Cfar

Signal Processing 91 (2011) 28–37

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Signal Processing journal homepage: www.elsevier.com/locate/sigpro

CFAR detection for multistatic radar Vahideh Amanipour, Ali Olfat n Signal Processing and Communication Systems Laboratory, School of Electrical and Computer Engineering, University of Tehran, Iran

a r t i c l e i n f o

abstract

Article history: Received 13 November 2009 Received in revised form 26 March 2010 Accepted 1 June 2010 Available online 16 June 2010

In this paper a multistatic radar system with n transmitters and one receiver is considered and several constant false alarm rate (CFAR) algorithms for detection are introduced. The decision statistics of the proposed detectors are the sum of the n largest returning signals in an array of N +n range cells. It is shown that the proposed decision statistic satisfies the CFAR condition and it is justified that the sum of the largest returning signals is the optimal statistic. The proposed CFAR detectors are simulated both in homogenous and non-homogenous backgrounds and their performances are compared with the performance of a monostatic radar of higher power. It is shown that a multistatic radar outperforms a monostatic radar under equal transmit power condition. & 2010 Elsevier B.V. All rights reserved.

Keywords: Multistatic radar CFAR detection Homogenous background Non-homogenous background

1. Introduction Bistatic radars are radar systems that use antennas at different locations for transmission and reception. They have been studied since the earliest days of radar. If further antennas, either transmitting or receiving or both, are added to the bistatic pair the system might be termed multistatic radar. However, other terminologies often include ‘‘netted radar’’, ‘‘multi site radar’’, ‘‘distributed radar’’ and ‘‘MIMO radar’’. Multistatic radars can provide improved performance against electronic countermeasures. Multistatic radars also have some inherent advantages. For example spatial distribution of the nodes of the network enables the area to be tailored according to the specific application of interest. Additionally, it is possible to increase sensitivity, as more of the scattered energy (in the different directions) can be collected and hence detection performance may be improved. Target classification and recognition can also be enhanced, as the target is observed from different perspectives. Multistatic radars n

Corresponding author. E-mail addresses: [email protected] (V. Amanipour), [email protected] (A. Olfat). 0165-1684/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2010.06.003

may also have a counter-stealth capability, since target shaping to reduce target radar cross section (RCS) will, in general, not reduce the bistatic cross section (BCS). Moreover, increased survivability and reliability is achieved because of the option of having ‘‘silent’’ or passive operation of the receivers. The receiving systems are undetectable and they are also potentially simple and cheap. They can improve the location accuracy of possible jammers by fusing the information from the network nodes. Finally, if a single node of the network is lost it can still provide a level of (reduced) performance and the network is said to exhibit graceful degradation [1–3]. On the other hand, increasing the number of transmitters comes with an increase in the cost of the radar system. High power transmitters are generally very expensive. An important question is if an expensive high power transmitter may be replaced by a number of lower power transmitters without losing the accuracy of detection. The important fact is that the economical price of a transmitter grows faster than linear with its power. Lower power transmitters are ‘‘much’’ cheaper than the high power transmitters. The system we would like to model in this paper is a system that consists of n transmitters and one receiver. In order to compare this system with the monostatic radar,

V. Amanipour, A. Olfat / Signal Processing 91 (2011) 28–37

we assume that the transmitted waveforms are the same. In comparison with a fixed monostatic radar, these n transmitters are assumed to be the same, except for their power. They are assumed to have a fixed fraction of the power of the monostatic transmitter. One difficulty in dealing with multistatic radar was the lack of a CFAR detection algorithm in the literature. The earlier work only includes the case where the radar system includes n spatially distributed detectors, or n receivers, such that each detector sends its information about the presence or absence of the target to a data fusion center. More specifically, among the earlier works on this subject we should mention that Barkat and Varshney [4,5], have developed the theory of decentralized CFAR detection. Elias-Fuste et al. [6] have extended the data fusion rule [5]. The performance of distributed CFAR detection in homogenous and non-homogenous backgrounds are analyzed in [7–9]. Liu et al. [10–12] proposed a novel data fusion by using genetic algorithm. For a monostatic radar CFAR algorithms are studied extensively [13–15] and also adaptive robust detectors are introduced [16]. Sheikhi and Zamani [17] have developed a CFAR detector for MIMO radars when the transmitters are assumed to transmit orthogonal waveforms. This assumption reduces the system of n transmitters and m receivers to mn independent bistatic radars, which may basically be studied independently. Their CFAR algorithm is a generalization of cell averaging CFAR algorithm which is simulated only in homogenous background. The main theoretical achievement of this paper is the introduction of a family of CFAR detection algorithms (called homogenous CFAR algorithms of order one) for a multistatic radar. In fact, we show that a family of statistics, called homogenous statistics of order one, may be used in our algorithm so that the result has the CFAR property. We then give some theoretic indications that one particular statistic should have a relatively better performance. We simulate the performance of a multistatic radar composed of n transmitters and one receiver using these CFAR detection algorithms and compare it with the performance of a monostatic radar as the power ratio varies. In particular, when there are three transmitters and the power ratio is equal to 1/3, or 1/2 we compare the performances in both homogenous and non-homogenous backgrounds for proposed algorithms. In reasonable signal to noise ratio the performance of the multistatic radar is better, even for power ratio equal to 1/3. Moreover we show, for a fixed signal to noise ratio, how the detection probability increases with the growth of power ratio. The remainder of the paper is organized as follows. In Section 2, the problem is formulated and we introduce our CFAR algorithms for detection, which includes a cellaveraging CFAR, an order statistic CFAR and a trimmed mean CFAR. Then in Sections 3 and 4 the probability of false alarm for these algorithms are computed. In Section 5, we give an argument why the chosen statistic (the sum of largest returning signals) in the algorithm is a logical choice. Simulation results are presented in Section 6

29

.Finally, Section7 summarizes the main results of the paper. 2. CFAR model We will assume that n transmitters and one receiver form our multistatic radar. Note that using three transmitters the exact location of the target may be detected. The transmitters and receiver are assumed to be synchronized and the square-law detector output for any range cell is exponentially distributed with probability density function (pdf) [11]: f ðxÞ ¼ ð1=2lÞexpðx=2lÞ,

xZ0

ð1Þ

The null hypothesis oH0 of no target in a range cell would mean that l is the total background clutter-plusthermal noise power, which will be denoted by m. Under the alternative hypothesis H1 of presence of a target, l is equal to m(1+ S), where S is the average signal-to total noise ratio (SNR) of a target. In other words, we are assuming a Swerling I model for radar returns from the target and Gaussian statistic for the background. Summarizing the above assumptions we have ( m under H0 l¼ ð2Þ mð1 þ SÞ under H1 We will test windows of N + n cells in order to detect the target. Depending on the locations of the transmitters and the receiver (i.e. the geometry of the multistatic radar) the signals sent by the n transmitters will return from the target in a particular pattern which also depends on the coordinates of the target. In particular, if two signals from two different transmitters are sent simultaneously, some delay between their returns will be observed by the receiver. The number N +n is chosen so that the maximum possible delay between any such two returning signals (and for any geometric location of the target) is at most (N + n)T, where T is the delay proportional to the radar range resolution. The observations in the N +n cells are assumed to be statistically independent, including the cell under test. Let li for i=1,2,y,N +n be the corresponding value of l for the ith cell. If in such a window the target is present (i.e. under H1 hypothesis) for n of the N + n cells we will have li = m(1+ Si) and for the rest of them li = m. Let X1, X2,y,XN + n be the exponential random variables which model the values of the range cells. We will sort these variables in the increasing order to obtain Y1,Y2,...,YN + n. We will assume that the n largest values, i.e. YN + 1,YN + 2,...,YN + n, correspond to the returns from the target. Fix a statistic Y= f (YN + 1,YN + 2,...,YN + n). The rest of Yi in CFAR processor form the statistic Z. A target is declared to be present if Y exceeds a threshold TZ. Here T is a constant scale factor used to achieve a desired constant false alarm probability. The n synchronized transmitters send identical signals simultaneously. So the receiver does not need to distinguish which returning signal is associated with what transmitter. Of course, we may send the signals using orthogonal codes in order to be able to distinguish them in the receiver. Also note that for each returning signal corresponding to a particular transmitter, we could have

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V. Amanipour, A. Olfat / Signal Processing 91 (2011) 28–37

chosen one cell from the window of length N and test all     N N the ... ¼ Nn possible n-component vectors to 1 1 form the ‘‘cell under test’’. So, the complexity and computational cost of the process would significantly increase. However, the proposed algorithm does not force such a complexity (and cost) for forming the ‘‘cell under test’’. Yet, combining both techniques (i.e. using orthogonal codes and the fast algorithm for forming the cell under test) we may locate the precise position of the target if the number of transmitters is more than two. Among the CFAR processors we will examine a class of CFAR processors called homogenous detectors of order one. Examples of such CFAR processors include OS-CFAR, CA-CFAR, TM-CFAR and Censor Mean Level detectors. OSCFAR has a relatively good performance, both in homogenous and non-homogenous environments for monostatic radar, while CA-CFAR is the optimal choice in homogenous environments. We will compare the performance of CA-CFAR, OS-CFAR and TM-CFAR algorithms for multistatic radar against monostatic radar both in homogenous and non-homogenous environments. We call a function f : RN -R homogenous of order p if it has the following homogeneity property:

fðay1 , ay2 ,. . ., ayN Þ ¼ ap fðy1 ,y2 ,. . .,yN Þ

ð3Þ N

If for a given homogenous function f : R -R of order one the statistic Z in the suggested CFAR algorithm is set to be Z =Zf = f(Y1,Y2,y,YN), the detector will be called a homogenous CFAR detector of order one, corresponding to the function f : RN -R. For example, if Z= f(Y1,Y2,y,YN) =Yk, the detector will be an OS-CFAR detector with parameter k. The block diagrams of CACFAR and general homogenous CFAR detectors of order one are illustrated in Figs. 1 and 2. In order to study the performance of the above processor, we need to compute the false-alarm probability and detection probability for an appropriate function f. We will see in the up-coming sections that an arbitrary

Fig. 2. Block diagram of proposed homogenous CFAR detector of order one.

choice of a homogenous function f : Rn -R of order one will result in a CFAR detector. It is important noting that OS-CFAR, CA-CFAR, TM-CFAR, and CMLD are all examples of such homogenous CFAR algorithms. Here is a list of homogenous functions corresponding to these detectors: OS-CFAR : fðy1 ,. . .,yN Þ ¼ yk CA-CFAR : fðy1 ,. . .,yN Þ ¼ y1 þ    þ yN TM-CFAR : fðy1 ,. . .,yN Þ ¼ yT1 þ 1 þ    þyNT2 CMLD : fðy1 ,. . .,yN Þ ¼ y1 þ    þ yk1 þ ðN þ 1kÞyk

3. False-alarm probability for proposed OS-CFAR In order to compute the false-alarm probability in OS-CFAR case, which is the easiest case to analyze, we will assume that the target is not present (i.e. we are under H0). This means that the random variables X1, X2,y,XN + n are all exponential random variables with parameter m, and independent from each other. The probability of false alarm is computed by the formula Z 1 Pfa ¼ EZ fP½9Y 4TZ9H0 g ¼ P½Y 4 Tz fZ ðzÞ dz ð4Þ 0

where fZ is the probability density function for the random variable Z. For the OS-CFAR processor with parameter k we have Z = Yk. The probability density function for this random variable is

Fig. 1. Block diagram of proposed CA-CFAR detector. The returning signals from the target are n of the N + n values X1, X2,yXN + n, which would be the n largest values YN + 1, YN + 2,y,YN + n under the assumption of the algorithm.

fk ðzÞ ¼ fZ ðzÞ  Nk þ n þ 1   k1   k Nþn z z ¼ 1exp exp 2m 2m 2m k ð5Þ

V. Amanipour, A. Olfat / Signal Processing 91 (2011) 28–37

parameter m such that

And using (5) the probability of false alarm is 3

2

31

 Z 1 Z1 Nþn 7 6 Pfa ¼ k ½expðtÞNk þ n þ 1 ½1expðtÞk1 4 fY ðyÞ dy5 dt k 0 2mtT

ð6Þ where fY(y) is the probability density function for the random variable Y= f (YN + 1,YN + 2,y,YN + n). In order for this problem to be CFAR, the value of this integral should be independent of m. We claim that for any homogenous function f : Rn -R of order 1, the probability of the false alarm given by Eq. (4) is independent of m. Examples of homogenous functions of order 1 that could possibly be used, are given below:

fY ðaÞ ¼ ð1=mÞffix ða=mÞ

ð11Þ

From this formula we may compute Z1

Z1

fY ðyÞ dy ¼

2mtT

1

m

ffix

  y

m

dy

Z1

x ¼ y=m

¼

2mtT

ffix ðxÞ dx

ð12Þ

2tT

If we replace (12) in (6), we obtain Pfa ¼ k



Nþn

Z

k

1 0

2 ½expðtÞNk þ n þ 1 ½1expðtÞk1 4

Z1

3 ffix ðxÞ dx5 dt

2tT

ð13Þ Clearly this last expression is independent of the parameter m, and the claim is proved.

f1 ðy1 ,. . .,yn Þ ¼ A1 y1 þA2 y2 þ    þ An yn f2 ðy1 ,. . .,yn Þ ¼ ðA1 y1 p þ A2 ðy2 y1 Þp þ    þAn ðyn yn1 Þp Þ1=p

4. False-alarm probability for proposed CA-CFAR

f3 ðy1 ,. . .,yn Þ ¼ maxfy1 ,y2 y1 ,. . .,yn yn1 g

In this section we will consider homogenous CFAR detectors of order one. In order to compute the falsealarm probability for CA-CFAR or more generally for a homogenous CFAR algorithm, again we assume that the target is not present. Once more, the random variables X1, X2,y,XN + n are all independent exponential random variables with parameter m. For the CA-CFAR processor, in equation (4) we should N P set Z ¼ Yk . For a general linear CFAR algorithm which

ð7Þ

Define the random variables W1,W2,y,Wn as follows: W1 ¼ YN þ 1 W2 ¼ YN þ 2 YN þ 1 ^ Wn ¼ YN þ n YN þ n1

ð8Þ

Since X1, X2,y,XN + n are independent exponential random variables with a fixed parameter (under H0 hypothesis), the ordered statistics Y1,Y2,y,YN + n are not independent random variables. However, the random variables W1,W2,y,Wn are independent positive random variables and their probability density functions are given by   N þn n   n   N N x x fW1 ðxÞ ¼ exp 1exp 2m 2m 2m   1 x ¼ g1 m m   n þ 1i ðn þ 1iÞx exp fWi ðxÞ ¼ 2m 2m   1 x , i ¼ 2,3,. . .,n ð9Þ ¼ gi

m

m

The functions g1, g2,y,gn are clearly independent of the parameter m. Note that the function f(y1,y2,y,yn) may be written as g(y1,y2  y1,y,yn  yn  1), where g is also a homogenous function of n variables. For any value a we can compute ZZ Z fY ðaÞ ¼ . . . fW1 ðy1 ÞfW2 ðy2 Þ. . .fWn ðyn Þ dA gðy1 ,:::,yn Þ ¼ a

ZZ

Z

...

¼

1

  y1

m

m

g n 1

g2

    y2 yn . . .gn dA

m

m

gðy1 ,...,yn Þ ¼ a

¼

1

m

ZZ

Z

:::

g1 ðs1 Þg2 ðs2 Þ. . .gn ðsn Þ dAu

ð10Þ

gðs1 ,:::,sn Þ ¼ a=m

Here si = yi/m is a change of variable, and dA, dA0 are the volume forms for the hyper-surfaces g(y1,y,yn)= a and g(s1,y,sn) = a/m, respectively. This means that there is a probability density function ffix, independent of the

k¼1

corresponds to the linear functionf : RN -R, we should consider Zf = f(Y1,Y2,y,YN) instead of Z. Note that we have Z ¼ NY1 þ

N X

ðN þ 1iÞðYi Yi1 Þ ¼

i¼2

N X

Vi

ð14Þ

i¼1

where in this equation Vi are independent exponential random variables defined as V1 ¼ NY1 Vi ¼ ðN þ 1iÞðYi Yi1 Þ,

i 41

ð15Þ

Similarly, if we define the linear function c : RN -R by

cðNx1 ,ðN1Þðx2 x1 Þ,. . .,xN xN1 Þ ¼ fðx1 ,. . .,xN Þ

ð16Þ

we may check that Zf = c(V1,V2,y,VN). The probability density functions for the random variables Vi will be     1 N þn þ1i ðN þn þ1iÞ exp x fVi ðxÞ ¼ 2m N þ 1i 2ðN þ 1iÞm   bi bi exp  x ð17Þ ¼ 2m 2m The density function for the random variable Zf would then be computed by the following integral: ZZ Z ff ðzÞ ¼ fZf ðzÞ ¼ . . . fV1 ðx1 Þ:fV2 ðx2 Þ. . .fVN ðxN Þ dA cðx1 ,...,xN Þ ¼ z

ZZ ¼

Z ...

cðx1 ,...,xN Þ ¼ z

! N 1 X exp  b x dA 2m i ¼ 1 i i ð2mÞN Q

bi

32

V. Amanipour, A. Olfat / Signal Processing 91 (2011) 28–37

¼

1

m

ZZ

Q

Z ...

N

2

cðx1 ,:::,xN Þ ¼ z

¼

1

m

ZZ

bi

Q

Z

... cðs1 ,:::,sN Þ ¼ z=m

exp 

bi

2N

 ! N 1X x dA bi i 2i¼1 m mN1

!   N 1X 1 z exp  bi si dAu ¼ hf 2i¼1 m m ð18Þ

where we have done a change of variable si = xi/m and dA, dA’ are the volume forms for the hyper-surfaces c(x1,y,xn) =z and c(s1,y,sn)= z/m, respectively. The functions hf(x) do not depend on the noise parameter m and depend only on the coefficients bi,i=1,y,N and the numbers N,n (and of course, on the linear function f). Using these formulas for ff(z)we can obtain the probability of false alarm in homogenous CFAR algorithm of order one, corresponding to the function f : RN -R: 2 3 Z 1 Z1 Z 1 Z1 6 7 Pfa ¼ hf ðtÞ4 fY ðyÞ dy5 dt ¼ hf ðtÞffix ðsÞ ds dt ð19Þ 0

2mtT

0

2tT

Here fY(y) is the probability density function for the random variable Y= f(YN + 1,YN + 2,y,YN + n), and ffix(s) is defined as before. Clearly, this integral is independent of the noise parameter. These computations show that if in the algorithm we use any linear combination of random variables Y1,Y2,y,YN to define the statistic Z, the result would be a CFAR algorithm. This proves the CFAR condition for CACFAR, TM-CFAR and CMLD, and more generally for any homogenous CFAR detector of order one.

S1,S2,y,Sn become larger, the subset A of H1 becomes bigger. In order to guess the best statistic Y = f (YN + 1,y,YN + n) we will focus our attention to this subset, where the computation is much simpler, i.e. we will compute ~ 1 ,y2 ,. . .,yn Þ ¼ Lðy

f-

ðy1 ,y2 ,. . .,yn Þ

f-

ðy1 ,y2 ,. . .,yn Þ

ðY 9AÞ

ðY 9H0 Þ

Since we are not able to distinguish the values S1,S2,y,Sn we make a further assumption that S1 ¼ . . . ¼ Sn ¼ S. Under these assumptions, there will be a choice of a set I =(i1 oi2 o?oin) of n indices such that {Xi1,Xi2,y,Xin} ={YN + 1,YN + 2,y,YN + n}. Let us denote the -

vector (Xi1,Xi2,...,Xin) by XI . The set A may be written as a disjoint union of the sets AI of events, where AI is the

KI ðr1 ,r2 ,. . .,rn Þ ¼

f-

ðr1 ,r2 ,. . .,rn Þ

f-

ðr1 ,r2 ,. . .,rn Þ

ðX I 9AI Þ

ðX I 9H0 Þ

Lðy1 ,y2 ,:::,yn Þ ¼

fððYN þ 1 ,YN þ 2 ,...,YN þ n Þ9H0 Þ ðy1 ,y2 ,. . .,yn Þ

Then, for any given threshold t, we should describe the inequality L (y1,y,yn)4 t as a relation of the form f (y1,y,yn)4 f(t) for an explicit function f (y1,y,yn) and some function f(t) of the threshold t. The statistic Y =f (YN + 1,YN + 2,y,YN + n) will then be an optimal statistic in our algorithm. Computing the above likelihood ratio L (y1,y,yn) is quite complicated for all possible events. However, on a relatively large subset of all events the outcome is relatively simple. Let A be the subset of H1 consisting of -

the events where Y ¼ ðYN þ 1 ,YN þ 2 ,. . .,YN þ n Þ corresponds to the signals returning from the target. As the values of

fðXi

ð2mÞn exp  ¼

9AÞ ðr1 Þ. . .fðXin 9AÞ ðrn Þ

1

fðXi

1

9H0 Þ ðr1 Þ. . .fðXin 9H0 Þ ðrn Þ

n P i¼1

!

ri 2mi

exp

n Q P ri 2 ð mi Þexp  2m

S 2mð1 þ SÞ

!¼

n

n P i¼1

ð1 þ SÞn

! ri ð23Þ

i¼1

Note that KI Z t if and only if n X

ri r 2mn

i¼1

1þS logðð1 þ SÞtÞ ¼ gðtÞ S

ð24Þ

Then we have f

-

ðY 9AI Þ

ð20Þ

ð22Þ

We have just changed the variables y1,y2,y,yn from Eq. (21) to r1,y,rn since these variables are no longer subject to the condition y1 oy2 o?oyn. We have KI ðr1 ,r2 ,. . .,rn Þ ¼

fððYN þ 1 ,YN þ 2 ,...,YN þ n Þ9H1 Þ ðy1 ,y2 ,. . .,yn Þ

-

-

subset of A where Y corresponds to XI . Under both hypothesis H0 and A, the random variables Xij are independent exponential random variables. In case of the hypothesis H0 the parameters of all of them are m and in the second case (under the hypothesis A) the parameters of these exponential random variables are equal to mi = m(1+ S), i= 1,y,n. We may thus define the following expression in place of

5. An optimal choice of the function f We would like to choose the function f (y1,y2,y,yn) so that the probability of detection, with a fixed probability of false alarm is maximized. We will need to make some extra assumptions in order to make the calculation possible. Let Xi1,Xi2,y,Xin, i1 oi2 o?o in be the n largest random variables among Xl, l =1,2,y,N +n. This means that as a set {Xi1,Xi2,y,Xin} ={YN + 1,YN + 2,y,YN + n} while the random variables Xij do not have an order, and are consequently independent. In order to use the Neyman–Pearson test [19] for finding the best possible statistic Y = f (YN + 1,YN + 2,y,YN + n) we need to compute the likelihood ratio:

ð21Þ

ðy1 ,. . .,yn Þ r tf -

ðY 9H0 Þ

ðy1 ,. . .,yn Þ

ð25Þ

for some given y1 oy2 o? oyn if and only if for all the subset I= (i1 oi2 o? oin) of indices (corresponding to the n largest values for the random variables) and some re-ordering r1,y,rn of y1,y,yn the following relation is satisfied: f-

ðX I 9AI Þ

ðr1 ,. . .,rn Þ r tf -

ðX I 9H0 Þ

ðr1 ,. . .,rn Þ

ð26Þ

This is equivalent to the simple relation: r1 þ r2 þ    þrn r gðtÞ

ð27Þ

Since r1 + r2 +?+ rn =y1 + y2 + ?+ yn, this is satisfied if we have y1 þ y2 þ    þ yn rgðtÞ

ð28Þ

V. Amanipour, A. Olfat / Signal Processing 91 (2011) 28–37

Thus over the subset A of the events the function f(y1 + ?+yn) =y1 + ?+ yn gives the optimal choice for the statistic Y =f (YN + 1,YN + 2,y,YN + n). 6. Simulation results In this section, we will give through simulations a comparison between the performance of multistatic radar, processed with either of OS-CFAR, CA-CFAR or TM-CFAR algorithms of this paper, and the performance of monostatic radar. We will consider both the homogenous and non-homogenous environments. In case of multistatic radar, we will assume n =3, i.e. there are three transmitters. The transmitters of multistatic radar and monostatic radar are the same except for their power. The power of each transmitter in multistatic radar will be assumed to be a fraction of the power of the monostatic radar. The positions of transmitters are assumed to be at the vertices of a regular triangle, and the receiver would be located at the center of this triangle (see Fig. 3). We will simulate the algorithm using two different functions f : R3 -R. First we consider the following statistic (multistatic 1): Y ¼ f ðYN þ 1 ,YN þ 2 ,YN þ 3 Þ ¼ YN þ 1 þYN þ 2 þYN þ 3 and then we use the second statistic(multistatic 2) defined as Y ¼ f ðYN þ 1 ,YN þ 2 ,YN þ 3 Þ ¼ maxðW1 ,W2 ,W3 Þ ¼ maxðYN þ 1 ,YN þ 2 YN þ 1 ,YN þ 3 YN þ 2 Þ We assume N=24, k=21 and false alarm probability is Pfa =10  4. The parameters of TM-CFAR are set T1 =20 and T2 =1. Setting these parameters, we compute the scale factor T for each of these two statistics (multistatics 1 and 2),

Fig. 3. Multistatic radar with three transmitters and one receiver.

33

and for either of multistatic OS-CFAR, CA-CFAR, and TM-CFAR algorithms, as well as for the monostatic radar. Throughout the simulations, a Swerling I model for fluctuating point target is used and for a comparison between multistatic radar and monostatic radar we assume that radar cross section (RCS) of target and bistatic cross section (BCS) of target are the same. 6.1. Homogenous background Figs. 4 and 5 show detection probability versus signal to noise ratio of target in homogenous background when the power of multistatic transmitters is 1/3 of the power of monostatic transmitter. In Fig. 6 power ratio is set to 1/2. The SNR quoted in the figures refers to the signal to noise ratio assumed for the monostatic radar. The corresponding SNR for the multistatic radar is typically much smaller. In fact according to the radar equations in bistatic and monostatic cases [18], the ratio of SNRmult(i) (the value of SNR for the ith transmitter in multistatic radar) to SNRmono (the value of SNR for the monostatic radar)can be expressed as SNRmult ðiÞ R4 R2 ¼ PRðiÞ 2 2 ¼ PRðiÞ 2 SNRmono RR RT ðiÞ RT ðiÞ

ð29Þ

where PR(i) is the power ratio between the ith multistatic transmitter and the monostatic transmitter, RR = R is the distance between the target and the receiver and RT(i) is the distance between the ith transmitter and the target. Thus, according to (29), the average SNR of multistatic radar (compared to monostatic radar) is proportional to 2 power ratio PR and the average of R RðiÞ2 . The average of T R2 depends on the geometry of the transmitters and the RT ðiÞ2 receiver. For the geometry shown in Fig. 3, that is used in 2 our simulations the average of R RðiÞ2 is approximately 0.6 T which was computed by the Monte-Carlo method. So, in the our simulations, for power ratios PR(i) equal to 1/3

Fig. 4. Detection performance of monostatic radar versus multistatic radars with power ratio 1/3.

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more transmitters, in comparison with both OS-CFAR and TM-CFAR. More importantly, using multistatic 1 and CA-CFAR algorithm, the performance of the multistatic radar with power ratio 1/3 becomes better than the monostatic radar when signal to noise ratio for the monostatic radar is 18 dB or more, while signal to noise ratio for the multistatic radar is more than just 6 dB. Multistatic radar with power ratio 1/2 (see Fig. 6) performs quite better than the monostatic radar when signal to noise ratio for the monostatic radar bigger than 14 dB (and this ratio for the multistatic radar is thus slightly over 7 dB), which is quite impressive. In Figs. 7 and 8, we illustrate detection probability versus power ratio when signal to noise ratio is 20 dB. When the power of the transmitters in the multistatic

Fig. 5. Detection performance of monostatic radar versus multistatic radars with power ratio 1/3.

Fig. 7. Detection performance of multistatic radars as the power ratio between multistatic and monostatic radar is between 10% and 60%. Fig. 6. Detection performance of monostatic radar versus multistatic radars with power ratio 1/2.

and 1/2 , the SNRmult is approximately 9.2 and 7.4 dB less than the SNRmono, respectively. This means the range of SNR in Figs. 5 and 6 (which is 15–30 dB for SNRmono) is equivalent to 5.8–20.8 dB for SNRmult. As we expected from the computations of the Section 5, the performances of CA-CFAR, TM-CFAR and OS-CFAR algorithms are better when multistatic 1 is used in the simulations. Some other homogenous functions of degree one have been tested as well and the simulations supported our claim that the sum of three largest signals is the optimal choice for the statistic Y in the algorithms. The performance of CA-CFAR and OS-CFAR algorithms do not have a significant difference for monostatic radar, while both of these algorithms have better performance in comparison with TM-CFAR. Note that it is well-known that CA-CFAR has the optimal performance in homogenous background. Moreover, CA-CFAR shows a notable superiority in performance when the radar system has

Fig. 8. Detection performance of multistatic radars as the power ratio is between 10% and 60%.

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radar is at least 25% of the power of the transmitter of monostatic radar, detection probability becomes larger for multistatic radar. The high probability of detection in low SNR is what should be regarded as a measure for a good and satisfying performance. As these figures suggest, starting from power ratio 1/2 the performance of the multistatic radar corresponding to multistatic 1 becomes better than the performance of a monostatic radar. When the power ratio is raised to 2/3 the probability of detection using a multistatic radar, even in low SNR, is almost 10% better than the probability of detection by a monostatic radar. 6.2. Non-Gaussian background In the study of a non-Gaussian background, we have also considered a clutter with heavy tailed distribution in Fig. 9. More precisely, the distribution of the background noise is assumed to be Weibull with shape parameter equal to 0.5, and the performance of multistatic and monostatic radars are compared. As it may be observed from Fig. 9, the performance of the multistatic radar in a Weibull clutter is similar to the performance of a monostatic radar, while the performance becomes better when the signal to noise ratio increases. 6.3. Non-homogenous background

Detection performance in Weibull clutter: N = 24, Pf = 1e-4

1

þ    þ PðFalse9An ÞP ðAn Þ

0.8 0.7 0.6 0.5 0.4 multistatic1-sum (os) multistatic2-max (os) monostatic (os) multistatic1-sum (ca) multistatic2-max (ca) monostatic (ca)

0.2 0.1 0 10

12

14

16

18 20 22 SNR (db)

24

26

28

30

Fig. 9. Detection performance of monostatic radar versus multistatic radars in Weibull clutter.

ð30Þ

where Ai is the event that i of the n cells YN + 1,YN + 2,y,YN + n are in the clutter edge. The total probability of the false alarm in the monostatic case may be computed from the following observation. When we test the ith cell   Xi 2 X1 ,X2 ,. . .,XN þ n in the monostatic radar, a threshold value ti = t(X1,y,Xi  1,Xi + 1,y,XN + n) will be computed and if Xi is larger than ti we will have a false alarm. Note that if Xi 4Xj we will have ti o tj. This implies that if after testing Xj a false alarm is declared, the same would happen after testing Xi. Thus the total probability of a false alarm is the probability of false alarm for the largest value YN þ n 2 fX1 ,X2 ,. . .,XN þ n g. Pfa ðMonoÞ ¼ Pfa ðYN þ n Þ

0.9

Detection Probability

modeled by exponential random variables with parameter m0 and in 24 m cells modeled by exponential random variables with parameter m1, with m1 4 m0. The target may be either in or outside the clutter region, and the returning signals may have parameters m1(1+ Si) or m0(1+ Si) accordingly. The expected noise parameter is estimated by examining the values of m random variables with parameter m0 and 24  m random variables with parameter m1, so the estimated parameter is expected to be some value m0 o mexp o m1 . If the cells under test correspond to the clutter region (i.e. their noise parameter is m1) both the probability of detection and the probability of false alarm will be increased, when compared with the homogenous background, since the actual noise parameter is larger than the estimated noise parameter mexp . As the number of cells in the clutter region increase, the value of mexp becomes closer and closer to m1, the false alarm probability will be decreased, and will converge to the false alarm probability in homogenous background. If the cell under test is outside the clutter region both the detection and false alarm probability will become less than the homogenous case, by a similar reasoning. In order to compute the probability of false alarm in the multistatic case, we will need to use the following formula for conditional probability: Pfa ðMultiÞ ¼ PðFalse9A1 ÞPðA1 Þ þ PðFalse9A2 ÞPðA2 Þ

In non-homogenous background, it is expected that OS-CFAR and TM-CFAR algorithms have better performance for monostatic radar when compared with CA-CFAR algorithms. In this section, we report our simulations in such backgrounds using our multistatic OS-CFAR, TM-CFAR and CA-CFAR algorithms. We will simulate two types of non-homogenous background. The first case is when reference window contains a clutter edge. In this case, we assume that n = 3, N = 24, k= 21, T1 = 20 and T2 = 1 as before. The noise is supposed to appear in our reference window as m cells

0.3

35

ð31Þ

Clearly, if the clutter edge is large and the number of cells in clutter is very small (a very unlikely event), it is natural that the radar, with either monostatic or multistatic structure, gets confused. Yet, our simulations using the above formulas, which are illustrated in Fig. 10, indicate that the performance of the multistatic radar under the assumption of a clutter edge is almost as good as the monostatic radar. In simulations shown in Figs. 11 and 12 a clutter edge of 10 dB is assumed (typically larger than the signal to noise ratio assumed for the radar in homogenous background). Fig. 13 illustrates the changes in detection probability as the number of cells in the clutter region increases, for different CFAR algorithms and fixed signal to noise ratio. When the clutter region is small, OS-CFAR has a relatively better performance for monostatic radar, while the performance of our multistatic CA-CFAR algorithm is

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false alarm in 10dB clutter edge: N = 24, Pf = 1e-4

Log (False alarm Probability)

0

-0.5

-1

-1.5 multistatic1-sum (os) multistatic2-max (os) monostatic (os) multistatic1-sum (ca) multistatic2-max (ca) monostatic (ca)

-2

-2.5 1

2

3

4 5 6 7 8 9 No. of cells in clutter edge

10

11

12

Fig. 10. Total false alarm probability versus the number of cells in the clutter region for 10 dB clutter edge.

Fig. 11. False alarm probability versus the number of cells in the clutter region for 10 dB clutter edge and CUT in the clutter region.

optimal for multistatic 1. Again, using TM-CFAR does not have any advantages. The second type of non-homogenous background considered in this paper is the presence of an interfering target in the reference window. We will assume that in our window of 24 cells, we will have the signals returning from one interfering Swerling I target, and the interference to signal ratio is 0.5 dB. Fig. 14 illustrates the performances of monostatic and multistatic radars (power ratio is 1/3) when OS-CFAR, TM-CFAR and CA-CFAR algorithms are used. It can be seen than monostatic OS-CFAR performs better than monostatic CA-CFAR, while for multistatic radar the probability of detection is better for CA-CFAR processing using multistatic 1. Once more, there is no advantage in using TM-CFAR. As signal to noise ratio becomes larger than 18 dB, multistatic radar begins to show better performance in this type of nonhomogenous background.

Fig. 12. False alarm probability versus the number of cells in the clutter region for 10 dB clutter edge and CUT in the clutter region.

Fig. 13. Probability of detection versus number of cells in the clutter region for 10 dB clutter edge and CUT outside the clutter region.

7. Conclusion We have proposed a class of CFAR algorithms for multistatic radar with n transmitters and one receiver in this paper. The simulation results suggest that the CACFAR detection algorithm using a particular statistic (which is the sum of the n largest signals in a test window) has the best performance. In both homogenous and non-homogenous environments we have compared the performance of multistatic and monostatic radars. When the power of each transmitter in the multistatic radar is a fraction of the power of the transmitter in the monostatic radar, our simulations indicate that the detection probability is better in multistatic radar if the signal to noise ratio is large enough. In particular, for three transmitters and power ratio equal to 1/3 and 1/2 we have observed that in reasonable SNR multistatic radar has better performance. Similar results are obtained

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Fig. 14. Detection probability versus signal to noise ration in presence of one interfering target with I/S= 0.5 dB.

in non-homogenous background. These observations strongly indicate that besides all the advantages of using multistatic radar, it is economical to replace an expensive high power transmitter by several lower power transmitters without losing on performance of radar system. References [1] H.D. Griffiths, Bistatic and Multistatic Radar, Military Radar Seminar, 7 September 2004. Available from: /http://www.iee.org/ oncomms/pn/radar/Griffiths%20Shrivenham.pdfS. [2] C.J. Baker, H.D. Griffiths, Bistatic and multistatic radar sensors for homeland security, 2005. Available from: /http://www.nato-asi. org/sensors2005/papers/baker.pdfS.

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[3] W. Beide, The nature of bistatic and multistatic radar, IEEE Proceedings, International Conference on Radar (2001) 882–884. [4] M. Barkat, P.K. Varshney, Decentralized CFAR signal detection, IEEE Transactions on Aerospace and Electronic Systems 25 (2) (1989) 141–149. [5] M. Barkat, P.K. Varshney, Adaptive cell-averaging CFAR detection in distributed sensor networks, IEEE Transactions on Aerospace and Electronic Systems 27 (3) (1991) 424–429. [6] A.R. Elias-Fuste, A. Broquetas-Ibars, J.P. Antequera, J.C.M. Yuste, CFAR data fusion center with inhomogeneous receivers, IEEE Transactions on Aerospace and Electronic Systems 28 (1) (1992) 276–285. [7] M.K. Uner, P.K. Varshney, Distributed CFAR detection in homogeneous and nonhomogeneous backgrounds, IEEE Transactions on Aerospace and Electronic Systems 32 (15) (1996) 84–91. [8] G. Jian, H. You, P. Ying-Ning, Distributed CFAR detector based on local test statistic, Signal Processing 80 (February 2000) 373–379. [9] H.A. Meziani, F. Soltani, Performance analysis of some CFAR detectors in homogeneous and non-homogeneous Pearsondistributed clutter, Signal Processing 86 (August 2006) 2115–2122. [10] W. Liu, Y. Lu, J.S. Fu, Data fusion of multiradar system by using genetic algorithm, IEEE Transactions on Aerospace and Electronic Systems 38 (2) (2002) 601–612. [11] W. Liu, Y. Lu, J.S. Fu, A novel threshold optimization for distributed OS-CFAR of multistatic radar systems by using genetic algorithm, IEEE Proceedings, IEEE Radar Conference (2001) 275–278. [12] W. Liu, Y. Lu, J.S. Fu, CFAR data fusion of multistatic radar system under homogeneous and nonhomogeneous backgrounds, IEE Proceedings, IEE Radar Conference (2002) 248–252. [13] P.P. Gandhi, S.A. Kassam, Analysis of CFAR processors in nonhomogeneous background, IEEE Transactions on Aerospace and Electronic Systems 24 (4) (1988) 427–445. [14] H. Rohling, Radar CFAR thresholding in clutter and multiple target situations, IEEE Transactions on Aerospace and Electronic Systems AES-19 (4) (1983) 608–621. [15] Amar Mezache, Faouzi Soltani, A novel threshold optimization of ML-CFAR detector in Weibull clutter using fuzzy-neural networks, Signal Processing 87 (September 2007) 2100–2110. [16] Muralidhar Rangaswamy, Freeman C. Lin, Karl R. Gerlach, Robust adaptive signal processing methods for heterogeneous radar clutter scenarios, Signal Processing 84 (September 2004) 1653–1665. [17] A. Sheikhi, A. Zamani, Coherent detection for MIMO radars, IEEE Radar Conference (2007) 302–307. [18] N.J. Willlis, in: Bistatic Radar, Artech House, 1991. [19] N. Levanon, in: Radar Principles, John Wiley & Sons, 1988.