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Analytical Performance Evaluation of Adaptive Detection
of Fluctuating Radar Targets
Mohamed B. El Mashade
Al-Azhar University, Cairo, Egypt
Received in final form June 14, 2011
Abstract—A radar target whose return varies up and down in amplitude as a function of time represents
the basis of a large number of real targets. This paper is intended to provide a complete analysis of CFAR
detection of fluctuating targets when the radar receiver post-detection integrates Mreturned pulses from
c2fluctuating targets with two and four degrees of freedom and operates in a non-ideal environment.
Owing to the importance of Swerling models in representing a large number of such type of radar targets,
we are interested here in adaptive detection of this class of fluctuation models. Swerling cases I and III
represent scan-to-scan fluctuating targets, while cases II and IV represent fast pulse-to-pulse fluctuation.
Exact expressions of detection probability are derived for all of these models. A simple and an effective
procedure for calculating the detection performance of both fixed-threshold and adaptive-threshold
algorithms is obtained. In the CFAR case, the estimation of the noise power levels from the leading and
the trailing reference windows is based on the CA technique. The performance of this detector is analyzed
in the cases when the operating environment is ideal and when it includes some of spurious targets along
with the target of interest. The primary and the secondary interfering targets are assumed to be fluctuating
in accordance with the four Swerling’s models cited above. The numerical results show that for strength
target return the processor detection performance is highest in the case of SWIV model while it attains its
minimum level of detection in the case of SWI model. Moreover, SWII model has higher performance
than the SWIII representation of fluctuating targets. For weak target return, on the other hand, the reverse
of this behavior is occurred. This observation is common for both fixed-threshold or for
adaptive-threshold algorithms.
DOI: 10.3103/S0735272713070017
I. INTRODUCTION
Radar is basically a means of gathering information about distant objects, or targets, by sending
electromagnetic waves at them and analyzing the echoes.
There are two aspects to the radar statistical problem. The first is concerned with the background noise,
which is random in character. In the absence of this background noise, detection poses no difficulty which
means that however small the reflected signal from a target is, it may be detected with sufficient gain in the
receiver. Background noise interference, however, imposes a limit on the minimum detectable signal. The
question of target existence is, in fact, a choice of deciding between noise alone or signal-plus-noise mixture.
Random noise interference arises from many sources including radiation from the external environment
and internal thermal noise. Generally, this noise is wideband with a uniform (white) or nearly uniform
spectral density.
In addition, there is another major background noise source, which is referred to as clutter. This type of
noise represents the aggregate radar return from a collection of many small scatterers, e.g., ground return,
sea return, reflection from rain, chaff, and decay clouds. Detection and estimation in a clutter environment is
a major problem in modern radars [1, 4, 11].
The second statistical aspect of the radar problem stems from the reflective properties of radar targets. If
the radar cross section of an aircraft, or other complex target structures, is observed as a function of the
aspect angle, the resulting pattern is characterized by rapid fluctuations in amplitude with minute changes in
the aspect angle value. In a typical radar situation, the target is observed many times. The aspect angle at a
particular time will govern the target cross section observed by the radar.
Since many targets have relative motion with respect to the radar, aspect angle changes on successive
observations alter the radio frequency phase relationships, thereby modifying the radar target’s cross
section. This change may be a slow variation and occur on a scan-to-scan basis (on successive antenna scans
321
ISSN 0735-2727, Radioelectronics and Communications Systems, 2013, Vol. 56, No. 7, pp. 321–334. © Allerton Press, Inc., 2013.
Original Russian Text © Mohamed B. El Mashade, 2013, published in Izv. Vyssh. Uchebn. Zaved., Radioelektron., 2013, Vol. 56, No. 7, pp. 3–17.
across a target) or it may be on a pulse-to-pulse basis (on successive sweeps). Because the exact nature of the
change is difficult to predict, a statistical description is often adopted to characterize the radar target’s cross
section [1].
Three families of radar target’s cross section fluctuation models have been used to characterize major
targets of interest; namely c2family, Rice family, and the log-normal family.
The c2models are used to represent complex targets, such as an aircraft, and have the characteristics that
the distribution is more concentrated around the mean as the value of its defined parameter is increased.
As special cases of these models, Swerling’s versions are derived. This Swerling representation of
fluctuating targets brackets the majority of the actual radar targets.
In SWerling model I (SWI) the echo pulses received from a target on any one scan are of constant
amplitude throughout the entire scan, but are independent (uncorrelated) from scan to scan. A target echo
fluctuation of this type is called scan-to-scan fluctuation. It is also known as slow fluctuations.
SWerling model II (SWII) has the same behavior as SWI except that the fluctuations are independent
from pulse to pulse rather than from scan to scan. It is sometimes called fast fluctuations.
In SWerling model III (SWIII) the radar cross section is assumed to be constant within a scan and
independent from scan to scan; but with a probability density function different from that of SWI. This
probability density function is representative of targets that can be modeled as one scatterer together with a
number of small scatterers.
Swerling model IV (SWIV) is characterized by fluctuations from pulse to pulse with the same probability
density function as in SWIII [4].
The major form of Constant False Alarm Rate (CFAR) has been the CA (Cell-Averaging) technique. It
uses the maximum likelihood estimate of the noise power to set the adaptive threshold. Although the
presence of interferers inside the reference window leads to an overestimate of the actual noise power and
this in turn gives rise to a masking of legitimate targets, it is still of major importance because it is the
optimum CFAR processor when the background noise is homogeneous and the reference cells contain
independently, identically, and exponentially distributed observations [3, 7, 12].
Analysis of adaptive threshold setting algorithms has generally relied on either Monte-Carlo simulation,
or closed form techniques. Monte-Carlo simulation has the drawbacks of both requiring substantial
computer time, and lacking precision. Closed-form analysis, when mathematically tractable, is preferable
because it yields more precise results in much less computation time.
This paper aims to obtain closed-form analysis for fixed threshold as well as CA adaptive threshold
schemes when they are used to detect c2models of fluctuating targets, especially, the four Swerling types
(SWI, SWII, SWIII, and SWIV).
The organization of this paper is as follows: section II describes the system model, formulates the
problem under consideration. and computes the characteristic function of the post-detection integrator
output for the case where the signal fluctuation obeys c2statistics. In section III, the performance of the
processors under consideration is analyzed in non-homogeneous background environment. Section IV
presents the numerical results, while section V contains a brief discussion along with conclusions.
II. GENERAL STATISTICAL MODEL
Detection of signals is equivalent to deciding whether the receiver output is due to noise alone or to
signal-plus-noise mixture. This is the type of decision made by a human operator from the information
presented on a radar display. When the detection process is carried out automatically by electronic means
without the aid of an operator, the detection criterion must be carefully specified and built into the decision
making device.
The radar detection process was described in terms of threshold detection. The level of this threshold
divides the output into a region of no detection (H0) and a region of detection (H1). By raising or lowering
this bias level, the number of times a noise pulse surpasses the bias level either decreases or increases,
respectively. The setting of this level depends on the number of times that noise is be permitted to exceed the
bias level during a given period of time. Mistakenly taking a noise pulse for a signal return is called false
alarm. Mistakenly taking a signal return for a noise pulse is called miss detection. Miss detection and false
alarm, therefore, are subject to the trade-off.
An adaptive threshold detector is an algorithm which provides a constant false alarm rate in varying
non-homogeneous clutter and noise interference environment by adaptively adjusting the detection
threshold. This procedure assumes that the general form of the interference’s probability distribution is
RADIOELECTRONICS AND COMMUNICATIONS SYSTEMS Vol. 56 No. 7 2013
322 MOHAMED B. EL MASHADE
known except for a small number of unknown parameters. These unknown parameters are estimated on a
cell-to-cell basis by examining reference cells surrounding the cell under test. The resulting estimated
interference probability distribution function is then used in each cell to obtain a threshold setting that
provides the desired probability of false alarm. Figure 1 is a useful way of depicting the CFAR detection
technique of a radar target in the case of non-coherent integration of M-pulses.
Under certain conditions, usually met in practice, maximizing the output peak signal-to-noise ratio
(SNR) of a radar receiver maximizes the detectability of a target. A linear network that does this is called a
matched filter. Thus, a matched filter is the basis for designing almost all radar receivers.
The square-law device demodulates the baseband signal and Mconsecutive sweeps are non-coherently
integrated to represent the input of the adaptive technique. The box labeled “cell under test” represents the
radar range cell that is currently being examined for the presence of a target. Buffer cells adjacent to the cell
under test can be used to avoid contamination with the edge of the matched filter output from the target
return. Two tapped delay lines sample echo signals in a number of reference cells located on both sides of the
range cell of interest. Spacing between the reference cells is equal to the radar range resolution which is
usually equal to the pulse width.
The statistic Zwhich is proportional to the estimate of total noise power is constructed by processing the
contents of Nreference cells surrounding the cell under investigation whose content is denoted by n. A target
is declared to be detected if nexceeds the threshold ZT, where Tis a constant scale factor used to achieve the
required rate of false alarm for a given window size when the background noise is homogeneous. The
processor configuration varies with different CFAR algorithms.
Our approach in analyzing a CFAR processor is to evaluate its probability of detection Pdwhich is
defined as
()
PTZ
HdPr 1
@n> |. (1)
This definition can be formulated in another simpler form as
()
PZT
HdPr 1
@<n/ |. (2)
In terms of the probability density functions (PDF) of nand Z(2) can be formulated as
Pfyfxxy
Z
yT
d
00
dd=
¥
òò
n() ()
/
. (3)
RADIOELECTRONICS AND COMMUNICATIONS SYSTEMS Vol. 56 No. 7 2013
ANALYTICAL PERFORMANCE EVALUATION OF ADAPTIVE DETECTION 323
Fig. 1.
Noise-level
estimator
Noise-level
estimator
Noise power
estimation
Matched
filter
Quadratic
rectifier
Noncoherent
integrator
Cell under
investigation
p
Z
T
Decision
Detection
threshold
It is important to note that the inner integral in the above expression represents cumulative distribution
function (CDF) of the random variable (RV) Z. Thus,
PfyFyTy
Zd
0
d=
¥
òn() ( / ) . (4)
Another important version of (4) has an expression given by
PTfTyFyy
Zd
0
d=
¥
òn() ()
. (5)
From the definition of Laplace transformation, Pdcan be put in a more simpler form as
PTG
t
d0
==
()|ww, (6)
where Gt(w) denotes the Laplace transformation of the integrand of (5)
GfTF
tZ
() ( ) ()wnnwnn
n
D
0
exp( )d
¥
ò-=*
1
TTZ
FY
nww(/) ()
. (7)
where “*” symbol represents mathematical convolution, Fn(.) denotes the characteristic function (CF) of
the RV n, and Y(.) represents the Laplace transformation of the CDF of the RV Z.
Finally, the detection probability takes the following analytical form
PuTuu
C
Zd
1
2j d=-
-
ò
=
pw
n
w
FY(/) ( )
0
. (8)
The contour of integration C–consists of a vertical path in the complex u-plane crossing the negative real
axis at the rightmost negative real axis singularity of Fn(.) and closed in an infinite semicircle in the left half
plane.
From (8) it is evident that the CF of the RV that represents the content of the cell under test plays an
important role in determining the processor detection performance. Let us now calculate this very interesting
function for the four cases of Swerling representation of fluctuating targets.
In the analysis that follows, it is assumed that the clutter background is homogeneous over the area
encompassed by the delay line. This means that the PDF of the output from any delay line tap with no target
return present is
fx xUx
nyy
() exp ()=-
æ
è
çö
ø
÷
1, (9)
where yis unknown average noise power and the unit-step function is denoted by U(.). The delay line
outputs are assumed to be statistically independent RV’s. When a nonfluctuating target return-plus-noise
mixture is present in any tap, on the other hand, the output of this tap has a PDF given by [9]
fx I xA x A Ux
nyy y
() ()=æ
è
ç
ç
ö
ø
÷
÷-+
æ
è
çö
ø
÷
12exp
0, (10)
RADIOELECTRONICS AND COMMUNICATIONS SYSTEMS Vol. 56 No. 7 2013
324 MOHAMED B. EL MASHADE
where A/ydenotes the SNR at the square-law detector input and I0(.) represents the modified Bessel
function of type 1 and of order 0.
When the target return present in one of the tap outputs comprising the sum of M-pulses, which are
non-coherently integrated, i.e.,
Ql
l
M
Dn
=
å
1
, (11)
it has a PDF of the form [2]
fx
Q()=æ
è
çö
ø
־
è
ç
ç
ö
ø
÷
÷-+
æ
è
çö
ø
÷
-
-
12exp
1
21
yyy
x
AIxA x A Ux
M
M()
. (12)
The CF associated with this PDF can be easily calculated and the result can be put as
FQ
M
AA
(/)wyw
w
yw
=+
æ
è
çö
ø
÷-+
æ
è
çö
ø
÷
1
1exp 1. (13)
The unconditional CF is now obtained by averaging the previous expression over the target fluctuation
distribution of A. For c2family of target models, the RV Ahas a PDF given by [12]
fAA K
K
AAK
A
AUA
A
KK
(/) () ()=æ
è
çö
ø
÷-
æ
è
çö
ø
÷
-
1exp
1
G, (14)
where Adenotes the average M-pulse SNR and K> 0 represents a fluctuation parameter.
The unconditional CF is then calculated as
FF
D
QQA
Af A A A() ( / ) ( / )ww d
0
¥
ò=+
æ
è
çö
ø
÷+
æ
è
çö
ø
÷
-
1
1
1
1yw bw
MK K
,
by y
D1+
æ
è
çö
ø
÷
A
K
/. (15)
Equation (15) is the fundamental expression from which the Swerling models are derived as special cases
as we shall see in the next section.
III. CALCULATION OF PROBABILITY OF DETECTION
OF FLUCTUATING TARGETS
Now, we are going to evaluate the detection performance of the adaptive threshold setting techniques for
the fluctuating targets of Swerling models.
Swerling I case (SWI)
In this case, the fluctuation parameter Khas a unit value. Letting K= 1 in (15) yields
FQ
M
()wy
wy
a
wa
=+
æ
è
çö
ø
÷+
æ
è
çö
ø
÷
-
1/
1/
1/
1/
1
,
RADIOELECTRONICS AND COMMUNICATIONS SYSTEMS Vol. 56 No. 7 2013
ANALYTICAL PERFORMANCE EVALUATION OF ADAPTIVE DETECTION 325
()()
ay y y
D11
1
+=+AMA/,
where A1represents the average per pulse SNR. The substitution of this expression into (8) gives
()
PTT
M
Zd
1
1/
1/ 1/
=-
æ
è
çö
ø
÷
-
a
y
ya aY /
++
æ
è
çö
ø
־
è
çö
ø
÷-
--
(/)
()
TTT
M
MM
y
aya
111
1G
d
d
1
2M
MZ
uT
uuT u
-
-=-
+
æ
è
çö
ø
÷-
ì
í
î
ü
ý
þ
2/()
/
ay
Y. (17)
Swerling II case (SWII)
This fluctuation model is characterized by K=M, the substitution of which into (15) gives
FQ
M
()wa
wa
=+
æ
è
çö
ø
÷
1/
1/ ,
()
ay yy
D11
1
+
æ
è
çö
ø
÷=+
A
MA
/. (18)
In this case, the processor detection performance takes the form
{}
PT
Mu
u
MMM
MZ
uT
d
11
d
d
=æ
è
çö
ø
÷---
-=
aa
()
() ()
/
1
1
GY. (19)
Swerling III case (SWIII)
By letting K= 2 in the general expression of (15), the resulting model is known as SWIII which has a CF
of the form
FQ
M
()wy
wy
a
wa
=+
æ
è
çö
ø
÷+
æ
è
çö
ø
÷
-
1/
1/
1/
1/
22
,
ay yy
D1212
1
+
æ
è
çö
ø
÷=+
æ
è
ç
çö
ø
÷
÷
AMA
/. (20)
The probability of detecting a fluctuating target of SWIII model becomes
PTT
M
d
22
=æ
è
çö
ø
־
è
çö
ø
÷
-
ay
d
d
12
uuT u
M
Z
uT
+
æ
è
çö
ø
÷-
é
ë
ê
ê
ù
û
ú
ú
ì
í
ï
î
ï
-
=-
/()
/
ya
Y
+-
1
2G( )M
d
d
1
32
M
MZ
uT
uuT u
-
-
=-
+
æ
è
çö
ø
÷-
é
ë
ê
ê
ù
û
ú
ú
ü
ý
ï
þ
ï
3/()
/
ay
Y. (21)
RADIOELECTRONICS AND COMMUNICATIONS SYSTEMS Vol. 56 No. 7 2013
326 MOHAMED B. EL MASHADE
Swerling IV case (SWIV)
The characterization of this fluctuation model is K=2M. Thus, the substitution of this value into (15)
yields
FQ
MM
()wy
wy
a
wa
=+
æ
è
çö
ø
÷+
æ
è
çö
ø
÷
-
1/
1/
1/
1/
2
,
ay yy
D11
2
1
+
æ
è
çö
ø
÷=+
æ
è
ç
çö
ø
÷
÷
A
M
A
/
2. (22)
The detection performance of the adaptive processor for this type of fluctuating targets is [13]
PTT
M
MM
d
1
=æ
è
çö
ø
־
è
çö
ø
÷
-
ay
2
2G()
d
d
1
21
21
M
M
M
Z
uT
uuT u
-
-
-
=-
+
æ
è
çö
ø
÷-
ì
í
ï
î
ï
ü
ý
ï
þ
ï
/()
/
ya
Y. (23)
In the absence of the radar target (a=y), the CF of the cell under test which represents the no target
hypothesis (null hypothesis) takes a very simplified expression of the form
FF
Q
M
C
() ()wyw w= +
æ
è
çö
ø
÷
1
1
@. (24)
The detection probability in this case converges to the probability of false alarm which becomes
{}
PT
Mu
u
MM
MZ
uT
f
fa
1
1d
d
=æ
è
çö
ø
÷-
-
-
=-
yy
G( ) Y
1()
/
. (25)
Equations (17), (19), (21), (23), (25) are the basic analytical formulas of our analysis in this manuscript.
These expressions are general for any CFAR detector.
Our aim in the remaining part of the paper is to evaluate performance of the fixed-threshold as well as one
of the most important adaptive-threshold (CA) detectors to determine their behavior against fluctuating
targets of Swerling models.
By examining the above formulas, it is obvious that they rely on the Laplace transformation of the CDF of
the noise power level estimate Zand its mathematical differentiation. Therefore, we are interested in
formulating this transformation when the detection scheme operates in an environment that contains several
extraneous targets along with the target under investigation.
IV. PROCESSOR DETECTION PERFORMANCE
Here, we are interested in applying the previously derived formulas to the optimum detector, against
which any proposed processor is compared, and one of the most popular and efficient scheme in maintaining
a constant rate of false alarm against environmental impairments which is known as cell-averaging (CA)
detector. We are going to evaluate their performance against fluctuating targets of Swerling models.
1. Fixed-Threshold Detector
A useful procedure for establishing the decision threshold at the output of a radar receiver is based on the
classical statistical theory of the Neyman–Pearson criterion which is described in terms of the two types of
errors that can be made in the detection decision process.
RADIOELECTRONICS AND COMMUNICATIONS SYSTEMS Vol. 56 No. 7 2013
ANALYTICAL PERFORMANCE EVALUATION OF ADAPTIVE DETECTION 327
– One type of these errors is to mistake noise for signal when only noise is present. It occurs whenever
noise at the receiver’s output is large enough to exceed the decision threshold. In statistics, this is called false
alarm.
– Another type of errors occurs when a signal is present but is erroneously considered to be noise. The
radar engineer would call such an error a missed detection.
It might be desired to minimize both errors, but they both cannot be minimized independently. In the
Neyman–Pearson theory, the probability of first type error is fixed and the probability of the second type
error is minimized. Since the threshold level is set in such a way that a specified probability of false alarm is
not exceeded, this is equivalent to fixing the probability of type one error and minimizing the type two error
(or maximizing the detection probability). This is the Neyman-Pearson test used in statistics for determining
the validity of a specified statistical hypothesis. It is employed in most radars for making the detection
decision.
The optimum detector sets a fixed threshold to determine the presence of a target under the assumption
that the total homogeneous noise power yis known a priori. In this case, the detection probability is given by
{}
PQQHfyy
Q
Q
d01
Pr d
0
@>=
¥
ò
|()
=-=-
¥
òò
fyy fyy FQ
QQ
Q
Q
() () ( )dd1
00
0
0
. (26)
From (26) it is obvious that the CDF of the content of the cell under test is the backbone of the analysis of
the fixed-threshold detector, Once the CF of Qis calculated, the Laplace transformation of its CDF is
consequently obtained as [8, 13]
YF
QQ
() ()/www=. (27)
Once the w-domain representation of CDF of Q is computed, its t-domain representation can be easily
obtained through the Laplace inverse technique. Consequently, the processor performance against SWI,
SWII, SWIII, and SWIV target fluctuation models becomes an easy task.
2. Adaptive-Threshold Detector
Since the clutter plus noise power is not known at any given location, a fixed-threshold detection scheme
can not be applied to the radar returns if the rate of false alarm is to be controlled. Moreover, radar
performance is often degraded by the presence of false targets. To reduce this effect, radar detection
processing can use an algorithm to estimate the clutter energy in the tested cell and then adjust the
constructed threshold to reflect changes in this energy at different test cell positions.
An attractive class of schemes that can be used to overcome the problem of clutter is that of CFAR type
which set the threshold adaptively based on local information of total noise power. A relatively simple
algorithm uses the average received energy in the Nnearby range cells to set the threshold. Such a processor
is known as cell-averaging (CA) detector. It is the optimum CFAR processor in a homogeneous background
when the reference cells contain independent and identically distributed observations governed by an
exponential distribution. As the size of the reference window increases, the detection probability approaches
that of the fixed-threshold (optimum) detector.
As we have previously shown in section III, the detection performance of an adaptive-threshold scheme
depends mainly on calculating the Laplace transformation of CDF of noise power level estimate Z. Hence,
all we need in the analysis of this type of detection techniques is to compute CF based on background noise
level estimate, which we aim to accomplish in the rest of this section.
The multiple target situation is frequently encountered in practice in which the reference window
contains nonuniform samples. This may occur in a dense environment where two or more potential targets
appear in the range cells surrounding the cell under test. The amplitudes of all the interfering target returns
that may be present amongst the candidates of the reference window are assumed to be of the same strength
and to fluctuate in accordance with Swerling models. The interference-to-noise ratio (INR) for each of the
RADIOELECTRONICS AND COMMUNICATIONS SYSTEMS Vol. 56 No. 7 2013
328 MOHAMED B. EL MASHADE
spurious targets is taken as a common parameter and is denoted by I. Thus, for reference cells containing
extraneous target returns, the total background noise power is y(1 + I), while the remaining reference cells
have the identical noise power of y.
Suppose that the reference subset of size n(n=N/2) contains rcells from interfering target returns with
background power level of y(1 + I) and the remaining q=n–rcells from clear background with noise power
y. Thus, the noise power level of this subset is estimated as
Zrq X
ll l i
i
r
j
j
q
ll
(, )@c
==
åå
+
11
,l= 1, 2. (28)
When the secondary interfering target fluctuates in accordance with Swerling models, the random
variable representing its return has a CF of the form:
FSWI
1
1/
1/
1/
1/
()wy
wy
a
wa
=+
æ
è
çö
ø
÷+
æ
è
çö
ø
÷
-M
,
ay
D()1+MI , (29)
FD
SWII
1/
1/ 1() , ( )wa
wa ay=+
æ
è
çö
ø
÷+
M
I, (30)
FSWIII
22
1/
1/
1/
1/
()wy
wy
a
wa
=+
æ
è
çö
ø
÷+
æ
è
çö
ø
÷
-M
,
ay
D12
+
æ
è
çö
ø
÷
MI, (31)
FSWIV
1/
1/
1/
1/
()wy
wy
a
wa
=+
æ
è
çö
ø
÷+
æ
è
çö
ø
÷
-MM2
,
ay
D12
+
æ
è
çö
ø
÷
I. (32)
Since the final noise power level estimate in mean detector is constructed by adding the noise power level
estimates of the two reference subsets, it has a Laplace transform for its CDF given by
{} { }
YF F
ZC
MN R
J
R
() () ()
()
wwww=-
1
SW ,
J@I, II, III, IV, (33)
where R=r1+r2represents the total interfering target returns that may be present amongst the contents of
the reference set and (N–R) denotes the number of reference cells containing clear background noise, the CF
of which has the form (24). Once the Laplace transformation of the CDF of the noise power level estimate is
formulated, the processor performance evaluation becomes an easy task [5–10].
RADIOELECTRONICS AND COMMUNICATIONS SYSTEMS Vol. 56 No. 7 2013
ANALYTICAL PERFORMANCE EVALUATION OF ADAPTIVE DETECTION 329
V. PROCESSOR PERFORMANCE ASSESSMENT
In this section, we are concerned with the numerical evaluation of the analytical expressions that we have
obtained in the previous subsections to assess the performance of adaptive and fixed threshold detectors for
c2fluctuating targets when the operating environment is free of any impurities (homogeneous) and when the
reference channels contaminated with returns from spurious targets.
The detection analytical expressions are programmed on a PC digital system for some parameter values
and the results of these programs are presented in figures below. The SNR in these figures means
signal-to-noise ratio of a primary target.
Figures 2, 3 depict the detection performance of the fixed threshold processor when the primary target
fluctuates following SWI, SWII, SWIII, and SWIV, in the case where radar receiver integrates 2 (Fig. 2) and
4 (Fig. 3) consecutive sweeps. The reference false alarm rate is chosen to be10 6-. Moreover, the monopulse
detection performance of the same scheme, for the same parameter values, is included in these families of
curves for the sake of comparison.
For low values of SNR, the detection performance for SWI is higher than that for SWIII which in turn
higher than that for SWII and it attains its lowest value for SWIV. When the SNR becomes high, on the other
hand, the processor performance is reversed which means that it attains its highest value for SWIV which in
turn higher than that for SWII and the worst detection curve is for SWI model. In the single sweep case, the
optimum detector behaves the same taking into account that SWII tends to SWI and SWIV tend to SWIII for
the case of monopulse detection.
It is important to note that for M= 2, the detection performance of the processor under consideration for
SWIII model is the same as its behavior against SWII model. In addition, we note that the curves of these
families are functions of the number of post-detection integrated sweeps and the target fluctuation model.
After getting an idea about the detection behavior of the fixed threshold scheme, let us go to show the
same behavior of the adaptive threshold detector against the same fluctuation models.
Figuress 4, 5 display a comparison between the reactions of adaptive- (CA) and fixed-threshold (Opt.)
procedures to the detection of the four cited target fluctuation models when the radar receiver
non-coherently integrates 2 (Fig. 4) and 4 (Fig. 5) consecutive sweeps, respectively, in a homogeneous
environment. As predicted, these figures illustrate superiority of the fixed-threshold scheme over the
RADIOELECTRONICS AND COMMUNICATIONS SYSTEMS Vol. 56 No. 7 2013
330 MOHAMED B. EL MASHADE
Fig. 2. Fig. 3.
–5 –1 3 7 11 15 19 23 SNR, dB –5 –1 3 7 11 15 19 23 SNR, dB
0
0.2
0.4
0.6
0.8
Pd
0
0.2
0.4
0.6
0.8
Pd
Ì=1 SWI
Ì=2 SWI
SWIII SWIV
SWII
SWIII
Ì=2
Ì=1 SWI
Ì=4 SWI
SWIII SWIV
SWII
SWIII
Ì=4
Fig. 4. Fig. 5.
–5 –1 3 7 11 15 19 23 SNR, dB –5 –1 3 7 11 15 19 23 SNR, dB
0
0.2
0.4
0.6
0.8
Pd
0
0.2
0.4
0.6
0.8
Pd
SWI
SWII
SWIII
SWIV
Opt CA
SWIV
SWIII
SWII
SWI
N= 24, M=2,Pfa =10
–6
SWI
SWII
SWIII
SWIV
Opt CA
SWIV
SWIII
SWII
SWI
N= 24, M=2,Pfa =10
–6
adaptive-threshold scheme in achieving the homogeneous detection of fluctuating targets under the same
parameter values and for the same underling target model.
For comparison, Fig. 6 depicts the receiver operating characteristics (ROC) of both detectors under
consideration for M-sweeps when the primary target fluctuates following SWIII and SWIV models and has a
strength of 5 dB. The behavior of the curves family confirms the previous conclusion that the fixed-threshold
procedure gives better performance than the adaptive-threshold one under the same conditions of operation.
It is interesting to note that, for SWI and SWII models, we are previously plotted the reaction of CA-CFAR
scheme against them and the results are shown in [12].
Now, let us turn our attention to the multiple-target situations and what happens to the performance of the
adaptive-threshold detector when the reference channel, from which the noise power level is estimated, is
contaminated with returns from extraneous targets.
These effects are illustrated in Figs. 7–10 for the different fluctuation models of Swerling. In calculating
the results of these figures, we assume that the size of the reference channel is 24 samples, design probability
of false alarm is 10 6-, and there is a single cell amongst the contents of the leading and trailing reference
channels which is contaminated with interfering target returns (r1=r2= 1).
RADIOELECTRONICS AND COMMUNICATIONS SYSTEMS Vol. 56 No. 7 2013
ANALYTICAL PERFORMANCE EVALUATION OF ADAPTIVE DETECTION 331
Fig. 6.
10–6 10–4 10–2 Pfa
0
0.2
0.4
0.6
0.8
PdCA M= 1, SWIII
M= 2, SWIII
M= 2, SWIV
M= 4, SWIII
M= 4, SWIV
Opt
M= 4, SWIV
M= 4, SWIII
M= 2, SWIV
M= 2, SWIII
M= 1, SWIII
Fig. 7. Fig. 8.
–5 –1 3 7 11 15 19 23 SNR, dB –5 –1 3 7 11 15 19 23 SNR, dB
0
0.2
0.4
0.6
0.8
Pd
0
0.2
0.4
0.6
0.8
Pd
Fig. 9. Fig. 10.
–5 –1 3 7 11 15 19 23 SNR, dB –5 –1 3 7 11 15 19 23 SNR, dB
0
0.2
0.4
0.6
0.8
Pd
0
0.2
0.4
0.6
0.8
Pd
N= 24, Pfa =10
–6
r1=r2=1
M=1
M=2
M=3
M=4
M=4
M=3
M=2
M=1
INR = SNR
INR=0
M=1
M=2
M=3
M=4
M=4
M=3
M=2
M=1
INR = SNR
INR=0
N= 24, Pfa =10
–6
r1=r2=1
N= 24, Pfa =10
–6
r1=r2=1
M=1
M=2
M=3
M=4
M=4
M=3
M=2
M=1
INR = SNR
INR=0
M=1
M=2
M=3
M=4
M=4
M=3
M=2
M=1
INR = SNR
INR=0
N= 24, Pfa =10
–6
r1=r2=1
Figure 7 depicts the detection performance of the adaptive-threshold scheme under consideration in
homogeneous and multiple-target environments when the primary as well as the secondary interfering
targets fluctuate following SWI model taking into account that the radar receiver post-detection integrates M
pulses. It is to be noted that the family of curves representing the processor performance in the absence of
interfering targets (homogeneous situation), are labeled as (INR = 0), while those depicting the multi-target
performance are labeled as (INR = SNR), which means that the interfering target return is of the same
strength as the primary target.
As predicted, there is an improvement in the detection probability as the number of non-coherently
integrated pulses increases and this is common to both homogeneous or multi-target situations.
Figure 8 shows the processor reaction against fluctuating targets of SWII model under the same
conditions as outlined in Fig. 7. The processor performance for this case is higher than in the previous case
and this result is also predicted as we have previously explained in [12].
Figure 9 illustrates the CA behavior in detecting SWIII fluctuating targets and Fig. 10 displays the same
thing for SWIV target fluctuation model. The comparison of these two figures with their corresponding
previous figures leads to a conclusion that there is an improvement in the SWIII model relative to SWI case
and in the SWIV model relative to SWII case for the same parameter values.
To verify this conclusion, Figs. 11, 12 show the CA detection performance against the four types of
Swerling in both kinds of operating environments (homogeneous and multiple-target) for a number of
integrated pulses equals 2 (Fig. 11) and 4 (Fig. 12).
The results plotted in these figures show that for low SNR values, the processor performance improves as
the model number decreases; i.e., SWI gives better performance that that for SWIII which in turn better than
that for SWII and SWIV has the worst detection performance. As the strength of the target return becomes
strong, the reverse of this action occurs; i.e., SWIV becomes the first one in this group which has the top
performance, SWII has the next higher detection performance, SWIII becomes next one and SWI has the
worst performance. This behavior is common irrespective of the number of integrated pulses or the
environment in which the radar receiver operates.
To show the effect of outlying target returns on the CFAR property of adaptive-threshold detector,
Fig. 13 shows the actual false alarm rate performance, as a function of the strength of the level of
interference, when the reference cells are contaminated with interfering target returns and the designed level
RADIOELECTRONICS AND COMMUNICATIONS SYSTEMS Vol. 56 No. 7 2013
332 MOHAMED B. EL MASHADE
Fig. 11. Fig. 12.
–5 –1 3 7 11 15 19 23 SNR, dB –5 –1 3 7 11 15 19 23 SNR, dB
0
0.2
0.4
0.6
0.8
Pd
0
0.2
0.4
0.6
0.8
Pd
N= 24, M=2,Pfa =10
–6
r1=r2=1
M=1
M=2
M=3
M=4
M=4
M=3
M=2
M=1
INR = SNR
INR=0
M=1
M=2
M=3
M=4
M=4
M=3
M=2
M=1
INR = SNR
INR=0
N= 24, M=4,Pfa =10
–6
r1=r2=1
Fig. 13.
–5 –1 3 7 11 15 19 23 INR, dB
10–18
10–14
10–12
10–10
10–8
Pfa N= 24, designed Pfa =10
–6
r1=r2=1
M=1
M=2
M=3
M=4
M=4
M=3
M=2
M=1
M=4
M=2
10–16
of false alarm is of the order of10 6-. The candidates of this set are labeled in the number of integrated pulses
Mand the fluctuation model of the outlying target. The label “M= 2, SWIV” on a specified curve means that
it is plotted when the spurious target fluctuates in accordance with SWIV model and for M=2.
The results of this figure show that as the interference level increases, the rate of false alarm decreases
and the rate of decreasing increases as Mincreases. In addition, the rate of decreasing of the false alarm in the
case of SWI fluctuation model is slower than that in the case of SWII model which in turn slower than SWIII
and SWIV model has the lowest behavior of false alarm rate which must be held constant.
To illustrate the effect of the secondary target fluctuation model on the processor detection performance
Figs. 14, 15 are concerned with the M-sweeps detection behavior of the adaptive-threshold scheme for all the
possible fluctuation models for the secondary interfering targets when the primary target fluctuates
following SWI (Fig. 14) and SWII (Fig. 15) models. The curves of these families are labeled to mention M,
the primary target fluctuation model, and the secondary target fluctuation model.
From the results of these figures, we note that there is a negligible improvement in the processor
performance as the model of the outlying targets varies from I to IV, given that the number of integrated
pulses as well as the model of the primary target are held constants.
Finally, Figs. 16, 17 display the required SNR, to achieve an operating point of Pd= 0.9, given that Pfa =
10 6-, of the procedure under consideration, as a function of the window size N, when this scheme operates in
an ideal environment and the radar receiver post-detection integrates 2 (Fig. 16) and 4 (Fig. 17) consecutive
pulses. For the sake of comparison, the SNR required by the fixed-threshold detector to achieve the same
operating point is also included in these figures under the same fluctuation model as the adaptive-threshold
scheme.
As expected, the SWIV model requires the minimum SNR to attain the requested values, the SWII comes
in the second class, the SWIII model needs higher values of SNR, and SWI case needs the highest values of
SNR to arrive to the same levels of detection and false alarm. This behavior is independent on the number of
integrated pulses.
These figures give us an indication about the behavior of the adaptive-threshold detector, relative to the
fixed-threshold scheme, against the fluctuating targets of Swerling models, under the condition that the
detection and false alarm probabilities are held constants. It is obvious that, as the number of integrated
pulses is increased, the greater the SNR and the more likely it is that a noise-submerged signal will be
detected. At the same time, however, it should be noted that as the number of pulses integrated is increased,
RADIOELECTRONICS AND COMMUNICATIONS SYSTEMS Vol. 56 No. 7 2013
ANALYTICAL PERFORMANCE EVALUATION OF ADAPTIVE DETECTION 333
Fig. 14. Fig. 15.
–5 –1 3 7 11 15 19 23 SNR, dB –5 –1 3 7 11 15 19 23 SNR, dB
0
0.13
0.26
0.39
0.52
Pd
0
0.17
0.34
0.51
0.68
Pd
N= 24, Pfa =10
–6
r1=r2=1
SWI
SWII
SWIII
SWIV
SWIV
SWIII
SWII
SWI
M=4,SWI
M=2,SWI
Fig. 16. Fig. 17.
048121620N04 8121620N
0
5
10
15
20
SNR, dB
0
4
8
12
16
SNR, dB
N= 24, Pfa =10
–6
r1=r2=1
SWI
SWII
SWIII
SWIV
SWIV
SWIII
SWII
SWI
M= 4, SWII
M= 2, SWII
25 20
SWI
SWII
SWIII
SWIV
SWIV
SWIII
SWII
SWI CA
Opt
M=2
SWI
SWII
SWIII
SWIV
SWIV
SWIII
SWII
SWI CA
Opt
M=2
the more sensitive the probability of false alarm to the bias level and the longer the period of time required
for determining if a target is detected.
VI. CONCLUSIONS
In this paper, we have given a detailed analysis of the detection performance calculation of the
fixed-threshold as well as the adaptive-threshold procedures under the condition that the primary and the
secondary outlying targets fluctuate following c2fluctuation model with two and four degrees of freedom.
The fluctuation rate may vary from essentially independent return amplitudes from pulse-to-pulse to
significant variation only on a scan-to-scan basis. A Swerling fluctuating target is a model which describes
the fluctuation in target amplitude caused by changes in target aspect angle, rotation, or vibration of target
scattering sources or changes in radar wavelength. This fluctuation model includes the classical models of
target echo fluctuation which are known as SWI, SWII, SWIII, and SWIV. The correlation coefficient
between the two consecutive echoes in the dwell-time is equal to unity for SWI and SWIII models and is zero
for SWII and SWIV models.
The adaptive-threshold processor is chosen to be the cell-averaging (CA) scheme since it is the only
procedure that has a detection performance which is closest to that of an optimum processor when the
operating environment is free of any impurities. The analysis illustrates the utility of the contour-integral
approach in determining the CFAR processor performance in the presence of interferers.
The analytical results have been used to develop a complete set of performance curves including
detection probability in homogeneous and multiple target situations, the variation of false alarm rate with the
strength of interfering targets that may exist amongst the contents of the estimation set, and the required SNR
to achieve a predetermined operating point of fixed levels for detection and false alarm rates. As expected,
lower threshold values and consequently higher detection performance is obtained as the number of
post-detection integrated pulses increases.
On the other hand, as the signal correlation increases from zero (SWII and SWIV) to unity (SWI and
SWIII), more per pulse SNR is required to achieve a prescribed probability of detection. In addition, the
processor performance for fluctuating targets of chi-square model with four degrees of freedom is higher
than that for chi-square model with two degrees of freedom, given that the same case of signal correlation is
held unchanged. On the other hand, the processor performance for SWIV model is higher than that for SWII
model and its behavior against SWIII model is higher than its reaction against SWI model.
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334 MOHAMED B. EL MASHADE
