Radial basis function neural network for pulse radar detection

Radial basis function neural network for pulse radar detection D.G. Khairnar, S.N. Merchant and U.B. Desai Abstract: A new approach using a radial basis function network (RBFN) for pulse compression is proposed. In the study, networks using 13-element Barker code, 35-element Barker code and 21-bit optimal sequences have been implemented. In training these networks, the RBFN-based learning algorithm was used. Simulation results show that RBFN approach has significant improvement in error convergence speed (very low training error), superior signal-to-sidelobe ratios, good noise rejection performance, improved misalignment performance, good range resolution ability and improved Doppler shift performance compared to other neural network approaches such as back-propagation, extended Kalman filter and autocorrelation function based learning algorithms. The proposed neural network approach provides a robust mean for pulse radar tracking. 1 Introduction The advantage of using narrow pulses in radar is superior range resolution. Due to maximum peak power limitations of the transmitter, pulse widths cannot be reduced indefinitely without deteriorating the detection performance. Pulse compression techniques utilise signal processing to provide the advantages of extremely narrow pulse width while remaining within the peak power limitations of the transmitter. Therefore pulse compression techniques are used to obtain pulse radar detection. In practice two different approaches are used to obtain pulse compression. The first one is to use a matched filter; here codes with small sidelobes in their autocorrelation function (ACF) are used [1, 2]. An interesting approach to real time correlation of pulse coded radar waveforms has been implemented using SAW convolver devices [3]. The method achieves correlation of 255-bit PSK sequence in 8 ns, which is much faster than any of the digital techniques discussed in this paper. Selviah and Stamos [4] have investigated a more advanced correlation high-speed learning technique than the basic correlation approach. The second approach to pulse compression is to use inverse filters of two kinds, namely, non-recursive time invariant causal filter [5] and recursive time variant filter [6]. Two different approaches using a multi-layered neural network, which yield better signal-to-sidelobe ratio (SSR) (the ratio of peak signal to maximum sidelobe) than the traditional approaches have been reported in [7 9]. In the first, a multi-layered neural network approach using back-propagation (BP) as the learning algorithm is used [7, 10]. Whereas in the second approach, the extended Kalman filtering (EKF)-based learning algorithm has been used [8, 11, 12]. In both these approaches, the 13-element Barker code f1, 1, 1, 1, 1, 21, 21, 1, 1, 21, 1 21,1g and the maximum length sequences (m-sequences) of lengths 15, 31 and 63 (all of them are # The Institution of Engineering and Technology 2007 doi:10.1049/iet-rsn:20050023 Paper first received 22nd March 2005 and in revised form 3rd March 2006 The authors are with the SPANN Laboratory, Indian Institute of Technology Bombay, India E-mail: dgk@ee.iitb.ac.in 8 single period) were used as the signal codes [6, 13]. The convergence speed of the BP- and EKF-based learning algorithm is inherently slow. In this paper, we propose a new radial basis function (RBF) neural network for pulse radar detection. Different signal codes are used as input to the neural network. It is found that this new algorithm has much better SSR, better noise rejection capability, superior range resolution ability and improved misalignment performance for duplicating and discarding some bits in the sequence than the BP- and EKF-based algorithms. Moreover, this algorithm has much faster convergence speed than the approaches based on BP or EKF. 2 Digital pulse compression and sequences used Pulse compression correlates the received signal to a delayed copy of that which was transmitted. This correlation is a cross-correlation because the echo is different from the transmitted waveform. Phase-coded waveforms are well adapted to digital pulse compression. Digital waveforms are usually bi-phase modulated sinusoids, with two possible phases being 08 and 1808. Bi-phase modulation is used because it yields the widest bandwidth for a given code sequence. Pulse compression waveform design is predicted on simultaneously achieving wide pulse width for detection and wide bandwidth for range resolution. The spectrum of a waveform is a critical parameter. The waveform s ACF determines its ability to resolve in range. Narrow autocorrelations, corresponding to wide bandwidths, are necessary for good range resolution. Optimal binary amplitude sequence is the one having an ACF with a peak (largest) sidelobe magnitude that is the smallest possible for a given sequence length. The optimal codes having peak sidelobe levels of one are called Barker codes. Some of the sequences used are 13-element Barker code, 35-element Barker code and 21-bit optimal sequences. In this paper, we have carried out investigation with these sequences because of the ease in implementation. Currently we are also investigating with larger length sequences and preliminary results clearly demonstrate that, in general, conclusions drawn from shorter length sequences are extendable to larger length sequences also. For relative comparison, the amplitude of the ACF of a 13-element IET Radar Sonar Navig., 2007, 1, (1), pp. 8 17

Fig. 1 Autocorrelation of different sequences a 13-element Barker b 35-element optimal c 35-element Barker d 13-element pseudo random Barker, 35-element optimal sequence, 35-element combination Barker and 13-element pseudo random sequence is shown in Fig. 1 [2, 14]. 3 Problem formulation and training using RBF algorithm In any neural network application, training of the network plays an important role [20]. In the pulse compression application under investigation, once the network is trained, it can distinguish between the transmitted signal and the other received signals, which could be external disturbances or time-shifted versions of the transmitted signal. The transmitted signals used are 13-element Barker code, 35-element Barker code and 21-bit optimal sequences [15]. The structure of the generalised RBF network for the Barker code of length 13 is shown in Fig. 2. The network consists of three layers. The first layer is composed of input (source) nodes whose number is equal to the dimension N of the input sequence vector x. The second layer is the hidden layer, composed of nonlinear units that are connected directly to all of the nodes in the input layer. In our simulations, we have experimented with different number of nodes in the hidden layer and based on these simulations, we have chosen the empirical value of hidden nodes to be 7. The activation functions of the individual hidden units in a generalised RBF network are defined by the Gaussian function [15, 16] referred to as RBF. An RBF is a multi-dimensional function that depends on the distance between the input vector and a centre vector. The input layer has neurons with linear functions that simply feeds the input signals to the hidden layer. Moreover, the connection between the input layer and the hidden layer are not weighted, that is, each hidden neuron receives each corresponding input value unaltered. The hidden neurons are processing units that implement the RBF. In contrast Fig. 2 Structure of the RBFN for 13-element Barker code IET Radar Sonar Navig., Vol. 1, No. 1, February 2007 9

to the MLP network, the RBF network usually has only one hidden layer. The transfer function of the hidden neurons in the generalised RBF network used is Gaussian function G(x 2 x i ) ¼ exp[2(1/2s i 2 )kx 2 x i k 2 ], where s is a real parameter, called a scaling parameter, and kx 2 x i k 2 is the distance between the input vector and the centre vector. The connections between the hidden layer and the output layer are weighted. The single neuron of the output layer has input output relationship that performs simple weighted summations. The output of the RBF network in Fig. 2 can be expressed as y ¼ FðxÞ ¼ XK i¼1 w i w i ðxþ where fw i (x), i ¼ 1, 2, 3,..., Kg is a set of basis functions that we assume to be linearly independent and w i constitute a set of connection weights for the output layer. When using RBFs, the basis w i (x) are chosen as Gaussian functions normally ð1þ w i ðxþ ¼Gðkx t i kþ; i ¼ 1; 2;...; K ð2þ Gðkx t i kþ ¼ exp 1 ðx t 2 i Þ T S 1 ðx t i Þ ð3þ where t i ¼ [t i1, t i2, t i3,..., t in ] T,witht i as unknown centres to be determined. S is a symmetric positive definite weighting matrix of size N N. G(.) represents a multivariate Gaussian distribution with mean vector t i and covariance matrix S. By using the above equations, we rewrite F(x) as FðxÞ ¼ XK i¼1 w i Gðkx t i kþ We determine the weight vector W ¼ [w 1, w 2, w 3,..., w K ] T, and the set t of vectors t i of centres such that the cost ð4þ functional jðw; tþ ¼ 1 2 X M i¼1 d i XK j¼1 w j Gðkx i t j kþ! 2 ð5þ is minimised. Here M is the number of training samples and d i is the desired response. The term on the right-hand side of the equation may be expressed as the squared Euclidean norm kd 2 GWk 2, where d ¼½d 1 ; d 2 ; d 3 ;...; d M Š T 2 3 Gðx 1 ; t 1 Þ Gðx 1 ; t 2 Þ Gðx 1 ; t K Þ Gðx 2 ; t 1 Þ Gðx 2 ; t 2 Þ Gðx 2 ; t K Þ G ¼ Gðx 3 ; t 1 Þ Gðx 3 ; t 2 Þ Gðx 3 ; t K Þ 6........ 7 4. 5 Gðx M ; t 1 Þ Gðx M ; t 2 Þ Gðx M ; t K Þ In this network, provision is made for a bias applied to the output unit. This is done simply by setting one of the linear weights in the output layer of the network equal to the bias and treating the associated RBF as a constant equal to þ1. In the RBFN of Fig. 2, the linear weights associated with the output layer, the positions of the centres of the RBFs and S 21 are all unknown parameters that have to be learnt. The different layers of an RBF network perform different tasks, and so it is reasonable to separate the optimisation of the hidden and output layers of the network by using different techniques, and by operating on different time scales. Here, we used supervised selection learning strategies for centre selection in the design of an RBF network. In this approach, the centres of the RBFs and all other free parameters of the network undergo a supervised learning process. A common approach for this is error-correction learning and it is based on gradient descent procedure. The first step in the development of a learning procedure Fig. 3 10 Error convergence trajectory of RBFN, BP and EKF for the 13-element Barker code IET Radar Sonar Navig., Vol. 1, No. 1, February 2007

is to define the instantaneous value of the cost function j ¼ 1 2 X M e 2 i i¼1 where M is the size of the training sample used to do the learning, and e i is the error signal defined by e i ¼ d i Fðx i Þ ¼ d i XK j¼1 w j Gðkx i t j kþ We assume the covariance matrix S to be diagonal, that is, S ¼ diagonal[s 1, s 2,..., s N ], where s j is the variance of the jth element of the input vector x and j ¼ 1, 2,..., N. The requirement is to find the free parameters w i, t i and S 21 so as to minimise j. Based on the results of this minimisation, it can be shown [17] that the cost function j is convex with respect to the linear parameters w i, but nonconvex with respect to the centres t i and s j 21. The search for the optimum values of t i and s j 21 may get stuck at a local minimum in parameter space. The RBF networks ð6þ ð7þ with supervised learning were able to exceed substantially the performance of multilayer perceptions. After updating at the end of an epoch, the training is continued for the next epoch and it continues until the maximum error among all M training patterns is reduced to a prespecified level. Simulation results show that the linear weights associated with the output unit of the network tend to evolve on a different time scale compared to the nonlinear activation functions of the hidden units. Thus, as the hidden layer activation functions evolve slowly in accordance with some nonlinear optimisation strategy, the output layer weights adjust themselves rapidly through a linear optimisation strategy. 4 Simulation results and performance evaluations Once the training is over, the neural network can be exposed to various sets of input sequences. This section illustrates the performances of the RBFN, which is then compared with BP-, ACF- and EKF-based algorithms. In all cases, we consider the 13-element Barker code, 35-element Barker code and 21-bit optimal code and, based on the Fig. 4 Example of compressed waveforms a Using ACF b Using BP c Using EKF d Using RBFN IET Radar Sonar Navig., Vol. 1, No. 1, February 2007 11

simulation results, the empirically chosen value of K ¼ 7. RBF is a special class of functions. Its characteristics feature is that its response decreases (or increases) monotonically with distance from a central point. Its important parameters are its centre c and its radius r. In our simulations, we have chosen c ¼ 0 and the radius of attraction r ¼ 1 around the training patterns. We apply a hard cut-off whenever the goal error is reached, so that the network distinguishes the correct sequence and it avoids convergence to random patterns. 4.1 Convergence performance As shown in Fig. 3, the convergence speed of the BP algorithm is inherently slow. Multi-layered neural network based on the EKF learning algorithm has better convergence speed than the BP algorithm. The proposed approach based on RBFN has much better convergence speed and very low training error compared to BP and EKF algorithms. 4.2 Signal-to-sidelobe ratio performance The SSR is defined as the ratio of peak signal amplitude to maximum sidelobe amplitude. Fig. 4 shows the compressed waveforms of 13-element Barker code using ACF, BP, EKF and RBFN approach. The results of the investigation are depicted in Table 1. It shows that the proposed RBFN approach achieved, higher output SSR compared to other approaches in all the cases. 4.3 Noise performance Since in real life, the signal from the target is corrupted by noise, it is important to test the algorithm by adding noise to the pulse. The input signals are corrupted by white Gaussian noise with different noise variance. The performance of the ACF, BP, EKF and RBFN for the noisy case is shown in Tables 2 4. The network was trained both without noise (for s n ¼ 0.0) and with noise (for s n = 0.0). From these tables, it is clear that the performance of RBFN is much better than any other approach. 4.4 Misalignment performance Simulations have been done for clock misalignment by duplicating (for fast clock) or by discarding (for slow clock), some bits in the input sequence. For the 13-element Barker code, the 7th bit, that is, the middle bit was duplicated in one simulation and the same bit was discarded in another simulation. The SNR obtained when the bit is discarded or duplicated is 49.27 and 42.21 db, respectively. Similarly, when the 11th bit in 21-bit optimal code was discarded or duplicated the SNR is 52.34 and 41.11 db, respectively, as shown in Tables 5 and 6. For the 35-element Barker code, the 18th bit, that is, the middle bit was discarded or duplicated and the SNR obtained are 41.23 and 37.29 db, respectively. In all the cases, there is significant improvement in misalignment performance using the proposed RBFN approach. 4.5 Range resolution The range resolution is the ability to distinguish between two targets solely by measurements of their ranges in radar systems. The signal received from the target is bound to be noisy. The network should be able to distinguish between two close-by targets and should be able to resolve two overlapped targets. To resolve two targets in range, the basic criterion is that the targets must be separated by at least the range equivalent to the width of the processed echo pulse. To make the comparison of the range resolution ability, we consider 13-element Barker code with two n-delay apart (DA) overlapping sequences having same and different magnitude ratios, where n ¼ 2, 3, 4, 5 and 15. Fig. 5 shows the added input waveform of equal magnitude Barker codes of 5-delay-apart used for the range resolution simulations. In Tables 7 and 8, the input magnitude ratio (IMR) is defined as the magnitude of the first pulse train over that of the delayed pulse train. The results in Tables 7 and 8 clearly indicate the superior performance of RBFN over other techniques. Figs. 6 and 7 show the examples of compressed waveforms of overlapped 13-element Barker codes using ACF, BP, EKF and RBFN approach. Table 1: Comparison of SSR in db 13-element Barker code 21-bit optimal code 35-element Barker code Table 3: Comparison under noisy condition for 21-bit optimal code s n ¼ 0.0 s n ¼ 0.1 s n ¼ 0.3 s n ¼ 0.5 s n ¼ 0.7 ACF 22.17 20.42 13.97 BP 45.58 31.19 35.67 EKF 48.46 36.22 42.78 RBFN 64.90 60.18 71.70 ACF 20.19 18.11 16.18 12.14 10.78 BP 43.21 40.32 33.57 25.19 19.45 EKF 47.20 42.09 38.43 30.81 20.16 RBFN 64.91 62.81 54.01 48.25 43.17 Table 2: Comparison under noisy condition for 13-element Barker code Table 4: Comparison under noisy condition for 35-element Barker code s n ¼ 0.0 s n ¼ 0.1 s n ¼ 0.3 s n ¼ 0.5 s n ¼ 0.7 s n ¼ 0.0 s n ¼ 0.1 s n ¼ 0.3 s n ¼ 0.5 s n ¼ 0.7 ACF 22.16 22.14 20.71 17.13 14.81 BP 44.11 41.38 29.31 20.13 16.71 EKF 47.13 43.17 33.71 24.18 20.61 RBFN 63.19 60.18 53.81 46.52 39.81 ACF 14.10 13.02 11.36 10.12 10.01 BP 36.16 33.18 30.26 28.30 25.29 EKF 43.10 41.21 39.23 31.21 30.01 RBFN 72.77 70.37 68.17 65.33 62.20 12 IET Radar Sonar Navig., Vol. 1, No. 1, February 2007

Table 5: Comparison of misalignment bit performance in db 13-bit Barker code 7th bit discarded 21-bit optimal code 11th bit discarded 35-bit Barker code 18th bit discarded ACF 9.55 13.98 4.87 BP 33 36.21 17.24 EKF 38.46 41.12 29.56 RBFN 49.27 52.34 41.23 Table 6: Comparison of misalignment bit performance in db 13-bit Barker code 7th bit duplicated 21-bit optimal code 11th bit duplicated 35-bit Barker code 18th bit duplicated ACF 16.30 14.41 5.06 BP 22.18 21.61 15.31 EKF 30.72 31.78 26.53 RBFN 42.21 41.11 37.29 Fig. 5 Input waveform on additional of two 5-delay-apart 13-element Barker sequence having same magnitude a Left shift b Right shift c Added input waveform d Waveform after flip about vertical axis IET Radar Sonar Navig., Vol. 1, No. 1, February 2007 13

Table 7: Comparison of 13-element Barker code for range resolution ability of two targets having same IMR and DA 2-DA 3-DA 4-DA 5-DA 15-DA ACF 16.90 22.3 16.9 22.3 22.3 BP 38.40 40.61 39.46 38.12 37.46 EKF 41.50 43.16 41.11 39.67 38.33 RBFN 62.40 68.13 61.20 66.87 58.13 Table 8: Comparison of range resolution ability of two targets having different IMR and DA for 13-element Barker code Methods 2-DA 3-DA 4-DA 5-DA 5-DA 2-IMR 3-IMR 4-IMR 5-IMR 15-IMR ACF 13.97 12.74 10.63 8.29 1.25 BP 39.01 31.10 26.11 14.02 6.04 EKF 40.31 36.13 28.11 23.46 8.18 RBFN 61.66 60.76 51.33 35.11 23.13 4.6 Doppler shift performance To check the Doppler tolerance of the pulse compression algorithms in this work, we consider 13-element Barker code. By shifting phase of the individual elements of the phase code, the Doppler sensitivity is caused. In the extreme, if the last element is shifted by 1808, the code word is no longer matched with the replica. That is, the used sequence of pulse compression is changed from f1, 1, 1, 1, 1, 21, 21, 1, 1, 21, 1 21, 1g to f21, 1, 1, 1, 1, 21, 21, 1, 1, 21, 1 21, 1g. The results of comparison of all algorithms are shown in Fig. 8. Results show that RBFN has significant advantage of robustness in Doppler shift interference. From Fig. 9, it is observed that the BP, ACF and EKF are sensitive to the Doppler shift produced by a moving target. 4.7 Computational time The algorithms have been implemented on a personal computer with Pentium III (1.88 GHz) processor with 256 KB of cache memory, 256 MB RAM and Linux operating system. CPU time for different algorithms and for different codes is presented in Table 9. Fig. 6 Example of compressed waveforms of overlapped 13-element Barker codes having same IMR and 5-DA a Using ACF b Using BP c Using EKF d Using RBFN 14 IET Radar Sonar Navig., Vol. 1, No. 1, February 2007

Fig. 7 Example of compressed waveforms of overlapped 13-element Barker codes having 5-IMR and 5-DA a Using ACF b Using BP c Using EKF d Using RBFN Fig. 8 Example of compressed waveforms of overlapped 13-element Barker codes having 15-IMR and 5-DA a Using ACF b Using BP c Using EKF d Using RBFN IET Radar Sonar Navig., Vol. 1, No. 1, February 2007 15

Fig. 9 Simulated results for Doppler shift a Using ACF b Using BP c Using EKF d Using RBFN Table 9: Comparison of CPU time for different algorithms 5 Conclusion CPU time, ms 13-bit Barker code 21-bit optimal code 35-bit Barker code ACF 0.06461 0.06530 0.06538 BP 1.5338 1.5690 1.6070 EKF 1.6119 1.6460 1.6530 RBFN 2.3076 2.3138 2.3146 In this work, radial basis function network (RBFN) is proposed for radar pulse compression. The simulation results for various cases indicate that the performance of the proposed RBFN algorithm is much better than techniques such as BP-, EKF-based learning algorithms and the conventional ACF approach. Simulations have illustrated that the convergence speed of the RBFN is superior to BP and 16 EKF algorithms. For RBFN selection of radius of attraction around the training patterns and application of a hard cut-off in radius are important parameters. It is also important to note that the tolerance to different effects also depends on what fraction of the total memory capacity [18, 19] for codes has been trained. The RBFN approach has very low training error compared to BP and EKF algorithms. This approach has higher SSRs in different adverse situations of noise, and with misalignment in the clock. It has better range resolution and robustness in Doppler shift interference. As shown in Fig. 8, for two overlapped sequences delay 5 with the input magnitude ratio higher than 15, the approach based on BP fails to detect, whereas the proposed RBFN approach not only detects the target but provides 14.95 db improvement over EKF-based approach. 6 References 1 Nathanson, F.E.: Radar design principles (McGraw-Hill, New York, 1969), pp. 452 469 2 Skolnik, M.I.: Introduction to radar systems (McGraw-Hill Book Company Inc., 1962) IET Radar Sonar Navig., Vol. 1, No. 1, February 2007

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