Synthesis of a radar recognition algorithm with ability to meet reliability of decisions
Journal of Science and Technique - N.203 (11-2019) - Le Quy Don Technical University
SYNTHESIS OF A RADAR RECOGNITION ALGORITHM WITH
ABILITY TO MEET RELIABILITY OF DECISIONS
Nguyen Hoang Nguyen*
Le Quy Don Technical University
Abstract
This paper is devoted to the variant of synthesis of a radar recognition algorithm with the
ability to meet reliability of decisions. The algorithm is based on the theory of sequential
analysis in combination with flexible change in the level of the classification detail when
the observation time cannot be increased. Compared with one-step algorithms, the proposed
algorithm allows guaranteeing “the posteriori probability of decisions is not smaller than
the set value”. The proposed algorithm can be used in radar target recognition systems.
Keywords: Radar target recognition; sequential analysis; multi-step decision.
1. Introduction
Radar target classes are modeled by a conditional probability density function
p (ξ) p ξ H
(CPDF) of radar portrait (RP)
, l 1L ( : RP; H : the l-class
ξ
l
l
l
existence hypothesis; L : the number of target’s classes). Here, the identification
features are the discrimination of CPDFs on their shapes or parameters and the one-step
decision algorithm [1, 4, 7, 9, 10], is given below:
“If k argmax P p ξ H
then H* H ”.
(1.a )
l
l
k
l1L
P P H ;l 1 L
H* Hk; k 1 L: decision on k-class target existence.
In which,
: the probability of Hl (priori probability);
l
l
If the priori probability is unknown, the distribution is hypothesized to be
1
P ;l 1L
uniform,
and (1.a) becomes:
l
L
“If k arg max p ξ H
then H* H ”.
(1.b)
l
k
l1L
The recognition quality is usually illustrated by a table of conditional decision
probabilities P P H* H ; k,l 1 L . Based on them, other parameters can be
l
k/l
k
evaluated, such as average conditional probabilities; a sum of non-conditional
probabilities (true, false decisions); … [7, 8, 9, 11].
* Email: [email protected]om
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Journal of Science and Technique - N.203 (11-2019) - Le Quy Don Technical University
Additionally, (1.a) is constrained by the criterion of maximum non-conditional
probability; (1.b) is the maximum likelihood criterion which makes the minimum false
conditional decision probability [7-12].
Due to the noise interference and the obscure priori information, the quality of
radar recognition systems is still poor and unacceptable [7, 8] while a false decision can
lead to serious outcomes. Hence, the problem of recognition quality guarantee is
important and necessary.
Some methods that try to improve the information collection and reduce the priori
inconsistence in radars demanding systematic complication and high price [1, 4, 7-9]. In
other words, by results of the limitation of technology those methods only support the
recognition improvement in general but cannot satisfy specific practical conditions.
Another approach that prolongs the observation cycle to gather a mass of information
for classification. Several previous works applied this way, for example, multi-step
decisions based on the sequential analysis theorem, however, their results have been
limited to the detection problem ( L 2 ), and the recognition problem ( L 2 ) needs to
be researched further [2, 3, 5, 6, 10, 13]. Besides, the observation cycle cannot be too
long; thus, that measure is also unfeasible. In that case, a common solution in most
researches is “if the observation cycle has reaches the limited value but no final result
returned by the sequential algorithm, then the unsatisfied quality acceptance are
informed and the regular one-step decision rule is applied”.
On purpose “guarantee reliability of decisions”, this paper focuses on:
- Synthesizing a sequential (multi-step) algorithm of radar recognition and
decision-making solution if the observation cycle is critical.
- Analyzing and evaluating the algorithm quality.
2. Algorithm synthesis
In the radar target recognition, the observation cycle extension is equivalent to the
ξ
increase in the number of contacts with targets. Call m to be RP at the cycle “ m ”
(n)
( m 1,2...), then after the n-cycle, there are a set of “n” RPs
ξ
.
ξ1 ,ξ2 ,,ξn
Generally, due to the large observation cycle compared with the signal fluctuation, the
(n)
RP can be considered independently and the PDF of the set ξ
is given by [7-9, 11]:
n
p(n)
ξ
H
p(1) (ξ )
,
l m
(n)
p ξ
(n)
(2)
l
l
m1
where pl(1) (ξm ) p(ξm Hl ); l 1 L : CPDF of the Hl -class of targets at the
m-observation cycle.
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Journal of Science and Technique - N.203 (11-2019) - Le Quy Don Technical University
The fundamental of the multi-steps decision algorithm is to divide the whole
n
*
(n)
dynamic of ξ
into “ L 1” separated regions [ ξ ]i ;
and make decision
i 0 L
in [2, 6, 13]:
*
H*(n) Hi*(n)
(n)
(n)
“With i 0L , if ξ [ ξ ]i then
”.
(3)
*(n)
k (l)
H *(n)
H
in which
the class
: decision at the n-cycle;
: decision “targets belonging to
; k,l 1 L
Hk (l) H0*(n) H *(n1)
”;
: decision “prolongs the observation cycle”.
Apart from the decision H0*(n) , the reliability of decisions Hk*(n)
P H / H*(n)
;
is the
(4)
k 1 L
posteriori probability
which satisfy the condition:
k
k
P P(n)
P H / H*(n)
P* ;
,
k
k/k
k 1 L
k
k
k
M
P P(n)
l
k/l
l1
where P* ;
: the required reliability of the decision H*(n) ; P(n) P H*(n) / H ;
k 1 L
l
k
k
k/l
k
k,l 1 L : conditional decision probability at the “n”-cycle.
2.1. Multi-steps recognition algorithm
n
Suppose that the set ξ is obtained after “n” observation cycles. At this time,
the posteriori probability of target classes is given by the formula:
P pk(n)
ξ
(n)
k
n
; l 1L .
(5)
P H / ξ
l
L
P p(n)
ξ
(n)
l
l
l1
Thus, the decision selection in Hl*(n) ; l 1 L can be carried out by the posteriori
probability maximum:
arg max P p(n)
ξ
then
H*(n) Hk*(n)
”. (6)
(n)
(n)
“If k argmax P H / ξ
l
l
l
l1L
l1L
Next, check the reliability of the decision if it is satisfactory then this decision is
final and vice versa switch to the adjacent cycle. Hence, the multi-steps decision
algorithm can be described as below:
H *(n) Hk*(n)
H*(n) H0*(n)
P*
(n)
and
(n)
“If k arg max P H / ξ
then
when
P H / ξ
k
l
k
l1L
P*
when
(n)
”. (7)
P H / ξ
k
k
The recognition program of the algorithm (7) comprises L processing channels
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Journal of Science and Technique - N.203 (11-2019) - Le Quy Don Technical University
corresponding to L target classes. A channel is responsible for a target class to calculate
the posteriori probability through the expression (5). The final decision is based on the
maximum of all the channels.
2.2. Decision-making solution in critical cycle
If the observation cycle reaches the critical point in the time extension ( n Nmax
)
while the algorithm (7) has not returned a decision, then the level of the classification
detail
should
be
max
reduced.
Accordingly,
a
decision
*(Nmax
)
)
)
H
H *k(N [Hl*(N ) ; l [l*k(N ) ]]
[l*k( N
max
max
max
“
- targets in a group of classes
” is
]
made instead of seeking a target class based on the maximum of a processing channel
)
)
) . The elements in the group
have to qualify the reliability
]
H*(N Hk*(N
[l*k( N
max
max
max
)
L*k(N
H*k(N
) . Here, the reliability of decision
is
max
max
with the minimum number of classes
a sum of posteriori probabilities:
(Nmax
(Nmax
)
)
.
(8)
P
Hl / ξ
P H / ξ
l
*( N
)
)
l[lk
]
l[l*k( N
]
max
max
Considering the initial reliability of single decisions, we give a condition for the
group decision:
)
(Nmax
P*k(N
P H / ξ
max P*;l [l*(Nmax ) ] P* .
(9)
)
max
l
l
k
k
)
l[l*k( N
]
max
The determination of classes in group decisions [lk*(N ) ] according to (9) is as the
max
following process:
)
k arg max
P
Hl /
ξ(N
(n)
;
max
1)
2)
;
P H / ξ
P*
k
k
l1L
)
)
max
*( Nmax )
H*( N [Hk*( N
]
[lk
max
; Put “k ” into the group
;
]
k
(Nmax
)
3) l argmax P H / ξ
;
i
)
i[l*k( N
]
max
i1L
*( Nmax
[lk
)
4) Add the class “l ” into the group
;
]
( Nmax
max P*;l [l*(Nmax ) ] then jump backward into
)
5) If
P H / ξ
l
l
k
)
l[l*k( N
]
max
the step (3);
6)
)
)
max
H*k( N [Hl*(N ) ; l [l*k(N ]]
max
max
.
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Journal of Science and Technique - N.203 (11-2019) - Le Quy Don Technical University
Hence, the multi-steps recognition algorithm with decision-making solution in
critical cycle has a following form:
(n)
“With k arg max P H / ξ
:
(10)
l
l1M
H *(n) Hk*(n)
P*
If
If
(n)
then
(single decision);
P H / ξ
k
k
H*(n) H*k(N [Hl*(N ) ; l [l*k(N ]]
)
)
max
P H / ξ
P*
(n)
then
when n N
max
max
max
k
k
H*(n) H0*(n)
(group decision)” or
when
n Nmax ”.
*( Nmax
[lk
)
Here,
(n) is calculated by (5) and (2); the group term
is defined by
]
P H / ξ
l
the above 6-steps process.
In order to implement the algorithm (10) in classic radar target recognition
systems with the one-step decision rule (1), some function blocks should be added:
- A processing channel needs a storage of observation cycles and a calculator of
conditional probability density function - formula (2);
- A calculator of posteriori probability for target classes - formula (5);
- A decision-making element - algorithm (10).
3. Algorithm analysis and evaluation
3.1. Quality analysis of single decisions
If omitting the observation time extension, then the single decision making in (10)
(n)
complies the criterion “maximum posteriori probability for the set ξ
”:
arg max P p(n)
ξ
then
H*(n) Hk*(n)
”. (11.a)
(n)
(n)
“If k arg max P H / ξ
l
l
l
l1L
l1L
Therefore, this is optimized through “minimum sum of non-conditional
probability”:
L
L
L
F(n)
P H , H*(n)
P H P H*(n) / Hl
l
P P(n) min .
VDK
l
k
k
l
k/l
lk
lk
lk
k,l1
k,l1
k,l1
P(n) P H*(n) / Hl
where
: decision probability of target class “k ” at the observation
k/l
k
cycle “n” in the existence of target class “l ”.
If “n 1” then (11.a) is not different from the one-step algorithm along with the
criterion (1.a). Obviously, if “n” increases then the sum of false probabilities F(n)
VDK
decreases. The reason is that the amount of information for the recognition is
(n)
accumulated in observation cycles (the algorithm employs CPDF of all the sets ξ
).
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Journal of Science and Technique - N.203 (11-2019) - Le Quy Don Technical University
1
P ;l 1L
When the priori probability is obscure, set
and (11.a) becomes:
H*(n) Hk*(n)
then ”. (11.b)
l
L
n
k arg max p(n)
ξ
argmax
p(1) (ξ )
(n)
“If
l
l
m
l1L
l1L
m1
This is the criterion “maximum likelihood function”; the optimization means
“minimum average of false conditional recognition probability”:
L
L
1
1
F(n)
P H*(n) / Hl
P(n) min .
L
L
TB
k
k/l
lk
lk
k,l1
k,l1
If “ n 1” then (11.b) is not different from (1.b). Apparently, if “n ” increases then
F(n)
the sum of false probabilities
decreases. Here, the information accumulation over
TB
observation cycles is explained as follows:
Suppose that at the first cycle ( n 1), priori probabilities are equivalently
1
P H
P(1) ; i 1 L
considered:
, in the next cycles ( n 2 ) the priori
l
l
n1
L
probabilities are taken to be the posteriori probabilities in the previous cycles
P H P(n) P H /ξ
; therefore, we have a formula:
l
n1
l
l
n
n
p(1) (ξ )
P(n) pl(1)
ξ
pi(1)
ξ
m1
L
l
m
n
n
m
l
P H / ξ
n
.
n
l
L
L
P(n) p(1)
ξ
P(m) p(1)
ξ
P(m) p(1)
ξ
m1
n
m
m
c
c
c
c
c
c
c1
c1
m1 c1
Thus, the algorithm (11.b) is equivalent to:
“If k argmax P H /ξ argmax P(n) p(1)
ξ
n
then
H*(n) Hk*(n)
”. (12)
l
n
l
l
l1L
l1L
It can be seen that (12) is the one-step algorithm with the criterion “maximum
posteriori probability - for ξn received at the decision moment”. Here, the information
P H / ξ
acquisition at the previous cycles is presented by the posteriori probability
n1
l
and applied for the priori probabilities P(n) for the decision-making in the next cycle.
l
3.2. Evaluation of required reliability
3.2.1. For single decisions
H *(n) Hk*(n)
Assume that at the n-cycle, the system makes the decision
;
(n)
ξ
(n) *
k 1M . According to the algorithm (2), this is equivalent to
. If that,
ξ
k
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Journal of Science and Technique - N.203 (11-2019) - Le Quy Don Technical University
from (10), we have:
n
P pk(n)
ξ
L
k
n
n
n
P*
P pk(n)
ξ
P* P p(n)
ξ
;
. (13)
and
P H / ξ
k
k
l
l
k
k
L
P p(n)
ξ
n
l1
l
l
l1
*
k
(n)
Take the integration for the two sides in the range
[ ξ
]
pl(n)
ξ
d ξ
P H*(n) / Hl P(n)
(n)
(n)
substitute
into that, we have:
k /l
k
( n) *
ξ
k
P P(n)
P H / H*(n)
P* .
(14)
k
k/k
k
k
k
M
P P(n)
l
k/l
l1
Thus, the condition (4) is always satisfied with each decision Hk*(n) ; k 1M .
3.2.2. For group decisions
n N
Suppose that at the cycle
max , the system makes a group decision
*
) . Set
to be a decision region H*k(N ) , according to (10),
)
H*(N H*k(N
(Nmax
)
max
max
max
ξ
k
along with the process of class determination in a group decision we have:
*
(Nmax
( Nmax
)
ξ
)
If ξ
then:
k
)
(Nmax
P pl(N
L
ξ
)
max
l
( Nmax
)
max
P H / ξ
max P*;l [l*(N ) ] P*
l
l
k
k
)
)
l[l*k( N
]
ll*k( N
]
max
max
)
)
max
max
P p( N
m m
ξ(N
m1
L
)
(Nmax
)
)
max
P pl(N
ξ
P*
P p(N
ξ(N
.
(15)
)
max
max
k
l
m
m
)
l[l*k(N
]
max
m1
*
(Nmax
)
Take the integration of the two sides in a range
, transform the
ξ
k
*
to be H*k(N ) , we have:
(Nmax
(Nmax
)
ξ
)
max
expression and replace ξ
k
)
*(Nmax
)
)
P H / H*(N
P
Hl / Hk
P* max P*;l [l*(N
]
max
max
l
l
k
k
k
)
*( N
)
l[l*k(N
]
l[lk
]
max
max
Thus, the reliability of a group decision is not smaller than the required value of
the posteriori probability all over the classes in the group.
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Journal of Science and Technique - N.203 (11-2019) - Le Quy Don Technical University
4. Conclusion
In this paper, a radar recognition algorithm with the ability to meet the reliability
of decisions has been synthesized. The analysis results show that:
- Compared with conventional multi-step algorithms, the synthesized algorithm
allows group decision-making to ensure the reliability of decisions when the
observation time cannot prolong. However, if the observation time is not limited, then
the conventional multi-step algorithm is capable of quality assurance without reducing
the level of the classification detail.
- Compared with one-step algorithms, the synthesized algorithm not only raises
the recognition quality in general but also allows guaranteeing the reliability of
decisions. The solution for this is the observation time extension or detailed
classification reduction.
- The algorithm is applicable for classic recognition systems by adding a small
number of hardware resources.
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Наука.
TỔNG HỢP THUẬT TOÁN NHẬN DẠNG MỤC TIÊU RA ĐA
VỚI KHẢ NĂNG ĐÁP ỨNG ĐỘ TIN CẬY CỦA CÁC QUYẾT ĐỊNH
Tóm tắt: Bài báo trình bày phương án tổng hợp thuật toán nhận dạng mục tiêu ra đa với
khả năng đáp ứng độ tin cậy của các quyết định. Thuật toán được xây dựng trên cơ sở lý thuyết
phân tích lần lượt kết hợp với việc thay đổi mức chi tiết phân lớp một cách linh hoạt khi không
thể kéo dài thời gian quan sát. So với các thuật toán nhận dạng một bước đã có, thuật toán đề
xuất cho phép đảm bảo xác suất hậu nghiệm của các quyết định đưa ra luôn lớn hơn giá trị cho
trước. Thuật toán này có thể áp dụng trong những hệ thống nhận dạng mục tiêu ra đa.
Từ khóa: Nhận dạng mục tiêu ra đa; phân tích lần lượt; ra quyết định nhiều bước.
Received: 20/5/2019; Revised: 21/11/2019; Accepted for publication: 22/11/2019
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