2.3 Chebyshev or equal ripple error approximation filters  (Page 7/9)

This algorithm is iterated as a multiple exchange, keeping the number of ripples in the pass and stop band constant, to give an optimal extraripple filter. The location and width of the transition band is controlled only by the choice of how the number of initial ripples aredivided between the pass and stop band. The final filter may not have the transition located where you want it. Indeed, no solution may exist withthe desired location of the transition band.

The designs produced by the HOS algorithm are always maximum ripple but this comes with a loss of accurate control over the location of thetransition band. The algorithm is not, strictly speaking, an optimization algorithm. It is an interpolation algorithm. The Chebyshev error is notminimized, the designed amplitude interpolates the specified error ripples. However, although not directly minimized, the transition bandwidth of these designs seems to be minimized [link] , [link] , [link] . Extra or maximum ripple designs seem to be efficient in using all the zeros toproduce small ripple size and narrow transition bands, however, the loss of accurate control over the location of the transition bands becomes evenmore problematic with multiple transition band designs. Perhaps some compromise methods can be devised that use some of the efficiency of themaximum ripple approximations with some of the control of other methods. The next two design methods are of that type.

The shpak and antoniou algorithm

Shpak and Antoniou [link] propose decoupling the size of the pass and stopband ripple sizes in order to have control over the pass and stopband edges and have an extra ripple design. The Parks-McClellan design has the ripple sizes related with a fixed weight ${\delta }_p=K{\delta }_s$ , the modified Parks-McClellan design fixes one ripple size and minimizesthe other, the Hoffstetter, Oppenheim, and Siegel design fixes both ripple sizes but cannot set the transition band edges. The Shpak-Antonioudesign fixes the transition band edges and gives a maximum ripple design with minimum ripple but the relationship of the pass and stopband rippleis uncontrolled.

This method has two ripple sizes, ${\delta }_p$ and ${\delta }_s$ , appended to the $a\left(n\right)$ vector similar to the single $\delta$ used in [link] or [link] . This allows controlling an additional extremal frequency and results in an extra ripple approximation. This can become somewhatcomplicated for multiple transition bands but seems very flexible [link] .

The new equal ripple design formulation and exchange algorithm

Because the arguments in the Weisburn, Parks, and Shenoy paper [link] require the assumption of no signal or noise energy in the transition band, it is nowmore obvious that a narrow transition band is very desirable. For this reason it may be better to fix the pass and stop band peak error, ${\delta }_p$ and ${\delta }_s$ and the transition band center frequency ${\omega }_o$ then minimize the transition band width rather than fixing the pass and stop band edges, ${\omega }_p$ and ${\omega }_s$ , then minimizing ${\delta }_p$ and ${\delta }_s$ . Two methods have been recently developed to address this point of view. The first is a new exchange algorithm that isin some ways a combination of the Parks-McClellan and Hofstetter-Oppenheim-Segiel algorithms [link] and the second is a limiting case for a constrained least squares method based on Lagrangemultipliers [link] , [link] , [link] , [link] using tight constraints.