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

Passive Immittances and Positive-Real Functions 5-7that the minimum value of Re F 1 (s) for all Re s 0 occurs on the jv-axis; but according to Equation 5.23,this value is nonnegative:Re F 1 ðjvÞ ¼ Re FðjvÞ 0 (5:24)Thus, the real part of F 1 (s) is nonnegative everywhere in the closed RHS orRe F 1 ðÞ0 s for Re s 0 (5:25)This, together with the fact that F 1 (s) is real whenever s is real, shows that F 1 (s) is positive real.Since each term inside the parentheses of Equation 5.23 is positive real, and since the sum of two ormore positive-real functions is positive real, F(s) is positive real. This completes the proof of the theorem.In testing for positive realness, we may eliminate some functions from consideration by inspectionbecause they violate certain simple necessary conditions. For example, a function cannot be PR if it has apole or zero in the open RHS. Another simple test is that the highest powers of s in numerator anddenominator not differ by more than unity, because a PR function can have at most a simple pole or zeroat the origin or infinity, both of which lie on the jv-axis.A Hurwitz polynomial is a polynomial devoid of zeros in the open RHS. Thus, it may have zeros on thejv-axis. To distinguish such a polynomial from the one that has zeros neither in the open RHS nor onthe jv-axis, the latter is referred to as a strictly Hurwitz polynomial. For computational purposes,Theorem 5.3 can be reformulated and put in a much more convenient form.THEOREM 5.4A rational function represented in the formFs ðÞ¼ Ps ðÞQs ðÞ ¼ m 1ðÞþn s 1 ðÞ sm 2 ðÞþn s 2 ðÞ s(5:26)where m 1 (s), m 2 (s), and n 1 (s), n 2 (s) are the even and odd parts of the polynomials P(s) and Q(s),respectively, is positive real if and only if the following conditions are satisfied:1. F(s) is real when s is real.2. P(s) þ Q(s) is strictly Hurwitz.3. m 1 (jv)m 2 (jv) n 1 (jv)n 2 (jv) 0 for all v.A real polynomial is strictly Hurwitz if and only if the continued-fraction expansion of the ratio of theeven part to the odd part or the odd part to the even part of the polynomial yields only real and positivecoefficients, and does not terminate prematurely. For P(s) þ Q(s) to be strictly Hurwitz, it is necessaryand sufficient that the continued-fraction expansionm 1 ðÞþm s 2 ðÞ s 1¼ a 1 s þn 1 ðÞþn s 2 ðÞ s1a 2 s þ 1.. .þ 1a k s(5:27)yields only real and positive a’s, and does not terminate prematurely, i.e., k must equal the degreem 1 (s) þ m 2 (s) orn 1 (s) þ n 2 (s), whichever is larger. It can be shown that the third condition of the