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People’s Democratic Republic of Algeria
Ministry of Higher Education and Scientific research
M’hamed Bougara University, Boumerdes
Institute of Electrical and Electronic Engineering,
Laboratory of Signals and Systems (LSS)
Volume : 2 Issue : 1 (April 2017)
Special Issue of the International Conference on
Technological Advances in Electrical Engineering
Skikda, Algeria, 24-26 October 2016
Laboratory of Signals and Systems
Address : IGEE (Ex-INELEC), Boumerdes University, Avenue de l’indépendance, 35000,
Boumerdes, Algeria
Phone/Fax : 024 79 57 66
Email : lss@univ-boumerdes.dz ; ajsyssig@gmail.com
©LSS/2017
ISSN
:
2
543
-
37
92
Title: Analysis Study of Radar Probability of Detection for Fluctuating and Non-
fluctuating Targets
Authors: Naceur AOUNALLAH(1)*, Ali KHALFA(2)
Affiliation:
(1) Department of electronic and telecommunications. Kasdi Merbah University
(UKMO), Ouargla 30000, Algeria
(2)Department of electronic. Mohamed Boudiaf University, M’sila 28000, Algeria
Page range: 12- 20
IMPORTANT NOTICE
This article is a publication of the Algerian journal of Signals and Systems and is protected by the
copyright agreement signed by the authors prior to its publication. This copy is sent to the author
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ALGERIAN JOURNAL OF SIGNALS AND SYSTEMS (AJSS)
Vol.2, Issue 1, April-2017| ISSN-2543-3792 12
Analysis Study of Radar Probability of Detection
for Fluctuating and Non-fluctuating Targets
Naceur AOUNALLAH(1)*, Ali KHALFA(2)
(1) Department of electronic and telecommunications. Kasdi Merbah University (UKMO), Ouargla
30000, Algeria
(2) Department of electronic. Mohamed Boudiaf University, M’sila 28000, Algeria
* aounallah.na@univ-ouargla.dz
Abstract: The radar analyst can develop and use mathematical and statistical techniques that lead to
accurate prediction or adapting models for estimating the target detection performance. In radar detection
theory, detection probability, false alarm probability, number of samples non-coherently integrated for a
detection test, and signal-to-noise ratio (SNR) are closely interrelated. The present paper is intended to
provide an overview of the calculations of radar probability of detection and its related parameters. The main
methods and procedures for predicting the detection performance of either non-fluctuating or fluctuating
targets are described. Performance’s analysis of the studied models is included, along with some graphical
simulation examples.
Keywords: Radar signal processing, radar detection, probability of detection, probability of false alarm,
Swerling target model.
1. INTRODUCTION
The first applications of radio were telecommunications and radio-navigation, but by the early 20th
century, precursors considered the ability to detect the presence of metal objects through the use
of electromagnetic waves. The word RADAR itself, which is now universally adopted to designate a
material satisfying these requirements for detecting, locating, and identifying reflecting objects over
long distances. The term RADAR was coined in 1941 as an acronym for Radio Detection and
Ranging. This acronym of American origin replaced the previously used British abbreviation RDF
(Radio Direction Finding). However, due to its wide use, the word has become a standard noun in
English, and almost all people have had an experience with radar [1-2].
In general, radar systems use modulated waveforms and directive antennas to transmit
electromagnetic energy into a specific volume in space to search for targets. Objects (targets)
within a search volume will reflect portions of this energy (radar returns or echoes) back to the
radar. These echoes are then processed by the radar receiver to extract target information such as
range, velocity, angular position, and other target identifying characteristics [3].
Basic Radar functions can be classified as detection, tracking, or imaging and the most
fundamental problem in radar is detection of an object or physical phenomenon. Nevertheless, one
can’t never determine the range or estimate the speed without detecting the target, and it is
necessary to distinguish the signal reflected from the target, from the signal containing only noise.
The target detection problem in a radar system is naturally a statistical problem that random
fluctuations, due to noise located at the receiver output, corrupt the target signal and influence the
detector performance. The receiver detection of a target refers then to a decision made by the
radar detector concerning presence or absence of the useful signal in the presence of additive
noise.
The first research in detection theory that has been used in radar signal was made by Marcum [4-
5]. He considered the detection of a completely known signal in white Gaussian noise using
multiple received samples. This research was continued by Swerling [6-8], and since then, radar
detection has been consistently developing. The fundamental theory behind detection and
classification was developed in mathematical statistics and decision theory, and signal detection is
a special case of hypothesis testing theory in statistical inference [9-10].
The organization of this paper is as follows: section 2 describes briefly the radar system and its
operational blocs and characteristics. Section 3 formulates the problem under consideration and
ALGERIAN JOURNAL OF SIGNALS AND SYSTEMS (AJSS)
Vol.2, Issue 1, April-2017| ISSN-2543-3792 13
presents the detection techniques. Section 4 deals with the numerical results and brief discussion
analysis, while section 5 contains our conclusions.
2. RADAR SYSTEM
Radar is an electromagnetic system that detects, locates, and recognizes target objects. It
transmits electromagnetic signal and then receives echoes from target objects to get their location
or other information. The received signal is frequently accompanied by noise and clutter. The
disturbances may cause serious performance issues with radar systems by concluding these
signals as targets. The basic parts of a radar system are illustrated in the simple block diagram of
fig.1. Radar equipment consists of a transmitter, an antenna, a receiver, and a signal processor.
Radar transmitters and receivers are usually located in the same place. Each block of radar system
has a specific operation:
The waveform generator is a unit which purpose is to create and control the waveform to
be modulated and transmitted by the transmitter.
The transmitter, for its side, produces powerful pulses and/or waveforms of
electromagnetic energy at precise time intervals and sends them to the antenna system.
The main role of the antenna is to provide a transducer between the free-space
propagation and the guided-wave propagation. The antenna system includes a transmitting
function and a receiving function. During transmission is to concentrate the radiated energy
into a shaped directive beam which illuminates the targets in a desired direction. During
reception the antenna collects the energy contained in the reflected target echo signals
and delivers it to the receiver.
The target reflected energy is received by the receiver from the antenna system. Then, the
receiver performs amplification, filtering, and demodulation on the received signal.
The computer/signal processor performs complex mathematical computations on the
demodulated signal to extract target velocity and/or range information.
The timing-and-control block affords timing information to synchronize various signals and
to control the operation of other radar components.
Fig. 1 Basic block diagram of a radar system
3. RADAR DETECTION PROBABILITIES
The purpose of this section is to illustrate the mathematical model that determines the generated
radar signal and to show mathematically how the detection problem can be described in terms of
two decision hypothesis (detection, or false alarm). Then, it gives different methods of radar
detection which used either for a fluctuating or non-fluctuating targets.
We consider the basic problem of detecting the presence or absence of a complex signal s(t) with
envelope A in a set of measurements y(t) = y(t) + i y(t) corrupted by a sum of independent
additive complex noises corresponding to the clutter echoes c(t) and white Gaussian thermal noise
n(t). Mathematically, we describe this problem in terms of a hypothesis test between the following
pair of statistical hypothesis [11]: H y
(
t
)=n
(
t
)+c(
t)
(1)
H y
(
t
)=s
(
t
)+n
(
t)+c
(
t
)
(2)
ALGERIAN JOURNAL OF SIGNALS AND SYSTEMS (AJSS)
Vol.2, Issue 1, April-2017| ISSN-2543-3792 14
If we note pH(r) the probability density of the noise envelope |n(t) + c(t)|, the detection threshold
T is fixed by the value of the given probability of false alarm P.
P=pH(r)dr
(3)
While, denoting pH(r) the probability-density function (PDF) of the envelope of the complex signal
embedded in noise |s(t) + n(t) + c(t)|, the detection probability P is classically given by :
P=pH(r)dr
(4)
Generally, target signal is modeled either as a random variable in the single pulse case or as a
very simple stochastic process in the pulse train case [12]. In the latter case only completely
correlated or completely uncorrelated pulse-to-pulse fluctuations can be considered. So, target
signals can be classified into fluctuating target models and non-fluctuating target models.
Target fluctuation models
Swerling extended Marcum’s works which are methods for predicting the detection performance of
non-fluctuating targets to incorporate what has become known as the four Swerling models which
concerned the fluctuating targets [13].
Detection of Swerling I
The Swerling I model signifies fluctuating amplitude, constant within a scan, but uncorrelated from
scan to scan. The probability of detection for this type targets was derived by Swerling as the
following formula: () = 1 (5)
=1
(
2)+1+ 1
.
1 + 1
2 () > 1
(6)
Where is the threshold, is the number of integrated pulses, and represent is the incomplete
gamma function defined as: ()=
!
(7)
Detection of Swerling II
The formula for the probability of detection for Swerling II type targets is given by:
=1
1+ ; 50 (8)
In this model, the PDF is as for Swerling I case, but the fluctuations are more rapid and are taken
to be independent from pulse to pulse.
Detection of Swerling III
In the case of Swerling III targets, the probability of detection is given by the following expression:
=1+ 2
1+
1 + 2
2(2
)
(
)
(9)
The fluctuations are independent from scan to scan as in case Swerling I.
Detection of Swerling IV
The probability of detection for Swerling IV model is given by the following expression:
=1+
+2
!
!()
!
2
.
2 + ,21
(10)
ALGERIAN JOURNAL OF SIGNALS AND SYSTEMS (AJSS)
Vol.2, Issue 1, April-2017| ISSN-2543-3792 15
The PDF is as for case Swerling III, but the fluctuations are independent from pulse to pulse.
Target non-fluctuation models
In this class of non-fluctuation we have two main models
Detection of Swerling V
A common, fifth, target type is a constant RCS (Radar Cross Section) target. This is termed a
Swerling 0 target by some and a Swerling V target by others. It is the simple case implies constant
amplitude or no fluctuation.
(10())
2
(11)
is the single pulse:
=1
2+
4 .
4 +1+2
16
(12)
Where ()=
is one form of the error function.
) is the modified Bessel function of the first kind and order .
Detection of Albersheim
Walter Albersheim [14] derived a simple formula for signal to noise ratio, which is required to
achieve a given level of performance for envelope detection of non-fluctuating signal, in narrow
band noise. The Albersheim equation is given as:
+6.2+ 4.54
+ 0.44(+ 0.12+ 1.7)
(13)
Where 0.62
, (()
), and means the false alarm probability.
However, it can be rearranged to obtain a solution for detection probability (P) in terms of the other
parameters using the following sequence of calculations [15]:
=
, and =
, then the Albersheim detection probability is:
=1
1+ (14)
4. NUMERICAL SIMULATIONS
The purpose of this section is to investigate the performances of radar system which are, mainly,
based on the probability of detection curves. We have used the Matlab software to program and
develop the different types of targets which are theoretically described and studied in the above
section. We can show firstly the plots of detection probabilities versus signal-to-noise ratio for given
false alarm probability for all the six target cases, then the plots of the probability of detection value
versus the signal to noise ratio for several values of probability of false alarm are depicted, and
finally, 3D representations are given in order to illustrate the variation of the SNR as function of
detection probability and number of integrated samples for a fixed value of false alarm probability.
ALGERIAN JOURNAL OF SIGNALS AND SYSTEMS (AJSS)
Vol.2, Issue 1, April-2017| ISSN-2543-3792 16
Fig. 2 Probability of detection as a function of SNR for different target models when N = 10, and Pfa = 10-6.
The results illustrated in fig.2 and fig.3 lead us to make the following observation; for high detection
probabilities, more signal-to-noise is required for all target models.
The fig.2 and fig.3 show also a comparison between the six detection probabilities with 10 and 100
pulses integrated respectively. For all these models, greater number of pulses integrated requires
low signal-to-noise ratio to yield a good probability of detection. For low number of pulses
integrated, on the other hand, the reverse of this relationship is correct. In addition, the comparison
between fig.2 and fig.3 indicate also that when the number of integrated pulses is larger, the more
likely it will be for the fluctuations to average out, and the curves of Swerling II and Swerling IV will
approach to the constant target case which represents non-fluctuating (Swerling V) and Albersheim
models.
Fig. 3 Probability of detection as a function of SNR for different target models when N = 100, and Pfa = 10-6.
-10 -5 0 5 10 15 20 25 30
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Signal to Noise Ratio (SNR (dB))
Probability of Detection (P.
D)
Comparison between detection probabilities
Nonfluctuating
Swerling I
Swerling II
Swerling III
Swerling IV
Albersheim
-10 -5 0 5 10 15 20 25 30
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Signal to Noise Ratio (SNR (dB))
Probability of Detection (P.
D)
Comparison between detection probabilities
Nonfluctuating
Swerling I
Swerling II
Swerling III
Swerling IV
Albersheim
ALGERIAN JOURNAL OF SIGNALS AND SYSTEMS (AJSS)
Vol.2, Issue 1, April-2017| ISSN-2543-3792 17
Fig. 4 Probability of detection for Swerling I versus SNR, for various values of Pfa when N=50.
Fig.4 shows plots for the probability of detection value of Swerling I as scan-to-scan fluctuation
versus the signal to noise ratio for several values of probability of false alarm. In other part, fig.5
shows plots for the probability of detection value of Swerling IV as pulse-to-pulse fluctuation versus
the signal to noise ratio for several values of probability of false alarm.
Fig. 5 Probability of detection for Swerling IV versus SNR, for various values of Pfa when N=50.
The performance comparisons showed through these figures allows us to say, for a given SNR, the
detection probability is inversely proportional to the value of the false alarm probability. This
statement is true either for scan-to-scan detectability (i.e. Swerling I) or pulse-to-pulse detectability
(i.e. Swerling IV).
The following figures show an example of the variation of the required SNR as a function of
probability of detection (Pd), probability of false alarm (Pfa), and number of integrated pulses (N)
for various target models (the four Swerling fluctuation models, the non-fluctuation model, and
Albersheim’s model) . In this example, the value of Pfa is fixed to 10-6 and Pd and N vary.
-10 -5 0 5 10 15 20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Signal to Noise Ratio (SNR (dB))
Probability of Detection (P.
D)
Swerling I
Pfa=1e-06
Pfa=1e-08
Pfa=1e-10
Pfa=1e-12
-10 -5 0 5 10 15 20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Signal to Noise Ratio (SNR (dB))
Probability of Detection (P.
D)
Swerling IV
Pfa=1e-06
Pfa=1e-08
Pfa=1e-10
Pfa=1e-12
ALGERIAN JOURNAL OF SIGNALS AND SYSTEMS (AJSS)
Vol.2, Issue 1, April-2017| ISSN-2543-3792 18
Fig. 6 Signal to noise ratio as a function of probability
of detection and number of integrated pulses for
Swerling I targets.
Fig. 7 Signal to noise ratio as a function of probability
of detection and number of integrated pulses for
Swerling II targets.
By visualizing fig.6 and fig.8, we can not notice a great difference because Swerling I and Swerling
III are both scan-to-scan fluctuating targets. The comparison between the figure of Swerling II and
that of Swerling IV allows us to deduce the similarity because Swerling II Swerling and IV are of
unique type which is pulse-to-pulse fluctuating targets. We cannot almost distinguish the difference
between the representations of fig.10 and fig.11 because both are from the same class of targets.
Fig. 8 Signal to noise ratio as a function of probability
of detection and number of integrated pulses for
Swerling III targets.
Fig. 9 Signal to noise ratio as a function of probability
of detection and number of integrated pulses for
Swerling IV targets.
00.2 0.4 0.6 0.8 1
0
50
100
-10
0
10
20
30
40
PD
Swerling I
N
S NR (dB )
00.2 0.4 0.6 0.8 1
0
50
100
-10
0
10
20
30
40
PD
Swerling II
N
SNR (dB)
00.2 0.4 0.6 0.8 1
0
50
100
-10
0
10
20
30
PD
Swerling III
N
S NR (d B)
00.2 0.4 0.6 0.8 1
0
50
100
-5
0
5
10
15
20
25
PD
Swerling IV
N
SN R (dB)
ALGERIAN JOURNAL OF SIGNALS AND SYSTEMS (AJSS)
Vol.2, Issue 1, April-2017| ISSN-2543-3792 19
Fig. 10 Signal to noise ratio as a function of
probability of detection and number of integrated
pulses for non-fluctuating (Swerling V) targets.
Fig. 11 Signal to noise ratio as a function of
probability of detection and number of integrated
pulses for Albersheim targets.
Finally, the following general remarks can be drawn through our simulation’s analytical work:
1. For detection probability upper than 50%, non-uctuating targets (Swerling V and
Albersheim) are easier to detect than any uctuating targets which can make detection
more difficult by requiring a higher SNR for example.
2. The way in detection of the fluctuating targets is not the same. In fact, pulse-to-pulse
uctuations (Swerling II and Swerling IV) are easier to detect than scan-to-scan
uctuations (Swerling I and Swerling III) for detection probabilities higher than 50%.
3. The converse of the above two remarks is occurred for detection probabilities less than
50%.
5. CONCLUSION
A radar tracks a target, measures its range and velocity, and sometimes can identify it, only
because there is, before all, a detection of an echo signal carried out by the system radar itself.
The radar detection probabilities can be calculated by Swerling for four different fluctuation models
of cross section. In two of the four cases (Swerling.I and Swerling.III), it is assumed that the
fluctuations are completely correlated during a particular scan but are completely uncorrelated from
scan to scan. In the other two cases (Swerling.II and Swerling.IV), the fluctuations are assumed to
be more rapid and uncorrelated pulse to pulse. The models presented and described in this paper
are simple algorithms which give quick solutions for the radar detection of targets. They are also
important due to their fast computation speed and simple solution performance.
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Vol.2, Issue 1, April-2017| ISSN-2543-3792 20
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