is most efficient only for the special case where the extremal<br />

error alternates. In general, nonlinear-phase filters are not guaranteed to have this strict alternating<br />

property, so the exchange method does not converge to the optimum. In order to get the optimum filter<br />

in the case of nonalternating extremal errors, a second stage is needed for CRemez. This second stage<br />

has to be a general optimization method that ends up being as inefficient as the SILM method. As a<br />

result, some filters are designed very quickly by CRemez, but others take a long time when the general<br />

optimization step must be invoked.<br />

Constrained Least-Squares<br />

Adams [27] suggested that Chebyshev digital filters do not always have the best overall characteristics.<br />

He found that by allowing the worst-case (Chebyshev) error to increase slightly, the least-squares error<br />

can be reduced significantly. To design this sort of filter, a constrained least-squares problem was introduced.<br />

The problem has been solved by [28–31] with an algorithm that is quite efficient for designing<br />

FIR filters.<br />

Generalized Remez Algorithm<br />

The constrained least-squares methods have two design drawbacks: (1) error constraints are required to set<br />

up the problem, and (2) the existing methods only handle the FIR case. The first drawback is not severe,<br />

but it reduces the design efficiency because prior information such as a prior filter design procedure is<br />

needed to estimate the constraints; however, both drawbacks can be eliminated by using a different norm<br />

(called the combined norm) and by minimizing via the iterative reweighted least-squares (IRLS) technique.<br />

© 2001 by CRC Press LLC<br />

minδ<br />

subject to<br />

ℜ E(ω )e jθ<br />

{ } <<br />

δ, for all ω and θ

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