P. L. Stocklin 345 
additive Gaussian noise is independent between sampling points in space 
and between sampling points in time. 
Il. (space only): Signal is known exactly in space. Noise is additive and 
Gauss—Markov between space sampling points. 
III. (space only): Signal wavefront is Gauss—Markov perturbed between 
space sampling points; noise is Gauss—Markoy between space sampling 
points. 
Example I: Suppose there are M hydrophones receiving either (a) a signal 
S known exactly in time and space plus independent Gaussian (in time and space) 
noise, or (b) the noise alone. What is the optimum space—time processor? If iXj 
is the input acoustic pressure amplitude at the ith hydrophone at the jth instant 
of time, i$; is the signal pressure amplitude of the jth hydrophone at the jth instant 
of time, and 
X= (iki, 1X2,-+-) 1X2wrs---3 mX1y +++) mX own) (8) 
then 
3 1 ou 21a 
fy(X) = Qnoy)-™"7 exp (-—5 >) axy (9) 
_ 2on j=l fal 
and 
y 1 M 2WT 
fsy(X) = (2r0y) "7 exp |~ —> De (ix; — ,s;)? (10) 
2on fz1 j=l 
where f,(X) is the probability that if noise alone is present then sample 
X =(1%1,---,yX2wr)Will occur; and f,,(X) is the same for signal-plus-noise con- 
dition. The space--time likelihood ratio for this example,/:(X), is 
_fsn (X) LS a 2 1 
hQ) = mo | dee x > Gsy = 21015) a 
In Eq. (11), we see that 
a. The basic processing operation is a space—time cross-correlation be- 
tween the signal known exactly ;s, and the input ix,, since the double 
summation over ,s; is just the total signal energy received at all the 
hydrophones, and so is a constant known exactly. 
b. It is immaterial whether the space summation (usually called "integra- 
tion") is done first. 
c. By "signal known exactly in space" we mean that we know the wavefront 
location. Thus, our signal-known-exactly assumption means that we know 
both the waveform and the wavefront. 
Two completely equivalent block diagrams for Eq. (11) are given in Fig. 18.2. 
As a final point, the ROC curve for this example is identical to Fig. 2 of [1], 
with 
2ME 
No 
ale 
where E is the signal energy as measured at one hydrophone output. The spatial 
