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Fig. 12. Performance comparison of FNN-CML-CFAR and CML-CFAR detectors with two interfering targets, N = 16, K = 12, Pra = 1077, b= 1 and c= 1.2. the best performance always appears when the FNN-CML-CFAR detector is used. Optimal weights of the FNN threshold estimator and fitness values are found by a GA learning tool as shown in Table 2. The choice of the order K affects slightly he performance of the FNN-CML-CFAR without interfering targets. Now, the detection performance is also examined when a number of interfering argets are present in the observation window. We assume that the interference-to-clutter ratio (ICR) is equal to 20dB and the optimal weights used are hose obtained in Table 2. The performance of he FNN-CML-CFAR detector is tested against the

Figure 12 Performance comparison of FNN-CML-CFAR and CML-CFAR detectors with two interfering targets, N = 16, K = 12, Pra = 1077, b= 1 and c= 1.2. the best performance always appears when the FNN-CML-CFAR detector is used. Optimal weights of the FNN threshold estimator and fitness values are found by a GA learning tool as shown in Table 2. The choice of the order K affects slightly he performance of the FNN-CML-CFAR without interfering targets. Now, the detection performance is also examined when a number of interfering argets are present in the observation window. We assume that the interference-to-clutter ratio (ICR) is equal to 20dB and the optimal weights used are hose obtained in Table 2. The performance of he FNN-CML-CFAR detector is tested against the

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Fig. 1. Fuzzy-neural network (FNN) estimator. There are various approaches to combine neural and fuzzy systems. These approaches can be classified according to the seeked mapping between a FIS and a neural network. In a very direct approach, the rules can simply be trained into a regular MLP network [6] as shown in Fig. 1. First, two logic-based neurons performing fuzzy “AND” and “OR” operations are introduced and _ the construction of networks using these neurons is discussed. The neural implementation for the evaluation of fuzzy sets and defuzzification is considered where a simplified input interface for triangular fuzzy sets is derived. The implementation of networks of arbitrary size (number of layers, number of neurons in each layer) is straightforward. triangular fuzzy sets is derived. The implementation
Fig. 1. Fuzzy-neural network (FNN) estimator. There are various approaches to combine neural and fuzzy systems. These approaches can be classified according to the seeked mapping between a FIS and a neural network. In a very direct approach, the rules can simply be trained into a regular MLP network [6] as shown in Fig. 1. First, two logic-based neurons performing fuzzy “AND” and “OR” operations are introduced and _ the construction of networks using these neurons is discussed. The neural implementation for the evaluation of fuzzy sets and defuzzification is considered where a simplified input interface for triangular fuzzy sets is derived. The implementation of networks of arbitrary size (number of layers, number of neurons in each layer) is straightforward. triangular fuzzy sets is derived. The implementation
Fig. 2. Triangular membership functions. The interpretation of the results of the network built by “AND” and “OR” neurons is significant only if the input signals contain membership values whose logical relationships are to be revealed. Nodes in layer three (the “AND” layer) perform precondition matching of rules of inferences using the algebraic product. The output of “AND” nodes
Fig. 2. Triangular membership functions. The interpretation of the results of the network built by “AND” and “OR” neurons is significant only if the input signals contain membership values whose logical relationships are to be revealed. Nodes in layer three (the “AND” layer) perform precondition matching of rules of inferences using the algebraic product. The output of “AND” nodes
Fig. 3. FNN-ML-CFAR and FNN-CML-CFAR detectors. In this section, we explain the training of the neuro- fuzzy threshold estimator for the ML-CFAR and CML-CFAR detectors in Weibull clutter with un- known shape parameter. The FNN-ML-CFAR and the FNN-CML-CFAR are shown in Fig. 3 where the observed signal under the two hypotheses, H, for target present and Hg for clutter only, is given by In [3], the ML estimates of b and c are given fron the N background samples x;, i= 1,...,N. Solvin; iteratively for ¢, we obtain the equation
Fig. 3. FNN-ML-CFAR and FNN-CML-CFAR detectors. In this section, we explain the training of the neuro- fuzzy threshold estimator for the ML-CFAR and CML-CFAR detectors in Weibull clutter with un- known shape parameter. The FNN-ML-CFAR and the FNN-CML-CFAR are shown in Fig. 3 where the observed signal under the two hypotheses, H, for target present and Hg for clutter only, is given by In [3], the ML estimates of b and c are given fron the N background samples x;, i= 1,...,N. Solvin; iteratively for ¢, we obtain the equation
Fig. 4. Performance comparison of FNN-ML-CFAR and ML- CFAR detectors with N = 16, Pra = 10-7, b= 1 and c= 08. Unfortunately, since a closed-form expression for Pra and the probability of detection, Pp, do not exist, the results will be carried out using Monte- Carlo simulations. In (13), the adaptive threshold T has a nonlinear characteristic with respect to b and c. In order to find the best estimation of T, we replace in our work the scale factor « by an intelligent system based on the neuro-fuzzy estima- tor outlined in the previous section which has weights to be adjusted. The FNN threshold estimator to be designed here has two inputs: the ML estimates of Weibull scale and shape para- meters from N background samples and one output: the estimated threshold to be compared with the
Fig. 4. Performance comparison of FNN-ML-CFAR and ML- CFAR detectors with N = 16, Pra = 10-7, b= 1 and c= 08. Unfortunately, since a closed-form expression for Pra and the probability of detection, Pp, do not exist, the results will be carried out using Monte- Carlo simulations. In (13), the adaptive threshold T has a nonlinear characteristic with respect to b and c. In order to find the best estimation of T, we replace in our work the scale factor « by an intelligent system based on the neuro-fuzzy estima- tor outlined in the previous section which has weights to be adjusted. The FNN threshold estimator to be designed here has two inputs: the ML estimates of Weibull scale and shape para- meters from N background samples and one output: the estimated threshold to be compared with the
Fig. 6. Performance comparison of FNN-ML-CFAR and ML- CFAR detectors with N = 16, Pra = 1073, b= 1 and c= 1.6. Fig. 5. Performance comparison of FNN-ML-CFAR and ML- CFAR detectors with N = 16, Pra = 107°, b= 1 and c = 1.2.
Fig. 6. Performance comparison of FNN-ML-CFAR and ML- CFAR detectors with N = 16, Pra = 1073, b= 1 and c= 1.6. Fig. 5. Performance comparison of FNN-ML-CFAR and ML- CFAR detectors with N = 16, Pra = 107°, b= 1 and c = 1.2.
Fig. 7. Performance comparison of FNN-ML-CFAR and ML- CFAR detectors with N = 16, Pra = 1077, b= 1 and c =2.
Fig. 7. Performance comparison of FNN-ML-CFAR and ML- CFAR detectors with N = 16, Pra = 1077, b= 1 and c =2.
Optimal parameters found for ML-CFAR and FNN-ML-CFAR detectors with a = 1077, b= 1 and N = 16 Table 1
Optimal parameters found for ML-CFAR and FNN-ML-CFAR detectors with a = 1077, b= 1 and N = 16 Table 1
Fig. 8. Performance comparison of FNN-CML-CFAR and CML-CFAR detectors with N = 16, K = 12, Ppa = 107°, b= 1 and c= 0.8. Step 1: At layer one, present the input ML estimates of Weibull parameters and create an initial population of chromosomes according to Eq. (15) where the values must be limited to the interval [0, 1].
Fig. 8. Performance comparison of FNN-CML-CFAR and CML-CFAR detectors with N = 16, K = 12, Ppa = 107°, b= 1 and c= 0.8. Step 1: At layer one, present the input ML estimates of Weibull parameters and create an initial population of chromosomes according to Eq. (15) where the values must be limited to the interval [0, 1].
Fig. 9. Performance comparison of FNN-CML-CFAR and CML-CFAR detectors with N = 16, K = 12, Ppa = 107°, b= 1 and c= 1.2.
Fig. 9. Performance comparison of FNN-CML-CFAR and CML-CFAR detectors with N = 16, K = 12, Ppa = 107°, b= 1 and c= 1.2.
Fig. 11. Performance comparison of FNN-CML-CFAR and CML-CFAR detectors with N = 16, K = 12, Prax = 107, b=1 and c = 2. Fig. 10. Performance comparison of FNN-CML-CFAR and CML-CFAR detectors with N = 16, K = 12, Ppa = 103°, b=1 and c = 1.6.
Fig. 11. Performance comparison of FNN-CML-CFAR and CML-CFAR detectors with N = 16, K = 12, Prax = 107, b=1 and c = 2. Fig. 10. Performance comparison of FNN-CML-CFAR and CML-CFAR detectors with N = 16, K = 12, Ppa = 103°, b=1 and c = 1.6.
Fig. 13. Performance comparison of FNN-CML-CFAR and CML-CFAR detectors with four interfering targets, N = 16, K = 12, Pex = 1073, b= 1 and c= 1.2.
Fig. 13. Performance comparison of FNN-CML-CFAR and CML-CFAR detectors with four interfering targets, N = 16, K = 12, Pex = 1073, b= 1 and c= 1.2.
Optimal parameters found for CML-CFAR and FNN-CML-CFAR detectors with a = 107,b=1,N=16, K=12andc=1.2 Table 3
Optimal parameters found for CML-CFAR and FNN-CML-CFAR detectors with a = 107,b=1,N=16, K=12andc=1.2 Table 3
Related topics:
EngineeringGeneticsTechnologySignal Processingadaptive neuro fuzzy inference system (ANFIS)Fuzzy Neural NetworkFuzzy SystemWeibull distributionFuzzy Neural Networks
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