Filter analysis by use of pencil of functions: Part I

A pair of functions, when linearly combined via a parameter, produces a mathematical entity called a pencil of functions. These pencils are especially interesting when a signal
g_{1} (t)
is processed by a cascade of simple operators such as first-order filters (FOF's)
1/(s +q_{l}), q_{l}
>
0, i = 1,{\cdots}, n,
because the pencils formed by pairs of the resulting signal ensemble
g_{l} + y_{l}g_{l+1}
possess some very useful properties. Most useful of these concerns the linear dependence of the set of pencils thus produced. It is shown in Parts I and II that a necessary condition for a set of pencil of functions to be linearly dependent is a polynomial equation that must be satisfied by their parameters. Applications of the result include linear system identification and rational modeling of the power density spectrum of a random signal. The former of these is discussed in Part I. System dynamics is estimated in closed form requiring no prior estimates. The estimated parameters coincide with true values in the event of noise-free data. Inner products are utilized for computations, and minimum variance corrections are made when the data are noisy.