Representation of Acoustical Space-Time Signals

Abstract

An acoustic field is fully described if it is specified at all points and at all times within a specified space. Such a situation may exist in some careful laboratory experiments, if the properties of the space are all known and if all sources of sound are under our control. The above case is theoretically trivial; the case of interest arises when the field is imperfectly known and we wish to arrive at an estimate of the “real” field. The function of four variables describing the field can be called a four-dimensional signal that we wish to “receive” with an imperfect receiver: there are certain advantages to the analogy with the theory of communication. The signal may be known only within a sub-space of the total four-dimensional space, be it a small geometric region, a short time interval, or a combination of such bounds; or it may be known only at a finite or denumerable collection of four-dimensional points—i.e., it is a sampled signal. Sources of unknown positions and waveforms usually exist outside the “receiving region,” and the properties of the medium and its boundaries are usually known as sampled data. The estimation of signal parameters at the inaccessible points involves an extrapolation; we discuss here several ways to do this, and their deficiencies. We reach the conclusion that the problem cannot have a unique solution, but that the use of representations for the signal that are orthogonal in four dimensions is at least necessary to prevent gross divergence in our estimate. We finally discuss the problem when the sources, medium, or boundaries are in unspecified motion.