CFAR Processors in Pulse Jamming*
CFAR Processors in Pulse Jamming* Ivan Garvanov1, Vera Behar2 and Christo Kabakchiev1 1 Institute
of Information Technologies, Bulgarian Academy of Sciences “Acad. G. Bonchev” Str., bl. 2, 1113 Sofia, Bulgaria e-mail: [email protected], [email protected] 2 Dep. of Biomedical Engineering, Institute of Technology, Haifa 3200 e-mail: [email protected]
Abstract
In this paper we study the efficiency of CA CFAR BI, EXC CFAR BI and Adaptive CFAR PI detectors in strong pulse jamming. We achieve new results for the average decision threshold (ADT) using the minimum detectable signal (Pd=0.5). For comparison we use also the approach with Monte-Karlo simulation for estimation of the ADT of the studied CFAR detectors. Differently from other authors, we consider the entire range (0 to 1) of the probability for the appearance of pulse jamming in range cells.
1
Introduction
Cell-Averaging Constant False Alarm Rate (CA CFAR) signal processing proposed by Finn and Johnson in [1]is often used for radar signal detection. The detection threshold is determined as a product of the noise level estimate in the reference window and a scale factor to achieve the desired probability of false alarm. The presence of strong pulse jamming (PJ) in both, the test resolution cell and the reference cells, can cause drastic degradation in the performance of a CA CFAR processor as shown in [12]. In such situations it would be desirable to know the CFAR losses, depending on the parameters of PJ, for rating the radar behavior. There are two approaches for the calculation of CFAR losses offered by Rolling and Kassam in [2,3]. The conventional method, used in [5,7-12], is to compute the additional SNR needed for the CFAR processing scheme beyond that for the optimum processor, to achieve a fixed detection probability (e.g. 0.5). For a particular CFAR scheme losses obviously vary with detection probability. Alternatively, the authors in [2,3] use another criterion based on the average decision threshold (ADT), since the threshold and the detection probability are closely related to each other. Then the difference between the two CFAR systems is expressed by the ratio between the two ADTs measured in dB, as shown in [2,3] The false alarm rate of the postdetection integrator (PI) is extremely sensitive to pulse jamming, and the binary integrator (BI) which uses a K-out-of-M decision rule is insensitive to at most (K-1) interfering pulses [7]. For keeping of constant false alarm rate in PJ the CA CFAR processor presented in [9,12]is used. But this method is not as effective as the conventional method for the calculation of CFAR losses. For the minimization of CFAR losses in case of pulse jamming *
This work is supported by IIT – 010044, MPS Ltd. Grant “RDR” and Bulgarian NF “SR” Grant ¹¿ I – 902/99.
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postdetection integration (PI) or binary integration (BI) is implemented in CFAR processors as shown in [5,8,10]. The use of excision CFAR detectors, supplemented by a postdetection integrator or a binary integrator as shown in [6,7,10], increases the CFAR losses. Minimum CFAR losses in PJ are obtained in [5,11] with a CFAR adaptive postdetection integration (API) processor with adaptive selection on PJ in reference windows and appriory selection in test windows as shown in [5], and adaptive censoring in reference and test windows as presented in [11]. We assume in this paper that the noise in the test cell is Rayleigh envelope distributed and target returns are fluctuating according to Swerling II model as in [3,5]. As a difference from the authors in [5], we assume that the samples of PJ are distributed according to the compound exponential law, where weighting coefficients are the probabilities of corrupting and non-corrupting of the samples. Differently from [7-11], we consider the entire range (0 to 1) of the probability for the appearance of pulse jamming in range cells. For values of the weighting coefficients higher than 0.3, the Poisson process model is used, but it is rough [15]. The binomial distribution is correct in this case. In this paper we study different CFAR techniques in the presence of strong pulse jamming, similarly to [5]. We use the average decision threshold (ADT) approach for comparison of the processors. The analytical expressions for the probability functions of CA CFAR BI, EXC CFAR BI and API CFAR detectors are achieved in [7, 8, 10, 11]. We achieve in these paper new results for the ADT using the SNR approach. The SNR of the minimum detectable signal (Pd=0.5) is approximately the same as the ADT of each CFAR system. For comparison we use also the approach with Monte-Karlo simulation for estimation of the ADT of the studied CFAR detectors. The experimental results show that the API CFAR processors are most effective for values of the probability of appearance of pulse jamming in the interval (0 to 0.5). For ε0 > 0.5 we recommend binary integration after the CFAR processor.
2
Performance of CA CFAR BI and excision CFAR BI processors in the presence of pulse jamming
2.1 Probability of detection and false alarm of CA CFAR BI detectors After filtration the signal is applied to a square-law detector and sampled in range by the (N+1) resolution cells resulting in a vector of (N+1) observations. The sampling rate is such that the samples are statistically independent. It is assumed that L broadband pulses hit the target and one sample of the cell under test suffices to detect a pulse of target returns. In conditions of pulse jamming the background environment includes the random interfering pulses and the receiver noise. Therefore the samples surrounding the cell under test (a reference window) may be drawn from two classes. One class represents the interference-plus-noise situation, which may appear at the output of the receiver with the probability ε0 . This probability can be expressed as ε0 = t c F j , where F j is the average repetition frequency of PJ and t c is the length of pulse transmission. The other class represents the noise only situation,
2
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which may appear at the outputs of the receiver with the probability ( 1 + ε0 ). The set of samples from the test resolution cell at the input of the binary integrator (xl, where l=1,..,L) is assumed to be distributed according to Swerling II case [9, 15] with the probability density function (pdf) given by:
f () x =
−x 1 − ε0 ε0 −x exp + exp ( ) λ0 ( 1 + s ) λ0 ( 1 + s ) λ0 ( 1 + rj + s ) λ 1 + r + s 0 j
(1)
where λ0 is the average power of the receiver noise, r j is the average interference-tonoise ratio (INR) of pulse jamming, s is the per pulse average signal-to-noise ratio (SNR) and N is the number of observations in a reference window. In this case the Poisson process model is used, and it is valid only for ε0 ≤0.3. The probability density function (pdf) of the reference window outputs can be defined as (1), setting s =0. All samples (xl)L are compared with the threshold HD according to the rule:
1, if x l ≥ H D Φ l (x l )= 0, otherwise
(2)
The binary integrator performs summing of L decisions Φ l . The target radar image detection is declared if this sum exceeds the second threshold M.
H1: (t arg et is present ) , if H 0 :( no t arg et ),
∑
L l =1
Φl ≥M
(3)
otherwise
In a conventional CA CFAR detector the noise level estimate is formed as a sum of all N the outputs of the reference window: V = x . In this case the mgf of the estimate
∑
i =1
i
V is defined to be M V (U ) = M xN (U ) , where M x (U ) is the mgf of the random variable
xi .The mgf of the estimate V M V (U ) = ∑
i =1
is obtained in [9]:
C ε( 1 − ε0 ) i i N 0 N −i
N
N −i
i (1 + λ0U ) ( 1 + λ0 ( 1 + rj ) U)
(4)
According to (3) the probability of target detection for CA CFAR BI processor as in [10] is computed by the expression: L
L −l
PD = ∑ C Ll Pd 1 ( 1 − Pd 1 ) l
(5)
l =M
where
T T + ε0 M V Pd1 = ( 1 − ε0 )M V λ ( 1 + s λ 1 + r + s ) 0 j 0 The probability of false alarm is evaluated by (5, 6, 4), setting s = 0 .
(
)
(6)
3
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2.2 Probability of detection and false alarm of excision CFAR BI detectors In an excision CFAR BI processor the noise level estimate V is formed as an
{yi }N ,
average mean of nonzero samples at the output of the excisor V=
1 k
K
∑y
i
that is:
. According to [6] the operation of the excisor is defined as follows:
i =1
x : xi ≤ B E yi = i 0 : othewise
(7)
where B E is the excision threshold.
The probability that a sample xi survives at the output of the excisor, is given as: (8) P =1 − ( 1 − ε )exp (− B / λ )− ε exp − B / λ 1 + r E
E
0
0
0
(
E
0
( )) j
The probability that k out of N samples of the reference window survive at the output of the excisor is given as: v(k )= C Nk PEk ( 1 − PE ) . In this article we use the moment generating function on the excision CFAR processor from [7] N −k
M V (U ) =
N
∑C
k N
PEk (1 − PE ) N −k M V (U , k )
(9)
k =1
where i
1− exp(R1 − BEU / k) ) (1−ε0 )(1−exp(R2 − BEU / k)) ε( MV (U, k) = ∑ C 0 1−exp(R1 ) )( 1+Uλ0 ( 1+ rj ) / k) 1−exp(R2 ) )(1+Uλ0 / k) i=0 ( ( k
k−i
i k
(10)
and R1 = −B E / λ0 (1 + r j ); R 2 = −B E / λ0 . The probability of target detection for excision CFAR BI processor in [7, 10] is computed by the expression: L
L −l
PD = ∑ C Pd 2 ( 1 − Pd 2 ) l
l L
l =M
(11)
where T T , k + ε M , k 0 V λ 1 + s λ 1 + r + s 0( ) j 0 k =1 The probability of false alarm is evaluated by (11, 12), setting s = 0 . Pd2 =
N
∑ C P (1 − P ) (1 − ε )M
3
k k N E
N −k
E
0
V
(
)
(12)
Performance of API CFAR processors in the presence of pulse jamming
Let us assume that L pulses hit the target, which is modeled according to Swerling II case. The received signal is sampled in range by using (M+1) resolution
4
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cells resulting in a matrix with (M+1) rows and L columns. Each column of the data matrix consists of the values of the signal obtained for L pulse intervals in one range resolution cell. Let us also assume that the first M/2 and the last M/2 rows of the data matrix are used as a reference window in order to estimate the “noise-plusinterference” level in the test resolution cell of the radar. In this case the samples of the reference cells result in a matrix X of the size M*L. The test cell or the target includes the elements of the (M/2+1) row of the data matrix and is a vector Z of length L. The elements of the reference window are independent random variables with the compound exponential distribution law (1), setting s = 0 . In the presence of a desired signal from a target the elements of the test resolution cell are independent random variables with distribution law (1). 3.1 Probability of detection and false alarm of API CFAR detectors We use the adaptive censoring algorithm, proposed by Himonas and Barkat in [4], before pulse-to-pulse integration for censoring of the elements of pulse jamming in the reference window and the test resolution cells, in order to form the detection algorithm. The expression for the probability of target detection for an API CFAR processor is achieved in [11, 14]. The authors in [11, 14] study the probability of target detection only for ε0≤0.5 and calculate only the first member of the expression (13). We calculate the probability of target detection for ε0∈(0;1) and calculate the value of PD by using the following expression: k N N L L l −1 k + i −1 T i ( 1+ s) k N −k l L −l ( ) ( ) PD = ∑ 1 − ε ε 1 − ε ε + ∑ ∑ 0 0 0 0 i k +i k =1 k l =1 l i =0 T +1 + s ) ( N N L −1 k + i −1 k N −k L k −(k +i ) ( 1 − ε ) ε ε Ti( 1 + r j + s )( T +1 + r j + s ) + ∑ ∑ 0 0 0 i k =1 k i =0 l −1 N + i −1 1 + s L (1 − ε0 )l ε0L −l ε0N ∑ T i ∑ l i l =1 i =0 1 + r j L
N
1 + rj + s N + i −1 i ε ε ∑ T i i =0 1+ r j N 0
L 0
4
L −1
N
T + 1 + s 1+ r j −(N +i )
−(N +i )
(13)
+
1+ rj + s T + 1+ rj
Average decision threshold of CFAR detectors
Deviating from the methods usually described in radar literature, we use the average decision threshold (ADT) for comparison of various CFAR processors. We compute the SNR , needed for the CFAR processing scheme, to achieve a fixed detection probability (e.g. 0.5). The losses or the difference between the two CFAR systems can be expressed by the ratio of the two ADT’s measured in dB [2].
5
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∆[ dB]=
5
E (TV ADT1 1 1) for Pfa1 = Pfa 2 , PD1 = PD 2 = 0.5 = 10log ADT2 E (T2V2 )
(14)
Numerical results
The experimental results are obtained for the following parameters: average power of the receiver noise λ0 =1, average interference-to-noise ratio (INR) rj=30 [dB], probability for the appearance of pulse jamming with average length in the range cells ε0 from 0 to 1. Probability of false alarm Pfa = 10
−6
and excision
threshold BE=2. The size of the testing sample is 16 and the reference window is of the size 16x16. The results for the ADT are received by using Monte-Karlo method and the probability functions (SNR). They are marked as follows: Monte-Karlo (*) and SNR (continuous line).
Fig. 1. CA CFAR BI processor M=16, L=16
Fig. 2. CA CFAR BI processor M=10, L=16
The ADT, T and V of CA CFAR BI processors with a binary rule M-out-ofL (16/16 and 10/16) are showed on Fig.1 and Fig.2. It can be seen that CFAR BI processors with the binary rule M-out-of-L=16/16 are better in cases of lower values ε0 ≤0.5 of the probability for the appearance of pulse jamming. For higher values of the probability for the appearance of pulse jamming ε0 >0.5, the using of the binary rule M-out-of-L=10/16 results in lower losses.
6
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Fig. 3. EXC CFAR BI processor M=16, L=16
Fig. 4. EXC CFAR BI processor M=10, L=16
The ADT, T and V of excision CFAR BI processors with a binary rule Mout-of-L (16/16 and 10/16) are showed on Fig.3 and Fig.4. It can be seen that excision CFAR BI detectors have the same behavior as CFAR BI detectors.
Fig. 5. API CFAR processor, M=16, L=16
Fig. 6. ADT from fig.1 to fig.5
The ADT, T and V of an API CFAR processor are shown on Fig.5. In this case the results for the ADT achieved by using the probability functions (SNR) are identical with the results achieved by using Monte-Karlo simulation for values of ε0 up to 0.4. The suggested algorithm is not working for higher values of ε0 due to the fact, that the hypothesis for censoring in the test cell is disturbed. In such cases the big difference in power between the background and the pulse jamming is disturbed and the automatic censoring of pulse jamming is impossible. The ADTs of all studied processors are shown on Fig.6. The numbers from 1 to 5 correspond to the detectors from Fig.1 to Fig.5. The API CFAR processor is the most suitable one to use for values of the probability for the appearance of pulse jamming ε0 ≤0.5. When the probability for the appearance of pulse jamming ε0 takes value between 0.5 and 1 both, CA CFAR BI and EXC CFAR BI processors with Mout-of-L=10/16 rule can be successfully used.
6
Conclusions
We investigate in this paper the efficiency of different CFAR techniques in the presence of strong pulse jamming by using the ADT approach suggested by Rohling. We consider the whole range (0 to 1) of the probability for the appearance of pulse jamming in range cells. The ADTs are determined by using probability functions and Monte-Karlo simulation. The experimental results show that API CFAR processors are most suitable for use when the probability for the appearance of pulse
7
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jamming takes values in the interval (0 to 0.5). In cases when the probability for the appearance of pulse jamming takes values in the interval (0.5 to 1), we recommend binary integration after the CFAR processor. A problem, concerning the improvement of the performance of excision CFAR and API CFAR processors when the probability for the appearance of pulse jamming takes values in the interval ( ε0 >0.5), can be solved by using Himonas approach [4]. In such cases the threshold estimation is achieved by using the cells with pulse jamming.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
12.
13.
Finn, H., Johnson, P.: Adaptive detection mode with threshold control as a function of spatially sampled clutter estimation, RCA Review, vol. 29, No 3, (1968) 414-464 Rohling, H.: Radar CFAR Thresholding in Clutter and Multiple Target Situations, IEEE Trans., vol. AES-19, No 4, (1983) 608-621 Gandhi, P., Kassam S.: Analysis of CFAR processors in nonhomogeneous background, IEEE Trans., vol. AES-24, No 4, (1988) 443-454 Himonas, S., Barkat, M.: Automatic Censored CFAR Detection for Non-homogeneous Environments, IEEE Trans., vol. AES-28, No 1, (1992) 286-304 Himonas, S.: CFAR Integration Processors in Randomly Arriving Impulse Interference, IEEE Trans., vol. AES-30, No 3, (1994) 809-816 Goldman, H.: Analysis and application of the excision CFAR detector, IEE Proceedings, vol.135, No 6, (1988) 563-575 Behar, V., Kabakchiev, C.: Excision CFAR Binary Integration Processors, Compt. Rend. Acad. Bulg. Sci., vol. 49, No 11/12, (1996) 45-48 Kabakchiev, C., Behar, V.: CFAR Radar Image Detection in Pulse Jamming, IEEE Fourth Int. Symp. ISSSTA'96, Mainz, Germany, (1996) 182-185 Behar, V.: CA CFAR radar signal detection in pulse jamming, Compt. Rend. Acad. Bulg. Sci., vol. 49, No 7/8, (1996) 57-60 Kabakchiev, C., Behar, V.: Techniques for CFAR Radar Image Detection in Pulse Jamming, IEEE Fourth Int. Symp. EUM'96, Praga,Cheh Republic, (1996) 347-352 Behar, V., Kabakchiev, C., Dukovska, L.: Adaptive CFAR Processor for Radar Target Detection in Pulse Jamming, Journal of VLSI Signal Processong, vol. 26, No 11/12, (2000) 383-396 Kabakchiev, C., Dukovska, L., Garvanov, I.: Comparative Analysis of Losses of CA CFAR Processors in Pulse Jamming, CIT, No 1, (2001) 21-35 Garvanov, I., Kabakchiev, C.: Average decision threshold of CA CFAR and excision
CFAR detectors in the presence of strong pulse jamming, German Radar Symposium 2002, (GRS 2002), Bonn, Germany, September (2002), (submitted for presentation) 14. Chakarov, V.: Adaptive CFAR processor for radar target detection design of SHARC signal processor, PhD thesis, Technical University - Sofia, (1999) 15. Akimov, P., Evstratov, F., Zaharov, S.: Radio Signal Detection, Moscow, Radio and Communication, 1989, pp. 195-203, (in Russian).
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of Information Technologies, Bulgarian Academy of Sciences “Acad. G. Bonchev” Str., bl. 2, 1113 Sofia, Bulgaria e-mail: [email protected], [email protected] 2 Dep. of Biomedical Engineering, Institute of Technology, Haifa 3200 e-mail: [email protected]
Abstract
In this paper we study the efficiency of CA CFAR BI, EXC CFAR BI and Adaptive CFAR PI detectors in strong pulse jamming. We achieve new results for the average decision threshold (ADT) using the minimum detectable signal (Pd=0.5). For comparison we use also the approach with Monte-Karlo simulation for estimation of the ADT of the studied CFAR detectors. Differently from other authors, we consider the entire range (0 to 1) of the probability for the appearance of pulse jamming in range cells.
1
Introduction
Cell-Averaging Constant False Alarm Rate (CA CFAR) signal processing proposed by Finn and Johnson in [1]is often used for radar signal detection. The detection threshold is determined as a product of the noise level estimate in the reference window and a scale factor to achieve the desired probability of false alarm. The presence of strong pulse jamming (PJ) in both, the test resolution cell and the reference cells, can cause drastic degradation in the performance of a CA CFAR processor as shown in [12]. In such situations it would be desirable to know the CFAR losses, depending on the parameters of PJ, for rating the radar behavior. There are two approaches for the calculation of CFAR losses offered by Rolling and Kassam in [2,3]. The conventional method, used in [5,7-12], is to compute the additional SNR needed for the CFAR processing scheme beyond that for the optimum processor, to achieve a fixed detection probability (e.g. 0.5). For a particular CFAR scheme losses obviously vary with detection probability. Alternatively, the authors in [2,3] use another criterion based on the average decision threshold (ADT), since the threshold and the detection probability are closely related to each other. Then the difference between the two CFAR systems is expressed by the ratio between the two ADTs measured in dB, as shown in [2,3] The false alarm rate of the postdetection integrator (PI) is extremely sensitive to pulse jamming, and the binary integrator (BI) which uses a K-out-of-M decision rule is insensitive to at most (K-1) interfering pulses [7]. For keeping of constant false alarm rate in PJ the CA CFAR processor presented in [9,12]is used. But this method is not as effective as the conventional method for the calculation of CFAR losses. For the minimization of CFAR losses in case of pulse jamming *
This work is supported by IIT – 010044, MPS Ltd. Grant “RDR” and Bulgarian NF “SR” Grant ¹¿ I – 902/99.
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postdetection integration (PI) or binary integration (BI) is implemented in CFAR processors as shown in [5,8,10]. The use of excision CFAR detectors, supplemented by a postdetection integrator or a binary integrator as shown in [6,7,10], increases the CFAR losses. Minimum CFAR losses in PJ are obtained in [5,11] with a CFAR adaptive postdetection integration (API) processor with adaptive selection on PJ in reference windows and appriory selection in test windows as shown in [5], and adaptive censoring in reference and test windows as presented in [11]. We assume in this paper that the noise in the test cell is Rayleigh envelope distributed and target returns are fluctuating according to Swerling II model as in [3,5]. As a difference from the authors in [5], we assume that the samples of PJ are distributed according to the compound exponential law, where weighting coefficients are the probabilities of corrupting and non-corrupting of the samples. Differently from [7-11], we consider the entire range (0 to 1) of the probability for the appearance of pulse jamming in range cells. For values of the weighting coefficients higher than 0.3, the Poisson process model is used, but it is rough [15]. The binomial distribution is correct in this case. In this paper we study different CFAR techniques in the presence of strong pulse jamming, similarly to [5]. We use the average decision threshold (ADT) approach for comparison of the processors. The analytical expressions for the probability functions of CA CFAR BI, EXC CFAR BI and API CFAR detectors are achieved in [7, 8, 10, 11]. We achieve in these paper new results for the ADT using the SNR approach. The SNR of the minimum detectable signal (Pd=0.5) is approximately the same as the ADT of each CFAR system. For comparison we use also the approach with Monte-Karlo simulation for estimation of the ADT of the studied CFAR detectors. The experimental results show that the API CFAR processors are most effective for values of the probability of appearance of pulse jamming in the interval (0 to 0.5). For ε0 > 0.5 we recommend binary integration after the CFAR processor.
2
Performance of CA CFAR BI and excision CFAR BI processors in the presence of pulse jamming
2.1 Probability of detection and false alarm of CA CFAR BI detectors After filtration the signal is applied to a square-law detector and sampled in range by the (N+1) resolution cells resulting in a vector of (N+1) observations. The sampling rate is such that the samples are statistically independent. It is assumed that L broadband pulses hit the target and one sample of the cell under test suffices to detect a pulse of target returns. In conditions of pulse jamming the background environment includes the random interfering pulses and the receiver noise. Therefore the samples surrounding the cell under test (a reference window) may be drawn from two classes. One class represents the interference-plus-noise situation, which may appear at the output of the receiver with the probability ε0 . This probability can be expressed as ε0 = t c F j , where F j is the average repetition frequency of PJ and t c is the length of pulse transmission. The other class represents the noise only situation,
2
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which may appear at the outputs of the receiver with the probability ( 1 + ε0 ). The set of samples from the test resolution cell at the input of the binary integrator (xl, where l=1,..,L) is assumed to be distributed according to Swerling II case [9, 15] with the probability density function (pdf) given by:
f () x =
−x 1 − ε0 ε0 −x exp + exp ( ) λ0 ( 1 + s ) λ0 ( 1 + s ) λ0 ( 1 + rj + s ) λ 1 + r + s 0 j
(1)
where λ0 is the average power of the receiver noise, r j is the average interference-tonoise ratio (INR) of pulse jamming, s is the per pulse average signal-to-noise ratio (SNR) and N is the number of observations in a reference window. In this case the Poisson process model is used, and it is valid only for ε0 ≤0.3. The probability density function (pdf) of the reference window outputs can be defined as (1), setting s =0. All samples (xl)L are compared with the threshold HD according to the rule:
1, if x l ≥ H D Φ l (x l )= 0, otherwise
(2)
The binary integrator performs summing of L decisions Φ l . The target radar image detection is declared if this sum exceeds the second threshold M.
H1: (t arg et is present ) , if H 0 :( no t arg et ),
∑
L l =1
Φl ≥M
(3)
otherwise
In a conventional CA CFAR detector the noise level estimate is formed as a sum of all N the outputs of the reference window: V = x . In this case the mgf of the estimate
∑
i =1
i
V is defined to be M V (U ) = M xN (U ) , where M x (U ) is the mgf of the random variable
xi .The mgf of the estimate V M V (U ) = ∑
i =1
is obtained in [9]:
C ε( 1 − ε0 ) i i N 0 N −i
N
N −i
i (1 + λ0U ) ( 1 + λ0 ( 1 + rj ) U)
(4)
According to (3) the probability of target detection for CA CFAR BI processor as in [10] is computed by the expression: L
L −l
PD = ∑ C Ll Pd 1 ( 1 − Pd 1 ) l
(5)
l =M
where
T T + ε0 M V Pd1 = ( 1 − ε0 )M V λ ( 1 + s λ 1 + r + s ) 0 j 0 The probability of false alarm is evaluated by (5, 6, 4), setting s = 0 .
(
)
(6)
3
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2.2 Probability of detection and false alarm of excision CFAR BI detectors In an excision CFAR BI processor the noise level estimate V is formed as an
{yi }N ,
average mean of nonzero samples at the output of the excisor V=
1 k
K
∑y
i
that is:
. According to [6] the operation of the excisor is defined as follows:
i =1
x : xi ≤ B E yi = i 0 : othewise
(7)
where B E is the excision threshold.
The probability that a sample xi survives at the output of the excisor, is given as: (8) P =1 − ( 1 − ε )exp (− B / λ )− ε exp − B / λ 1 + r E
E
0
0
0
(
E
0
( )) j
The probability that k out of N samples of the reference window survive at the output of the excisor is given as: v(k )= C Nk PEk ( 1 − PE ) . In this article we use the moment generating function on the excision CFAR processor from [7] N −k
M V (U ) =
N
∑C
k N
PEk (1 − PE ) N −k M V (U , k )
(9)
k =1
where i
1− exp(R1 − BEU / k) ) (1−ε0 )(1−exp(R2 − BEU / k)) ε( MV (U, k) = ∑ C 0 1−exp(R1 ) )( 1+Uλ0 ( 1+ rj ) / k) 1−exp(R2 ) )(1+Uλ0 / k) i=0 ( ( k
k−i
i k
(10)
and R1 = −B E / λ0 (1 + r j ); R 2 = −B E / λ0 . The probability of target detection for excision CFAR BI processor in [7, 10] is computed by the expression: L
L −l
PD = ∑ C Pd 2 ( 1 − Pd 2 ) l
l L
l =M
(11)
where T T , k + ε M , k 0 V λ 1 + s λ 1 + r + s 0( ) j 0 k =1 The probability of false alarm is evaluated by (11, 12), setting s = 0 . Pd2 =
N
∑ C P (1 − P ) (1 − ε )M
3
k k N E
N −k
E
0
V
(
)
(12)
Performance of API CFAR processors in the presence of pulse jamming
Let us assume that L pulses hit the target, which is modeled according to Swerling II case. The received signal is sampled in range by using (M+1) resolution
4
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cells resulting in a matrix with (M+1) rows and L columns. Each column of the data matrix consists of the values of the signal obtained for L pulse intervals in one range resolution cell. Let us also assume that the first M/2 and the last M/2 rows of the data matrix are used as a reference window in order to estimate the “noise-plusinterference” level in the test resolution cell of the radar. In this case the samples of the reference cells result in a matrix X of the size M*L. The test cell or the target includes the elements of the (M/2+1) row of the data matrix and is a vector Z of length L. The elements of the reference window are independent random variables with the compound exponential distribution law (1), setting s = 0 . In the presence of a desired signal from a target the elements of the test resolution cell are independent random variables with distribution law (1). 3.1 Probability of detection and false alarm of API CFAR detectors We use the adaptive censoring algorithm, proposed by Himonas and Barkat in [4], before pulse-to-pulse integration for censoring of the elements of pulse jamming in the reference window and the test resolution cells, in order to form the detection algorithm. The expression for the probability of target detection for an API CFAR processor is achieved in [11, 14]. The authors in [11, 14] study the probability of target detection only for ε0≤0.5 and calculate only the first member of the expression (13). We calculate the probability of target detection for ε0∈(0;1) and calculate the value of PD by using the following expression: k N N L L l −1 k + i −1 T i ( 1+ s) k N −k l L −l ( ) ( ) PD = ∑ 1 − ε ε 1 − ε ε + ∑ ∑ 0 0 0 0 i k +i k =1 k l =1 l i =0 T +1 + s ) ( N N L −1 k + i −1 k N −k L k −(k +i ) ( 1 − ε ) ε ε Ti( 1 + r j + s )( T +1 + r j + s ) + ∑ ∑ 0 0 0 i k =1 k i =0 l −1 N + i −1 1 + s L (1 − ε0 )l ε0L −l ε0N ∑ T i ∑ l i l =1 i =0 1 + r j L
N
1 + rj + s N + i −1 i ε ε ∑ T i i =0 1+ r j N 0
L 0
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L −1
N
T + 1 + s 1+ r j −(N +i )
−(N +i )
(13)
+
1+ rj + s T + 1+ rj
Average decision threshold of CFAR detectors
Deviating from the methods usually described in radar literature, we use the average decision threshold (ADT) for comparison of various CFAR processors. We compute the SNR , needed for the CFAR processing scheme, to achieve a fixed detection probability (e.g. 0.5). The losses or the difference between the two CFAR systems can be expressed by the ratio of the two ADT’s measured in dB [2].
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∆[ dB]=
5
E (TV ADT1 1 1) for Pfa1 = Pfa 2 , PD1 = PD 2 = 0.5 = 10log ADT2 E (T2V2 )
(14)
Numerical results
The experimental results are obtained for the following parameters: average power of the receiver noise λ0 =1, average interference-to-noise ratio (INR) rj=30 [dB], probability for the appearance of pulse jamming with average length in the range cells ε0 from 0 to 1. Probability of false alarm Pfa = 10
−6
and excision
threshold BE=2. The size of the testing sample is 16 and the reference window is of the size 16x16. The results for the ADT are received by using Monte-Karlo method and the probability functions (SNR). They are marked as follows: Monte-Karlo (*) and SNR (continuous line).
Fig. 1. CA CFAR BI processor M=16, L=16
Fig. 2. CA CFAR BI processor M=10, L=16
The ADT, T and V of CA CFAR BI processors with a binary rule M-out-ofL (16/16 and 10/16) are showed on Fig.1 and Fig.2. It can be seen that CFAR BI processors with the binary rule M-out-of-L=16/16 are better in cases of lower values ε0 ≤0.5 of the probability for the appearance of pulse jamming. For higher values of the probability for the appearance of pulse jamming ε0 >0.5, the using of the binary rule M-out-of-L=10/16 results in lower losses.
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Fig. 3. EXC CFAR BI processor M=16, L=16
Fig. 4. EXC CFAR BI processor M=10, L=16
The ADT, T and V of excision CFAR BI processors with a binary rule Mout-of-L (16/16 and 10/16) are showed on Fig.3 and Fig.4. It can be seen that excision CFAR BI detectors have the same behavior as CFAR BI detectors.
Fig. 5. API CFAR processor, M=16, L=16
Fig. 6. ADT from fig.1 to fig.5
The ADT, T and V of an API CFAR processor are shown on Fig.5. In this case the results for the ADT achieved by using the probability functions (SNR) are identical with the results achieved by using Monte-Karlo simulation for values of ε0 up to 0.4. The suggested algorithm is not working for higher values of ε0 due to the fact, that the hypothesis for censoring in the test cell is disturbed. In such cases the big difference in power between the background and the pulse jamming is disturbed and the automatic censoring of pulse jamming is impossible. The ADTs of all studied processors are shown on Fig.6. The numbers from 1 to 5 correspond to the detectors from Fig.1 to Fig.5. The API CFAR processor is the most suitable one to use for values of the probability for the appearance of pulse jamming ε0 ≤0.5. When the probability for the appearance of pulse jamming ε0 takes value between 0.5 and 1 both, CA CFAR BI and EXC CFAR BI processors with Mout-of-L=10/16 rule can be successfully used.
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Conclusions
We investigate in this paper the efficiency of different CFAR techniques in the presence of strong pulse jamming by using the ADT approach suggested by Rohling. We consider the whole range (0 to 1) of the probability for the appearance of pulse jamming in range cells. The ADTs are determined by using probability functions and Monte-Karlo simulation. The experimental results show that API CFAR processors are most suitable for use when the probability for the appearance of pulse
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jamming takes values in the interval (0 to 0.5). In cases when the probability for the appearance of pulse jamming takes values in the interval (0.5 to 1), we recommend binary integration after the CFAR processor. A problem, concerning the improvement of the performance of excision CFAR and API CFAR processors when the probability for the appearance of pulse jamming takes values in the interval ( ε0 >0.5), can be solved by using Himonas approach [4]. In such cases the threshold estimation is achieved by using the cells with pulse jamming.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
12.
13.
Finn, H., Johnson, P.: Adaptive detection mode with threshold control as a function of spatially sampled clutter estimation, RCA Review, vol. 29, No 3, (1968) 414-464 Rohling, H.: Radar CFAR Thresholding in Clutter and Multiple Target Situations, IEEE Trans., vol. AES-19, No 4, (1983) 608-621 Gandhi, P., Kassam S.: Analysis of CFAR processors in nonhomogeneous background, IEEE Trans., vol. AES-24, No 4, (1988) 443-454 Himonas, S., Barkat, M.: Automatic Censored CFAR Detection for Non-homogeneous Environments, IEEE Trans., vol. AES-28, No 1, (1992) 286-304 Himonas, S.: CFAR Integration Processors in Randomly Arriving Impulse Interference, IEEE Trans., vol. AES-30, No 3, (1994) 809-816 Goldman, H.: Analysis and application of the excision CFAR detector, IEE Proceedings, vol.135, No 6, (1988) 563-575 Behar, V., Kabakchiev, C.: Excision CFAR Binary Integration Processors, Compt. Rend. Acad. Bulg. Sci., vol. 49, No 11/12, (1996) 45-48 Kabakchiev, C., Behar, V.: CFAR Radar Image Detection in Pulse Jamming, IEEE Fourth Int. Symp. ISSSTA'96, Mainz, Germany, (1996) 182-185 Behar, V.: CA CFAR radar signal detection in pulse jamming, Compt. Rend. Acad. Bulg. Sci., vol. 49, No 7/8, (1996) 57-60 Kabakchiev, C., Behar, V.: Techniques for CFAR Radar Image Detection in Pulse Jamming, IEEE Fourth Int. Symp. EUM'96, Praga,Cheh Republic, (1996) 347-352 Behar, V., Kabakchiev, C., Dukovska, L.: Adaptive CFAR Processor for Radar Target Detection in Pulse Jamming, Journal of VLSI Signal Processong, vol. 26, No 11/12, (2000) 383-396 Kabakchiev, C., Dukovska, L., Garvanov, I.: Comparative Analysis of Losses of CA CFAR Processors in Pulse Jamming, CIT, No 1, (2001) 21-35 Garvanov, I., Kabakchiev, C.: Average decision threshold of CA CFAR and excision
CFAR detectors in the presence of strong pulse jamming, German Radar Symposium 2002, (GRS 2002), Bonn, Germany, September (2002), (submitted for presentation) 14. Chakarov, V.: Adaptive CFAR processor for radar target detection design of SHARC signal processor, PhD thesis, Technical University - Sofia, (1999) 15. Akimov, P., Evstratov, F., Zaharov, S.: Radio Signal Detection, Moscow, Radio and Communication, 1989, pp. 195-203, (in Russian).
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