Of Poles and Zeros

Seismologists try to obtain information about physical processes within the earth by recording and analysing seismic ground motion. Historically, their attention has been focused on the earthquake source. Since, however, the ground motion recorded at a seismic station on the earth’s surface differs considerably from seismic signals generated by the earthquake source a major problem in seismology is the separation of the original source signal from other effects.

A very simple filter will highlight some basics from the theory of systems. This filter consists of a capacitor C and a resistor R in series (Fig. 2.1).

We studied the simple RC circuit in great detail because the concepts we used for its analysis are also valid for more complicated systems. In the following we will make the transition to general systems with only the restrictions of linearity and time invariance (LTI system). By the end of this chapter we will have progressed to frequency response functions relevant to seismological systems.

So far we have encountered several methods for describing linear time invariant systems. We have learned to design and analyse simple filters in terms of the poles and zeros of the transfer function. We have also seen some relationships between the different approaches.

Digital seismograms are sequences of numbers which in general have been obtained from the continuous output voltage of a seismic sensor by the procedures of sampling and analog to digital conversion. We will now consider some of the characteristic properties of these processes. We will simulate the sampling of analog data and its reconstruction from sampled values. We will find that an analog signal can only be reconstructed from its sampled values, if the frequency content of the signal to be sampled contains no energy at and above half of the sampling frequency (sampling theorem). We will investigate what happens if we deliberately violate this rule (aliasing effect).

In this chapter we will look at the properties of analog-to-digital converters (ADC) and the limitations they introduce into sampled data. We will discuss the relationship between the resolution and the dynamic range of ADCs and end this chapter by simulating techniques for improving the dynamic range of ADCs by gain ranging and oversampling.

In previous chapters we developed concepts which describe the properties of continuous-time LTI systems, such as is shown in the analog part of Fig. 6.19. On the other hand, when we used DST to demonstrate these properties we were, of course, working with sequences of numbers. This was acceptable in the context of the examples we used. In a general context, however, we must be well aware of several important differences between discrete and continuous systems. Our intuitive grasp of the more important system properties must by now be firm enough that we can extend our view using a more formal approach. In this fashion we will not only acquire additional tools for data processing, but also we will gain some insight into the links between systems defined for infinite continuous-time signals and systems defined for finite discrete-time signals. The focus of this chapter will be on consequences of this transition for the concepts which we developed so far. We will see for example that the transition from continuous-time to discrete-time systems corresponds to a transition from aperiodic to periodic Fourier spectra. This property for example will allow us to take a new look at the aliasing problem. Furthermore, we will meet the z-transform for discrete sequences, the discrete counterpart of the Laplace transform. We will see that the concepts introduced for continuous-time systems, such as the transfer function or the concept of poles and zeros, remains valid for discrete-time systems as well. Even more important, we will see that most of the relevant system properties can intuitively be understood from their analog counterparts. In the following the sequence notation of Oppenheim and Schafer (1989) will be used.

High performance seismic recording systems that make use of oversampling and decimation techniques reduce the requirements on the analog anti-alias filters because most of the filtering is performed in the digital domain (cf. Fig. 6.19). The quantization noise level of such systems decreases within the passband of interest with the ratio of the original Nyquist frequency to the final Nyquist frequency (oversampling ratio). On the other hand, it is desirable to keep the usable frequency range for the signal (passband of the anti-alias filter) as wide as possible. For this reason, digital anti-alias filters with very steep transition bands are needed to obtain the best resolution for a given frequency band of interest. The filter should also leave a band-limited signal which falls completely within the passband as unaffected as possible, causing neither amplitude nor phase distortions.

We have now looked at all the essential building blocks of modern seismic recording systems (cf. Fig. 6.19) and seen how they affect recorded seismic signals. We have learned to model the individual components (seismometer, ADC, etc.) using the concepts of continuous and discrete linear time invariant systems (transfer function, frequency response function, impulse response function) and how their interactions can be described by convolution in both the time and frequency domains (convolution theorem). We have become acquainted with the elegant and powerful concept of poles and zeros and learned what they can tell us about important system properties such as minimum phase, maximum phase, etc. In addition, we have seen how we can use poles and zeros to design simple, special-purpose digital filters. After reviewing the theoretical background of the Fourier-, Laplace-, and z-transforms, we have taken a detailed look at the digital antialias filter. We have seen how we can cure the precursory ringing problem of high frequency signals and discussed the design and implementation of a recursive correction filter.

Standard observational measurements from seismograms only include a relatively small number of signal parameters such as onset times, amplitudes, rise times, pulse widths, signal moments and onset polarities of particular wave groups (Fig. 10.1).