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An in-depth analysis of the performance evaluation of optimal detectors in cauchy noise using the direct method. The basic performance measures for a binary decision rule, the computation of cumulative distribution functions (cdfs) of the decision statistic under each hypothesis, and the calculation of false-alarm probability (pf) and probability of detection (pd) for a correlation detector in cauchy noise.
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ECE642: Detection and Estimation Theory
Dr. Sudharman K. Jayaweera
Assistant Professor
Department of Electrical and Computer Engineering
University of New Mexico
Lecture 15 - October
th
, Thursday
Fall 2007
ECE642: Detection and Estimation Theory
Basic Performance Measures for a Binary Decision Rule
δ˜
Optimum rules based on likelihood-ratio tests are of the form:
δ˜ T (^) ( y )
γ
if
y )
τ
where
is a mapping from
to
probabilitiesDetector performance is primarily given by the two conditional error
F
and
M
:
F
( δ˜ )
0 ( δ˜
chooses
1 )
and
M
(
δ˜ )
1
( δ˜
chooses
0 )
ECE642: Detection and Estimation Theory
Direct Performance Computation
Denote by
T , (^) j
the
cumulative distribution function
(cdf) of
under the hypothesis
j , where
is the decision statistic of a
detector of the form of (1). i.e.
T , (^) j ( t ) = P j
t ) =
t | H
j )
P Then, for a detector of the form of (1): F
(
δ˜ T )
τ | H
0 ) +
(^) γ P (^) ( T (^) ( Y (^) ) =
τ | H
0 )
τ | H
0 ) +
(^) γ
[ P
(^) ( T (^) ( Y (^) )
≤ τ | H 0 )
lim
t →
τ − (^) P
(^) ( T (^) ( Y (^) )
≤
t | H
(since the cdf is right continuous)
T , 0 ( τ ) +
(^) γ
[ F T , 0 ( τ ) (^) −
lim
t →
τ − (^) F
T , 0 ( t ) ]
(from (4))
4
ECE642: Detection and Estimation Theory
Direct Performance Computation (ctd...)
Similarly,
M
δ˜ T )
τ | H
1 ) + (
(^) γ ) P
(^) ( T (^) ( Y (^) ) =
τ | H
1 )
≤ τ | H 1 )
(^) γ P
(^) ( T (^) ( Y (^) ) =
τ | H
1 )
T , 1 ( τ ) (^) −
(^) γ
[ F T , 1 ( τ ) (^) −
lim
t →
τ − (^) F
T , 1 ( t ) ]
of the form of (1) is possible if the cdf’sObserve from (5) and (6) that the performance evaluation of any LRT
T , (^) j
of
under
0
and
1
can be determined easily in a neighborhood of the threshold
τ .
ECE642: Detection and Estimation Theory
Recall that the characteristic function (ch.f.) of a random variable
is,
φ X (^) ( u )
e iuX
for
u
∈
Define,
φ T , (^) j ( u )
j^ {
e iuT
(^) ( Y ) }
ch.f. of
under
j
and
φ g k , (^) j ( u )
j^ {
e iug
k ( Y k ) }
ch.f. of
g k ( Y k )
under
j
where
j^ { . }
denotes the expectation under
j
ECE642: Detection and Estimation Theory
Characteristic Function of the Decision Statistic for iid Observations
From the independence of the
k ’s, (from (9))
φ T , (^) j ( u )
j^ {
e iuT
(^) ( Y ) }
j^ {
e iu
∑^ k n = 1 (^) g k ( Y k ) }
(from (7))
j^ {
n
= 1 e iug
k ( Y k ) }
n
= 1 E
j^ {
e iug
k ( Y k ) }
(because
k ’s are independent)
φ T , (^) j ( u )
n
= 1 φ g k ,
(^) j ( u )
(from (10))
ECE642: Detection and Estimation Theory
pdf of the Decision Statistic with iid Observations
When
y )
is continuous under
j , F T , (^) j
has a corresponding pdf
p T , (^) j
given by
p T , (^) j ( t )
π
Z
∞
− ∞
φ T , (^) j ( u ) e − iut
du^
Since from (9):
φ T , (^) j ( u ) = Z ∞
− ∞
e iut
p^ T , (^) j ( t ) dt
Note that in this case
φ
T
, (^) j
and
p
T , (^) j
are a
Fourier transform pair
ECE642: Detection and Estimation Theory
Performance Analysis with iid Observations (ctd...)
Thus, if observations are
independent
and
is a continuous
random variable, then from (5), (11) and (13) we have,
F
( δ˜ T )
π
Z
∞
τ
Z
∞
− ∞
[
n
= 1 φ g k ,
0 ( u ) ] e −
iut
dudt^
(since
is continuous, the boundary has probability zero)
Similarly,
M
δ˜ T )
π
Z
τ
− ∞
Z
∞
− ∞
[
n
= 1 φ g k ,
1 ( u ) ] e −
iut
dudt^
ECE642: Detection and Estimation Theory
Correlation Detection in Cauchy Noise
Suppose that we employ a
correlation detector
for detecting the
coherent signal
s k } k n = 1
in this additive Cauchy noise.
Appendix K)Note: This may not be the optimal detector for this case (see
The decision statistic
y )
of a correlation detector is
y )
n
= 1 s k y k
Hence, from (7),
g k ( y k ) = s k y k
for
k
=
(^) n
ECE642: Detection and Estimation Theory
Correlation Detection in Cauchy Noise (ctd...)
Hence, the characteristic function
φ g k , 0 ( . )
of
g k ( Y k )
is (from (10),
φ g k , 0 ( u ) = E 0 { e
iug
k ( y k ) }
(from (10))
0 { e ius
k Y k (^) }
(from (20))
e ius
k N k (^) }
(from (17))
φ N k (^) ( us
k )
where
φ N
k (^) (
. )
is the ch.f. of Cauchy noise
k .
ECE642: Detection and Estimation Theory
Under
0 : ch.f. of the Decision Statistic of Correlation Detector in
Cauchy Noise
From (11) and (23): φ T , 0 ( u ) = n
= 1 φ g k ,
0 ( u )
(from (11))
n
= 1 e −|
us
k |
(from (23))
n
= 1 e −|
u || s k |
=
e −|
u | ∑^ k n = 1 (^) | s k |
=
e −|
u | n n^1 (^) ∑
1 (^) | s k |
Define,
s |
n 1
n
= 1 | s k |
Then, (24) becomes
φ
T , 0 ( u ) = e − n ¯
| s || u |
ECE642: Detection and Estimation Theory
Under
0 : pdf of the Decision Statistic of Correlation Detector in
Cauchy Noise
Clearly (26) shows that
φ T , 0 ( u )
is
absolutely integrable
. Hence,
y )
is continuous under
0 , and from (1),
p T , 0 ( t ) = 1
π
Z
∞
− ∞
e − n || (^) ¯s || u | e − iut
du^
(from (1) and (26))
π
Z
0
− ∞
e n
¯
| s | u − iut
du^
π
Z
∞
0
e − n
¯
| s | u − iut
du^
π
Z
∞
0
e − n ¯ | s | u
iut
du^
π
Z
∞
0
e − n ¯ | s | u − iut
du^
π
Z
∞
0 e − ( n ¯
| s |−
it ) u du
π
Z
∞
0 e − ( n ¯
| s |
it ) u du
2 π [ − e − ( n ¯
| s |−
it ) u
n
s | −
(^) it
0 ∞
2 π [ − e − ( n ¯
| s |
it ) u
n
¯
s | (^) +
(^) it
0 ∞
ECE642: Detection and Estimation Theory
False-alarm Probability of Correlation Detector in Cauchy Noise
is, Hence, the false alarm probability of the correlator in Cauchy noise
F (^) ( δ˜ T (^) )
0 ( T (^) ( Y
)
≥
τ )
(since
) is continuous)
n
¯
| s | π Z ∞
τ
t
n
¯
| s | ) 2 (^) dt
(from (27))
π 1
[ tan
− 1 (
t
n
s | )]
τ ∞
π 1
[
2 π
−
(^) tan
− 1 (
τ
n
¯
s | )]
π 1
tan
− 1 (
τ
n
s | )
ECE642: Detection and Estimation Theory
Under
1 : ch.f. of the Decision Statistic of Correlation Detector in
Cauchy Noise
φ Similarly, from (10), g k , 1
1 {
e iug
k ( Y k ) } = E 1 { e
ius
k Y k (^) }
e ius
k ( s k + N k ) }
(from (17))
e ius
k 2 E^
(^) {
e ius
k N k (^) }
e ius
k 2 φ^ N k (^) ( us
k ) =
e ius
k 2 e^ −|
us
k |
Hence, from (11)
φ T , 1 ( u ) = n
= 1 φ g k ,
1 ( u ) =
n
= 1 e ius
k 2 e^ −|
us
k |
(from (29))
e iu
∑^ k n = 1 (^) s k 2 e^ −|
u | ∑^ k n = 1 (^) | s k |
=
e iun
¯ s 2 e − n ¯
| s || u |
where we have defined,
s 2
n 1
n
= 1 s k 2
20
