Passive, active, and digital filters (3ed., CRC, 2009) - tiera.ru

FIR Filters 18-23Rejection of superfluous potential extremals. The solution of Equation 18.50 can be obtained only ifprecisely r þ 1 extremals are available. By differentiating E(v), one can show that in a filter with onefrequency band of interest (e.g., a digital differentiator) the number of maxima in jE(v)j (potentialextremals in step 2 of Algorithm 18.1) can be as high as r þ 1. In the weighted-Chebyshev method, bandedges at which jE(v)j is maximum or jE(v)jjdj are treated as potential extremals (see Algorithm 18.2).Therefore, whenever the number of frequency bands is increased by one, the number of potential˘extremals is increased by 2, i.e., for a filter with J bands there can be as many as r þ 2J 1 frequenciesv i and a maximum of 2J 2 superfluousv i may occur. This problem is overcome by rejecting r r of thepotential extremals v i ,ifr > r, in step 4 of the algorithm.A simple rejection scheme is to reject the r r frequencies v i that yield the lowest jE( v i )j and thenrenumber the remaining v i from 0 to r [8]. This strategy is based on the well-known fact that themagnitude of the error in a given band is inversely related to the density of extremals in that band, i.e., alow density of extremals results in a large error and a high density results in a small error. Conversely,a low band error is indicative of a high density of extremals, and rejecting superfluousv i in such a band isthe appropriate course of action.A problem with the scheme just described is that whenever a frequency remains an extremal in twosuccessive iterations, jE(v)j assumes the value of jdj in the second iteration by virtue of Equation 18.49.In practice, there are almost always several frequencies that remain extremals from one iteration tothe next, and the value of jE(v)j at these frequencies will be the same. Consequently, the rejectionof potential extremals on the basis of the magnitude of the error can become arbitrary and may lead tothe rejection of potential extremals in bands where the density of extremals is low. This tendsto increase the number of iterations, and it may even prevent the algorithm from converging onoccasion. This problem can to some extent be alleviated by rejecting only potential extremals that arenot band edges.An alternative rejection scheme based on the aforementioned strategy, which gives excellent results fortwo-band and three-band filters, involves ranking the frequency bands in the order of lowest averageband error, dropping the band with the highest average error from the list, and then rejecting potentialextremals, one per band, in a cyclic manner starting with the band with the lowest average error [11].The steps involved are as follows.˘˘˘˘˘˘ALGORITHM 18.3:Rejection of Superfluous Potential Externals1. Compute the average band errorsE j ¼ 1 v jX v i ) ^v i 2V jE(˘for j ¼ 1, 2, ..., Jwhere V j is the set of potential externals in band j given bynV j ¼ v j : v Lj ˘˘v j v Rjov j is the number of potential externals in band j, and J is the number of bands.2. Rank the J bands in the order of lowest average error and let l 1 , l 2 ,...,l J be the ranked list obtained,i.e., l 1 and l J are the bands with the lowest and highest average errors, respectively.3. Reject one v i in each of bands l 1 , l 2 ,..., l J 1 , l 1 , l 2 ,..., l 1 until r r superfluous v i are rejected.In each case, reject the v i , other than a band edge, that yields the lowest jE( v i )j in the band.˘˘˘˘