Extended Range Profiling in Stepped-Frequency Radar with Sparse Recovery
aa r X i v : . [ c s . I T ] N ov Extended Range Profiling in Stepped-FrequencyRadar with Sparse Recovery
Yang Hu, Yimin Liu, Huadong Meng, Xiqin Wang
Department of Electronic Engineering, Tsinghua University, BeijingEmail: [email protected]
Abstract —The newly emerging theory of compressed sensing(CS) enables restoring a sparse signal from inadequate numberof linear projections. Based on compressed sensing theory, anew algorithm of high-resolution range profiling for stepped-frequency (SF) radar suffering from missing pulses is proposed.The new algorithm recovers target range profile over multiplecoarse-range-bins, providing a wide range profiling capability.MATLAB simulation results are presented to verify the proposedmethod. Furthermore, we use collected data from real SF radarto generate extended target high-resolution range (HRR) profile.Results are compared with ‘stretch’ based least square methodto prove its applicability.
I. I
NTRODUCTION
The range resolution of a radar system is determined by thebandwidth of the transmitted signal. Stepped-frequency (SF)pulse train obtains large signal bandwidth by linearly shifting,step-by-step, the center frequencies of a train of pulses. Itis widely used in high-resolution radar systems and welldocumented in the literature [1],[2]. In SF radars, the ‘stretch’processing method [2], based on inverse discrete Fouriertransform (IDFT) technique, can acquire high-range resolution(HRR) profiles with narrow instantaneous bandwidth andlow system complexity. However, SF radar suffers greatlyfrom missing-pulse problems due to interference or jammingimpinging on the receiver, since SF technique occupies a largebandwidth. While some pulses missing are and hence must bediscarded, the IDFT based stretch processing will inevitablyleads to high sidelobes, thus undermining the profiling quality.Various methods have been proposed to interpolate the missingdata (see [3] and reference therein). Theoretical analysis andexperience indicate that the longer the signal interpolationlength is, the larger the interpolation error is. If the missingpulse number becomes bigger, the performance of existedmethod will reduce rapidly [3].Besides the missing-pulse problem, SF radar suffers from‘ghost image’ phenomenon. This problem, mainly caused byrange ambiguity among adjacent ‘coarse-range-bins’, is deli-cately addressed in [4], where the author solved the problemby least square (LS) technique. But this method is applicableon the assumption that full pulses are well received, and thefoundation of it is still IDFT. Therefore, missing-pulses alsodeteriorate the profiling results. Recently, the new emergingtheory of compressed sensing (CS) [5],[6] that achieves highresolution has been widely used in radar applications [7].The main advantage of this theory is that, with sub-Nyquist samples, sparse signal can still be reconstructed perfectly. CStheory was introduced in the signal processing for SF radarby Sagar Shah et al. [8]. With reduced number of transmittedpulses in one coherent processing interval (CPI), their methodprovides super-resolution ability in both range and Doppler do-main. It also indicted that missing-pulse problem can be solvedwith their method. However, they only discussed profilingrange of only one coarse-range-bin, limiting their applicationon narrow-range-gate profiling.This paper introduces a new profiling algorithm for SFradar with missing pulses, and the profiling range gate extentsfor multiple coarse-range-bins. We focus on profiling of astationary object. Unavailable data from missing pulses arediscarded; sparse recovery is used to obtain extended syntheticrange profile. We demonstrate that new algorithm can solve themissing-pulse problem, it also has a wide profiling range gate.The remainder of this paper is organized as follows. In SectionII, the signal model of HRR profiling for SF radar is stated.CS based profiling with missing pulses is described in sectionIII. Simulation results are presented in section IV. Section Vconcludes the new approach.II. S
YSTEM M ODEL
In SF radar, a pulse train of N pulses are transmitted withstepped carrier frequencies. For the n th pulse, the carrierfrequency is f n = f c + n ∆ f , where f c is the initial frequencyand ∆ f the frequency step. The complex profile of themeasured scene can be represented by system function H ( t d ) ,as has been derived in [1]. t d is the time domain variable,and H ( t d ) describes the complex reflectivity of measuredscene corresponding to time delay t d . For the convenienceof signal modeling and derivation, it is assumed that onetarget falls in the range gate [ R , R + D ] over the wholecoherent processing interval, where R = cQ/ f , and D = cL/ f ( Q , L are nonnegative integers and c is the speedof light). In the ‘stretch’ processing [1], the range resolutionis c/ N ∆ f [2]. Choosing this resolution as the samplingperiod, the p th high-resolution range cell, which representsthe complex amplitude of the scatterer located in the range R + ( cp/ N ∆ f ) , is written by h p = H ( p/N ∆ f ) . Thus,the HRR profile of the target can be expressed by the vector h = [ h , h , , h NL − ] T .A target response matrix (TRM) [2] was used to organize theecho signal of the pulse train. The TRM contains N rows and columns. The n th row consists of S uniformly sampled time-domain data from the baseband echo signal of the n th pulse(If the transmitted baseband waveform is pulse compressingwaveform, the ‘baseband echo signal’ refers to the pulse-compressed echo signal). The elements in the same column arebaseband samples of the same coarse range cell. The columnnumber is S = 2 D/c ∆ t , where ∆ t is the sampling interval.The TRM of a target can be denoted by E = E (0) E (∆ t ) · · · E ( S ∆ t − ∆ t ) E (0) E (∆ t ) · · · E ( S ∆ t − ∆ t ) ... ... .. . ... E N − (0) E N − (∆ t ) · · · E N − ( S ∆ t − ∆ t ) . (1)Here, E n ( τ ) is the baseband echo signal of the n th pulse,and τ is the baseband sampling instant. As derived in [1], thebaseband echo signal from stationary target is E n ( τ ) = NL − X p =0 h p R X ( τ − R c − pN ∆ f ) e − j π npN + u n ( τ ) (2)where R X ( τ ) is the baseband pulse shape and u n ( τ ) isadditive noise.In the ‘stretch’ processing method, the IDFT is applied toeach TRM column to form a HRR profile in one coarse-range-bin [1]. Missing pulse problem means data from some rows ofTRM are not available. If the missing pulse number is large,profiling quality is greatly decreased using IDFT. Our newmethod solve this problem by sparse recovery based on CStheory, which can provide a better profiling quality. Based onthe observation that discrete system function vector h is sparse,we propose a new scheme for HRR profiling based on sparserecovery in the next section.III. HRR PROFILING WITH MISSING PULSES
We now introduce the new CS based HRR profiling method,on condition that some pulses are missing. Suppose only M pulses ( M < N ) from N transmitted carrier frequencies arevalid, that the carrier frequency of the m th valid pulse is F m = f c + C m ∆ f , where m is an integer between and M − , C m is an integer between and N − . Substituting pulse numberindex n in equation (2) by C m , we derive sample output forthe m th pulse at sampling instance τE C m ( τ ) = NL − X p =0 h p R X ( τ − R c − pN ∆ f ) e − j π CmpN + u m ( τ ) . (3)We rewrite (3) in vector multiplication form: E C m ( τ ) = ϕ ( C m , τ ) h + u m ( τ ) . (4) ϕ ( C m , τ ) is a row vector of length N × L , the p th element ofthe vector is ϕ p ( C m , τ ) = R X ( τ − R c − pN ∆ f ) e − j π CmpN . (5) Deleting the invalid data in the TRM, the row numberdecreases to M . ˜ E = E C (0) E C (∆ t ) · · · E C ( S ∆ t − ∆ t ) E C (0) E C (∆ t ) · · · E C ( S ∆ t − ∆ t ) ... ... . . . ... E C M − (0) E C M − (∆ t ) · · · E C M − ( S ∆ t − ∆ t ) . (6)The new TRM includes all available information we received.Note that each element of the TRM is a linear projection ofsystem function h . By vectorizing this matrix, we may writethe following equation Y = vec (˜ E ) = Φ h + U. (7)The observation vector Y is of length M × S . Matrix Φ is theprojection matrix of M × S rows and N × L columns, each rowof Φ is corresponding to an observation. For an instance, therow corresponding to pulse number C m and sampling instance s ∆ t is ϕ ( C m , s ∆ t ) . U the noise vector for all observations.We have established a linear projection for complex profile h . While pulses are missing, M < N holds, and inequality
M N < SL holds. Therefore, (7) becomes an underdeterminedequation. According to CS theory, recovering a sparse signalfrom insufficient observation is possible by ℓ minimization[6]: min k ˜ h k s.t. k Y − Φ˜ h k ≤ ǫ (8)where ˜ h is an reconstruction of h and ǫ is an estimation errorthat is determined by received signal noise.IV. R ESULTS
We show some primary results of simulation. The HRRrange profile of a real aircraft (Fig.1(a)) was measured bya wideband C-band chirp radar. The chirp bandwidth was512MHz, providing a range resolution of about 0.3m. Thismeasured range profile is used as the scatterer truth. ForSF radar simulation, 32 LFM pulses are transmitted in acoherent pulse train. The frequency step size is 16MHz; andthe total effective bandwidth is N ∆ f = f s equals single-pulse bandwidth. The profile range gate covers coarse-range-bins. We simulate the missing pulse conditionby discarding data received from randomly selected 12 pulses,the left 20 pulses are valid. White Gaussian noise was addedto the received data, SNR is approximately 15dB. A. Simulated Data
The results of simulation data obtained via different meth-ods are compared in Fig.1. Fig.1(b) show the result obtainedfrom LS method [4], Fig.1(c) demonstrate the result from newapproach. From which it can be noted that LS method hascreated high sidelobe, and the result by new method is moresimilar to original target range profile.To analyze the profiling results of the two methods quan-titatively, we measure the similarity between the simulatedtarget and the reconstruction profile by normalized crosscorrelation. Similarity equals means perfect reconstruction.
10 20 30 40 50 60 70 8000.51
Range,Meter N o r m a li z ed m agn i t ude Simulated target (a)
Range,Meter N o r m a li z ed m agn i t ude LS Result (b)
Range,Meter N o r m a li z ed m agn i t ude Sparse reconstruction (c)Fig. 1. Comparison of result obtained via different methods for simulativedata. (a) Model of scatters. (b) LS result of missing data. (c) Sparse recoveryof missing data.
Missing pulse number S i m il a r i t y m ea s u r e Reconstruction quality comparison
LS methodSparse Reconstruction
Fig. 2. Similarity measure.
Fig.2 illustrate the comparison. We increase the number ofmissing pulses from 0 to 20. The line marked by “ △ ” denotesthe similarity by LS method, and line marked by “ (cid:3) ” denotesthe similarity by sparse recovery. Sparse recovery has anobvious advantage over the LS counterpart. B. Real Radar Data
We use real radar data obtained from SF radar. An ex-periment was carried out in a wide and flat field. A singlemetal reflector was placed 1010m away from the radar an-tenna. Experimental data of I/Q channels was collected fromthe baseband of the radar receiver. All parameters in theexperiment were equal to the simulated data. We discard 12pulses randomly to simulate the missing data condition. Newapproach is applied to the missing data. Profiling results arecompared to IDFT based LS method.Fig. 3(a) shows the profiling result of LS method with fulldata. Fig. 3(b) and 3(c) compare the profiling results to themissing data by LS method and the new method respectively.LS method exhibits high sidelobe as predicted, while theprofiling result by new method is similar to full data profiling.
995 1000 1005 1010 1015 1020 1025 1030 103500.51
Range,Meter N o r m a li z ed m agn i t ude LS result, Full data (a)
995 1000 1005 1010 1015 1020 1025 1030 103500.51
Range,Meter N o r m a li z ed m agn i t ude LS result, Missing data (b)
995 1000 1005 1010 1015 1020 1025 1030 103500.51
Range,Meter N o r m a li z ed m agn i t ude Sparse reconstruction, Missing data (c)Fig. 3. Comparison of result obtained via different methods for real radardata. (a) Full pulse recovery. (b) LS result of missing data. (c) Sparse recoveryof missing data.
Sparse recovery outperforms LS using real radar data.V. C
ONCLUSION
The application of sparse recovery in extended HRR profil-ing for SF radar is illustrated. The simulated data and real dataexperiments prove that the proposed method is an appropriatetool to deal with missing data problem. Profiling quality of thenew method has an obvious advantage over IDFT based leastsquare method, if some pulses are missing. Moreover, it canprofile multiple coarse-range-bins simultaneously, indicatinga wide profiling range. The profiling result is not corruptedby ghost images. Further work should consider reducingcomputational load for real-time implementations.R
EFERENCES[1] EINSTEIN T.H., “Generation of high resolution radar range profiles andrange profile autocorrelation functions using stepped frequency pulsetrains,” Project Report TT-54, Massachusetts Institute of Technology,Lincoln Laboratory, 18 October 1984 (AD-A149242)[2] WEHNER D.R,
High resolution radar , 2nd ed. Artech House, Norwood,MA, 1995.[3] L. Zhang, M. Xing, C. Qiu, J. Li, Z. Bao, “Achieving higher resolutionISAR imaging with limited pulses via compressed sampling,”
IEEEGeoscience and Remote Sensing Letters , vol. 6, no. 3, pp. 567-571,July 2009.[4] Y. Liu, H. Meng, H. Zhang and X. Wang, “Eliminating ghost imagesin high-range resolution profiles for stepped-frequency train of linearfrequency modulation pulses,”
IET Radar Sonar Navig. , 2009, Vol. 3,Iss. 5, pp. 512-520.[5] E. Cands, J. Romberg, and T. Tao, “Robust uncertainty principles: Exactsignal reconstruction from highly incomplete frequency information,”
IEEE Trans. Inform. Theory , vol. 52, no. 2, pp. 489-509, Feb. 2006.[6] D. Donoho, “Compressed sensing,”
IEEE Trans. Inform. Theory , vol.52, no. 4, pp. 1289-1306, Apr. 2006.[7] R. Baraniuk, P. Steeghs, “Compressive radar imaging,”