Specifying a system

When P(s) is a polynomial of s and P(s₀)=0, then s₀ is called a zero, or a root, of the polynomial. A polynomial of order n has n complex zeros $s_i, i=1 \dots n$, and can be factorized as $P(s)=p\cdot\prod (s-s_i)$. 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).