342 Lecture 18 
other the band-limited, time-limited acoustic field. Their importance lies in 
determining the number and the geometry of the samples in space and time 
necessary to reconstruct completely—or, in the case of finite v, almost com- 
pletely—the space—time acoustic field. Using these theorems, a finite-order 
joint probability density function may be constructed over V and T for each pos- 
sible (mutually exclusive) event. These density functions are in fact the state-of- 
knowledge descriptions we seek. To get on with efficient processor design, the 
possible events are considered two at a time. For each pair of events, the ratio 
is formed of their conditional-probability density functions and is called the 
likelihood ratio for this pair of events. The likelihood ratio for any pair of 
events is a mathematical description of the processor which is (a) optimum in 
the sense of minimun average risk and (b) most efficient statistically of all the 
processors that could be built given this state of knowledge. 
The likelihood-ratio processor for the entire set of events is formed by the 
linear superposition of the processor for each pair of events. If each event is 
considered relative to a common event (such as noise alone), each individual 
processor receives a weight in the over-all processor corresponding to the 
a priori probability of occurrence of the noncommon event. For the examples 
at the end of this paper, we will consider only the simplest possible set of 
events, the binary set, and the two events will be denoted by SN (the event is that 
both signal and noise are present) and N(the event is that noise alone is present). 
18.3.2. Sampling Theorems 
Much of the subsequent material is developed from [2], where in particular 
the following two space—time sampling theorems are proved. 
18.3.2.1. Theorem I (Uniform Space Sampling of a Monochromatic Field 
If a sound field p(x,y,z;t) consists entirely of acoustic radiation of a single 
frequency f, formed from the superposition of fields from an arbitrary number 
of single-frequency sources of arbitrary phase and amplitude, then at any instant 
of time t, p(x,y,z;t) is given within the volume not containing these sources as a 
function of x, y, and z by its complex values, at points spaced a half wavelength 
apart in x, y, and z, times a three-dimensional Nyquist—Shannon sampling func- 
tion, such sampling extending throughout space: 
p(x,y,2;t) = ef Ee p( a 5" BA, 2.) S(l,m, n; x, y, Z) (2) 
l,m,n=— 
where the function p(JA/2, md/2, nd/2) is the complex amplitude of the sound field 
at x=IA2, y=m\/2, z=nd/2, t=0; and 
S(x,y, z; 1, m,n) = spatial sampling function 
_ sinz(2x/A — 1), sinw(2y/A- m); sin(2z/A — n) e)) 
~ “a (2x/N— D) m2y/A — m) 1] Tz/h—n) 
