Principles of Modern Radar, Basic Principles 11. Your enemy, 20 km distant, fire

Question

Principles of Modern Radar, Basic Principles

11. Your enemy, 20 km distant, fires a mortar round in your general direction. The round has a vertical component of velocity of 200 meters per second. (Assume that this does not change during the engagement.) You are searching the area using a weapon-locating radar. What must your maximum scan time be to ensure at least four opportunities to cetect the target before it passes through your"search fence," which is the elevation sector extending from 0 to 4 degrees above the horizon?

Explanation / Answer

We assume a parametric measurement model
p(x|)[orp(x;) , which is the notation that we sometimes use in the
classical setting

In point estimation theory, we estimated the parameter

Suppose now that we choose
0 and 1 that form a
partition
of the parameter space
:01= , 01=.

In detection theory, we wish to identify which hypothesis is true
(i.e. make the appropriate decision):
H0:0, null hypothesis H1:1,
alternative hypothesis
.
Terminology:
If can only take two values, ={0, 1},0={0},1={1}
we say that the hypotheses are
simple
. Otherwise, we say that they are composite

Composite Hypothesis Example:
H0:= 0 versus H1:(0,).

Composite Hypothesis Example:
H0:= 0 versus H1:(0,).We wish to design a decision rule (function)
(x) :X (0,1):(x) ={1,decideH1,0,decideH0.
which partitions the data space
X[i.e. the support of p(x|)] into two regions:
Rule (x):X0={x:(x) = 0},X1={x:(x) = 1}.
Let us define probabilities of false alarm and miss:
PFA= Ex|[(X)|] =X1px|)dxforin0PM= Ex|[1(X)| = 1-X1p(x|)dx=X0p(x|)
dx for in 1.
Then, the probability of detection (correctly decidingH1) is
PD= 1PM= Ex|[(X)|] =X1p(x|)dxforin1.
EE 527, Detection and Estimation Theory, # 5
5
Note:
PFA and PD/PM
are generally functions of the parameter(where
0
when computing
PFAand 1 when computing PD PM).
More Terminology.
Statisticians use the following
terminology:
•
False alarm“Type I error”
•Miss“Type II error”
•Probability of detection“Power”
•Probability of false alarm“Significance level.”
EE 527, Detection and Estimation Theory, # 5

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