Specifying a system

When *P*(*s*) is a polynomial of *s* and *P*(*s*_{0})=0, then *s*_{0} is called a zero, or a root, of the polynomial. A polynomial of order *n* has *n* complex zeros
,
and can be factorized as
.
Thus, the zeros of a polynomial together with the constant *p* determine the polynomial completely.
Since our transfer functions *H*(*s*) are the ratio of two
polynomials *G*(*s*) and *F*(*s*) as in Eqs. (10) and (11),
they can be specified by their
zeros (the zeros of the numerator *G*(*s*)), their poles (the
zeros of the denominator *F*(*s*)), and a gain factor
(or equivalently the total gain at a given
frequency). The whole system - as long as it
remains in its linear operating range, and does
not produce noise - can thus be described by
a small number of discrete parameters.

Transfer functions are usually specified according to one of the following concepts:

- 1.
- The real coefficients of the polynomials in the numerator and denominator are listed.
- 2.
- The denominator polynomial is decomposed into
normalized first-order and second-
order factors with real coefficients (a total
decomposition into first-order factors would
require complex coefficients). The factors can
in general be attributed to individual
modules of the system. They are preferably given
in a form from which corner periods and
damping coefficients can be read, as in Eqs. (16)
to (18). The numerator often reduces to a
gain factor times a power of
*s*. - 3.
- The poles and zeros of the transfer
function are listed together with a gain
factor. Poles
and zeros must either be real or symmetric to
the real axis, as mentioned above. When
the numerator polynomial is
*s*^{m}, then*s*=0 is an*m*-fold zero of the transfer function, and the system is a high-pass filter of order*m*. Depending on the order*n*of the denominator and accordingly on the number of poles, the response may be flat at high frequencies (*n*=*m*), or the system may act as a low-pass filter there (*n*>*m*). The case*n*<*m*can occur only as an approximation in a limited bandwidth because no practical system can have an unlimited gain at high frequencies. High-frequency poles are often ignored in seismic systems.

In the header of the commonly used SEED data format, the gain factor is split up into a normalization factor that brings the gain to unity at a specified normalization frequency in the passband of the system, and another factor representing the total gain at this frequency. A simple BASIC program and Windows executable POL_ZERO is available for an interactive interpretation of the "analog" part of SEED headers (section 9).

2002-11-08