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Analysis Study of Radar Probability of Detection for Fluctuating and Non-fluctuating Targets

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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.
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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
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
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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
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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)
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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 + ,21
(10)
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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.
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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
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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
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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)
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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.
References
[1] M. A. Richards, Fundamentals of Radar Signal Processing, New York: McGraw- Hill, 2005, pp. 1–22.
[2] E. Fishler, A. Haimovich, R. Blum, et al, "Spatial diversity in radars-models and detection performance",
IEEE Transactions on Signal Processing, vol. 54, no 3, pp. 823-838, 2006.
[3] B.R. Mahafza, Radar Systems Analysis and Design Using MATLAB, Chapman & Hall/CRC 2000.
[4] J. I. Marcum, “A statistical theory of target detection by pulsed radar,” IRE Transactions on Information
Theory, vol. 6, no.2, pp.59-267, 1960.
[5] J.H. Lee, and H.T. Kim, “Radar target recognition based on late time representation: Closed form
expression for criterion," IEEE Transactions on Antennas and Propagation, vol. 54, no. 9, pp. 2455-2462,
2006
[6] P.Swerling, “Detection of fluctuating pulsed signals in the presence of noise,” IRE Transactions on
Information Theory, vol. 3, no.3, pp.175-178, 1957.
[7] P. Swerling, “Probability of detection for fluctuating targets,” IRE Transactions on Information Theory, vol.
6, no. 2, pp.269-308. 1960.
[8] M. I. Skolnik, Introduction to radar systems, 4th Edition, McGraw-Hill, New York, 2004.
[9] T. S. Ferguson, Mathematical Statistics: A Decision Theoretic Approach, New York: Academic Press,
1967.
00.2 0.4 0.6 0.8 1
0
50
100
-5
0
5
10
15
PD
Non-fluctuating
N
S NR (d B)
00.2 0.4 0.6 0.8 1
0
50
100
-5
0
5
10
15
PD
Albersheim
N
S NR (d B)
ALGERIAN JOURNAL OF SIGNALS AND SYSTEMS (AJSS)
Vol.2, Issue 1, April-2017| ISSN-2543-3792 20
[10] E. L. Lehmann, Testing Statistical Hypotheses, New York: Wiley, 1959.
[11] O.Jean-Philippe, and J. Emmanuelle, “New methods of radar detection performances analysis,” In
Acoustics, Speech, and Signal Processing, Proceedings, vol. 3, pp. 1181-1184, IEEE conference,
March1999.
[12] A.Farina, and A. Russo, “Radar detection of correlated targets in clutter,” IEEE transactions on aerospace
and electronic systems (5), pp.513-532, 1986.
[13] M.A.Richards, J.A.Scheer, and W.A. Holm, Principles of modern radar, SciTech Pub , 2010.
[14] W.J. Alberhseim, “A closed-form approximation to Robertson's detection characteristics,” Proceedings of
the IEEE, vol. 69, no. 7, pp.839-839, 1981.
[15] M. A. Richards, Fundamentals of Radar Signal Processing, second edition, McGraw-Hill, 2014.
... Note that the choice of Alberhseim's model is not unique, more complex models of fluctuating targets can be used [16]. However, at large integration times, fast fluctuating target models, such as Swerling II and IV, agree with a very low error with the Albersheim's model [17]. These models allow to precisely estimate the effect of incoherent integration, which is not possible using SNRs. ...
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Stealth targets have extremely small Radar Cross Section (RCS) and fluctuating echoes. For these kinds of fluctuating targets, the detection performance of non-coherent detection is more and more difficult to meet the requirements. The coherent detection with better performance than non-coherent detection, can detect slow fluctuation target, but can't detect fast fluctuation target. In view of the insufficient performance of radar, based on the previous work, this paper applies the coherent demodulation technology in communication systems to radar detection for fluctuating targets. The theoretical expressions of fluctuating targets coherent demodulation detection (FTCDD) method are derived, and the detection probability of Swerling targets is calculated. The correctness of the theoretical derivation is verified by simulation, and the radar detection performance is analyzed under different false alarm probability and Barker code lengths. Proposed method has better performance than non-coherent detection and can meet all fluctuation models.
Chapter
The paper concerns research on the efficiency of the communication system and MIMO radar cooperating with each other. For this system a measure of efficiency was determined, which is the speed of radar estimation and the speed of data transfer. The combination of these measures made it possible to set a performance limit for this system. The original the solution is to optimize the efficiency of the communication system cooperating with MIMO radar. Using derivative-free optimization method has been maximized radar estimation speed and information transfer speed. This makes it possible to find optimal parameters for both systems, which is helpful in its design or analysis. Simulation tests confirmed the correctness of the analysis.
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Original methods for radar detection performance analysis are derived for a fluctuating or non-fluctuating target embedded in additive and a priori unknown noise. This kind of noise can be, for example, the sea or ground clutter encountered in surface-based radar for the detection of low grazing angle targets and/or in high-resolution radar. In these cases, the spiky clutter tends to have a statistic which strongly differs from the Gaussian assumption. Therefore, the detection theory is no longer appropriate since the nature of statistics has to be known. The new methods proposed here are based on the parametric modelling of the moment generating function of the noise envelope by Padi approximation, and lead to a powerful estimation of its probability density function. They allow to evaluate the radar detection performances of targets embedded in arbitrary noise without knowledge of the closed form of its statistic and in the same way to take into account any possible fluctuation of the target. These methods have been tested successfully on synthetic signals and used on experimental signals such as ground clutter.
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In the previous paper, we have defined the normalized estimation error and presented the radar target recognition scheme in frequency domain. A time-domain version of the previous paper is considered and a closed-form expression for the defined normalized estimation error in terms of Z-plane natural frequencies and transient response is derived. Evaluation of the closed-form expression of the normalized estimation error is shown in the numerical results, where it is shown that the performance of the radar target recognition scheme improves with an increase of the number of the natural frequencies and with an increase of signal-to-noise ratio. Furthermore, the mean and the variance of the square of the numerator of the normalized estimation error are obtained, and the validity of the statistics is shown using the numerical results
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Principles of Modern Radar: Basic Principles is a comprehensive and modern textbook for courses in radar systems and technology at the college senior and graduate student level; a professional training textbook for formal in-house courses for new hires; a reference for ongoing study following a radar short course; and a self-study and professional reference book. Principles of Modern Radar focuses on four key areas: Basic concepts, such as the the radar range equation and threshold detection; radar signal phenomenology, such as radar cross section models, clutter, atmospheric effects, and Doppler effects; descriptions of all major subsystems of modern radars, such as the antenna, transmitter, receiver, including modern architectural elements such as exciters, and advanced signal processors; and signal and data processing basics, from digital signal processing (DSP) fundamentals, through detection, Doppler processing, waveforms and pulse compression, basic imaging concepts, and tracking fundamentals. While several established books address introductory radar systems, Principles of Modern Radar differs from these in its breadth of coverage, its emphasis on current methods (without losing sight of bedrock principles), and its adoption of an appropriate level of quantitative rigor for the intended audience of students and new professional hires. The manuscript for this book was reviewed by over 50 professionals in academia, military, and commercial enterprises. These reviewers were among thousands of potential users approached by the publisher and asked to share their expertise and experience in radar training and instruction. Their extensive comments, corrections, and insights ensure that Principles of Modern Radar will meet the needs of modern radar educators and students around the world. Written and edited by world-renowned radar instructors and critically reviewed by users before publication, this is truly a radar community-driven book. © 2010 by SciTech Publishing, Raleigh, NC. All rights reserved.
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From the Publisher: Written for researchers and engineers working in the field, Radar Systems Analysis and Design Using MATLAB provides fundamental radar principles, advanced topics, and rigorous but easy-to-follow mathematical derivations.
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This report considers the probability of detection off a target by a pulsed search radar, when the target has a fluctuating cross section. Formulas for detection probability are derived, and curves off detection probability vs, range are given, for four different target fluctuation models. The investigation shows that, for these fluctuation models, the probability of detection for a fluctuating target is less than that for a non-fluctuating target if the range is sufficiently short, and is greater if the range is sufficiently long. The amount by which the fluctuating and non-fluctuating cases differ depends on the rapidity of fluctuation and on the statistical distribution of the fluctuations. Figure 18, p. 307, shows a comparison between the non-fluctuating case and the four fluctuating cases considered.
Article
This report presents data from which one may obtain the probability that a pulsed-type radar system will detect a given target at any range. This is in contrast to the usual method of obtaining radar range as a single number, which can be taken mathematically to imply that the probability of detection is zero at any range greater than this number, and certainty at any range less than this number. Three variables, which have so far received little quantitative attention in the subject of radar range, are introduced in the theory: l.The time taken to detect the target. 2.The average time interval between false alarms (false indications of targets). 3.The number of pulses integrated. It is shown briefly how the results for pulsed-type systems may be applied in the case of continuous-wave systems. Those concerned with systems analysis problems including radar performance may profitably use this work as one link in a chain involving several probabilities. For instance, the probability that a given aircraft will be detected at least once while flying any given path through a specified model radar network may be calculated using the data presented here as a basis, provided that additional probability data on such things as outage time etc., are available. The theory developed here does not take account of interference such as clutter or man-made static, but assumes only random noise at the receiver input. Also, an ideal type of electronic integrator and detector are assumed. Thus the results are the best that can be obtained under ideal conditions. It is not too difficult, however, to make reasonable assumptions which will permit application of the results to the currently available types of radar. The first part of this report is a restatement of known radar fundamentals and supplies continuity and background for what follows. The mathematical part of the theory is not contained herein, but will be issued subsequently as a Separate report(2a)
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This paper treats the detection of pulsed signals in the presence of receiver noise for the case of randomly fluctuating signal strength. The system considered consists of a predetection stage, a square law envelope detector, and a linear postdetection integrator. The main problem is the calculation of the probability density function of the output of the postdetection integrator. The analysis is carried out for a large family of probability density functions of the signal fluctuations and for very general types of correlation properties of the signal fluctuations. The effects of nonuniform beam shape and of nonuniform weighting of pulses by the postdetection integrator are also taken into account. The function which is actually evaluated is the Laplace transform of the probability density function of the integrator output. In many of the cases treated, the resulting Laplace transform has an inverse of known form. In such cases the evaluation of the probability density function would require the computation of a finite number of constants; in practice this would usually require the use of computing machinery, but would be perfectly feasible with presently available computing machinery.
Article
Inspired by recent advances in multiple-input multiple-output (MIMO) communications, this proposal introduces the statistical MIMO radar concept. To the authors' knowledge, this is the first time that the statistical MIMO is being proposed for radar. The fundamental difference between statistical MIMO and other radar array systems is that the latter seek to maximize the coherent processing gain, while statistical MIMO radar capitalizes on the diversity of target scattering to improve radar performance. Coherent processing is made possible by highly correlated signals at the receiver array, whereas in statistical MIMO radar, the signals received by the array elements are uncorrelated. Radar targets generally consist of many small elemental scatterers that are fused by the radar waveform and the processing at the receiver, to result in echoes with fluctuating amplitude and phase. It is well known that in conventional radar, slow fluctuations of the target radar cross section (RCS) result in target fades that degrade radar performance. By spacing the antenna elements at the transmitter and at the receiver such that the target angular spread is manifested, the MIMO radar can exploit the spatial diversity of target scatterers opening the way to a variety of new techniques that can improve radar performance. This paper focuses on the application of the target spatial diversity to improve detection performance. The optimal detector in the Neyman-Pearson sense is developed and analyzed for the statistical MIMO radar. It is shown that the optimal detector consists of noncoherent processing of the receiver sensors' outputs and that for cases of practical interest, detection performance is superior to that obtained through coherent processing. An optimal detector invariant to the signal and noise levels is also developed and analyzed. In this case as well, statistical MIMO radar provides great improvements over other types of array radars.