346 Lecture 18 
GRATION 
BEAM 
FORM 
J | 
THRESHOLD 
THRESHOLD 
Fig. 18.2. Example I, space—time processor. 
processing gain in db, then, is 10 log M for this case, and the temporal process- 
ing gain is 10 log (2E/No). 
Example II: The following example is restricted to space statistics only, to 
bring out the idea that for several quite likely statistical situations pattern 
formation in the usual sense is not optimum likelihood-ratio processing in 
space. This example illustrates the departure from pattern formation due to a 
space-correlated noise field, specifically, a first-order Gauss—Markov noise 
field. In this example, a particular instant of time is chosen and space statistics 
only will be discussed. For the noise field alone, with M (regularly spaced) point 
hydrophones, the Mth-order spatial-probability distribution function f,(X) is, 
under the Markovian assumption, 
fy(X1) = fwGixa- ++) m1) = fy (1X1) + fy, 1x1 (2¥1) fy, yx, (3%1) +**five m—1%1 (1) 
M 
fy(X1) = fy(x1) pun fy, pax 4X1) (12) 
and together with the Gaussian assumption, Eq. (12), gives 
2 M 2 
BG ell ral lee ale eg ex) |e (oS ews (13) 
2oy f22 2(1-pron 
where o, is the noise power per hydrophone, and py is the normalized noise 
(spatial) correlation function between adjacent hydrophones. 
Now, if signal is known exactly—in this example, such a state of knowledge 
means knowing the signal wavefront at the chosen instant of time—then fsy(X) is | 
x1 = 181) a yes — $1) — py(i-1x4+ y= 181)? 
faw(X1) = (2a 1-™/? [1 — pZ1--Y/? exp .— 
w(X1) = [2no% PN P 2oz, a 2(1- prow 
Using Eqs. (7), (13), and (14): 
= fy (X) a 1 2 2 ba 2 y2 
In(X1) = Fy = “lors -(1- py) st - > GS1— Pw -i-181) 72a Pw) 1X1-181 
M M 
+ 2 yX1.mS1 +2(1+ py) D> X1181 + 2 py DY Gx1s-181 + veel (15) 
i=2 i=2 
