Matrix-Pencil Approach-Based Interference Mitigation for FMCW Radar Systems
SSUBMITTED TO IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES 1
Matrix-Pencil Approach-Based InterferenceMitigation for FMCW Radar Systems
Jianping Wang,
Member, IEEE,
Min Ding, and Alexander Yarovoy,
Fellow, IEEE
Abstract —A novel matrix pencil-based interference mitigationapproach for FMCW radars is proposed in this paper. Theinterference-contaminated segment of the beat signal is firstlycut out and then the signal samples in the cut-out region arereconstructed by modeling the beat signal as a sum of complexexponentials and using the matrix pencil method to estimatetheir parameters. The efficiency of the proposed approach for theinterference with different parameters (i.e. interference duration,signal-to-noise ratio (SNR), and different target scenarios) isinvestigated by means of numerical simulations. The proposedinterference mitigation approach is intensively verified on ex-perimental data. Comparisons of the proposed approach withthe zeroing and other beat-frequency interpolation techniquesare presented. The results indicate the broad applicability andsuperiority of the proposed approach, especially in low SNR andlong interference duration situations.
Index Terms —FMCW radar, interference mitigation, matrixpencil, signal fusion.
I. I
NTRODUCTION F REQUENCY modulated continuous-wave (FMCW)radars are widely used in both civilian and militaryapplications due to its simple processing method, highaccuracy and high reliability. With the explosive increaseof wireless radio and sensing applications, FMCW radarsface increasingly severe interference from other devices. Forinstance, modern cars are equipped with multiple FMCWradars to assist drivers and improve transportation safety,where the radars inevitably cause strong interference amongeach other. Moreover, FMCW weather radars also sufferfrom the radio frequency interference from the surroundingenvironment. In these situations, the strong interferenceleads to reduced radar sensitivity and resolution, weak targetmasking and probably ghost target detection. Therefore,to overcome these problems and alleviate performancedegradation of the radar systems, it is crucial to take properinterference mitigation in practice.So far, a number of approaches have been proposed forinterference migration, which can be mainly classified intotwo categories: (i) system-level approaches; (ii) post-signalprocessing techniques. System-level approaches exploits tem-poral, spatial, polarization, frequency and code diversities inradar system, antenna array and waveform design. In [1], acircular polarized antenna architecture is design to combatthe linear polarized interference. Meanwhile, the frequencyhopping technique learned from bats is also generally used to
The authros are with the Faculty of Electrical Engineering, Mathematics andComputer Science (EEMCS), Delft University of Technology, Delft, 2628CDthe Netherlands. e-mail: [email protected], [email protected],[email protected]. counteract various interference caused by spectrum congestion[2]. To identify mutual interference, the predefined orthogonalpatterns [3] are imposed on the frequency modulation slopesof each FMCW burst which consists of hundreds of sweeps.Medium Access Control (MAC)-like approach is proposed toregulate transmission time of the multiple radars in the samearea [4], [5]. These approaches provide effective solution tointerference mitigation, but they increase the complexity ofradar system or antenna design for implementation and leadto costly systems.On the other hand, the post-signal processing techniquesutilize a range of digital signal processing approaches tomitigate interference probably at the expense of increasedcomputational load. The signal processing methods can befurther divided into three classes: filtering approaches [6], [7],signals separation [8], [9], and suppression and reconstructionapproaches [10]–[12]. In [6], weighted-envelope normalizationapproaches are proposed to deal with strong spiky mutualinterference by detecting the envelope variations within asliding time window and inversely normalizing the detectedinterference. In [7], an adaptive noise canceller is devisedfor mutual interference suppression by exploiting the differ-ent distributions of frequency spectra of target’s signals andmutual interference in the frequency domain. However, bothfiltering approaches are only applicable to tackle certain typeof interference or point-like targets scenario, which limits theirwide applications. Meanwhile, the stability of the adaptivefilter is hard to guarantee.The signals separation methods generally exploit differentfeatures, i.e., distinct sparsity of targets’ signals and the inter-ference in different transform domains to separate them [8],[9]. So these methods require some prior information about thesparsity of the desired signal and the related interference toconstruct proper bases for optimal separation. However, if the“off-grid” problem between the bases (e.g., the discrete Fourierbasis and short-time Fourier transform basis [8]) and the signalto be represented exists, it would lead to some loss of thedegree of sparsity, thus degrading the separation performance.By contrast, as long as the extension of the interference islimited in a certain domain, the simplest but effective methodto suppress the interference is, in practice, to directly cutthe interference-contaminated samples out of the signal withvarious windows (e.g., zeroing and inverse cosine window)[13], [14]. However, the interference cutting-out not justeliminates the interference but also suppresses part of theuseful signal of targets, which reduces the signal to noise ratio(SNR) of the targets after coherent processing and decreasesthe range resolution. To deal with the SNR loss problem, a r X i v : . [ ee ss . SP ] F e b UBMITTED TO IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES 2
FMCW WaveformGenerator LO LNAPA T x R x LPF Beat Frequency Signal f WF f RF Reference SignalReceivedSignalInterferencesignalScatteredsignalTransmittedsignalTarget A/Dconverter f Sampling
Fig. 1. General block diagram of mono-static linear FMCW radar system. a Burg method-based interpolation was used to extrapolatethe useful signal samples in the cut-out region in the time-frequency ( t - f ) domain [11]. It uses the signal samples on bothsides of the cut-out gap to separately extrapolate the cut-outdata forward and backward. Then, the forward- and backward-extrapolated samples in the cut-out region are summed upwith weights by a specifically designed cross-fading window.This method is generally applicable to mitigate various in-terference for FMCW radars (as indicated in Fig. 2 later).But its extrapolation accuracy degrades dramatically whenthe number of the cut-out samples of signals increases. In[12], the signal extrapolation with AR model was suggestedusing the instrumental variable method (IVM). However, thismethod is not very stable and cannot always get proper signalreconstruction.To accurately extrapolate the cut-out data after cut-outoperation (i.e., zeroing), we propose an iterative matrix-pencil(MP) method-based extrapolation for interference mitigation.Similar to the Burg method-based approach, the proposedapproach first cuts the interference-contaminated samples outof the signals and then reconstruct/extrapolate the clippedsamples of the useful signals. But the proposed approachsimultaneously accounts for the signals before and after theclipped samples by using a unified all-pole model whichis derived from the analytical model of the beat signals oftargets. So it provides the potential to get more accurateextrapolation of the non-contaminated signal in the cut-outregion. Before the extrapolation, the all-pole model is first es-timated based on the interference-free samples with the matrix-pencil method [15], [16]. However, in practice, the noiseand the possible discontinuity of the interference-free sampleswould impact the accuracy of the estimated signal model,thus resulting in less accurate reconstruction of the cut-outsamples of useful signals. To alleviate this effect, an iterativescheme is introduced to refine the model estimation and theextrapolation, which significantly improves the accuracy of thesignals in the cut-out region. Moreover, we want to mentionthat a method similar to the one presented in this paper hasbeen used for multi-band signal fusion for high-resolutionimaging in [17], [18]. Actually, for interference mitigation,the measured signals become two or more separate segmentsafter interference suppression. So, using the interference-freesignal segments to reconstruct/extrapolate the cut-out regionis in essence a signal fusion problem. The main differenceis absence of the incoherence-correction between differentsignal segments needed for interference mitigation. Note this paper focuses on interference mitigation on sweeps in thetime domain which would be flexible to be followed by otherfurther processing. Nevertheless, we should mention that in thecase of interference mitigation followed by some specific twodimensional (2-D) processing (e.g., range-Doppler processing,range-DOA estimation), the proposed interference mitigationapproach could also be extended and implemented in the high-dimensional space by exploiting the 2-D or high-dimensionalMP approaches [19], [20], which would be considered infuture.The rest of this paper is organized as follows. Section IIformulates the basic models of the signals received by FMCWradars. In Section III, the proposed iterative matrix-pencilmethod based interference mitigation approach is presented.Then, its performance of interference mitigation is demon-strated in different scenarios through the numerical simulationsin section IV and the experimental results in section V. Finally,conclusions are drawn in section VI.II. FMCW R ADAR S YSTEM M ODEL
A. Transmitted and received signals
The system diagram of an FMCW radar system is shownin Fig. 1. The transmitted FMCW signal can be expressed as p ( t ) = A tx exp (cid:20) j π (cid:18) f + 12 Kt (cid:19) t (cid:21) , (1)for < t < T / , where A tx is the amplitude of thetransmitted signal, and f is the starting frequency of anFMCW sweep. K = B/T is the chirp rate defined by theratio of the signal bandwidth B and the sweep time T .The transmitted electromagnetic (EM) signal is interceptedby targets and scattered back to the receiver. Consideringthe quasi-monostatic configuration of the transmit and receiveantennas and assuming single scattering process for eachtarget, the back-scattered signal can be represented as s r ( t ) = M (cid:88) i =1 A rx,i exp (cid:20) j π (cid:18) f ( t − t i ) + K t − t i ) (cid:19)(cid:21) (2)where t i = 2 d i /c is round-trip time delay of the scatteredsignal related to the i th target at a distance of d i , and A rx,i is the corresponding amplitude of the signal which subsumesthe scattering coefficient and propagation loss. c is the speedof light and M is the number of targets. B. Dechirp on receiver
In FMCW radar system, dechirp processing is commonlyused due to its simple operation and low requirement ofsampling rate for the Analog to Digital Converter (ADC). It isimplemented by mixing the received signals with the conjugateof the transmitted one, which leads to beat signals.Considering the occurrence of strong interference s int , thebeat signal after demodulating and filtering can be formulatedas (3) on the top of next page, where the superscript ∗ denotescomplex conjugate and F lp is the low-pass filter operator. ˜ A r,i is the amplitude of the received signal of the i th target and M (cid:48) ( ≤ M ) is the number of observed scatterers within the UBMITTED TO IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES 3 ˜ s ( t ) = F lp { [ s r ( t ) + s int ( t )] · p ∗ ( t ) } = F lp (cid:0) s int ( t ) · p ∗ ( t ) (cid:1) + F lp (cid:40) M (cid:88) i =1 A tx A rx,i exp (cid:20) − j π (cid:18) f t i − Kt i (cid:19)(cid:21) · exp ( − j πKt i t ) (cid:41) = F lp (cid:0) s int ( t ) · p ∗ ( t ) (cid:1) + M (cid:48) (cid:88) i =1 ˜ A r,i exp (cid:20) − j π (cid:18) f t i − Kt i (cid:19)(cid:21) exp ( − j πKt i t ) (3)desired unambiguous range. As exp (cid:104) − j π (cid:16) f t i − Kt i (cid:17)(cid:105) isa constant phase term related to the i th target which can besubsumed by the amplitude of the signal, one can present a i = ˜ A r,i exp (cid:104) − j π (cid:16) f t i − Kt i (cid:17)(cid:105) as a new complex signalamplitude. Then, (3) can be rewritten as a sum of complexexponential functions ˜ s ( t ) = F lp (cid:0) s int ( t ) · p ∗ ( t ) (cid:1) + M (cid:48) (cid:88) i =1 a i exp ( − j πf b,i t ) (4)where f b,i = Kt i is the beat frequency corresponding to the i th target. For moving targets, t i = 2 d i /c = 2( d i + v i t ) /c canbe used to account for the Doppler shift, where v i and d i arethe velocity and the initial distance of the i th target relativeto the radar. Generally, as v i (cid:28) c , it has negligible impact onthe target’s beat frequency within a short FMCW sweep. Aftergetting beat frequencies, the ranges of different targets can becalculated as d i = c · f b,i K (5)As thermal noise and measurement errors always exist dueto physical limitation of the practical radar system, the signalmeasurements can be modeled as s ( t ) = ˜ s ( t ) + n ( t )= M (cid:48) (cid:88) i =1 a i exp( − j πf b,i t ) + F lp (cid:0) s int ( t ) · p ∗ ( t ) (cid:1) + n ( t )= ˜ s tar ( t ) + ˜ s int ( t ) + n ( t ) (6)where s ( t ) represents the measured signal, n ( t ) denotes thenoise and measurement errors, ˜ s int ( t ) = F lp (cid:0) s int ( t ) p ∗ ( t ) (cid:1) is the signal resulting from the interference, and ˜ s tar ( t ) = (cid:80) M (cid:48) i =1 a i exp( − j πf b,i t ) is the beat signal of targets withinthe desired detection range. Equation (6) gives the generalmodel of the FMCW radar measurements contaminated bystrong interference. C. Interference
Nowadays, radar systems face various types of interferencedue to the rapid increase of radio wireless applications. Inparticular, for FMCW radar systems, the related interferencecan be classified as the following four cases [21]–[23]: 1)FMCW interference with the same chirp rate; 2) FMCWinterference with a different chirp rate; 3) CW interference;and 4) transient interference. These cases are illustrated inFig. 2. In Case 1), the FMCW interference would result in astrong ghost target if it appears within the reception windowof the system determined by the maximum detection range. In
InterferenceTarget echoTimeRF Transmitted signal InterferenceTarget echoTimeRF Transmitted signalInterference Target echoTimeRF Transmitted signal
Case 1 Case 2Case 3
TimeBeat Frequency TimeBeat FrequencyInterferenceTarget TargetInterferenceTimeBeat Frequency TargetInterference
Anti-aliasing filter boundariesAnti-aliasing filter boundaries f LP f LP f LP -f LP Anti-aliasing filter boundaries -f LP -f LP Interference Target echoTimeRF Transmitted signal
Case 4
TimeBeat Frequency TargetInterference
Anti-aliasing filter boundaries f LP -f LP Fig. 2. Four cases of interference which corrupt the FMCW radar system.Case 1: chirp interference with the identical sweep parameters as the victimradar; Case 2: chirp interference with different sweep parameters from thevictim radar; Case 3: sinusoidal/narrowband continuous interference; and Case4: instantaneous wideband interference.
Cases 2) and 3), the FMCW and CW interference have a longtime duration and lead to the non-constant beat frequency afterthe dechirp processing. Thanks to the low-pass filtering, theiroccurrences are confined in a short time around the frequencyintersecting moment. In Case 4), the spectrum of the transient(or pulse) interference with a rectangular amplitude in a shorttime can be considered as equidistant lines with a sin( x ) /x envelope. Some of these frequency lines intersect with thereference FMCW signal of dechirp operation and then, as inCase 3), result in the short interference after low-pass filtering[22].The above analysis indicates that the interference in Cases2), 3) and 4) all cause contaminated measurements in certaintime period within an FMCW sweep duration, which inprinciple can be tackled using the method described in thispaper (Note the interference with a very small sweep slopedifference from that of the victim radar (i.e., extreme situationsin case 2) could make all the signal samples contaminated,in which case the proposed approach and other zeroing plusreconstruction methods would not be applicable). Without loss UBMITTED TO IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES 4 of generality, we consider the FMCW signal was contaminatedby an FMCW interference with a different frequency slope,i.e., Case 2) in the following sections.Assuming an interfering FMCW radar is located at a dis-tance d I away from the transceiver, the interference signalarriving at the receiving antenna can be expressed as s int ( t ) = A I exp (cid:20) j π (cid:18) f I, ( t − t I ) + K I t − t I ) (cid:19)(cid:21) (7)for t I < t < T I + t I , where A I is the amplitude of theinterference. t I = d I /c is the time delay of the interferencesignal relative to the starting time of the transmission of thevictim radar. f I, is the starting frequency of the interferencesignal and K I = B I /T I is the chirp rate of the interferencesignal with the bandwidth B I and the sweep duration T I .Then, the interference signal ˜ s int ( t ) obtained after dechirp-ing and low-pass filtering can be explicitly expressed as ˜ s int ( t ) = F lp ( s int ( t ) p ∗ ( t )) = F lp { a I exp [ j Φ( t )] } (8)where Φ( t ) = 2 π (cid:20)(cid:18) K I − K (cid:19) t + ( f I, − f − K I t I ) t (cid:21) (9) a I = A I A tx exp (cid:20) j π (cid:18) K I t I − f I, t I (cid:19)(cid:21) (10)Taking the first derivative of the phase Φ( t ) with respect totime, one can get the instantaneous beat frequency f b,I ( t ) = − π ∂ Φ I ( t ) ∂t = ( K t + K ) (11)where K = ( K − K I ) and K = ( f − f I, + K I t I ) areconstant coefficients. According to (11), the beat frequenciesresulting from the interference are time-varying. After the low-pass filtering in (8), its frequency bandwidth and the timeof occurrence are confined but the time-varying property isnot affected. By contrast, the beat frequencies of targets areconstant, as shown in (6). This difference between the beatfrequencies of targets and interferer makes the interferencemitigation can be done in either time or time-frequency ( t - f )domain [13].III. M ATRIX P ENCIL M ETHOD B ASED I NTERFERENCE M ITIGATION
A model-based interference mitigation approach for theFMCW radar system is presented in this section. This ap-proach can operate in either the time domain or the time-frequency domain. Without loss of generality, its details areillustrated through the time-domain processing for the inter-ference mitigation in the following sections.
A. Discrete signal in the time domain
From (6), the discrete signal measurements can be writtenas s [ k ] = ˜ s tar [ k ] + ˜ s int [ k ] + n [ k ]= M (cid:48) (cid:88) i =1 a i z ki + ˜ s int [ k ] + n [ k ] (12) where z i = exp ( j πf b,i ∆ t ) , ∆ t is the sampling interval and k = 0 , , ..., N − is the sampling indices of the N time-domain samples in an FMCW sweep. As analyzed above,the interference component ˜ s int appears in a short period ina sweep; thus, only some of the measured signal samples, e.g.from N to N are contaminated, where ≤ N < N ≤ N − . Since the desired targets’ signal ˜ s tar is a sum of exponentialcomponents, it is natural to suppress the interference by cuttingout the contaminated samples from the measurements and thenreconstructing the cut-out samples with the uncontaminatedmeasurements and the model of the desired signal. As theclipped sample reconstruction is generally converted to anestimation problem of exponential components, it can beimplemented with root-MUltiple SIgnal Classification (root-MUSIC), Prony’s method [24], etc. To more efficiently andaccurately reconstruct the cut-out samples, we suggest usingmatrix pencil method in this paper, which leads to the proposedmatrix-pencil method based interference mitigation. B. Interference mitigation
The flowchart of the matrix-pencil method based interfer-ence mitigation for FMCW radars is shown in Fig. 3. Thedetailed processing involves two main steps:
1) Interference detection and cutting out:
Based on theanalysis in the previous section, the beat frequencies of targetsare generally constant in a sweep while the interference afterde-chirping and low-pass filtering still exhibits non-stationaryspectral property within its duration. Taking advantage of thisspectral difference, the interference and its duration can bedetected with many approaches, such as energy spikers de-tection [25], Constant False Alarm Rate (CFAR) thresholding[26], complex baseband oversampling [27] or other methods intime or time-frequency domain. After determining the locationof the interference, the contaminated signal samples can becompletely removed for interference suppression. However, italso eliminates part of the energy of the desired signals, whichwould cause signal to noise ratio (SNR) degradation of theresultant range profiles.
2) Signal extrapolation:
To overcome the SNR degradationof the targets’ signals caused by the interference suppression,the removed signal samples can be reconstructed by using theinterference-free samples and the corresponding signal model ˜ s tar . Generally, the all-pole signal model ˜ s tar [ k ] is unknownand has to be estimated from the interference-free samples.In this paper, matrix-pencil method is applied to estimate themodel parameters (i.e., model order, signal poles and the coef-ficients) by simultaneously accounting for the interference-freesamples in front of and behind the clipped ones. Moreover, toalleviate the impact of the noise and signal discontinuity of theinterference-free samples on the estimation of signal model oftargets, an iterative fusion process is introduced to minimizethe estimation error of the signals on the both sides of theclipped region relative to the interference-free measurements.If the estimation error fulfills a desired requirement after a fewiterations, the signals in the cut-out region are reconstructed. UBMITTED TO IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES 5
Time-domainmeasurementDetect and remove the interference-contaminated signal samplesEstimate model orderEstimate the fullsamples based onestimated signalmodelReconstructedsignal NoYesInputInterferencesuppressionSignalextrapolation
TimeTime A m p lit ud e Time A m p lit ud e A m p lit ud e S S Estimate the signal model(poles andcoefficients) based on the data segmentsUpdate the data andre-estimate the signalmodel based on fulldata (i.e., poles andcoefficients)
Fig. 3. Flowchart of the proposed MP-based interference mitigation approach.
C. Signal fusion and reconstruction
After cutting out the interference-contaminated samplesindexed from N to N , the interference-free measurementsin (12) can be represented as s [ k ] = M (cid:48) (cid:88) i =1 a i z ki + n [ k ] (13)where k = 0 , , · · · , N − , N + 1 , N + 2 , · · · , N − .Therefore, a gap is formed between the two signal samplesegments from 0 to N − and from N + 1 to N , asillustrated in the second plot on the right side of Fig. 3. Asthe useful signals in this gap are also eliminated due to theinterference clipping, it would cause some SNR loss of thefinal coherent processing results (e.g., range profile, range-Doppler map, etc.). To overcome this problem, in the next stepwe try to reconstruct the useful signals in the gap based onthe signal model (13) and the interference-free measurementson the both sides.As mentioned in the introduction, here the signal recon-struction can be converted to a signal fusion problem. Wesuggest using the matrix-pencil based fusion method in [17],[18] to implement the signal reconstruction but no incoherencecorrection between different signal segments is needed.For the convenience of description, we denote the signalsbefore and after the clipped region as s and s , given by (cid:40) s [ k ] = s [ k ] , k = 0 , , · · · , N − s [ k ] = s [ k + N + 1] , k = 0 , , · · · , N − N − (14)Then, the detailed steps of the signal reconstruction arepresented as follows.(1) Estimate the all-pole signal model (13) with the matrixpencil method based on the front and back signal segments,i.e., s and s . Generally, the signal model order M (cid:48) is estimated accordingto the Akaike Information Criterion (AIC), Bayesian Informa-tion Criterion (BIC), subspace-based automatic model orderselection (SAMOS) [28], [29], etc. As SAMOS is consideredto be one of the most general and robust approach to modelorder selection and outperforms the aforementioned methodsbased on the information theoretic criterion, it is used in thispaper. The signal poles can be estimated with the matrixpencil method. Different from the signal pole estimation withcontinuous uniform signal samples, the Hankel matrices basedon the discontinuous signals s and s are constructed in aslightly different way [17], [18]. Firstly, two Hankel matricesare constructed as H i = [ D i , D i , · · · , D iL − ] H i = [ D i , D i , · · · , D iL ] , (15)with D ik = [ s i [ k ] , s i [ k + 1] , · · · , s i [ M i − L − k ]] T , i = 1 , . (16)where T denotes the transpose operation, M = N and M = N − N − are the lengths of s and s , respectively. L is thematrix pencil parameter and ˆ M (cid:48) < L < min( M − ˆ M (cid:48) , M − ˆ M (cid:48) ) , where ˆ M (cid:48) is the estimated signal model order (Withoutexplicit statement, the ˆ · notation represents the estimated valueof a corresponding parameter).The Hankel matrices constructed above can be verticallystacked as X = (cid:20) H H (cid:21) , X = (cid:20) H H (cid:21) . (17)Then the matrix pencil L ( λ ) = X − λ X can be evaluatedto get the estimates the signal poles z i in (13) [17], [18]. Toget the eigenvalues of this matrix pencil, we take advantageof the singular value decomposition (SVD)-based method in[15]. Taking the SVD of the matrix X and X , we get X = [ U , U (cid:48) ] (cid:20) Σ , ˆ M (cid:48) Σ ,L − ˆ M (cid:48) (cid:21) [ V , V (cid:48) ] H (18) X = [ U , U (cid:48) ] (cid:20) Σ , ˆ M (cid:48) Σ ,L − ˆ M (cid:48) (cid:21) [ V , V (cid:48) ] H (19)where H denotes the conjugate transpose of a matrix, Σ , ˆ M (cid:48) and Σ , ˆ M (cid:48) are the diagonal matrices containing ˆ M (cid:48) dominantsingular values of X and X , respectively. The columns of U , U , V and V are the left and right singular vectorsrelated to the dominant singular values. ( U , Σ , ˆ M (cid:48) , V ) and ( U , Σ , ˆ M (cid:48) , V ) are the singular value systems related to thesignal subspace in X and X ,respectively. The rest terms in(18) and (19) form the corresponding singular value systemsrelated to the so-called noise subspace.To suppress the impact of the noise on the signal poleestimation, X and X can be approximated by their truncatedSVD as X T and X T X ≈ X T = U Σ , ˆ M (cid:48) V H (20) X ≈ X T = U Σ , ˆ M (cid:48) V H (21)Then the signal poles z i can be estimated by solving thegeneralized eigenvalue problem det ( L ( λ )) = 0 of the matrix UBMITTED TO IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES 6 pair { X ; X } , which is equivalent to the ordinary eigenvalueproblem det (cid:16) Σ − , ˆ M (cid:48) U H U Σ , ˆ M (cid:48) V H V − λ I (cid:17) = 0 (22)The signal pole estimations ˆ z i = λ i , i = 1 , , · · · , ˆ M (cid:48) areobtained.After that, using the estimated signal model order ˆ M (cid:48) andthe signal poles ˆ z i , the complex amplitude a i can be cast asthe least-square problem m = Za , where m = [ s , s ] T isthe measured interference-free data, Z is the matrix formedby signal poles and a = [ a , a , · · · , a ˆ M (cid:48) ] is the vector of thecoefficients. Explicitly, it is represented as s [0] s [1] ... s [ M − s [0] ... s [ M − = · · · z z · · · z ˆ M (cid:48) ... ... . . . ... z N − z N − · · · z N − M (cid:48) z N z N · · · z N ˆ M (cid:48) ... ... . . . ... z N − z N − · · · z N − a a ... a ˆ M (cid:48) (23) (2) After inserting the estimated signal poles ˆ z i and thecoefficients ˆ a i into (13), the full beat signal in the sweep canbe estimated by ˆ s [ k ] = ˆ M (cid:48) (cid:88) i =1 ˆ a i ˆ z ki , k = 0 , , · · · , N − (24)The estimated full beat signal indicates ˆ s [ k ] = ˆ s [ k ] , k ∈ [0 , N − s g [ k − N ] = ˆ s [ k ] , k ∈ [ N , N ]ˆ s [ k − N −
1] = ˆ s [ k ] , k ∈ [ N + 1 , N − (25)(3) To improve the estimation of the full beat signal, wereplace the ˆ s and ˆ s parts in ˆ s with the measurements s and s . Then the reconstructed full beat signal can be modified as ˆ s [ k ] = s [ k ] , k ∈ [0 , N − s g [ k − N ] , k ∈ [ N , N ] s [ k − N − , k ∈ [ N + 1 , N − (26)Next, the reconstructed signal ˆ s in (26) are used as a set ofcontiguous samples to re-estimate the signal poles z i and thecoefficients a i in (13) by using the traditional matrix-pencilmethod [15].(4) Repeat steps (2) and (3) to update the reconstructedresults. After the step (2) in each iteration, the l -norm of thedifferences between the estimated signals and their measuredcounterparts is examined to quantify the signal estimationaccuracy (cid:15) i = (cid:107) ˆ s ( i )1 − s (cid:107) + (cid:107) ˆ s ( i )2 − s (cid:107) , (27)where ˆ s ( i )1 and ˆ s ( i )2 are the estimated counterparts of themeasurements s and s in the i th iteration. If the signaldifference in the i th iteration satisfies the requirement (cid:15) i > (cid:15) i − , (28)then iteration will stop. Otherwise, it continues to improve theestimated model parameters. TABLE IP
ARAMETERS USED FOR SIMULATIONS FOR POINT - LIKE ANDDISTRIBUTED TARGET SCENARIOS
Parameter Value Unit
Center frequency 3 GHzBandwidth 40 MHzFMCW sweep duration 500 µs Sweep slope × Hz/sTransmit Power 1 WattSampling frequency 12 MHzMaximum unambiguous range 8 km
Point target scenario
Distances of three targets 2, 5, and 5.1 kmInterference duration 10-50% N/A
Extended targets scenario
Number of point targets 15 N/ADistance between adjacent targets < d < . mInterference duration relative to thesweep duration . N/A
After several iteration cycles, we get the most accurate re-covery of the full beat signal. Finally, by taking correspondingoperations on the reconstructed full beat signal, the rangeprofile and Doppler information of targets can be obtained withsubstantially improved dynamic range and suppressed “noise”floor. IV. N
UMERICAL S IMULATIONS
To analyze performance of the proposed MP-based methodto interference mitigation, several sensing scenarios have beensimulated. Its results are also compared with the traditionalzeroing and two of the state-of-the-art methods, i.e., Burg-based approach [11] and the IVM-based method [12].
A. Evaluation metric
To facilitate quantitative evaluation of the accuracy of thereconstructed beat signals by different methods, we introducetwo evaluation metrics: the Relative Signal-to-Noise Ratio(RSNR) and the correlation coefficient ρ . The RSNR and thecorrelation coefficient are defined asRSNR ( s , ˆ s ) = 20 log (cid:107) s (cid:107) (cid:107) s − ˆ s (cid:107) (29) ρ s , ˆ s = ˆ s H s (cid:107) s (cid:107) · (cid:107) ˆ s (cid:107) (30)where s is the vector of a clean reference beat signal (withoutinterferences and noise) and ˆ s is the beat signal formed bythe measured interference-free samples and the reconstructedsignal samples in the cut-out region. (cid:107)·(cid:107) denotes the (cid:96) norm operator. If the signal samples in the cut-out region arereconstructed with sufficient accuracy, a RSNR larger than theSNR of the input signal can be obtained according to (29).So the larger the obtained RSNR is, the more accurate therecovered signal samples are.The correlation coefficient is commonly used to evaluatethe similarity of two signals. Its formulation in (30) is anormalized inner product between the reconstructed signal andthe reference one, which specifically represents the rotation UBMITTED TO IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES 7 angle between the two signals. The correlation coefficientsatisfies ≤ | ρ s , ˆ s | ≤ . If | ρ s , ˆ s | = 1 , then the reconstructedsignal ˆ s is a linear function of the reference signal s withphase difference of (cid:54) ρ s , ˆ s (i.e., argument of ρ s , ˆ s ). That isto say, a correlation coefficient with a larger modulus and asmaller argument indicates a better recovery performance. B. Point target scenario
Firstly, we demonstrate the performance of the proposedMP-based interference mitigation approach in the point targetscenario. The parameters of the FMCW radar system used forthe simulation are shown in Table I. Three point targets areplaced at a distance of , and . , respectively,away from the transceiver. The amplitudes of the scatteredsignals from the three targets from the near to further distancesare set to be 1, 0.2, and 0.1, respectively.The victim FMCW radar system suffers from a stronginterference from an aggressor FMCW radar with the sameoperational center frequency but an opposite sweep slope anda time advancement of µs relative to the starting time of thevictim sweep. After dechirping, the interference-contaminatedbeat signal is acquired and illustrated in Fig. 4(a). The stronginterference appears at the interval from µ s to µ s (indicated by the red solid-line rectangle), which still exhibitsas a chirp-like signal (see the bottom-right inset in Fig. 4(a)).Meanwhile, for clarity, part of the interference-free beat signal(from µ s to µ s indicated by the blue dash-dottedrectangle) is zoomed in and shown in the top-right inset. Itis clear that the beat signal of targets is composed of thesinusoidal components. Moreover, white Gaussian noise withthe SNR of
15 dB is added to the signal to account for thethermal noise and measurement errors of the radar system.The interference-contaminated beat signal produces a rangeprofile with significantly increased noise floor (see “sig Int”in Fig. 5(a) where the two targets at the further distancesare almost shadowed by the raised noise floor) if the rangecompression is performed directly by using the fast Fouriertransform (FFT). To mitigate the interference by using theproposed MP-based approach, the interference-contaminatedsamples of the signal are firstly detected and cut out (i.e., zero-ing with a rectangular window [14]). Zeroing the interference-contaminated samples results in two separate signal segmentswith a gap inbetween (see the top panel in Fig. 4(c)), whichcauses not only power loss of targets’ signals but also highsidelobes of the range profile, thus degrading the performanceof target detection. To overcome these effects, the proposedMP-based interference mitigation method is used to recon-struct the signal samples in the cut-out gap based on the signalmodel (24) and the rest interference-free ones in front andback. Before reconstruction, the model order was estimatedto be three by using the SAMOS method (see Fig. 4(b)),which agrees with the true value. Then, by exploiting theproposed iterative scheme, the signal samples in the gap wererecovered with sufficient accuracy, as shown in the middle plotand a close-up of them in the bottom panel in Fig. 4(c). Forcomparison, the interference-free reference signal (with thenoise) and the recovered signals with the Burg-based method A m p li t ude Beat signal after reception
350 360 370-101 180 200 220 240 260-20020 (a)
Model order k M e t r i c o f SA M O S (3, 3373.9104) (b)
100 150 200 250 300 350 400 450-101 A m p li t ude Signal after Zeroing
170 180 190 200 210 220 230 240 250 260-101 A m p li t ude Reconstructed signal in the cut-out region
Ref sigBurgIVMMP
208 209 210 211 212 213-101 A m p li t ude Close-up of the reconstructed signal (c)Fig. 4. Numerical simulation for interference mitigation in the point targetscenario. (a) shows the interference-contaminated beat signal. (b) presents themetric values of SAMOS for model order estimation. (c) displays the resultsafter interference mitigation. and the IVM, which used the same model order as that of theMP-based method, are also shown in the middle and bottompanels. One can see that both the signal recovered with theproposed MP-based method has the best agreement with the
UBMITTED TO IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES 8
Range [km] -60-40-200 A m p li t ude [ d B ] Range Profile sig_IntRefZeroingBurgIVMMP (a)
Range [km] -40-30-20-100 A m p li t ude [ d B ] (b) Range [km] -50-40-30-20-10010 A m p li t ude [ d B ] (c)Fig. 5. (a) displays the range profiles of the targets obtained with theinterference-contaminated signal, interference-free reference signal, and thesignals processed with the zeroing, Burg- and MP-based methods; (b) and(c) are the close-ups of the range profile around the distances of targets,respectively. reference one. Meanwhile, the IVM method achieves moreaccurate reconstruction of the signals in the cut-out regionthan the Burg-based method in this case.To further examine the accuracy of the reconstructed signals,the range profiles of targets are constructed by taking theFFT of them and shown in Fig. 5. For comparison, therange profiles obtained with the interference-contaminated[i.e., “sig Int” in Fig. 5(a)] and interference-free reference beatsignals [i.e., “Ref” in Fig. 5(a)] are also presented. Note allthe range profiles in Fig. 5 are normalized by the maximum ofthe range profile acquired with the interference-contaminatedsignal.According to Fig. 5(a), all the interference mitigationmethods, i.e., zeroing, Burg-, IVM- and MP-based methods,significantly reduce the “noise” floor of the range profileand thus increase its dynamic range compared to the oneobtained with the interference-contaminated signal. Amongthem, the zeroing method is computationally most efficientby simply replacing the interference-contaminated sampleswith zeros, however, resulting in a gap between the frontand rear signal samples. Consequently, it causes high side-lobes and some SNR loss in the range profile compared tothat obtained with the reference signal. Specifically, fromthe insets in Fig. 5(b) and (c), the peaks of targets’ rangeprofiles obtained after zeroing are . lower than thoseformed with the reference signal and the signal reconstructedwith the MP-based method. Although the Burg- and IVM-based method efficiently interpolate the samples in the cut-outgap and result in comparable/identical range profiles as thereference signal for the target at the short distance, they failto overcome the power loss for the two weak targets at thefurther distances and get range profiles close to that of thezeroing method (see the insets in Fig. 5(c)). By contrast, the MP-based method not only conquers the power loss of therange profile for all the targets but also accurately reconstructstheir range profiles in terms of both main lobe and the side-lobes. Quantitatively, for the beat signals in Fig. 4(c) recoveredwith the Burg-, IVM- and MP-based methods, their RSNRsare .
55 dB , .
86 dB and .
68 dB , and the correspondingcorrelation coefficients are . e − j . , . e j . and . e j . , respectively, relative to the clean referencesignal. Therefore, it recovers the signal samples in the cut-out region more accurately than the Burg- and IVM-basedmethods. C. Extended target scenario
Here we consider the applicability of the proposed methodto extended target scenarios. The parameters used for thesimulation are shown in Table I. An extended target formedby 15 point scatterers with adjacent inter-distances less thanthe range resolution of the radar system (i.e., .
75 m in ourssimulation) was simulated. The target was located at the rangeof – .
025 km away from the transceiver. The amplitudesand phases of the scattered signals from these closely spacedscatterers were random values with uniform distribution in [0 , . and uniform distribution in [0 , π ] , respectively. Abeat signal with the SNR of
15 dB was synthesized by addingwhite Gaussian noise to consider measurement errors andthermal noise of the system and also contaminated by a stronginterference with the same center frequency but a sweep slopeof -0.98 times of that of the victim radar. The resultant beatsignal is illustrated in Fig. 6(a).Similar to the point target scenario, the interference-contaminated samples are first detected and cut out. The resultis shown in the top panel in Fig. 6(d). Then, the signalmodel order was estimated by using the SAMOS methodbased on the other interference-free samples. However, dueto the strong correlation among the beat signals scattered bythe closely spaced scatterers, the model order was selected tobe two by using the SAMOS method, which is significantlydifferent from the theoretical value fifteen (see Fig. 6(b)). Sothe SAMOS method cannot work properly in such scenarios.To investigate the reason of the failure of the SAMOS, wechecked the singular value distribution of the matrix usedfor model order selection, as shown in Fig. 6(c). Based onFig. 6(c), it is obvious that a proper model order should be notsmaller than four. Taking the model order of four, the signalsamples in the cut-out region were recovered by using theBurg-, IVM- and MP-based methods, which are shown in thetwo bottom plots in Fig. 6(d) and Fig. 6(e), respectively. It isclear that the IVM-based interpolation is not stable and a blow-up is observed in Fig. 6(e). Meanwhile, compared to the Burg-based method, the proposed MP-based method reconstructedthe signal samples with the best agreement with the referencesignal (see the bottom plot in Fig. 6(d)). Taking the FFT of thesignal obtained after zeroing and the recovered signals withBurg- and MP-based methods, the related range profiles oftargets were constructed and shown in Fig. 7(a). As expected,the range profile of the targets constructed with the signalrecovered with the MP-based method has the best agreement
UBMITTED TO IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES 9
100 200 300 400 500-2-1012 A m p li t ude Real part of the beat signal (a) model order M e t r i c o f SA M O S (b) Index -30-25-20-15-10-50 N o r m a li z ed SV s [ d B ] Singular value distribution (c)
100 150 200 250 300 350 400 450 500-0.2-0.100.1 A m p li t ude Signal after Zeroing
260 280 300 320 340 360-0.100.1 A m p li t ude Recovered signal: model order=4
Ref sigBurgMP
301 302 303 304 305 306 307 308 309-0.0500.05 A m p li t ude close-up of the reconstructed part (d)
100 150 200 250 300 350 400 450 500-505 A m p li t ude IVM: model order=4 real partimag part (e)
260 280 300 320 340 360-0.100.1 A m p li t ude Recovered signal: model order=15
Ref sigBurgMP
301 302 303 304 305 306 307 308 309-0.0500.05 A m p li t ude close-up of the reconstructed part (f)Fig. 6. Numerical simulation for interference mitigation in the extended target scenario. (a) shows the interference-contaminated beat signal of an extendedtarget. (b) shows the metric values of SAMOS approach for model order selection while (c) presents the singular values distribution of the matrix constructedfor model order selection. (d) shows the results after interference mitigation, where the top panel gives the beat signal after zeroing; the middle panel presentsthe reference beat signal and the beat signals recovered with the Burg- and MP-based methods with the model order of four; and the bottom panel is theclose-up view of the recovered samples in the cut-out region. (e) shows the beat signal recovered by the IVM with the model order of four and (f) displaysthe recovered beat signals with the model order of 15. with that formed using the reference signal. For quantitativeevaluation, the RSNRs of the beat signals recovered with theBurg- and MP-based methods are obtained as .
07 dB and .
66 dB , respectively. Their correlation coefficients relativeto the reference signal are . e − . and . e . .So the RSNRs and correlation coefficients confirm that theMP-based method gets more accurate signal reconstruction inthe cut-out region than the Burg-based method.Moreover, we also reconstructed the signal samples in thecut-out region using the three methods by setting the modelorder to be fifteen. Again, a blow-up as in Fig. 6(e) isobserved in the recovered signal by the IVM-based method(here the figure is omitted for conciseness). So it indicatesthat the instability of the IVM-based method may not becaused by the underestimation of the signal model order.Meanwhile, the recovered signal by the Burg-based method isstill less accurate than that obtained with the MP-based method(see Fig. 6(f) and Fig. 7(b)). The RSNR and correlationcoefficient of the recovered signal by the Burg-based methodare .
98 dB and . e . and their counterparts for the signal reconstructed with the MP-based method are .
48 dB and . e . , which further confirms that the MP-basedmethod is superior to the Burg-based one in term of the signalreconstruction accuracy.Finally, we want to mention that when multiple pointtargets in the same range bin are very close to each other,the Burg-based method could occasionally outperform theproposed MP-based method (for conciseness, we do not showit here). As the close targets in a range bin results in highlycorrelated beat frequencies, the characteristic polynomial ofthe corresponding AR model has many closely spaced roots.The proposed MP-based method tends to estimate some dom-inant sinusoidal components (i.e., roots) that are close to thereal roots in the mean square error sense while the Burg-based method attempts to estimate the coefficients of thecharacteristic polynomial of the AR model. Apparently, thelatter operation is easier in such cases; thus, the Burg-basedmethod results in more accurate signal estimation. UBMITTED TO IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES 10
Range [km] -40-30-20-100 A m p li t ude [ d B ] RP with model order=4 sig_IntRef ZeroingMP Burg (a)
Range [km] -40-30-20-100 A m p li t ude [ d B ] RP with model order=15 sig_IntRef ZeroingMP Burg (b)Fig. 7. Range profile of the extended target obtained with the reference signal,interference-contaminated beat signal, the signals obtained with zeroing, andthe signals recovered by the Burg- and MP-based methods with (a) the modelorder of four and (b) the model order of fifteen.
D. Effect of the length of interferences and SNR
The impact of the interference duration (equivalently, thesize of the cut-out gap caused by interference suppression)and the SNR on the performance of the proposed MP-basedmethod for signal recovery is investigated in this section. Forgenerality, the size of a cut-out gap is denoted by the ratiobetween the number of the removed interference-contaminatedsamples and the number of all signal samples in a sweep. Theparameters for section IV-B point target scenario simulationwere used here. In the simulation, the SNR changes from −
30 dB to
10 dB with steps of
10 dB and at each SNR theinterference duration increases from to with steps of . To investigate the statistical performance of the proposedMP-based approach, 100 Monte Carlo runs were conducted ateach SNR. The average RSNRs and correlation coefficientsof the signals recovered by Burg- and MP-based methods areshown in Fig. 8 (due to the blow-ups of signals recovered theIVM-based method, the corresponding RSNRs and correlationcoefficients cannot be computed and are omitted here).From Fig. 8(a), one can see that the RSNRs of the signalsreconstructed with the Burg- and MP-based methods are al-most identical when the SNR is smaller than . Meanwhile,they gradually improve and are larger than the SNRs withthe increase of the size of the cut-out region. By contrast,when the SNR is equal to/larger than the RSNRs ofthe signals obtained with the Burg- and MP-based methodsshow different changing trends (i.e., increase for MP-basedmethod while keep steady/decrease for Burg-based method)with the widening of the cut-out gap. This is because that the cut-out operation eliminates not only the interference butalso the noise in the interference-contaminated signal samples.When SNR < , the eliminated noise power is larger thanthat of the useful signals; thus, the RSNR would be largerthan the SNR as long as the useful signal samples in the cut-out region can be recovered with certain accuracy with eitherBurg- and MP-based methods. However, when SNR ≥ ,more signal power is suppressed than the noise power. TheMP-based method jointly uses the signal samples at bothsides of the gap to accurately recover the data in the cut-out region via an iterative scheme. The recovered signal couldbe equivalently regarded as the filtered samples, getting higherRSNR than the SNR of the original signal. In particular, whenthe cut-out gap occupies of the whole sweep, almost halfof the noise power is suppressed; thus, improvementof RSNR relative to the SNR of the input signal can beobtained as long as the useful signal samples in the cut-outregion are accurately reconstructed (see Fig. 8(a)). On theother hand, the Burg-based method separately extrapolatesthe signal samples in the cut-out gap from both sides. Itsextrapolation accuracy degrades rapidly with the wideningof the cut-out region, which causes larger signal difference(especially, large phase differences) between the reconstructedsignal and the reference and thus makes its RSNR even worsethan the original SNR. Therefore, in terms of the RSNR of therecovered signal, the Burg- and MP-based obtain comparableresults when SNR < while the latter one outperformsthe former one when SNR ≥ .However, Fig. 8(b) shows that the MP-based method con-stantly obtains comparable/better signal reconstruction com-pared to the Burg-based method regarding the modulus of thecorrelation coefficient. Moreover, with the increase of the SNRand the interference duration, the performance advantage ofthe MP-based method to the Burg-based one becomes larger.However, the phase of the correlation coefficient between therecovered signals with both methods and the reference arecomparable when the interference duration is smaller than (Fig. 8(c)). It gradually reduces to zero with the increase of theSNR of the original signal. Therefore, according to the aboveanalyses, the MP-based method generally gets more accuratesignal reconstruction than the Burg-based method in terms ofboth RSNR and correlation coefficient of the recovered signal. E. Computational Efficiency
Both Burg- and IVM-based methods are very computationalefficient as they just separately extrapolate the data in the gapfrom both sides. By contrast, the proposed MP-based methoduses the SVD and an iterative scheme to jointly recover thesignal in the cut-out region. So its computational load isslightly heavier than that of the Burg- and IVM based methods,which depends on the number of iterations in practice. For ascenario with moderate interference duration (20%-30%) andSNR, the MP-based method generally needs several iterations.Specifically, for the simulation in section IV-B, it took .
02 s , .
15 s and .
05 s for the Burg-, IVM- and MP-based methods,respectively, when they were implemented in MATLAB andrun on a computer with Intel Core i5-3470 Central Unit Pro-cessor (CPU) @ 3.2GHz and 8GB Random Access Memory
UBMITTED TO IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES 11
10 20 30 40 50
Percent of Gap [%] -40-30-20-10010 R S NR [ d B ] ,,,,, SNR=10dBSNR=0dBSNR=-10dBSNR=-20dBSNR=-30dB (a)
10 20 30 40 50
Percent of Gap [%] ,,,,, SNR=10dBSNR=0dBSNR=-10dBSNR=-20dBSNR=-30dB (b)
10 20 30 40 50
Percent of Gap [%] -0.0200.020.040.060.08 (c)Fig. 8. Impact of gap duration and SNR on the accuracy of the reconstructed signals with the Burg- (dashed blue lines) and MP-(solid red lines) basedmethods. (a) shows the RSNRs with different gap durations. (b) and (c) show the moduli and phases of the correlation coefficients with respect to differentgap durations. (RAM). In this case, four iterations were executed in the MP-based method. To accelerate the MP-based method, Lanczositeration [30] or randomized algorithm [31] for the SVD couldbe exploited in future.V. E
XPERIMENTAL RESULTS
In this section, experimental results with radar observationsof an industrial chimney and raindrops are presented to demon-strate the effectiveness and accuracy of the proposed MP-basedinterference mitigation method.
A. Experimental Setups
The experiments used the TU Delft PARSAX [32] S-band (3.1315 GHz) radar system which is a full-polarimetricFMCW radar with two independent highly linear polarimetricRF channels in both transmitter and receiver. In the ex-periments, we consider the interference problem among thedifferent polarimetric signals scattered from targets when thefull-polarimetric radar simultaneously emits both horizontallyand vertically polarized signals through the two transmit-ting channels and simultaneously acquires the scattered full-polarimetric signals. Specifically, we use the up- and down-chirp signals for simultaneous transmission on the horizon-tal (H-pol) and vertical (V-pol) polarization channels of thePARSAX radar, respectively. Then, the HV- (H-pol trans-mission, V-pol reception) and VV-(V-pol transmission, V-polreception) polarimetric signals scattered from the same targetwould arrived at the V-pol receiving antenna at the same time.Although the up- and down-chirp waveforms are of great helpto distinguish the scattered HV-pol and VV-pol signals, thestrong VV-pol signal would still cause strong interference inthe output of the HV-pol receiving channel. This kind of theinterferences is categorized as Case 2 in Fig. 2.In Experiment 1, we considered an industrial chimneyas a stationary target and took measurements for a singlesweep. The chimney is about .
07 km away from PARSAXradar, which is installed on the roof of the building of thefaculty of Electrical Engineering, Mathematics & ComputerScience (EEMCS), TU Delft. The PARSAX radar is shownin Fig. 9(a) and an image of the chimney captured by acamera with the same orientation as the radar is presentedin Fig. 9(b). In Experiment 2, we observed a rain storm,which can be considered as a distributed target, by pointing
TABLE IIE
XPERIMENTAL SETUP P ARAMETERS FOR E XPERIMENT AND E XPERIMENT Parameter Value
Center frequency 3.1315 GHzBandwidth 40 MHzTime duration of a sweep 1 msNumber of samples per sweep 16384Maximum range 18.75 kmNumber of sweeps per CPI 512Waveform Simultaneous up- and down-chirps on the H-pol andV-pol polarization channels(a) (b)Fig. 9. Experimental measurement setup. (a) shows PARSAX radar on theroof of EEMCS Faculty building and (b) the industrial chimney used as astationary target. the PARSAX radar vertically. The parameters for experimentalmeasurements are listed in Table II.
B. Experiment 1: Stationary isolated target (Chimney)
Fig. 10(a) shows the acquired HV-pol beat signal (i.e.,“sig Int” in the solid red line) when the transmitter si-multaneously emitted the up- and down-chirp signals withopposite chirp rates through the two transmitting channelswith horizontal and vertical polarizations, respectively. Theacquired HV-pol beat signal was polluted by the strong VV-pol signal arrived together at the receiving antenna, and theinterference-contaminated samples are indicated by the dashedred rectangle in Fig. 10(a). For comparison, the reference HV-pol signal (i.e., “ref sig” in the dashed blue line) acquired bytransmitting a single H-pol up-chirp signal is also presented.To suppress the VV-pol interference, the received signal wasprocessed by using the zeroing, Burg-, IVM- and the proposedMP-based interference mitigation methods and the results areshown in Fig. 10(c) and (d). Comparing the signals obtained
UBMITTED TO IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES 12 A m p li t ude sig_Intref sig (a) Index -50-40-30-20-100 N o r m a li z ed SV s [ d B ] Singular value distribution (b)
350 400 450 500 550-4000-2000020004000 A m p li t ude ref sig zeroing Burg MP (c) A m p li t ude Beat signal--IVM (d)Fig. 10. The beat signal acquired in one FMCW sweep for the chimney observation. (a) shows the measured beat signals with (i.e., “sig Int” in the solidred line) and without the cross-polarimetric interference (i.e., “ref sig” in the dashed blue line). (c) presents the signals around the interference-contaminatedregion after interference mitigation using zeroing, Burg- and MP-based methods while (d) shows the recovered beat signal with the IVM-based method.
Range [km] -60-40-200 A m p li t ude [ d B ] Range Profile sig_IntrefZeroingBurgMP (a)
Range [km] -60-50-40-30-20-10010 A m p li t ude [ d B ] Range Profile (b)
Range [km] -100-90-80-70-60-50-40 A m p li t ude [ d B ] Range Profile (c)Fig. 11. The range profiles of the Chimney scenario obtained with the signals before and after interference mitigation. (a) shows the range profiles of thescenario within
10 km from the radar. (b) and (c) are the zoomed-in views of the range profiles of the targets at the distances of .
07 km and . fromthe radar, respectively. by all four interference mitigation methods with the referencesignal, the MP-based method almost accurately reconstructsthe clipped samples in the interference-contaminated regionwhile the Burg-based method recovers these samples with un-derestimated amplitudes. By contrast, the IVM-based methodleads to a blow-up in the recovered beat signal (Fig. 10(d)),which again shows its instability. In addition, before applyingthe Burg-, IVM- and MP-based methods to recover the signalsamples in the cut-out region, SAMOS was used to estimatethe signal model order and a model order of two was selected,which is highly underestimated considering the complex en-vironment surrounding the chimney. Hence, we decided toselect the model order empirically based on the normalizedsingular value distribution of the matrix used by SAMOS (seeFig. 10(b)). With a threshold of − (i.e.,
20 dB ) for thenormalized SVs, a model order of 40 was selected and usedby the three methods for signal reconstruction.Moreover, the range profiles constructed with theinterference-contaminated signal, reference signal, thesignals acquired after interference mitigation are displayed inFig. 11(a) (due to invalid signal recovery of the IVM-based method, its RP is omitted). It is clear that the range profileobtained with the interference-contaminated signal has higher“noise floor” in contrast to that formed with other signals,which would mask weak targets. For the convenience ofcomparison, the close-ups of the range profiles of the chimneyat the distance of .
07 km and some weak targets at thedistance of . in Fig. 11(a) are shown in Fig. 11(b) and(c). From Fig. 11(c), a clear peak for a weak target at thedistance of .
24 km can be observed in the range profilesgenerated with the reference signals and the signals acquiredafter interference mitigation. By contrast, a deep null is seenat the same position in the range profile formed with theinterference-contaminated signal, which could be caused bythe destructive interference between the interference and thetarget’s signal. Moreover, the range profiles obtained withsignals after mitigating the interference by using Burg- andMP-based methods are comparable to the reference oneand have lower sidelobes for the weak targets around thedistance of . . On the other hand, the range profile ofthe chimney acquired after processing with the proposed UBMITTED TO IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES 13
Fig. 12. The signals of all the sweeps scattered from rain droplets.
Time[ms] -0.500.5 A m p li t ude Real part of the signal (a)
Time[ms] -0.1-0.0500.050.1 A m p li t ude ZeroingBurgIVMMP (b)Fig. 13. The time signals at a Doppler bin after taking FFT along the slow-time dimension. (a) and (b) show the time signal before and after interferencemitigation.
MP-based interference mitigation is almost identical to theone formed with the reference signal. However, the zeroingcaused a void of signal samples and the Burg-based methodunderestimated signal amplitude in the cut-out region; thus,they cause higher sidelobes and power loss in the constructedrange profiles (see the insets in Fig. 11(b)).
C. Experiment 2: Distributed target (Rain)
In this experiment, we used 512 sweeps as a CoherentProcessing Interval (CPI) for full-polarimetric measurementsof rain droplets. After simple preprocessing to suppress thedirect coupling, the acquired HV-pol signals in all the sweepsare shown in Fig. 12, where the interference-contaminatedsamples are located in the time interval from . to . .The interference was caused by the VV-pol signals, which aregenerally much stronger than the desired HV-pol signals (seethe much larger amplitudes of the interference-contaminatedsamples relative the rest ones). So after the range-Doppler (R-D) processing, the formed R-D map of the rain dropletsis completely overwhelmed by the interference, as shown inFig. 14(a).As the raindrops are moving targets, we suggest first takingthe FFT with respect to the slow time in a CPI and thenperforming the interference mitigation to the time signal alongeach Doppler bin to avoid the possible detrimental impactof errors caused by interference mitigation on the Dopplerinformation. Fig. 13(a) shows the time signal in a Doppler binafter taking the FFT along the slow time and the interferenceis still observed in the interval from . to . . Apply-ing the proposed MP-based interference mitigation method,zeroing, Burg- and IVM-based methods to this time signal,the resultant signals are presented in Fig. 13(b). The MP-based method successfully recovers the missing signals in thegap resulting from interference suppression while the Burg-and IVM-based methods reconstruct only the missing sampleswhich are close to the front and rear available measurementswith underestimated amplitudes. Note that for the rain dataset, the SAMOS method could not estimate proper modelorders, either. So we empirically determine the model orderof the signal in each Doppler bin based on the normalizedsingular value distribution of the matrix used by SAMOS witha threshold of − . The estimated signal model order wasused by the Burg-, IVM- and MP-based methods to reconstructthe signal in the cut-out region.After mitigating the interferences for the time signals inall Doppler bins, an FFT is taken along the fast time to getthe R-D map of the rain drops. Fig. 14(b)-(e) present theobtained R-D maps in logarithmic scale of the moduli ofsignals after interference mitigation with zeroing, the Burg-, IVM- and MP-based methods, respectively. Except the R-Dmaps obtained with the IVM-based method, the other threeR-D maps are visually almost identical and their qualitiesare noticeably improved compared to that obtained withoutinterference mitigation (Fig. 14(a)).Due to the lack of ground truth reference, we alternativelyassess the improvement of the R-D maps obtained with theBurg-, IVM- and MP-based methods relative to the one gotwith zeroing by computing the power differences betweenthe pixels of the R-D maps of the three signal reconstructionmethods and zeroing method. The results are shown in linearscale in Fig. 14(f)-(h). One can see that the power differencebetween the R-D maps of MP-based method and zeroing inFig. 14(h), compared with that in Fig. 14(f), presents a patternmuch closer to the R-D maps in Fig. 14(b), (c), and (e). Asin the rain data set the strong VV-pol interferences appearat the similar time interval in all the sweeps within the CPI,the zeroing method eliminates the signal samples within thistime interval (i.e., between about . to . ) in allthe sweeps. So the power difference of the R-D maps ofzeroing and the other three methods are determined by thecontribution of the beat signal samples in the cut-out region.Theoretically, the beat signals of rain droplets in the cut-outtime interval in a CPI can be considered as the acquired data byusing an FMCW radar with narrower bandwidth (i.e., shorterFMCW sweep duration) but keep other system parametersunchanged; thus, they can form a similar R-D map as that UBMITTED TO IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES 14 (a) (b) (c) (d)(e) (f) (g) (h)Fig. 14. The range-Doppler processing results of the rain data. (a) is the RD map obtained with the original interference-contaminated signals. (b), (c), (d),and (e) are formed by the signals after interference mitigation by using the zeroing, Burg-, IVM- and MP-based methods, respectively. (f), (g), and (h) showthe corresponding power differences between the RD maps in (c)-(e) and (b). constructed with the full-sweep signals in the CPI but withlower range resolution. Namely, the more accurate the signalsamples recovered by the Burg-, IVM- and MP-based methodsin the cut-out region are, the closer to the actual R-D mapthe pattern of the power difference between the R-D maps ofthese methods and the zeroing approach. Therefore, the MP-based method gets more accurate estimation of the signals inthe cut-out region than the Burg-based method. Furthermore,large portions of the positive power difference in Fig. 14(f)and (h) reveal that compared to the zeroing technique, bothBurg- and MP based method improve the signal powers byreconstructing the missing signals in the cut-out region. Inaddition, due to the instability of the IVM-based method, theblow-ups in its reconstructed signals cause the streaks withvery large amplitudes in many Doppler bins (Fig. 14(d)).So the accuracy of the recovered signals by the IVM-basedmethod is worse than that of the Burg- and MP-based methods.VI. C
ONCLUSION
In this paper, we present a matrix-pencil based interferencemitigation method for FMCW radar systems. The proposedmethod exploits the feature of the desired beat signals as asum of exponential sinusoidal components, which is differentfrom the chirp-like waveforms of interferences after dechirpingon reception, for interference suppression. The method isimplemented in two steps by first detecting and cutting outthe interference-contaminated samples and then recovering thesignal samples in the cut-out region based on the exponentialsinusoidal model of desired beat signals. It addresses thediscontinuity of the signals caused by traditional zeroingtechnique and overcomes the power loss of useful signals.Meanwhile, it results in lower sidelobes of the range profileof a target. Moreover, compared to the Burg-based method, itsignificantly improves the accuracy of the estimated signals in the cut-out region by an iterative estimation scheme, whichhas demonstrated through both numerical simulations andexperimental results. The numerical simulations also revealthat the proposed method can robustly work in the scenarioswith a low signal to noise ratio (down to 0dB) and witha long interference duration (up to 50% of a sweep). Inaddition, the proposed MP-based method can be extendedto 2D or high-dimensional cases to mitigate interferencesdirectly in a higher dimensional space (e.g., RD or range-DOAdomains), especially for point-target scenarios, which wouldbe considered in future work.A
CKNOWLEDGMENT
The authors acknowledge the contribution of N. Cancrinusto this research by testing applicability of the matrix pencilmethod to the beat signal reconstruction.R
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