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

Approximation 2-29Recall, now, the objective of exploring the preceding generalization of the sinusoid: one is looking fora transformation that will convert the characteristic function given by Equations 2.115 and 2.116into equivalent waveforms having the equiripple property. As one might suspect (since so much timehas been spent on developing it), the elliptic sine function is precisely the transformation desired.The crucial aspect of showing this is the application of a fundamental property of sn(u, k), known asan additional formula:sn(u þ a, k)sn(u a, k) ¼ sn2 (u, k) sn 2 (a, k)1 k 2 sn 2 (u, k)sn 2 (a, k)(2:122)The right-hand side of this identity has the same form as one factor in Equations 2.115 and 2.116, that isof one zero factor coupled with its corresponding pole factor. This suggests the transformationThe passband zeros then are given byv ¼v i ¼and the factors mentioned previously map intop ffiffik sn(u, k) (2:123)p ffiffik sn(ui , k) (2:124)v 2 iv 21 v 2 ¼ k ½ sn2 ðu i , kÞ sn 2 (u, k) Ši v2 1 k 2 sn 2 ðu i , kÞsn 2 (u, k) ¼ ksn ð u þ u iÞsnðu u i Þ (2:125)For specificity, the even-order case will be discussed henceforth. The odd-order case is the same if minornotational modifications are made. Thus, one sees thatR 2n (v) ¼Pi¼1,3,...,2n 1v 2 iv 21 v 2 i v2 ¼ Pi¼1,3,...,2nksn ð u þ u iÞsnðu u i Þ (2:126)1The u i are to be chosen. Before doing this, it helps to simplify the preceding expression by definingu i ¼ u i and reindexing. Calling the resulting function G(u), one hasG(u) ¼i¼1,3,...,2n Y 1i¼ 1, 3,..., (2n 1)ksnðu þ u i Þ (2:127)Refer now to Figure 2.24. Each of the sn functions is periodic with period 4K and is completely defined byits values over one quarter-period [0, K]. Suppose that one definesu i ¼ i K 2n(2:128)Figure 2.25 shows the resulting transformationpcorrespondingffiffi pffiffito Equations 2.123 and 2.124. As uprogresses from K to þK, v increases from k to þ k as desired. Furthermore, because of thesymmetry of sn(u), the set {u i } forms an additive group—adding K=2n to the index of any u i results inanother u i in thepset.ffiffiThispmeansffiffithat G(u) in Equation 2.127 is periodic with period K=2n. Thus, as vincreases from k to þ k , R2n (v) achieves 2n þ 2 extrema, that is, positive and negative peak values.But this is sufficient for R 2n (v) to be the Chebyshev rational function of best approximation.