Characteristic equation and parametric behaviour of single-pumping periodic systems

It is shown that the characteristic equation (CE) of an nth-order, ´single-pumping´, continuous-time periodic system is an nth-order polynomial equation in the z-plane, whose coefficients (for zero-mean, symmetric pumping of magnitude K) are power series in K². For cisoidal cases, the expressions for the K², K⁴ and K⁶ terms have been obtained. Further, for non-cisoidal systems, the K² term has also been obtained. Such CEs allow one, for the first time in literature, to study analytically the behaviour of periodic systems for any pumping frequency and even for large values of K. The K behaviour of several frequency-normalised, cisoidal systems has been studied to illustrate this point. The study further reveals that, (i) particularly for high-frequency systems, the K² terms of the power series are alone capable of describing the system behaviour over a wide domain, and (ii) as the value of K is increased, the roots, in general, move in an oscillatory fashion.