P. L. Stocklin 343 
18.3.2.2. Theorem II (Band-Limited Frequency Spectrum, Finite Time) 
If a sound field p(xy,z;t) of interest for finite time T consists almost com- 
pletely [3] of radiation within the band offrequencies (0,W), then pis given every- 
where by its real amplitude values at discrete sampling points times a sampling 
function: 
WT 5 
p (x,y,z;t) = (2WT)~* s cca aK(ays)em team 
q=- WT e=- WT 
co 
IX, mA, nA, s 
x D3 p(s, As, at 2.9, (m,n; x,y,z) (4) 
where @q = 27q/T 
Aq = 21c/w, 
7(~s +WT) 
K (q,s) il [ein v/v) etait] ag (5). 
(= s =WT) 
and 
; sin7(2x/A, -1) sinz(2y/A, —m) sinz(2z/A, —n) 
= SaeetiNG Alig ed 
Sq (1, m,n; x,y,z) 7 (2x/A, =T 7(2y/A, — m) ” a7(Qz Ng 2 (6) 
and p(IA,/2, mA,/2, nd,/2; s/2W) is the real amplitude of p (x,y,z;t) at point (x =1A,/2, 
y= mA,/2, A= nd, /2) at time t= s/2W. 
18.3.3. Space-Time Likelihood Ratio for Detection 
Let us consider the binary set of possible events SN (signal and noise are 
present) and WN (noise alone is present). The decision to be made is, which of 
these events has occurred. This is usually called the detection decision or 
simply detection. Following the outline given in the Section 18.1 and using the 
space—time sampling theorem for a series band-limited, time-limited, volume- 
limited (actually, M spatial sampling points) acoustic field, we have for the de- 
tection likelihood ratio: 
Jr = 288) _ few Oakae iho ees akoweh 2a css voXow eh oy ss emtawn) (7) 
fy(X) fyGiX1, 1%2>- + +>1Xawrs 2X1 + + + 2Xawrs «> mXi» ++ «mX2wr) 
where ,x; is the pressure at the th space sampling point and th time sampling 
point. Both f,,(X) and f,(X) have to do with the joint occurrence of the 2WT'M 
samples (;x,,...,,,xX2wr) in space—time. Under certain conditions [Eq. (2)], 1(X) is 
separable into space and time components; generally, it is not. 
Substitution of the specific probability density functions into Eq. (7) gives 
the specific processor design for efficient detection. This is done in the next 
section for one space—time and two space-only examples. While this is the pri- 
mary application of the space—time likelihood ratio, a second and equally inter- 
esting use of the likelihood ratio is as a gauge of the efficiency of existing 
or proposed processors, or of the effect of different states of knowledge upon 
the capability to make accurate or, more generally, minimum average risk 
descriptions. 
