where p is the complex frequency variable, h(p) is an arbitrary polynomial and g(p) is a strictly Hurwitz polynomial
. The polynomial f(p) is monic and represents the transmission zeros of the network .
First the authors present some results about the abscissa of stability of characteristic polynomials from linear differential equations systems; that is, they consider Hurwitz polynomials
. The starting point is the Gauss-Lucas Theorem, the authors provide lower bounds for Hurwitz polynomials
, and by successively decreasing the order of the derivative of the Hurwitz polynomial
one obtains a sequence of lower bounds.
As expected, the interlacing property of a Hurwitz polynomial is utilized to find all stabilizing controllers for the closed-loop characteristic equation [DELTA](s):
Hurwitz polynomial: A polynomial is termed Hurwitz if the real part of all its roots are negative.
Furthermore, if |DQGP - DKP - DHR QJR| is a Hurwitz polynomial, then the closed loop system depicted in Fig.1 will remain asymptotically stable and [lim.sub.s[right arrow]0][[??].sub.w](s) = [[??].sub.w](0) = 0
To show that (9) holds, since |[bar.P](s)| and |Q(s)| are both Hurwitz polynomials, the final value theorem can be employed; i.e.,
[sigma] is a unimodular constant; |[sigma]| = 1, g(p,[lambda]) is a scattering Hurwitz polynomial
If p(t) is a Hurwitz polynomial and [z.sub.1], [z.sub.2], ...,[z.sub.n] are its zeros then [sigma]p the abscissa of stability of p(t) is defined by
+ [a.sub.1]t + [a.sub.0] is a Hurwitz polynomial (n [greater than or equal to] 2) and [[sigma].sub.p] and [[sigma].sub.p'] are the abscissas of stability of p and p' = dp/dt, respectively, then [[sigma].sub.p'] [less than or equal to] [[sigma].sub.p].
Amax = the passband ripple (dB) n = the network order or number of poles s = the frequency variable (s = j [OMEGA]) [C.sub.n](s) = the generalized Chebyshev polynomial p(s) = the Hurwitz polynomial
The insertion phase response of the lowpass filter now can be determined by considering the LHS poles of the Hurwitz polynomial. From the poles' locations given in Equations 4 to 7, the insertion phase as a function of frequency can be computed by Equation 8.
of Notre Dame) are on the mathematics of circuits and filters, with discussion of Fourier methods, z-transforms, wavelet transforms, graph theory, and the theory of two-dimensional Hurwitz polynomials
, among other topics.