Resolution

  In locating an earthquake or a petroleum drilling site there will be an uncertainty in location, say $(\Delta x, \Delta y, \Delta z)$ caused by measurement errors and the physical size of the target. In measuring a voltage there will be a measuring accuracy $\Delta v$.The frequency of useful seismic waves will have a bandwidth $\Delta\omega$.The time at which an earthquake occurs will have an uncertainty given by the duration of shaking $\Delta t$.A telescope of diameter $\Delta d$ has at best a resolving power measured by a certain angular range $\Delta \theta$.It is often desirable to make measurements in such a way as to reduce the quantities $\Delta x$, $\Delta y$, $\Delta z$, $\Delta v$, $\Delta\omega$, $\Delta t$,$\Delta d$, and $\Delta \theta$ to values as small as possible. These measures of resolution (which are called variances, tolerances, uncertainties, bandwidths, durations, spreads, spans, etc.) sometimes interact with one another in such a way that any experimental modification which reduces one must necessarily increase another or some combination of the others. The purpose of this chapter is to discuss some of the commonly occurring situations where such conflicting interactions occur.

In this chapter we use $\Delta t$ to denote the time duration of a signal. We use $\tau$ to denote the amount of time which passes between sample points. In other chapters, $\Delta t$ is synonymous with $\tau$, the sample interval.