Li Douzhe,Wu Zhijun,b,*

aSchool of Electronic and Information Engineering,Tianjin University,Tianjin 300072,China

b School of Electronic and Information Engineering,Civil Aviation University of China,Tianjin 300300,China

DME Interference mitigation for L-DACS1 based on system identification and sparse representation

Li Douzhea,Wu Zhijuna,b,*

aSchool of Electronic and Information Engineering,Tianjin University,Tianjin 300072,China

bSchool of Electronic and Information Engineering,Civil Aviation University of China,Tianjin 300300,China

L-band digital aeronautical communication system 1(L-DACS1)is a promising candidate data-link for future air-ground communication,but it is severely interfered by the pulse pairs(PPs)generated by distance measure equipment.A novel PP mitigation approach is proposed in this paper.Firstly,a deformed PP detection(DPPD)method that combines a filter bank,correlation detection,and rescanning is proposed to detect the deformed PPs(DPPs)which are caused by multiple filters in the receiver.Secondly,a finite impulse response(FIR)model is used to approximate the overall characteristic offilters,and then the waveform of DPP can be acquired by the original waveform of PP and the FIR model.Finally,sparse representation is used to estimate the position and amplitude of each DPP,and then reconstruct each DPP.The reconstructed DPPs will be subtracted from the contaminated signal to mitigate interference.Numerical experiments show that the bit error rate performance of our approach is about 5 dB better than that of recent works and is closer to interference-free environment.

1.Introduction

L-band data-link has been selected to support next-generation aeronautical terrestrial air-ground communication in next 30 years.1Wherein,the L-band digital aeronautical communication system 1(L-DACS1)scheme based on the orthogonal frequency division multiplexing(OFDM)technology gains widespread attentions because it can provide high spectral eff iciency and high data rate in high-mobility environment.2To use spectral resources efficiently,inlay-deployment is often adopted,3i.e.,L-DACS1’s band is embedded in the middle oftwo adjacentbandsofdistancemeasureequipment(DME).However,inlay-deployment leads to the overlapping of parts of their spectrum,and DME causes severe disturbance on L-DACS1.

In time domain,the interference caused by DME,which consists of a series of pulse pairs(PPs),belongs to the category of impulsive noise(IN)interference.4Existing IN mitigation solutions can be divided into two categories,and some related works are listed below.

One category is based on the well-known pulse-blanking(PB)which is a type of memoryless nonlinear mapping,i.e.,the signal samples whose magnitude is higher than a given threshold will be forced to be zero;otherwise,they will be retained.A detailed mathematical analysis and performance evaluation of PB based on Gauss Bernoulli model were discussed in Refs.5,6.A gradually decreasing threshold scheme was proposed in Ref.7.For compensating the side effect caused by PB,some pilot-aided methods were compared in Ref.8.Epple et al.proposed a method in which the PB processed signal and the originally received signal were optimally combined to maximize the signal-to-interference-and-noise ratio.9A retransmission scheme was proposed in Ref.10in which two copies of the received OFDM symbols with interference were suitably combined after the PB operation to improve the bit error rate(BER)performance,but its data rate was reduced due to the redundant transmission.Iterative PB compensation methods were proposed in Refs.11,12.Generally,the above methods will dramatically damage useful signals and degrade system performance even though some compensation schemes are used,since the width of PP is much larger than random impulsive noise.

The other category is modeling IN by a proper statistical model and reconstructing it by the estimated model parameters.A factor graph-based scheme was proposed in Ref.13.It uses a Bernoulli-Gaussian mixture model for emission noise and a Bernoulli-Gaussian hidden Markov model for burst noise.Time-domain interleaving can effectively transform burst noise into random noise,14,15but it is not compatible with L-DACS1 which only uses frequency-domain interleaving.An algorithm that utilizes the null subcarriers in guard band for IN estimation and cancelation was proposed in Ref.16.It jointly exploits the specific structure of IN and the available a priori information for sparse signal recovery.However,a statistical model is not suitable for our environment since each DME PP has a relatively fixed waveform that cannot be modeled by statistical methods.

In the PP detection stage,PPs should be detected before they are mitigated.In Ref.4,several methods such as correlation,power detection,and filter-based methods were compared,and a double-filter scheme was proposed.However,they were only tested in a relatively ideal environment.In this paper,more real situations such as the variations of interference power and PPs’rate(measured by pulse pair per second,ppps)will be considered.

It is widely known that in aeronautical environment,the number of DME stations is finite,and the air route of a specific civil aircraft is also fixed,so the original waveform of PP(OWP)can be easily measured or obtained from DME’s manufacturers.Therefore,the OWP can be assumed to be known.This motivates us to use the OWP to reconstruct deformed PPs(DPPs)and then subtract them from contaminated signal for achieving an interference-free environment.The advantage of the proposed approach is that interference can be suppressed before any other successive signal processing such as synchronization and demodulation,and it will greatly enhance the BER performance.

The contribution of this paper can be summarized into the following three aspects:

(1)A filter bank(FB)is introduced to separate different DME bands.In addition,correlation and rescanning scheme are jointly used for enhancing DPP detection accuracy.

(2)A system identification method based on a finite impulse response(FIR)model is used to approximate the overall characteristic offilters in the receiver,and then the waveform of the DPP could be directly obtained by the OWP and the FIR model.

(3)Sparse representation is used to estimate the necessary parameters for reconstructing each DPP(even they are densely overlapped).

2.System model

Consider a baseband model which is compatible with LDACS1 specification,2,17,18as shown in Fig.1(a).The original information bits enter into an OFDM transmitter,which incorporates channel coding,modulating,and interleaving,inserts pilot symbols,and finally forms frequency-domain symbol Xk(the subscript k denotes the index of a subcarrier).Then Xkis transformed to time-domain symbol xnby inverse discrete Fourier transform(IDFT),i.e.,

Fig.1 System model.

where N is the number of subcarriers.A cyclic prefix is added to each xn.These symbols containing cyclic prefix will be inserted into a data frame sn,which is the information bearing signal to be transmitted(hereinafter the signal that carries information is called useful signal).

Denote the OWP by m.Usually,it can be modeled as19

where k ∈ [1，Lm].Δkis the time difference between two identical pulses of m.β is the width of each pulse.In practical situations,the OWP of DME stations from different manufacturers may have some slight differences,but Δkis constant and fixed.

Denote the originally received signal by.The observation model ofcan be described as

where?denotes circular convolution.hnis the channel impulse response and is constructed by a wide sense stationary uncorrelated scattering model.20Essentially,hnis a two-ray time variant Rician channel.The first ray is a strong line-ofsight(LOS)component,while the second ray is a superposition of several reflected non-line-of-sight(NLOS)echoes.The power ratio between LOS and NLOS components is called Rician factor.wnis additive white Gaussian noise(AWGN).inis DME interference and is constructed by superimposing numerous PPs,i.e.,

where P is the number of PPs,and m（p）denotes the p th parameterized version of m.The kth sample(k ∈ [1，Lm])of m（p）can be written as

where αpand τpare the amplitude and position of m（p）,respectively.αpis complex valued,and its phase is uniformly distributed on[0，2π]while its magnitude depends on the power of DME station.τpis modeled as a Poisson process of rate θ.(Note that through adjusting its noise level,a practical DME ground station keeps a constant PP rate,ppps=2700,no matter how many aircraft interrogating it.When there exists no aircraft,some filler PPs with random interval will be used to fill the DME transmitter duty cycle.A typical value of the minimum time interval between two PPs is 100 μs.21)

fpis the carrier frequency of m（p）.fponly has two possible values(i.e.,+0.5 MHz or-0.5 MHz)because an intermediate frequency filter(IFF)has suppressed the PPs with a carrier frequency greater than+0.5 MHz and less than-0.5 MHz,as shown in Fig.1(b).Here the IFF is a baseband equivalent representation of the passband filter in intermediate frequency.(Note that in practice fpmay deviate from exact±0.5 MHz since it is influenced by the Doppler effect during transmission.This phenomenon will be considered in Section 4.2.)

After passing through the IFF,the received signal rnis processed by the proposed approach(thick border),and then we get the interference-free signal~sn.Each OFDM symbol ynwill be extracted from~sn.By removing cyclic prefix from ynand using DFT,we can get the frequency-domain symbol Yk.Then channel estimation is performed and we get the estimated frequency-domain symbol X′k.Finally,after de-interleaving,demodulation,and decoding,the output information bits are used for BER performance evaluation.

According to the above-mentioned system model,in this paper, αpand τpare the unknown parameters that should be estimated for reconstructing each DPP.

3.Proposed PP mitigation approach

In this section,the proposed PP mitigation approach will be described and analyzed.It mainly includes the following three steps:

Step 1.Correlation detection is used for detecting DPPs’rough positions,and an FB and rescanning scheme are also jointly used to avoid false detection.Based on these detected rough positions,each segment that contains one or more DPPs is extracted from the received signal rnand used for simplifying the further DPP reconstruction.

Step 2.An odd-symmetric FIR model is used to approximate the overall characteristic of all the filters in the receiver.The model coefficients can be obtained by a linear least square(LS)estimator,and then the waveform of the DPP can be acquired through the known OWP and the FIR model.

Step 3.Based on the calculated DPP waveform,sparse representation is used to estimate the accurate position and amplitude of each DPP in each signal segment extracted from Step 1.Finally,the reconstructed DPPs are removed from the contaminated signal.

3.1.PP detection

Before our PP detection method is presented,first of all,the conventional detection method and the drawback of applying it to our system will be described.

3.1.1.Conventional correlation-based PP detection

Since the time difference(Δk)between two identical pulses in each PP is constant and fixed,calculating the normalized correlation between the two pulses of one PP4and then finding peaks above a threshold can readily detect the position of PPs,as shown in Eq.(6):

where Dnis the correlation result,rnis the received signal,Lcis the correlation length, （·）*denotes the conjugate,and|·|denotes the magnitude.

Fig.2 Drawback and improvement of the conventional PP detection method.

Fig.2(a)demonstrates an example of rnwhich contains six PPs(PP 1-PP 6),and a dashed vertical line is used to mark the starting position of each PP.PP 3amp;PP 4 are overlapped and have the same carrier frequency.PP 5amp;PP 6 are also overlapped but have different carrier frequencies.Fig.2(b)shows the result of applying Eq.(6)to the signal in Fig.2(a).The drawback is that when PP’s power is relatively low,such as those of PP 1amp;PP 2,the correct correlation peak will be submerged in some spurious peaks generated by the useful signal.Moreover,overlapped PPs such as PP 3amp;PP 4 and PP 5amp;PP 6 lead to the result of inaccurate correlation peaks and will cause wrong detection.As a comparison,Fig.2(c)gives the results of our proposed PP detection method.Obviously,our results are not affected by the spurious correlation peaks.

Fig.3 Filter bank structure.

3.1.2.Improved PP detection

To overcome aforementioned drawbacks,a deformed PP detection(DPPD)method is proposed.In DPPD,an FB containing four filters is developed to pre-process the received signal rn.The aim of the FB is separating different DME bands,so that the overlapped PPs with different carrier frequencies(i.e.,PP 5 and PP 6)can be separated.Meanwhile,rescanning can avoid wrong detection caused by the useful signal(i.e.,PP 1 and PP 2 can be detected effectively).

The transfer functions of the four filters are denoted by h1n-h4n,respectively,as shown in Fig.3.h1nand h3nare low-pass filters,while h2nand h4nare high-pass filters,and fsdenotes the sample rate.

The corresponding signal processing flow is shown in Fig.4.Before rnpasses through the FB,its carrier frequency should be up-converted from baseband to a proper frequency fcinstead of being directly processed in baseband(in a practical system,the carrier frequency of rncan be down-converted from intermediate frequency to fcfirstly).The reason is that the magnitude response of a digital filter with real coefficients is always symmetric about zero frequency(ω=0),and the carrier frequencies of two adjacent DME bands are also symmetric about zero frequency(see Fig.1(b)).Therefore,if the carrier frequency of rnis zero,different DME bands cannot be separated.

Then,the output of the FB will be down-converted to baseband for subsequent DPP detection and mitigation.The resulting signals are denoted by r1n-r4n, respectively. In conjunction with Fig.3,we can learn that r1nand r2ncontain interference and useful signal,while r3nand r4nmainly contain interference but also a small part of useful signal.

The FB has two features for facilitating the receiver design.One is that h1nand h2nare symmetric filters,i.e.,if M coeff icients of h1nare denoted by c0，c1，...，cM-1,the coefficients of h2nare 1-c0，-c1，...，-cM-1.Consequently,h1nand h2nalso have the same cut-offfrequency,leading to the relationship of r1n+r2n=rn,i.e.,r1nand r2ncan be processed individually and then combined together.The other feature is that the orders of h1n-h4nare equal,and thus the positions of DPPs in r1n（r2n）can be detected from r3n（r4n）.According to these two features,the following processing is only applied to r1nand r3n,and the same operation can be applied to r2nand r4n.

Fig.4 Block diagram of improved PP mitigation.

If Eq.(6)is directly applied to r3nfor detecting DPPs’positions,Dnwill still contain spurious peaks,since there exists residual useful signal in r3n.Therefore,a proper threshold Tbis given for removing the residual useful signal from r3n,i.e.,

Tbcan be determined by Tb=E（|r1n|）-E（|r3n|）,where E（|r1n|）denotes the averaged magnitude of the signal r1nand can be calculated byin a sliding window with a length of Lw.This setting strategy of Tbexcludes the power of interference originated from r3nusing the subtraction between r1nand r3n,and thus Tbcan give a rough empirical value of the averaged magnitude of the useful signal.

When applying Eq.(6)to~r3n,if the denominator of Eq.(6)is zero,Dnshould be directly set to zero,since this part of signal has been fully removed by the threshold Tb.After we get Dn,a threshold Tcis applied to Dnfor detecting correlation peaks,and Tcwill be determined in Section 4.1 through simulation.

To avoid false detection or missing any DPP,a rescanning scheme is used.Denote the magnitude of Q detected DPPs in~r3nby Aq,q ∈ [0，Q-1],and then the minimum Aqis chosen as the threshold Tr.This is reasonable because the DPPs whose magnitudes are lower than Trcannot be detected.Trcan be dynamically updated when more DPPs are detected.When rescanning~r3n,if there is a segment whose magnitude is higher than Trand whose width is larger than the width of the OWP(m),it should be labeled as containing DPPs;otherwise,not labeled as containing DPPs.

According to the labeled segments in r3n,the same segments in r1nwill be extracted,and each segment contains one or more DPPs.Fig.2(c)shows the output of h1nwhen the DPPD method is applied to the signal in Fig.2(a).The output of h2nis not presented for clarity.It is obvious that the DPPD method effectively avoids the spurious peaks and shows better performance.

3.2.Approximation of the overall characteristic offilters

The construction of an FIR model can be treated as a calibration stage which is used to identify the overall transfer function offilters in the receiver.Once model coefficients have been determined,the function is also determined,and the coeff icients will be stored in the receiver for further use.When the OWP has changed,users of the receiver can immediately get the corresponding waveform of the DPP by the model itself.

The waveform of the DPP is denoted by~m and the nth(n ∈ [1，L~m])sample of~m can be described as

where ciis tap coefficient,δ is the Kronecker delta function,and M is the order of htotalnand should be pre-selected according to the specific system.In Section 4.2,we will give the impact of different model orders on the model approximation accuracy.

The estimation of cican be obtained from a stimulus signal pnand its corresponding system response qn.pncan either be the model of PP(i.e.,the signal m)or some other known signal.The estimated version of htnotalis denoted byh^tnotaland can be obtained by minimizing the sum of squared error as follows:

Taking into account the time delay caused byh^tnotal,the position of the corresponding response qnshould be determined by correlation detection.Assuming that the length of pnis Lp,then the length of qnwill be Lp+M-1.Substituting Eq.(9)into Eq.(10)and expanding the convolution,one obtains

Now~m(i.e.,the waveform of the DPP)can be obtained through Eqs.(8),(9)and(14).

There are two factors that affect the approximation precision.One is order mismatch.Consider that some filters in the receiver are composed of analog devices(e.g.,IFF and other radio frequency filters),but signal has been totally digitized in baseband,so the order of the FIR model may be unequal to the actual order of overall system filters.In this case,the length of the FIR model output Q is different from the actual system output.Fortunately,the estimation of cican be performed off-line,and a proper order of the FIR model can be adjusted and chosen to minimize the error.The other factor is the inherent residual of LS estimation,i.e.,substituting the solution of W into Eq.(13)and normalizing by the power of the stimulus signal P,then we can obtain

In Section 4.2,the simulation results will show the impacts of the above two factors on approximation precision.

3.3.Mitigating DPPs by sparse representation

In this subsection,the DPPs will be reconstructed through sparse representation which is solved by a proximal gradient(PG)algorithm.The formulation of our algorithm will be described first,and then the parameter selection scheme and convergence property will be discussed.

3.3.1.Algorithm description

Assume that bn,n ∈ [0，Ls-1]is a segment of r1nand contains I DPPs.The representation of bnis shown as

where~m is the waveform of the DPP with length L~m(L~m＜Ls)that has been obtained in Section 3.2. αiand τi,τi∈ [0，Ls-L~m]are the amplitude and position of the i th~m,respectively.εnincludes the useful signal and AWGN.

For convenience,we rewrite Eq.(16)in matrix form as b=Ax,where b=[b0，b1，...，bLs-1]T∈ CLs×1is the observation vector.Each non-zero entry of x,x ∈ CLx×1specifies the position and amplitude of a specific~m and Lx=Ls-L~m+1.A ∈ CLs×Lx, and the jth column of A isEach column of A is a possible position of~m,in other words,all possible positions of~m form a dictionary A.

According to Ref.12,in an area with the highest density of DME stations(e.g.,Charles-de-Gaulles airport in Paris),the L-DACS1 receiver can be simultaneously interfered by three DME stations with the same carrier frequency(see Table 5).In our simulation based on this interference scenario,bnseldom contains more than three DPPs.Therefore,most entries in x will be zero,i.e.,x is a sparse vector which can be described by ?0norm(the number of non-zero entries in x).This problem is extremely difficult(NP-hard in general)to solve,so ?0norm is relaxed to ?1norm,resulting in the following optimization problem:

wheref（x）= （1/2）‖Ax-b‖22is the squared-error loss,g（x）= ‖x‖1is the ?1-norm regularization term that enforces sparsity,and λ is the positive smoothing parameter.f（·）is a smooth convex function and g（·）is a continuous non-smooth convex function.To solve Eq.(17),a PG algorithm,which is suitable for complex valued problems,is adopted.

Unlike the basic gradient method,PG iteratively approximates F around a given point y by a quadratic model Q（x，y）,and then minimizes Q（x，y）rather than the original objective function F.The definition of Q（x，y）22is given as

where ?f（x）=AT（Ax-b）is the gradient off（x）and Lfis a positive constant that indicates the width of Q（x，y）.Given a proximal point y,the closed form solution for x is given by the soft-thresholding operator,i.e.,

where Lfis unknown,but can be found by a line search method(e.g.,the backtracking step size rule,23please see lines 6–10 in Algorithm 1)in each iteration.The algorithm will stop when the final change in the sum of squares relative to its initial value is less than the default value of the function tolerance.The detailed steps of PG algorithm are listed in Table 1.

3.3.2.Convergence property and parameter analysis

In the PG algorithm,x achieves an accelerated non-asymptotic convergence rate of O（k-2）,24where k is the number of iterations.Actually,we found in most situations that the value off（x）tends to be stabilized almost in less than 15 iterations,since each extracted segment of r1nis usually short and contains only a few DPPs.In Section 4.3,an experiment will show that 20 iterations can lead to a good BER performance which is very close to interference-free environment.

λ can provide trade-off between the fidelity of measurements and noise sensitivity.While λ is too small or too large,the optimal solution of x is useless.When λ → 0,we are almost left with argand the regularization term is lost and will cause over-fitting or failing to converge.If λ is too large(larger than 2‖ATb‖∞),all the components of the solution of x will be zero.Somewhere in between these twovalues,there may be a set of values of λ by which the solution easily leads to the exact value of x.

Table 1 PG-based estimation.

Considering the system stability requirement in practical situations,λ is determined in advance.The related factors that affect the value of λ are analyzed based on a Bayesian scheme25which formulates the sparse representation as a maximum a posterior(MAP)estimation problem.According to Bayesian law,the MAP estimation of x with respect to the observation b can be written as

The sum of the useful signal and AWGN can be treated as still following a Gaussian distribution with variance σ2,so the probability density function of P（b|x）is

Since the amplitude of xiis always nonnegative,i.e.,xi≥0,the prior distribution of P（x）can be described by following an i.i.d.exponential distribution probability model:

where σxis the standard deviation of xi.Substituting the above Eqs.(21)and(22)into Eq.(20),λ can be calculated as

According to Eq.(22),the maximum likelihood estimation of σxis

From Eqs.(23)and(24),we could learn that the value of λ is related to two factors.One is σ2and the other is xi,i.e.,the amplitude of each non-zero entry of x.Unfortunately,xiis unobtainable,since it is what we want to estimate through sparse representation,so a fixed λ should be set in advance.It can be seen from Section 4.2 that the reasonable value of λ only varies in a narrow range,even if a fixed value can lead to almost the same DPP reconstruction performance.

3.4.Complexity analysis

In this section,we will analyze the complexity of our proposed approach and compare it with two recently published methods based on PB,i.e.,optimal combining(OC)9and subbandfilter-bank(SFB).11Note that all these three methods need the same FFT operation with a complexity of O（Nlg（N））,where N is the number of subcarriers.The tested received signal rnhas a length of Lr.

The computational overhead of our proposed approach falls in two parts,i.e.,PP detection and sparse representation.In PP detection,each filter in the FB has a complexity of O（Lr）.The correlation operations contain 3Lc+2 complex operations in one sample(see Eq.(6))when the correlation length is Lc,and thus the complexity of correlation is O（3LcLr）.Usually we have Lr? Lc,so the overall complexity is O（Lr）.However,it can be greatly reduced when ppps is lower,since most of the samples will be forced to be zero by Eq.(7).In sparse representation,the complexity of each iteration is mainly produced by Eq.(19)in which the gradient?f（y）should be calculated.Thus,in each extracted signal segment,the complexity is O（L2x）,where Lxis the length of the sparse vector x.

OC is a non-iterative PB compensating method.The overall complexity of OC is no more than O（Lr）,because rnis composed by multiple OFDM symbols and OC mitigates interference in each symbol individually.For each OFDM symbol,the complexity is O（N）9when perfect DPPs detecting is assumed.SFB is an iterative PB compensating method.It firstly uses a filter bank with several overlapped filters to constrain PB in each subband other than the whole spectrum,and each filter has a complexity of O（Lr）.Secondly,it iteratively mitigates interference in each symbol of each subband individually.In each iteration,it needs to reconstruct the inter-carrierinterference matrix of each subband,and its complexity is O（N2）.

OC is easy to implement but has a relatively lower performance.Comparing SFB to our approach,its complexity is almost on the same level as ours,since in the experiments we found Lxvaries between 30 and 150,and N is 64.Please refer to Section 4 for more details.

4.Experiment and analysis

In this section,the performance of the proposed DPP mitigation approach is evaluated using Monte-Carlo simulations,and compared with existing works.

Transceiver and channel parameters are given in Table 2 while OFDM parameters are given in Table 3.According to an L-DACS1 prototype,3the magnitude response of IFF(gn)should be-40 dB at±0.59 MHz,so a 64-order FIR filter is used to satisfy the requirement.h1n-h4nare all 32-order FIR filters.

The received signal r（nrx）is over-sampled by a factor of 4;3therefore,fc(i.e.,the cut-offfrequency of h1nand h2n)is set at fs/4 to avoid violating Nyquist theory.Considering that the bandwidth of rnis about 0.5 MHz(see Table 3),in experiments we found that almost the same DPP detection performance can be achieved when setting the cut-offfrequency of h3nand h4nto be fc-(300±25)kHz and fc+(300±25)kHz respectively.The used parameters are summarized in Table 4.

Table 2 Transceiver and channel parameters.

Table 3 OFDM parameters.

4.1.DPP detection performance

The DPP detection correctness ratio is denoted by Rcorrect.Rcorrect∈ [-1，1]and is defined as

where Ntotaldenotes the quantity of originally existing DPPs,and Ncorrectand Ndetecteddenote the quantities of correctly detected and all detected DPPs,respectively.If all DPPs are correctly detected,Rcorrectis equal to 1.While in the worst situation,all detected DPPs are false,then Rcorrectis equal to-1.

The choice of Tcis based on simulation.The interference source is one DME station,and the signal-to-interference ratio(SIR)of r（

nrx）is changed in-5 to 0 dB.Different ppps are also considered.As shown in Fig.5,when the SIR is less than-3 dB,Rcorrectis almost equal to 1 for all threshold values,so Tccan be selected in a relatively wider range,such as Tc=0.7 in this experiment.When the SIR is fixed,larger ppps will cause a decline of Rcorrect,since each DPP has relatively lower power and is harder to be detected.

The performances of three DPP detection methods are compared in Fig.6.The low-and high-pass method is the best method reported by Epple.4The energy-based method was recently proposed by Bartoli.10The last method is our proposed DPPD method.Two DME stations with different car-rier frequencies are used as the interference sources.The SIR of each DME interference source is-5 to 5 dB.

Table 4 Other parameters.

Fig.5 PPs detection correctness ratio for different interference power and threshold,SNR=4 dB.

Fig.6 PerformancesofdifferentPPsdetection methods,SNR=4 dB.

Obviously,the DPPD method shows more accuracy and stability,and is almost unaffected by the changes of ppps and interference power.When the SIR is lower than 1 dB,the correctness ratio of ours is almost equal to one.Lowand high-pass can also keep relatively high performance,but is not stable enough.The correctness ratio of the energybased method declines especially when the power of the interference is close to that of the signal.Furthermore,an obvious drawback of the low-and high-pass and energy-based methods is that they have to take extra operations to obtain the correct signal power for threshold determination since signal and interference are mixed in time domain and are time-variant.Although the DPPD method shows better performance,but as a compromise,its complexity,which is caused by correlation operation,is slightly higher.However,the complexity can be greatly reduced when there are only a few DPPs;in this situation,most of the signal samples in r3nwill be forced to be zero according to Eq.(7),and then the correlation result of an all-zero signal segment can be directly set to be zero.

4.2.DPP mitigation performance

Three main factors that affect DPP mitigation performance will be evaluated and analyzed in this subsection,i.e.,the fitting precision of the FIR model,the in fluence of Doppler effect on DPP waveform,and sparse representation-based DPP reconstruction.

The fitting precision of the FIR model with different orders(8–64)is shown in Fig.7,wherein,Fig.7(a)demonstrates the fitting error which includes order mismatch and LS estimation residual.The error is signi ficant when the order is 8 and the waveform of model generated DPP severely deviates from the original waveform of DPP.When the order is increased up to 24,the generated DPP is well matched with the original DPP.Fig.7(b)gives the quantitative analysis between the theoretical residual calculated by Eq.(15)and the actual fitting error.For calculating the actual error,the generated DPP should be aligned with the original DPP(like Fig.7(a),the overlapped waveform).Normalized mean squareerror(NMSE)is used to calculate the deviations between these two types of DPPs,as shown in Eq.(26):

where DPPgand DPPodenote the generated and original DPPs,respectively.It can be seen that when the order is increased,both the actual and theoretical errors decrease rapidly,and the difference between them is about 8 dB.

Fig.7 System identification error.

Another factor is that PP’s carrier frequency is affected by Doppler effect and will not be exactly±0.5 MHz.The Doppler shift is between-1.25 kHz and+1.25 kHz for L-DACS1’s band(960–1164 MHz)since the speed of a civil aircraft is often lower than 1000 km/h.In the experiment,the DPP waveform affected by Doppler is compared with(1)un-affected original DPP and(2)un-affected DPP generated by the FIR model.Fig.8 shows the errors between degraded DPP waveform and unaffected ones((1)and(2))calculated by the NMSE criterion.We only plot the negative Doppler shift since the NMSEs of positive and negative Doppler shifts are almost symmetric about zero frequency shift.Apparently,a larger Doppler shift causes a higher NMSE error for all the four cases.Moreover,the NMSE is influenced by the order of the FIR model.When the order of the FIR model is properly selected according to Fig.7(b)(e.g.,56 is optimal in our experiments),the NMSE curves of the above(1)and(2)are almost overlapped.When an improper order(such as 24 or 32)is selected,the NMSE value cannot keep declining continuously as the Doppler shift decreases.

In the rest of this subsection,the factor of sparse representation-based DPP reconstruction will be presented.4 DME stations with parameters given in Table 5 are combined into four interference scenarios:(1)Scenario 1,only#3;(2)Scenario 2,only#1;(3)Scenario 3,#1 and#4;and(4)Scenario 4,from#1 to#4.We will firstly analyze λ selection and DPP reconstruction performancewithoutconsideringDoppler effect,and then evaluate the DPP reconstruction performance by considering the influence of Doppler effect on the DPP waveform.

In order to evaluate how the optimal value of λ changes under the influence of σ2and σx(see Eq.(23)),cross validation is used to acquire the proper λ.In a specific interference scenario,each extracted segment will be separately processed using several different values of λ,and the λ corresponding to the best NMSE performance will be recorded and averaged.Here,the waveform of the DPP is assumed to be perfectly obtained.

As shown in Fig.9,when the SNR increases,σ2will decrease and λ shows a downward trend,which is consistent with Eq.(23)that σ2is directly proportional to λ.In Scenario 1,the interference power is lower,and thus λ has a relatively higher value.The interference power of Scenarios 2–4 is much higher than that of Scenario 1,and their corresponding λ values are lower.This is consistent with Eq.(23)that σxis inver-sely proportional to λ.In general, λ varies between 0.05 and 0.3.

Fig.8 Influence of Doppler effect on the waveform of the DPP(FIR DPP:un-affected DPP generated by the FIR model,Doppler DPP:DPP waveform affected by Doppler).

Table 5 Interference scenario.

DPP reconstruction performance without considering Doppler effect is shown in Fig.10 and 100 iterations are used for each reconstruction.The contaminated signal will subtract the reconstructed DPPs,and then be compared with the original useful signal under NMSE criterion.(Note that for excluding the impacts of different ppps,the NMSE is only calculated in the signal segments that contain DPPs.)Two λ selecting schemes are used in DPP reconstruction,one is the abovementioned optimal λ selecting scheme and the other is giving λ a fixed value(here λ=0.2 is used).It is observed that the difference between the resulting NMSEs of these two λ selecting schemes is almost negligible(less than 1 dB),so a fixed λ can be used for simplifying receiver design.

If only one interference source exists(Scenario 1 or 2),the best NMSE is presented.Performance degrades when two interference sources exist(Scenario 3),but they can still be separated by the FB if they are on different carrier frequencies.4 interference sources(Scenario 4)cause the worst result,and actually,overlapped PPs with the same carrier frequency are the major factor that affects performance.

The DPP reconstruction performance considering the in fluence of Doppler shift on the DPP waveform is evaluated in two scenarios(3 and 4)and shown in Fig.11.Also 100 iterations are used in each reconstruction.Each scenario includes three different SNR settings,i.e.,0,5,and 20,respectively.All the DME sources in each scenario have the same Doppler shift.It is clear that,for all the cases,a larger Doppler shift leads to a worse NMSE performance especially when the frequency shift is larger than 200 Hz,and otherwise,the NMSE performance has no signi ficant change when compared to that with zero Doppler shift.This demonstrates that when Doppler shift is less than 200 Hz,the in fluence of Doppler shift on the DPP waveform can be ignored.In Section 4.3,we will see that the BER performance does not have a notable deterioration and is still better than that of previous works even though the DPPs suffer from the most severe Doppler shift.

Fig.9 Optimal value of λ under different SNRs and scenarios.

Fig.10 Reconstruction performance comparison between using fixed and optimal selected λ through cross validation(CV).

Fig.11 Reconstruction performance considering the influence of Doppler effect on DPP waveform(λ=0.2).

4.3.BER performance

In this subsection,the BER performance of our proposed approach will be evaluated.For each extracted segment that contains DPPs in our method,only 20 iterations are taken to speed up the DPP reconstruction.As a comparison,the conventional pulse blanking(PB)12and recently published optimal combining(OC)9and subband-filter-bank(SFB)11will also be evaluated.

The start and Doppler shift of a data frame are estimated through a synchronization sequence in front of the frame.Linear interpolation is used for channel estimation since it does not depend on any channel priori knowledge.Although Wiener interpolation12can improve channel estimation performance,it is subject to higher computational complexity and needs the prior of channel autocorrelation which is not practical,since channel status often changes in different flight stages20such as taking off or approaching.Additionally,iterative decoding is not applied to our simulation for simplicity.

Fig.12 BER performance with coded transmission.

As shown in Fig.12,the bottom BER curve corresponds to the DME-free environment,and its decline rate gradually slows down since the NLOS component of channel cannot be synchronized.Obviously,both OC and PB methods that are applied to Scenario 2 do not bring any advantage even there exists only a single DME interfere source.This is because the width of each PP(almost one-fifth width of an OFDM symbol)is much wider than that of the random noise based on the Gaussian-Bernoulli model.Simply blanking such a large part of signal will greatly damage the useful information and cause serious inter carrier interference.Our proposed approach is evaluated in Scenarios 2–4.Note that more interference sources will degrade performance,e.g.,the performance of Scenario 4 is worse than that of Scenario 2.Ours in Scenario 4 brings significant improvement and is better than others,and also the BER of ours is closer to DME-free environment.

For a more practical aeronautical environment,the Doppler shift of DME interference is considered,and our approach will be compared with the SBF method.In this Doppler shifted case,the interference signal inof Scenario 4 will firstly be sent through the channel,and then be added to the useful signal for forming the received signal r（nrx）.For constructing the SFB method,nine overlapped filters with a width of each subband 78 kHz are used,and three iterations are used for reconstructing the inter carrier interference matrix in each subband.It can be seen from Fig.12 that a Doppler shift of inwill further degrade the performance,but the BER of our approach is still about 5 dB better than that of SFB.

5.Conclusions

A novel DME interference mitigation approach is proposed in this paper and evaluated in relatively real environment.Our approach includes DPP detection,obtaining the waveform of the DPP through an FIR model,and then estimating the amplitude and position of each DPP.The superiority of this approach is that only time-domain methods are adopted to mitigate PP interference,so the more accurately we could measure the original waveform of PP(OWP),the more excellent the performance can be achieved.Furthermore,our approach does not affect the subsequent processing such as channel estimation or decoding.Some improvements could be made in future works,such as automatic detection of DPPs’waveform which can further enhance the practicability of our proposed method.

Acknowledgments

This work was supported in part by the National Natural Science Foundation(Nos.U1533107 and U1433105),the Civil Aviation Science and Technology Innovation Foundation(No.MHRD20130217),and the Fundamental Research Funds for the Central Universities of CAUC(No.3122016D003).

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Li Douzhe is a Ph.D.candidate in Information and Communication System at Tianjin University.The topic of his study is the related key technologies about aeronautical telecommunication network.His interested area includes mobile wireless communication signal processing and mobile communication network optimization.

Wu Zhijun received his B.S.and M.S.degrees in Information Processing from the Department of Electronics Engineering at Xidian University,Xi’an,China,in 1988 and 1996,and his Ph.D.degree in Cryptography from the School of Electronics and Information Engineering at Beijing University of Posts and Telecommunications,Beijing,China,in 2004,respectively.After being a postdoctoral research associate in the Department of Electronics at Tsinghua University,Beijing,China,from 2004 to 2006,he has been with the School of Electronics and Information Engineering at Civil Aviation University of China,Tianjin,China,where he is a professor and Ph.D.advisor.His current research interests include aeronautical telecommunication network and information security.

17 February 2016;revised 24 June 2016;accepted 18 July 2016

Available online 21 October 2016

DME interference;

L-DACS1;

Least square approximations;

Proximal gradient algorithm;Sparse representation

?2016 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.This is an openaccess article under the CCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

*Corresponding author.Tel.:+86 22 24092827.

E-mail address:zjwu@cauc.edu.cn(Z.Wu).

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http://dx.doi.org/10.1016/j.cja.2016.09.011

1000-9361?2016 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.

This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).