Targetdetection for low angle radar based on multi-frequency order-statistics

National Laboratory of Radar Signal Processing,Xidian University,Xi’an 710071,China

Targetdetection for low angle radar based on multi-frequency order-statistics

Yunhe Cao?,Shenghua Wang,Yu Wang,and Shenghua Zhou

National Laboratory of Radar Signal Processing,Xidian University,Xi’an 710071,China

Forradartargets fl ying atlow altitude,multiple pathways produce fade or enhancement relative to the level that would be expected in a free-space environment.In this paper,a new detection method based on a wide-ranging multi-frequency radar for low angle targets is proposed.Sequentialtransmitting multiple pulses with differentfrequencies are fi rstapplied to decorrelate the coherence ofthe directand re fl ected echoes.After receiving allechoes, the multi-frequency samples are arranged in a sortdescending according to the amplitude.Some high amplitude echoes in the same range cell are accumulated to improve the signal-to-noise ratio and the optimalnumber ofhigh amplitude echoes is analyzed and given by experiments.Finally,simulation results are presented to verify the effectiveness ofthe method.

multipath,signal detection,order statistic,multifrequency,low angle.

1.Introduction

The problem of detecting and tracking low angle targets in the presence of multipaths has attracted lots of interest in the radarcommunity formany years[1-10,14-20].In lowangle targetdetection[3,5-7,9],radars receive two ormore coherentechoes via multipaths(typically four propagation paths),resulting in fade or enhancementof the targetecho amplitude.Severe signal cancellation arises and the target cannotbe detected effectively when the directand re fl ected components are nearly outof phase.

To alleviate multipath effects,the multi-frequency technique is used in some literature[3,7].The technique is effective to decorrelate the coherence of the direct and re fl ected echoes.The carrier frequency of radar changes within a large bandwidth from pulse to pulse.This kind of multi-frequency radar provides some characters of robustness to the signal fades and makes a proper use of multipath propagation to improve the target detection capability[8].Considering fourpropagation paths(one direct path and three re fl ected paths),ifthe directpath and the refl ected paths are in phase atsome frequency point,the echo amplitude in a multipath environmentshould be four times greater than that in the free space(only direct path)when the amplitude of the ground re fl ection coef fi cient is 1.0. Consequently,the signal-to-noise ratio(SNR)of the received signal is expected to get a gain of 12 dB.If the carrier frequency changes uniformly,the differentialphase between the directpath and the specularre fl ection path can be regarded as a uniformly distributed random variable and the gain of the received amplitude can range from 0 to 4. Consequently,the SNR of the return is bene fi cialin statistics for the detection of targets in multipaths[3].

The noncoherent accumulation method is common in radarsignalprocessing.However,during the low angle targetdetection,because of the severe multipath cancellation atsome frequencies,extremely low SNR echoes should be excluded from the accumulation.A new detection method, which is based on the order statistics and noncoherentaccumulation,is described in this paper.In the method,only some test cells with strong energy are employed in the detection procedure and the optimal number of test cells is given accurately.With regard to the detection performance, ourmethod achievesa higherdetection probability than the M-out-of-N detectorproposed by Wilson and Carlson[3].

This paper is organized as follows.In Section 2,a traditionalmultipath modeland the probability density function(PDF)ofthe targetecho forfourpropagation paths are given.In Section 3,the proposed multi-frequency orderstatistics method is described in detail.The expression of optimal parameter K is also introduced in this Section. Performance comparison of this method with the M-outof-N detector and the coherent accumulation method are presented in Section 4.Finally concluding remarks are given.

2.Multipath model

A simple fl at-earth multipath modelwith perfectre fl ection is used as shown in Fig.1.The radarreceives the echo signals from a low elevation targetand four components are taken into account.The fi rstcomponentgoes straightto the target and returns straight to the antenna,whereas the second componentgoes straight to the target and returns to the antenna via the re fl ecting plane.The third component travels to the target via the re fl ecting plane and returns straight to the antenna,whereas the last component travels to the target via the re fl ecting plane and so does the return[10].

Fig.1 Plat-earth modelin multipath

Suppose thatthe radar cross section(RCS)of the target is the same for the four propagation paths and the direct echo amplitude is normalized.The diffuse re fl ection is neglected for simplicity.Then,the voltage multipath factor of the received signalis given by

whereα=2πfδ/c,which is converted from the distance differenceδbetween the directand the specularre fl ections, denotes the phase difference,ρis the complex re fl ection coef fi cient,f and c are the carrier frequency and the speed of light,respectively.

Assuming thatthe frequency f is hopped uniformly over the radar work bandwidthΔf.It is expected to enable the echo amplitude G to traverse a cycle within the bandwidth Δf(and soαundergoes a phase shiftofπ),i.e.

whereδcan be estimated as follows:

where Rois the distance between the radar and the target, R1is the distance between the radar and the targetimage, haand htare the heightofthe radarantenna and the height of the target,respectively,and D is the ground range ofthe target.

For a low angle target,ha,ht＜＜D.Using Taylor expansion,(3)can be written approximately as

In this case,D≈R0.Equation(4)is often written[13]as

For example,assuming that ha=20 m,ht=50 m, Ro=10 km,the work frequency bandwidthΔf is expected to be 0.75 GHz.In this case,phase differenceαcan be treated as a uniformly distributed random variable over [0,π].Therefore,W=α/2πis uniformly distributed over [0,1/2],and the PDF of W can be expressed as

The phase of re fl ection coef fi cientρis approximately constant in a certain frequency range for horizontal polarization[11].Hence,we consider thatρis a complex constantnumberfordifferentcarrier frequencies(in the whole radar work bandwidth).Then we have

After substituting(6)and(8)into the following PDF formula,we obtain the PDF of G

3.Multi-frequency order-statistics method

In this section,the expression of multi-frequency orderstatistics false alarm probability function is deduced and a detection threshold is given.Subsequently,the detection probability of the multi-frequency order-statistics method in the multipath environment is presented.Furethermore, the optimal number of high amplitude echoes is analyzedby experiments.Severalassumptions are made in the algorithm as follows:i)the noise is temporally white and circularly symmetric zero-mean complex Gaussian random variable;ii)the noise in different cells is mutually statistically independent;iii)all the multipath delayed signals from the target lie within the range cell under consideration;iv)the sample data in varied frequency pulses are mutually statistically independent since the frequency bandwidth is wide enough.

3.1 Statisticalanalysis of test cells

Assuming thata frequency agile radar has received N differentfrequency echoes Xt(t=1,2,...,N),the testcells are chosen from the same range cellof the N echoes.The sample data obtained through the linear-law detector in the testcells can be denoted as Yt=|Xt|.

In the hypothesis H0,there is only noise in the testcell. Thus,the PDF of Yt,which is the Rayleigh PDF,can be written as

In the multipath environment,the effects of multipath fl uctuation and target fl uctuation need to be taken into deduction of the PDF of Yt.Here,we consider the Swerling III target model for the following analysis,and the amplitude of targetecho A＇is

where A represents the amplitude of the target fl uctuation. The PDF of A for Swerling IIItargetis given by

where f(a,g)is the joint PDF of a and g.The PDF of A＇can be obtained by the derivative of FA＇(a＇)with respect to a＇:

Since the target fl uctuation A and the target multipath echoes amplitude G are independent,the PDF of A＇can be rewritten as

Substituting(9)and(12)into(15),and noticing that g∈is a positive number,then

In the hypothesis H1,the samples in test cells contain the target signal.The PDF of the output signal of linearlaw detector can be expressed as

where I0(·)is the modi fi ed Bessel function of the zero order.

By(16)and(17),we can get the PDF of Ytin the hypothesis H1

3.2 Detection probability of the proposed method

Due to the intense variation of the return power of the different frequency echoes in the multipath environment,it is not optimal to accumulate all the received frequency echoes.It is feasible to choose only K strong frequency echoes to accumulate in the detection procedure.

Let Y(t)(t=1,2,...,N)be a descending order,i.e.

In this way,only K test cells with strong energy are involved in the non-coherent integration.De fi ne the teststatistic as

According to the detection theory,the targetis detected when the test statistic Z exceeds the threshold value ZT, where the decision hypotheses are

Since Y1,...,YNare independent identically distributed(IDD),we obtain from[12]that the joint PDF of the fi rst K large values after sorting is

where y(t)is the order statistics,satisfying y(1)≥y(2)≥...≥y(K),fY(y(t))is the individual PDF of Y, and FY(y(K))denotes the CDF of Y(FY(y(K))=By using the iterated integraltransformation,we can getthe PDF of Z

The function fZ(z)is complicated in calculation,so the numerical integration is generally needed.The function fY(·)in the hypothesis H1and H0have been given in (10)and(18),respectively.

Given that the probability of false alarm Pfais known previously,the voltage threshold ZTcan be obtained by the following equation

where fZ(z|H0)represents the PDF of Z in the hypothesis H0,which can be obtained by substituting(10)into(23).

By using the PDF of Z in the hypothesis H1and the voltage threshold ZT,the detection probability Pdis

where fZ(z|H1)is the PDF of Z in the hypothesis H1, which can be obtained by substituting(18)into(23).

3.3 The optimum value ofK

In what follows,the important problem is that how to judiciously choose the value of K and substantially enhance the multipath detection performance.For speci fi ed values of N,Pfaand Pd,the optimal value of K is expected to be picked outaccording to the minimum SNR required.

Here,a point-by-pointgraphicalprocedure has been carried out by using the Monte-Carlo simulations based on 108independenttrials fora Swerling IIItarget.The simulation results of minimum SNR required are shown in Fig.2 to Fig.5,for N=5,11,21,and 31,respectively,and for the various indicated combinations of Pfaand Pdat two differentvalues.

Fig.2 Minimum SNR required per pulse as a function of K for N=5

Fig.3 Minimum SNR required per pulse as a function of K for N=11

Fig.4 Minimum SNR required per pulse as a function of K for N=21

Fig.5 Minimum SNR required per pulse as a function of K for N=31

It can be seen from Fig.2 to Fig.5 that the value of K willaffectthe performance of the proposed method and there is an optimalvalue of K fora fi xed N attwo differentvalues.They also show that the traditional noncoherent accumulation method using allfrequency echoes(K=N) is notthe bestchoice in the multipath environment.

For a particular N with re fl ection coef fi cient amplitudespeci fi ed,the optimal value of K can be reasonably considered to be independent of Pfaand Pdwhen 10?6≤Pfa≤10?4and 0.6≤Pd≤0.9.We can see that the optimal value of K seems to shift slowly with increasing N and be approximately proportional to√N(yùn).Thus, the region of optimal K may be expressed as

where Z+denotes the positive integer.Fig.6 plots K versus the number of N.It can be seen from Fig.6 that the optimal value of K lies in the region(26).Additionally, the value ofˉρhas slightin fl uence on the choice of K.Generally speaking,the larger the value ofˉρ,the smaller the optimal K.That is to say,the value ofˉρis close to 0.6, we can selecta big integer satisfying(26),whereas a small integer is chosen when the value ofis close to 0.9.

Fig.6 Parameter K versus N

4.Performance comparison

In this section,we perform Monte Carlo simulations based on 106independenttrials to realize the performance comparisons with the following speci fi cations:the number of carrier frequencies N=11,the number of reference cells L=64,false-alarm probability Pfa=10?4.The target’s RCS fl uctuation is assumed to obey a Swerling III distribution.

Fig.7 depicts the variations of detection probability as a function of SNR per echo for the proposed detector and the M-out-of-N detector attwo differentˉρvalues.

Fig.7 Detection probability oftwo methods atdifferentvalues

According to the reference[3],the optimum of M for the M-out-of-N detector is 2,while Fig.3 shows thatthe optimum of K for the proposed method is differentat two differentˉρvalues.Itis obvious thatthe performance of theproposed detectoris betterthan thatofthe M-out-of-N detectorregardlessofthe value ofThe required SNRofthe proposed method is decreased by up to 2 dB for Pd=0.9 compared with that of the M-out-of-N detector for two cases plotted0.9.The value ofhas little in fl uence on the gain of SNR,thatis,the performance of the proposed algorithm is stable and reliable in multipath detection.

Now we compare the proposed algorithm with the multiple pulse coherent accumulation method in a low angle circumstance.For the sake of fairness,the transmit power is the same for the two methods.The proposed method sequentially transmits 11 pulses with different frequencies and then four(i.e.K=4)large amplitude echoes are noncoherently integrated as test statistic.The coherent accumulation method transmits 11 pulses with the same frequency and then all the pulses are coherently integrated as test statistics.Fig.8 depicts the detection performance of the proposed method and the coherent accumulation method.The fi gure includes two group curves where differentre fl ection coef fi cientamplitudevalues are used.It can be seen from the fi gure thatthe proposed method has a good performance when asked for a high detection probability in the multipath environment.For example,the value ofrequested SNRmay reduce from 20 dBto 0 dBatdetection probability 0.9 when re fl ection coef fi cient amplitudeis equavalentto 0.9.When the re fl ection coef fi cien tis 0.6,the value of requested SNR may reduce from 13 dB to 2 dB atdetection probability 0.9.

Fig.8 Effects of different SNR values on detection probability

5.Conclusions

In order to improve targetdetection performance at a low angle multipath environment,this paper proposes an order statistics detection method based on the multi-frequency technique.Only K large amplitude frequency samples rather than all are involved into the noncoherent integration,achieving a high detection probability at the same SNR.Moreover,a simple statistic model is given to obtain the optimalvalue of K in the multipath environment. Moreover,the proposed algorithm is robust to the multipath re fl ection coef fi cient.The simulation results demonstrate thatthe proposed method is feasible and effective.

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Biographies

Yunhe Cao was born in 1978.He received his Ph.D.degree from Xidian University in 2006.He is now an associate professor at National Laboratory of Radar Signal Processing,Xidian University.His research interests include radar array signalprocessing,wideband signalprocessing and MIMO radar signalprocessing.

E-mail:cyh xidian@163.com

Shenghua Wang was born in 1979.She received her B.S.degree and M.S.degree from Xidian University in 2001 and 2004,respectively.She is now a Ph.D.candidate ofthe National Laboratory of Radar Signal Processing,Xidian University.Her research interests include radar array signal processing and low angle tracking technique.

E-mail:wshh 2011@163.com

Yu Wang was born in 1991.He received his B.S. degree from Xidian University in 2013.He is now a Ph.D.candidate ofthe National Laboratory of Radar Signal Processing,Xidian University.His research interests include radar signal processing,parameter estimation and wideband array signalprocessing.

E-mail:xdwangyu@163.com

Shenghua Zhou was born in 1981.He received his Ph.D.degree from Xidian University in 2011.He is now an associate professor at National Laboratory of Radar Signal Processing,Xidian University.His research interests include multi-station radar signal processing,parameter estimation and target detection.

E-mail:shzhou@mail.xidian.edu.cn

10.1109/JSEE.2015.00032

Manuscriptreceived July 04,2014.

*Corresponding author.

This work was supported by the National Natural Science Foundation of China(61372136;61372134;61172137),the Fundamental Research Funds for the Central Universities(K5051202005),and the China Scholarship Council(CSC).