A Novel Application of Support Vector Machines to Detect Targets

Aditya V. Padaki† and Koshy George‡† ‡P.E.S. Centre for Intelligent Systems,

‡Dept. of Telecommunication Engineering, P.E.S. Institute of Technology,

100 Feet Ring Road, BSK III Stage, Bangalore 560 085, India. Email: kgeorge@pes.edu

Abstract—In this paper we develop a novel pulseradar detection scheme using Support Vector Ma-chines (SVMs). SVMs are a powerful tool for patternclassification. Exploiting this, we design a SVM forpulse radar detection of targets embedded in noise.An adaptive pre-processing stage is included whenthe echoes are very weak. It is observed that thesignal-to-sidelobe ratio obtained is much higher com-pared to neural network based pulse radar detectionschemes. We further examine the noise tolerance,range resolution ability and Doppler tolerance of thenew algorithm for pulse compression, and compare itwith the ones available in literature.

Keywords-Target detection, support vector ma-chines, binary codes, pattern classification.

I. Introduction

Pulse compression is a technique introduced in radarto resolve closely placed targets while maintaining a lowpeak transmitted power [1]. Classical matched filteringfor pulse compression typically results in large rangesidelobes, resulting in poor performance. Sidelobe sup-pression techniques have been proposed in the past; theseinclude the method of least squares [2] and the minimummean square estimation [3]. Despite very encouragingresults, the principal drawback of these techniques is theinversion of considerably large matrices, posing a hugecomputational burden for real-time implementation.

Artificial Neural Networks (ANNs) are universal ap-proximators in the sense that given any continuousfunction f(·) defined on a compact set, there exists anANN represented by F (·) that can approximate f(·) toany desired accuracy (e.g., [4]). This property was firstexploited in [5], and subsequently by several researchers[6]–[10]. The objective in these papers is to make theANN approximate an ideal autocorrelation sequence;i.e., peak when the autocorrelation lag is zero, andzero elsewhere. Accordingly, such a trained ANN hasconsiderably smaller sidelobes.

ANNs for pulse compression are trained using time-shifted sequences of the adopted codes, and tested bythese sequences corrupted with white Gaussian noise ofdifferent noise intensities to simulate different signal-to-noise ratios (SNRs). In practice, however, the noisepower across the range cells remain largely the samewith the probable value depending on the specific radar

system and its environment. Moreover, the principalobjective in practical applications is to detect targetsat the farthest distance in addition to targets withsmall radar cross sections. Both imply a requirement ofdetecting rather weak echoes. The immense importanceof extracting rather weak echoes of targets embeddedin noise appears to have been overlooked. Indeed, asfirst illustrated in [11], the performance of ANNs in thepresence of weak echoes is quite unsatisfactory whenthey are trained ignoring this reality.

Radar is used to detect the presence of targets. There-fore, if the objective is to train a neural network todetect the presence or absence of targets, then it becomesa pattern classifier. It turns out that training Feedfor-ward Neural Networks (FFNNs) with this viewpoint issufficient not only to detect a target with weak echoesbut as well achieve significant reduction in the sidelobes[11]–[13]. Other factors that contributed to better androbust performance of FFNNs in these references are thechoice of network architecture and training sequences.An additional adaptive gain was introduced in [13] tofurther enhance the performance.

Support Vector Machines (SVMs) are essentially pat-tern classifiers that improve the generalisation perfor-mance. That is, given a finite amount of training data,the objective is to find an optimal trade-off between theattained accuracy on the training set, and the capacityto learn without error. Providing more capacity is equiv-alent to memorising the training data. A measure of thiscapacity is the Vapnik-Chervonenkis (VC) dimension. InSVMs, the trade-off is achieved by designing the hyper-planes that separates the classes to have maximal separa-tion from the classes themselves. Accordingly, they offerimproved generalisation capability compared to otherneural network architectures. In this paper, we use SVMfor pulse radar detection of targets with extremely weakechoes (of the order of −160 dB) embedded in noise.

This paper is organised as follows: We briefly intro-duce SVM as a pattern classifier for radar detection inSection II, and design a two-class pattern classifier forBarker-13 code in Section III. Detection of targets withextremely weak echoes are dealt with in Section IV. Wepresent simulation results in Section V and compare theresults obtained with that available in the literature.

Second International Conference on Computational Intelligence, Modelling and Simulation

978-0-7695-4262-1/10 $26.00 © 2010 IEEE

DOI 10.1109/CIMSiM.2010.17

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Second International Conference on Computational Intelligence, Modelling and Simulation

978-0-7695-4262-1/10 $26.00 © 2010 IEEE

DOI 10.1109/CIMSiM.2010.17

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Second International Conference on Computational Intelligence, Modelling and Simulation

978-0-7695-4262-1/10 $26.00 © 2010 IEEE

DOI 10.1109/CIMSiM.2010.17

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II. Support Vector Machines

Pioneered by Vapnik [14], [15], Support Vector Ma-chines (SVMs) have been widely used for pattern recog-nition and regression. Suppose that {xi}, xi ∈ IRn,1 ≤ i ≤ N be a set of N patterns that belong to oneof two classes C1 and C2. The corresponding class labelsdi take one of two possible values; for instance di = +1if xi ∈ C1 or di = −1 if xi ∈ C2. The basic idea is todetermine a separating hyperplane (if it exists) defined

by f(x) = wT x + b, where w =(

w1 w2 · · · wn

)T

are the weights, b is the bias, and AT denotes thetranspose of the matrix A. For any pattern xi, the signof f(xi) indicates on which side of the hyperplane xi ispositioned, that is, the class of xi.

The discriminant f(x) = 0 defines the hyperplanethat separates the two classes C1 and C2; if the biasb = 0, the hyperplane passes through the origin ofIRn. The closest data point is referred to as the marginof separation. An SVM linear classifier maximises thismargin of separation. Therefore, the separating hyper-plane is determined such that f(xi) ≥ η if xi ∈ C1 andf(xi) ≤ −η if xi ∈ C2 for some fixed η. Without lossof generality, η = 1; otherwise, the weight vector w andthe bias b can always suitably be scaled. The problemof finding the optimal hyperplane that maximises themargin of separation can be reformulated as the follow-ing quadratic optimisation problem: min 1

2wT w subject

to the constraints di(wT xi + b) ≥ 1, 1 ≤ i ≤ N . This

leads to the following solution: w =∑N

i=1αidixi. These

concepts can be extended to the non-separable case byintroducing the so-called slack variables.

Support vector machines are based on the principleof structural risk minimisation. If α represent the set ofparameters defining the trained SVM, di the class labelassociated with a training sample xi, k(x, α) a functionthat maps the training samples to class labels, andP (x, d) the unknown probability distribution associatinga class label with each training sample, then, the goal inSVM is to minimise an upper bound on the expectedrisk defined by R(α) = 1

2

∫

|y − k(x, α)| dP (x, y). IfN denotes the number of training samples, then withprobability 1 − η, 0 ≤ η ≤ 1, an upper bound forthe expected risk R(α) is given by R(α) ≤ Re(α) +√

hN

(

log 2Nh

+ 1)

− 1

Nlog α. Here, Re is the empirical

risk measured on the training set, and h is the Vapnik-Chervonenkis (VC) dimension. The second term is calledthe VC confidence. In the linearly separable case, thehyperplane minimises the VC dimension. In order tominimise the upper bound, one can follow one of thefollowing two strategies: keep the empirical risk fixedand minimise the VC confidence, or minimise the em-pirical risk keeping the VC confidence fixed. The formerstrategy is used as a basis for Support Vector Machines.

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Range Cell

Pow

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dB)

Matched FilterNeural Network

Figure 1. Performance of a neural network not trained for weakechoes.

SVM for Pulse Radar Detection: As mentionedearlier, the essential idea in this paper is to use an SVMto differentiate between the following two classes: Thepresence of at least one target is denoted C1, and theabsence of all targets is denoted C2. In this paper weconsider an SVM with a linear kernel described earlierin this section, and hence rather amenable for hardwareimplementation. The class labels are unity for C1, andnull for C2. From the simulation results presented inSection V, it will become evident that such a viewpointensures the detection of targets with weak echoes.

III. SVM Design for Target Detection

We design a two-class pattern classifier to detecttargets from echoes when the pulses transmitted bya radar have been modulated by the Barker-13 code:S = {1, 1, 1, 1, 1,−1,−1, 1, 1,−1, 1,−1, 1}. A typicallyused architecture in the literature is an FFNN with aninput layer of 13 nodes (corresponding to the length ofthe code), a hidden layer of 3 neurons, and one neuronin the output layer. Such a network is generally trainedwithout taking into account echoes that are weak. Onan average, the performance of such a trained networkis rather poor in that the target is not detected [11].On the contrary, the performance of a matched filter isquite satisfactory. Fig. 1 illustrates this for a receivedradar signal power of −20 dB with an SNR of 40 dB.

In order to detect targets with weak echoes, theSVM is trained with the following target-return powers:0 dB, −6 dB, −12 dB, −18 dB, −24 dB, and −30 dBcorresponding to

(

1

2

)qS, q = 0, 1, . . . , 5. (For simplicity,

if a target-return has a power level of x dB, that targetis referred to in the sequel as a ‘x dB target.’) For

244

each power level, time shifted codes are presented to theSVM resulting in 26 input patterns. Thus, N = 156,which is much larger than typically used in the literature.Therefore, it is expected that the trained SVM exhibitbetter generalisation and hence better robustness. Tofurther improve generalisation capability of the SVM,these training patterns are corrupted by noise.

The ideal autocorrelation function is one that peakswhen the autocorrelation lag is zero (presence of atarget), and zero elsewhere (absence of all targets). Ac-cordingly, C1 denotes the presence of at least one target,and C2 the absence of all targets. The corresponding classlabels are respectively 1 and 0. Note that of the 156training sequences, only 6 belong to C1. Experimentalresults show that such a trained SVM can detect targetsup to approximately −22 dB.

IV. Pre-Processing for Weak Power Returns

As mentioned earlier, the SVM detects targets withecho returns of up to −22 dB. However practical situ-ations demand that targets with power levels of up to−160 dB also be detected. To facilitate this, we pre-process the target returns that are lower than −20 dBas follows before presenting the returns to the trainedSVM: Returns in the range −40 dB and −20 dB areamplified in the digital domain with a gain of 20 dB;those in the range between −60 dB and −40 dB with again of 40 dB; and so on. Finally, returns in the rangebetween −160 dB and −140 dB are amplified with again of 140 dB. Accordingly, all the input samples are inthe range between −20 dB and 0 dB, and hence can bedetected by the SVM designed earlier.

Naturally, one may question the requirement for train-ing the SVM for multiple power levels when the conceptof adaptive gain is being used. It may be argued thattraining the SVM for one specific power level and thenjust boosting the echoes to that power level using adap-tive gain may suffice. We note that despite the use ofsuch a gain, there is still a need for training the SVM formultiple power levels in the context of detecting multipletargets with overlapping echoes but differing powers.(Otherwise, the mismatch in the gain results in a failureto detect the second target.)

Further, it follows that noise is equally boosted in thisscheme. If the echo and the noise remain in the sameband, the amplification is the same resulting in the sameSNR. Those cases wherein they are in different powerranges may be addressed by making inactive one or moresuch amplifiers depending on the noise floor specific tothe radar system.

V. Simulation Results

The performance of the trained SVM is discussed here.For brevity, the target is assumed to be in the 46th

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Pow

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Figure 2. Comparison of amplitude plots for a 0 dB target.

range cell; similar results are obtained when a targetis present in any other range cell. With the noise floorset at −25 dB, the amplitude and power of its outputare compared in Figures 2 and 3 with those obtainedusing a matched filter. The matched filter has six equalsidelobes that correspond to −22.3 dB to either sideof the peak; i.e., the signal-to-sidelobe ratio (SSR) is22.3 dB. On the contrary, by virtue of the chosen trainingsequences, the SVM reduces the sidelobes to a perfectnull. One may view this as a situation with an infiniteSSR. In order to provide a meaningful visual comparisonof the power plots, we replace the null in the output bythe arbitrarily chosen number 10−20. Clearly, the SSRobtained when using our SVM is higher compared to42.73 dB, 139.2 dB, 188.2 dB and 100 dB obtained usingFFNNs respectively in [5], [12], [13] and [16]; 63.19 dBusing a radial basis function neural network (RBFNN)in [9]; or other techniques such as least squares inversefilter [2]. We emphasise that except for [11]–[13], only a0 dB target is considered in all these references.

This scenario is further tested for various power re-turns in the range between −10 dB and −22 dB with thenoise floor set at −25 dB. Alternatively, this correspondsto an input SNR ranging from 15 dB to 3 dB. In eachof these cases, the sidelobe obtained is a perfect null.This corresponds to an ‘ideal’ matched filter, which, tothe best knowledge of the authors has not been achievedpreviously. A comparison of the amplitudes of the outputof a matched filter with that obtained using the trainedSVM for a −22 dB target at 3 dB SNR is shown inFig. 4. Similar results are obtained for targets of differentpowers. Evidently, the trained SVM outperforms thematched filter in each scenario. Moreover, the SVM

355

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−250

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−100

−50

0

50

Range Cell

Pow

er(d

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Matched FilterSVM

Figure 3. Comparison of power plots for a 0 dB target.

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Matched FilterSVM

Figure 4. Comparison of amplitude plots for a −22 dB target.

performs better than the FFNNs in in [12], [13].

Performance with Weak Power Returns: We con-sider the following situations: A −120 dB target witha noise floor of −125 dB (Case A); and, a −160 dBtarget with a noise floor of −165 dB (Case B). (Ob-serve that the SNR is 5 dB.) As mentioned earlier, forsuch situations we consider pre-processing of the returnsusing different gains. It may be noted that six bandsof adaptive gains were used. The adaptive amplifiers inthe bands that are lower than the noise floor are madeinactive, or, in effect, switched off. The results for CasesA and B are respectively presented in Figures 5 and6. Clearly, the targets are detected, and the SSR levels

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Am

plit

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Matched FilterSVM

Figure 5. Comparison of amplitude plots for a −120 dB target.

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Figure 6. Comparison of amplitude plots for a −160 dB target.

quite ample. We note that the amplitude of the outputsof the matched filter for such targets is of the orderof 10−9. Accordingly, in the corresponding comparisonplots, the amplitude of the matched filter outputs cannoteasily be distinguished as it appears as a straight line at0. Fig 7 clearly illustrates the situation.

Discussions: Evidently, the performance of our SVMtrained for different target-return powers is more thansatisfactory. It can be observed that the output powerlevel when a target is present is the same irrespectiveof the power in the echoes. This is due to the mannerin which the SVM is trained to be a two-class pat-tern classifier. Obviously, the probability of detection

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5

10

15x 10−9

Range Cell

Am

plitu

de

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0.2

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1

Range Cell

Am

plitu

de

Matched Filter

SVM

Figure 7. Comparison of amplitude plots for a −160 dB target.

is considerably higher, and hence such an approach isadvantageous.

To the best knowledge of the authors, there are nosimilar results reported in the literature for neural-network based pulse compression. Further, when theneural network is not trained with weaker echoes, as isgenerally reported, the performance in the presence ofsuch weak echoes is rather unsatisfactory [11].

Noise Tolerance: The trained SVM is robust towardnoise. The SSR is largely the same for different powerreturns from targets. Previously, RBFNNs have beenconsidered for the development of robust noise-tolerantnetworks [8]–[10]. In contrast, our SVM is able to pro-vide a substantial noise tolerance, enabling the targetdetection at not only various SNRs but also for variouspowers of radar returns.

Range Resolution: We examine here the range reso-lution ability of the designed SVM. Two targets witha relative power ratio of unity placed at even adjacentrange cells are detected. Other cases where the targetsare detected are as follows: (i) Two targets with powerratio of 12.5 at 5 dB SNR, placed two range cells apart.(ii) Two targets with power ratio of 12.5 at 4 dB SNR,placed three range cells apart. (iii) Two targets withpower ratio of 12.5 at 3 dB SNR, placed four or higherrange cells apart.

It was reported in [9] that two targets placed 5 rangecells apart with a power ratio of 15 could be detectedusing RBFNN. In [12] and [13] two targets with powerratio of 100 also could be detected. Using SVM how-ever, targets with power ratio higher than 12.5 are notdetected. This is one drawback of SVM as comparedto the FFNN-based schemes in [12], [13]. In Fig. 8 two

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Figure 8. Range Resolution for Barker-13 codes. Second Targetis easily detected.

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Figure 9. Doppler Tolerance of Neural SVM

targets of 0 dB and −22 dB are placed at 46th and 51strange cells respectively (5 range cells apart) with a noisefloor of −25 dB. Evidently, both the targets are clearlydetected.

Doppler Tolerance: The phase of the received wave-form is altered and posed as an input to the SVM. Inan extreme situation, there is a 180◦ shift on the firsttransmitted bit of the code, resulting in a very differentpattern. In addition, the power of the received signal is−22 dB with a noise floor of −25 dB. From the resultsshown in Fig. 9, it is clear that the developed SVM isfairly Doppler tolerant.

A Stressing Scenario: Finally, we subject the devel-

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Figure 10. Stressing Scenario

oped SVM to the following scenario: A −18 dB targetand a −22 dB target are assumed to be present atrespectively the 46th and 51th range cells. Further, itis assumed that there is a Doppler shift of 180◦ in thelast bit for each target, and that the noise floor is setto −25 dB (i.e., 3 dB SNR). Fig. 10 shows the outputof the SVM for this situation. Clearly, both the targetsare detected. This demonstrates the robustness of thedeveloped SVM.

VI. Conclusions

In this paper, we developed a technique for pulsecompression based on support vector machines. Thesupport vector machine is trained as a two-class patternclassifier wherein the presence of at least one targetis one class, and the absence of all targets, the otherclass. An adaptive gain is included for very weak targets.Extensive simulations indicate the efficacy of our trainedsupport vector machine. It was observed that such atrained support vector machine can detect up to −22 dBtargets at 3 dB SNR. An additional pre-processing stagefacilitates the detection of targets with returns as lowas −160 dB at 5 dB SNR. In addition, it exhibitssatisfactory noise-toleration, range-resolution ability andDoppler tolerance. A stressing scenario demonstrates therobustness.

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