Research Article Sparse Representation Denoising for Radar …downloads.hindawi.com/journals/ijap/2014/875895.pdf · 2019. 7. 31. · Research Article Sparse Representation Denoising

Research ArticleSparse Representation Denoising for Radar High ResolutionRange Profiling

Min Li,1 Gongjian Zhou,2 Bin Zhao,3 and Taifan Quan4

1 Harbin Institute of Technology, Room 807, Building 20, 92 West Dazhi Street, Harbin 150001, China2Harbin Institute of Technology, Room 705, Building 20, 92 West Dazhi Street, Harbin 150001, China3Harbin Institute of Technology, Room 802, Building 20, 92 West Dazhi Street, Harbin 150001, China4Harbin Institute of Technology, Room 708, Building 20, 92 West Dazhi Street, Harbin 150001, China

Correspondence should be addressed to Taifan Quan; [email protected]

Received 15 November 2013; Accepted 11 February 2014; Published 17 March 2014

Academic Editor: Margaret Cheney

Copyright © 2014 Min Li et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Radar high resolution range profile has attracted considerable attention in radar automatic target recognition. In practice, radarreturn is usually contaminated by noise, which results in profile distortion and recognition performance degradation. To deal withthis problem, in this paper, a novel denoising method based on sparse representation is proposed to remove the Gaussian whiteadditive noise. The return is sparsely described in the Fourier redundant dictionary and the denoising problem is described asa sparse representation model. Noise level of the return, which is crucial to the denoising performance but often unknown, isestimated by performing subspace method on the sliding subsequence correlation matrix. Sliding window process enables noiselevel estimation using only one observation sequence, not only guaranteeing estimation efficiency but also avoiding the influenceof profile time-shift sensitivity. Experimental results show that the proposed method can effectively improve the signal-to-noiseratio of the return, leading to a high-quality profile.

1. Introduction

Radar high resolution range profile (HRRP) contains richinformation for target feature analysis, such as target struc-ture and scatterer distribution, and has received considerableattention in radar automatic target recognition (ATR) inrecent years [1–7]. HRRP feature selection and extractionhave been investigated in [1, 2]. Parametric statistical modelshave been applied to HRRP ATR [3–5]. High order spectraand complex HRRP features have been studied for targetclassification [6, 7]. In these works, high signal-to-noise ratio(SNR) is a necessary condition. However the target returnis usually contaminated by noise in practice. This resultsin HRRP signature distortion and recognition performancedegradation [8].

Some methods have been proposed to handle this prob-lem in target recognition stage. The noise-robust bispectrumfeatures have been extracted for target recognition [9]. Mod-ifying the statistical model parameters according to the noise

level of test samples is developed in [10], but this methodsuffers from a high computational burden.

Noise-robust imaging is another effective approach. In[11], higher order statistics is presented for imaging at lowSNR, which however has some information loss. Recently,noise-robust compressive sensing (CS) imaging is developed[12, 13]. In these methods, a key problem is how to estimatethe noise level, which impacts the quality of image because ofthe noise sensitivity of CS. In [12] the noise level is estimatedby computing the energy of the noise cells, which are selectedby an energy-based threshold.The estimation accuracy of thismethod relies on threshold selection and number of noisecells. In [13], the noise variance is estimated by employingthe median value of finest scale wavelet coefficients. Usually,HRRP is obtained from one observation sequence, in whichthe number of samples is limited. In this scenario, thesemethods cannot guarantee the estimation accuracy of noiselevel.

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2014, Article ID 875895, 8 pageshttp://dx.doi.org/10.1155/2014/875895

2 International Journal of Antennas and Propagation

Another natural choice is to remove noise in radar return.Wavelet shrinkage was shown to be effective when SNR isbeyond a certain threshold [14, 15]. However its performancedegrades largely in the presence of severe noise contami-nation. Combined bispectrum-filtering is used to suppressnoise in HRRP to improve recognition performance [16, 17],but this method requires a large number of observations toaverage bispectrum estimates. In addition, for some commonprofiling processes, for example, dechirping for widebandsignal, the dechirped output of target return containsmultiplefrequency components, which change with observationsdue to HRRP time-shift sensitivity [18]. In this case, therequirement of a large number of observations cannot be sat-isfied.

Sparse representation (SR) theory shows that, in sparseand redundant dictionary, signal energy concentrates onminority atoms, whereas noise energy spreads over almost allthe atoms, and the signal can be stably recovered from noisymeasurements [19–21]. Based on this, in this paper, we pro-pose a sparse representation denoising method to improvethe SNR of noisy return for profiling. A Fourier redundantdictionary is established to sparsely describe the return andthe denoising problem is described as a sparse representationmodel. Noise reduced return is recovered by solving thesparse representation problem. The noise level, which is akey parameter for denoising by sparse representation, is esti-mated by preforming subspace method on the subsequencecorrelationmatrix. In this way, the noise level can be obtainedonly using one observation sequence, not only guaranteeingthe efficiency but also avoiding the influence of HRRP time-shift sensitivity. Simulated and real data experiments areprovided to verify the effectiveness of the proposed denoisingmethod.

The rest of this paper is organized as follows. The basictheory of radar high resolution range profiling is introducedin Section 2. Radar return denoising by sparse representationis discussed in Section 3. The simulated and real data exper-iments are provided to verify the denoising performance inSection 4, followed by conclusions in Section 5.

2. Radar High Resolution Range Profiling

Usually, HRRP is obtained by wideband signal, such aslinear frequency modulation (LFM) signal and stepped-frequency (SF) signal. In this paper, we take the LFM signalas an example to introduce the basic theory of radar highresolution range profiling by dechirping process [22, 23].Thetransmitted wideband LFM signal can be represented as

𝑠 (𝑡) = rect( 𝑡𝑇𝑝

) exp {𝑗2𝜋 (𝑓𝑐𝑡 +1

2

𝛾𝑡2)} , (1)

where

rect (𝑢) ={{{

{{{

{

1 |𝑢| ≤

1

2

0 |𝑢| >

1

2

(2)

denotes complex signal envelope, 𝑓𝑐 denotes carrier fre-quency, 𝑇𝑝 is the pulse width, and 𝛾 is the frequency mod-ulation slope.

Given reference range 𝑅ref, the dechirping reference sig-nal can be represented as

𝑠ref (𝑡) = rect(𝑡 − 2𝑅ref/𝑐

𝑇ref)

× exp{𝑗2𝜋[𝑓𝑐 (𝑡 −2𝑅ref𝑐

) +

1

2

𝛾(𝑡 −

2𝑅ref𝑐

)

2

]} ,

(3)

where𝑇ref is the pulse width of reference signal, usually largerthan 𝑇𝑝. Assuming a scatterer locating at range 𝑅𝑡, the returnfrom the scatterer is

𝑠𝑟 (𝑡) = 𝐴 ⋅ rect(𝑡 − 2𝑅𝑡/𝑐

𝑇𝑝

)

× exp{𝑗2𝜋[𝑓𝑐 (𝑡 −2𝑅𝑡

𝑐

) +

1

2

𝛾(𝑡 −

2𝑅𝑡

𝑐

)

2

]} ,

(4)

where𝐴 denotes the return amplitude.The dechirping outputis

𝑠𝑖𝑓 (𝑡) = 𝑠𝑟 (𝑡) ⋅ 𝑠∗

ref (𝑡) , (5)

where ∗ stands for complex conjugate. Let 𝑅Δ = 𝑅𝑡−𝑅ref, andafter some manipulations, (5) can be rewritten as

𝑠𝑖𝑓 (𝑡)

= 𝐴 ⋅ rect(𝑡 − 2𝑅𝑡/𝑐

𝑇𝑝

)

× exp {−𝑗4𝜋𝑐

𝛾 (𝑡 −

2𝑅ref𝑐

)𝑅Δ − 𝑗4𝜋

𝑐

𝑓𝑐𝑅Δ + 𝑗

4𝜋𝛾

𝑐2𝑅2

Δ}


𝑇𝑝

)

×exp(−𝑗4𝜋𝑐

𝛾𝑅Δ𝑡 + 𝑗8𝜋𝑅ref𝑐2

𝛾𝑅Δ− 𝑗4𝜋

𝑐

𝑓𝑐𝑅Δ − 𝑗4𝜋

𝑐

𝑓𝑐𝑅Δ) .

(6)

The later three phase terms in (6) are constants, with nocontribution to HRRP. Denoting the sum of the later threephase terms by 𝜑, (6) can be rewritten as

𝑠𝑖𝑓 (𝑡) = 𝐴 ⋅ rect(𝑡 − 2𝑅𝑡/𝑐

𝑇𝑝

) exp (−𝑗4𝜋𝑐

𝛾𝑅Δ𝑡 + 𝑗𝜑)


𝑇𝑝

) exp (−𝑗2𝜋𝑓𝑑𝑡 + 𝑗𝜑) ,

(7)

where 𝑓𝑑 = 2𝛾𝑅Δ/𝑐. Formula (7) shows that the dechirpedoutput of the return from one scatterer is a complex sinu-soidal signal with a frequency proportional to its relativerange.

International Journal of Antennas and Propagation 3

3. Denoising by Sparse Representation

In this section, we start the presentation of the proposeddenoising method by first discussing how redundant dic-tionary is established for sparsely describing the dechirpedreturn. Once the sparse representation model is established,the noise level is estimated by subspace method, followedwith the reconstruction of the denoised signal by solvingsparse representation.

3.1. Dechirped Return Sparse Representation. In noisy cir-cumstance, assume a target contains 𝐾 scatterers locating atdifferent ranges and there are𝑀 sampling points in a singlepulse. According to (7), the time domain sampling sequenceof the dechirping output pulse can be represented as

𝑦 (𝑚) = 𝑠 (𝑚) + 𝑛 (𝑚)

=

𝐾

∑

𝑘=1

𝐴𝑘 ⋅ exp (−𝑗2𝜋𝑓𝑘𝑚 + 𝑗𝜑𝑘) + 𝑛 (𝑚) ,

𝑚 = 0, 1, . . . ,𝑀 − 1,

(8)

where 𝐴𝑘, 𝑓𝑘, and 𝜑𝑘 are the amplitude, relative frequencynormalized by sampling rate, and constant phase of the returnfrom the 𝑘th scatterer, respectively. 𝑠(𝑚) and 𝑛(𝑚) denotesignal and the Gaussian white noise sampling sequences,respectively. Let y = [𝑦(0), 𝑦(1), . . . , 𝑦(𝑀 − 1)]𝑇, s =[𝑠(0), 𝑠(1), . . . , 𝑠(𝑀−1)]

𝑇, andn = [𝑛(0), 𝑛(1), . . . , 𝑛(𝑀−1)]𝑇;then (8) can be rewritten as

y = s + n =𝐾

∑

𝑘=1

𝑢𝑘 ⋅ k𝑘 + n, (9)

where 𝑢𝑘 = 𝐴𝑘 exp(𝑗𝜑𝑘) and k𝑘 = [1, exp(−𝑗2𝜋𝑓𝑘), . . . ,exp{−𝑗2𝜋𝑓𝑘(𝑀 − 1)}]

𝑇. Formula (9) indicates that thedechirped output sequence contains multiple complexsinusoidal components superposed with noise. Usually,the number of the main scatterers of a target is much lessthan that of the range cells in the pulse. Thus s is sparse infrequency domain and can be sparsely represented by thecomplex Fourier redundant dictionary, which is constructedas

A = {𝜙1,𝜙2, . . . ,𝜙

𝑁} ∈ C𝑀×𝑁

, (10)

where

𝜙𝑖= exp {−𝑗2𝜋 ⋅ f𝑁 (𝑖) ⋅m} , 𝑖 = 1, . . . , 𝑁, (11)

m = [0, 1, . . . ,𝑀 − 1]𝑇, and 𝑁 > 𝑀. In (11), f𝑁 = [0, 1/𝑁,. . . , (𝑁 − 1)/𝑁] is the normalized frequency. Then (9) can berepresented as

y = Ax + n, (12)

where x is a sparse vector, composed of the decompositioncoefficients of signal s in A. Sparse representation theoryshows that if x satisfies ‖x‖0 < (1/2)spark(A), x can be stablysolved by the following 𝑙0 optimization problem [21]:

x̂ = argmin ‖x‖0 s.t.y − Ax

2≤ 𝛿, (13)

where ‖ ⋅ ‖0 stands for the 𝑙0 normof a vector (i.e., the numberof its nonzero elements), spark(A) denotes the minimumnumber of linearly dependent columns of A, and 𝛿 is thenoise level. It is hard to solve (13). Approximated solution canbe acquired by greedy algorithms, for example, orthogonalmatching pursuit (OMP) [24]. In this paper, OMP is utilizedto solve (13) because of its simplicity and efficiency. Whenobtaining x̂ from (13), the denoised signal can be acquired byŝ = Ax̂.

3.2. Estimation of Noise Level. In solving (13) by OMP, thenoise level 𝛿 is a crucial parameter, which is the iterationterminal condition of OMP. However, in most cases, 𝛿 isunknown and thus needs to be estimated. An underestimated𝛿will introduce extra noise components in the recovered sig-nal, and the overestimated one will cause some scattererinformation loss. Hence precise estimation of noise level isnecessary.

Subspace method combined with minimum descriptionlength (MDL) criterion is an effective method for detect-ing the number of signals and estimating the signal-to-interference ratio (SIR) [25–27]. To guarantee the estima-tion performance, a number of observations are needed toevaluate sample covariance matrix; meanwhile the signalfrequencies should not change during these observations.However, the frequency components in the dechirped radarreturn vary significantly with observations due to the HRRPtime-shift sensitivity [18]. As a result, it is hard to incorporatemultiple consecutive observations to estimate noise power as[27]. On the other hand, collecting multiple observations toestimate noise power impairs the efficiency. In this paper, thenoise level is estimated by subspace method with only oneobservation sequence rather than multiple observations.

In general, the number of sampling points within a singlepulse is much more than that of the target scatterers and thedechirped return can be assumed to be a stationary signal.Sliding window process can be adopted to obtain subse-quences, which are used to construct subsequence correlationmatrix. According to (8), assuming one observation sequenceof the dechirped signal is 𝑦(𝑚), 𝑚 = 0, 1, . . . ,𝑀 − 1, subse-quences are constructed as

y𝑖 = [𝑦(𝑖), 𝑦(𝑖 + 1), . . . , 𝑦(𝑖 + 𝐿 − 1)]𝑇= s𝑖 + n𝑖 = A𝛼𝑖 + n𝑖,𝑖 = 0, 1, . . . ,𝑀 − 𝐿,

(14)

where A = [A1,A2, . . . ,A𝐾] denotes subsequence represen-tation basis matrix,A𝑘 = [1, exp(−𝑗2𝜋𝑓𝑘), exp(−𝑗2𝜋2𝑓𝑘), . . . ,exp(−𝑗2𝜋(𝐿−1)𝑓𝑘)]

𝑇, 𝑘 = 1, 2, . . . , 𝐾 is the 𝑘th basis ofmatrixA, and 𝛼𝑖 = [𝐴1 exp(−𝑗2𝜋𝑖𝑓1 + 𝑗𝜑1), 𝐴2 exp(−𝑗2𝜋𝑖𝑓2 + 𝑗𝜑2),. . . , 𝐴𝐾 exp(−𝑗2𝜋𝑖𝑓𝐾+𝑗𝜑𝐾)]

𝑇 is the representation coefficientvector of y𝑖. The subsequence correlation matrix is given by

̂R = 1𝑀 − 𝐿 + 1

𝑀−𝐿

∑

𝑖=0

y𝑖y𝐻

𝑖, (15)

where𝐻 denotes conjugate transpose. Let 𝐿 > 𝐾. The expec-tation of ̂R is

̂R𝐸 = ̂R𝑆 + ̂R𝑛 = ̂R𝑆 + 𝜎2Ι, (16)


where R̂𝑛 = 𝜎2Ι denotes the noise correlation matrix, 𝜎2 is ascalar constant denoting noise power, Ι is the identify matrix,and

̂R𝑆 =1

𝑀 − 𝐿 + 1

𝑀−𝐿

∑

𝑖=0

s𝑖s𝐻

𝑖

=

1

𝑀 − 𝐿 + 1

𝑀−𝐿

∑

𝑖=0

A𝛼𝑖𝛼𝐻

𝑖A𝐻 = 1

𝑀 − 𝐿 + 1

AΡA𝐻(17)

with P = ∑𝑀−𝐿𝑖=0𝛼𝑖𝛼𝐻

𝑖= 𝛽𝛽𝐻, 𝛽 = [𝛼0,𝛼1, . . . ,𝛼𝑀−𝐿]. ̂R𝑆

stands for the signal correlationmatrix. Let𝑀−𝐿+1 > 𝐿 > 𝐾.Thematrix 𝛽 is row linearly independent; namely, the rank of𝛽 is 𝐾. It follows that the rank of R̂𝑆 is 𝐾; equivalently, the𝐿−𝐾 smallest eigenvalues of ̂R𝑆 are all equal to zero.Thus ̂R𝑆can be diagonalized by

U𝐻̂R𝑆U = Λ𝑆, (18)

where U is an orthogonal matrix and Λ𝑆 = diag[𝜆1, 𝜆2, . . . ,𝜆𝐾, 0, . . . , 0], 𝜆1 ≥ 𝜆2 ≥ ⋅ ⋅ ⋅ ≥ 𝜆𝐾. In the same way, ̂R𝐸 can bediagonalized as

U𝐻̂R𝐸U = Λ𝑌 = Λ𝑆 + 𝜎2Ι, (19)

where Λ𝑌 = diag[𝜆1 + 𝜎2, 𝜆2 + 𝜎

2, . . . , 𝜆𝐾 + 𝜎

2, 𝜎2, . . . , 𝜎

2]

is the eigenvalue matrix of R̂𝐸. It seems that the noise powercan be estimated by observing the rest of the smallest 𝐿 − 𝐾eigenvalues of ̂R𝐸. However, in practice, because of the finitelength of sampling sequence, the smallest 𝐿 − 𝐾 eigenvaluesare usually not equal to each other. In this case, we can utilizeminimum description length (MDL) criterion to detect thenoise components and the noise power can be estimated byaveraging the rest of the smallest 𝐿 − 𝐾 eigenvalues [27].

According to the above analysis, for a 𝑀 × 1 noisysequence y, the procedure of estimating noise level 𝛿 can besummarized as follows.

(1) Estimate the subsequence correlation matrix by (15).(2) Perform eigenvector decomposition of the correlation

matrix as ̂R = UΛU𝐻, where

Λ = diag (𝑙𝑖) , 𝑖 = 1, 2, . . . , 𝐿, 𝑙1 ≥ 𝑙2 ≥ ⋅ ⋅ ⋅ ≥ 𝑙𝐿. (20)

(3) For 𝑝 = 0, 1, . . . , 𝐿 − 1, compute the MDL function:

MDL (𝑝)

= − (𝑀 − 𝐿 + 1) (𝐿 − 𝑝) log(∏𝐿

𝑖=𝑝+1𝑙1/(𝐿−𝑝)

𝑖

(1/ (𝐿 − 𝑝))∑𝐿

𝑖=𝑝+1𝑙𝑖

)

+ 0.5𝑝 (2𝐿 − 𝑝) log (𝑀 − 𝐿 + 1) .(21)

(4) Find the index �̂� that minimizes the MDL function;that is,

�̂� = argmin𝑝=0,1,...,𝐿−1

MDL (𝑝) . (22)

0 50 100 150 200 2500

10

20

30

40

50

Range cell

Am

plitu

de (d

B)

Figure 1: Noiseless HRRP of simulated target.

(5) Estimate the noise power by averaging the rest of thesmallest 𝐿 − �̂� eigenvalues as

�̂�2=

1

𝐿 − �̂�

𝐿

∑

𝑖=�̂�+1

𝑙𝑖. (23)

(6) Calculate the noise level as

̂𝛿 = √𝑀�̂�

2. (24)

In the above procedure, the noise level is estimated usingonly one-observation data and thus this denoising methodcan be used to deal with observations at any unknown SNR.We refer to this denoising method as sparse representa-tion denoising with noise level estimation by subsequence(SRDN-NS).

SRDN-NS contains two stages: noise level estimationand the sparse-representation solving. Hence its computa-tional complexity can be divided into parts: (i) noise levelestimation, mainly solving eigenvector decomposition ofsubsequence correlation matrix, requires 𝑂(𝐿3) operations;(ii) the sparse-representation solving, if (13) is solved byOMP, requires 𝑂(KMN) operations [24].

4. Results and Discussion

In this section, both simulated experiments and real data areexplored to verify the denoising performance of SRDN-NS.

4.1. Simulated Experiments Setup. We simulate the noisyradar return of a target with 5 scatterers to verify the effective-ness of SRDN-NS.The scatterer distribution and radar systemparameters are shown in Tables 1 and 2, respectively.

Under the above system parameters, every samplingsequence of a single pulse is a 256-length vector. Assumingthe reference range for dechirping is 12010m, the noiselessHRRP obtained by FFT is illustrated in Figure 1.

The number of scatterers is 5, much less than the 256range cells. Thus the sampling sequence is a sparse vector


Table 1: Scatterer distribution.

Scatterer Range (m) Amplitude of return1 12000 0.82 12002.1 0.53 12005.3 0.44 12006.7 15 12010 0.7

Table 2: Radar system parameters.

Parameter name ValueCarrier frequency 5.52GHzBandwidth 400MHzPulse width 25 usReference pulse width 25.6 usSampling rate for dechirped return 10MHz

in frequency domain. To produce noisy return, we add thecomplex Gaussian white noise (CGWN) into the noise-freereturn with SNR defined as

SNROrig =∑𝐾

𝑘=1𝐴2

𝑘

𝜎2

, (25)

where 𝐴𝑘 is the amplitude of the return from 𝑘th scatterer,𝜎2 is the variance of CGWN, and 𝐾 denotes the number of

scatterers.

4.2. Noise Level Estimation. Noise level estimation is a criticalproblem for SRDN-NS. In the first experiment, we surveyedthe estimation accuracy of 𝛿 as discussed in Section 3.2. Thelength of sliding window is 60. As a comparison, the selectingnoise cells method [12] and wavelet method [13] are also uti-lized to estimate the noise level. In wavelet method, the “db8”wavelet basis is chosen for its experimental superiority overother “db”wavelet bases.Thedechirped return is decomposedinto 8 layers of wavelet. To verify the estimation performanceof these methods, the absolute estimation error |̂𝛿−𝛿|, wherê𝛿 is the estimation value of noise level 𝛿, is investigated bya Monte Carlo experiment. 100 trails are performed at everySNR, and the results with SNR varying from −10 dB to 20 dBare shown in Figure 2. It can be seen that, in general, theproposed sliding subsequence method has better estimationperformance than selecting noise cells and wavelet. Noticethat the estimation error of selecting noise cells method isnot stable reducing with SNR increasing. That is because,with SNR increasing, the sidelobe of HRRP introduces mucherror; besides the threshold which determines noise cells isnot easy to select.

4.3.Denoising Performance. In this subsection,we investigatethe denoising performance of SRDN-NS. The redundantdictionary is established according to (10) with𝑀 = 256 and𝑁 = 1024, 2048, respectively, for different experiments. OMPis employed to solve (13) for recovering the denoised return.For comparing denoising performance, SRDN with noise

0

5

10

15

Estim

atio

n er

ror o

f noi

se le

vel

Selecting noise cellsWaveletSliding subsequence

−10 −5 0 5 10 15 20

Original SNR (dB)

Figure 2: Estimation error of noise level by various methods withSNR varying from −10 dB to 20 dB.

0 2 4 6 8 10 12 14 16 18 205

10

15

20

25

30

Original SNR (dB)

Den

oise

d SN

R (d

B)

DWTDNDWTDN with 𝜎2 knownSRDN-NS: N = 1024

SRDN-NS: N = 2048SRDN-𝛿: N = 1024

SRDN-𝛿: N = 2048

Figure 3: SNR of denoised return by various methods with originalSNR varying from 0 dB to 20 dB.

level known (referring to it as SRDN-𝛿) and the traditionalDWTDNas discussed in [14, 28, 29] are also utilized to reducenoise in the return. InDWTDN, “db8” wavelet basis is chosenand the dechirped return is decomposed into 8 layers ofwavelet, and then the Heursure threshold is used to rejectnoise. For full comparison, DWTDN with noise varianceknown is also presented, and the soft threshold is selected as𝜆 = 𝜎√2 ln𝑀.

To verify the denoising performance, we define SNR ofthe denoised return as

SNR𝑑 = 10 log10‖s‖22

‖ŝ − s‖22, (26)


0

10

20

30

40

50A

mpl

itude

(dB)

(I)

(II)

0 50 100 150 200 2500

10

20

30

40

50

Am

plitu

de (d

B)

Noiseless HRRPNoisy HRRP

(a)

0 50 100 150 200 250

Noiseless HRRPHRRP with DWTDN

(b)

Noiseless HRRPHRRP with SRDN-NS

0 50 100 150 200 250

(c)

Figure 4: Profiling results of simulated return: (a) noiseless HRRP and noisy HRRP at 10 dB (I) and noisy HRRP at 1 dB (II); (b) noiselessHRRP and HRRP with DWTDN; (c) noiseless HRRP and HRRP with SRDN-NS.

where s denotes the original noise-free return and ŝ denotesthe denoised return. ‖ŝ − s‖2

2stands for the noise components

and information loss in the denoised return. 100Monte Carlotrails are performed to compute SNR𝑑 at every SNR. Thedenoising results of these methods with original SNR varyingfrom 0 dB to 20 dB are showed in Figure 3.

It can be seen from Figure 3 that SRDN presents muchbetter denoising performance than DWTDN.This is becausethe dechirped radar return is composed of complex sinu-soidal components. It is more sparsely represented in theFourier redundant dictionary than in wavelet basis; it followsthat the signal energy is more concentrated in the Fourierredundant dictionary, thus leading to a better denoisingperformance. Note that the denoising performance of SRDNimproves with 𝑁 increasing. The reason is that, with 𝑁increasing, A gradually improves frequency resolution, andthe dechirped signal can be represented more exactly withless atoms in A. But 𝑁 cannot increase infinitely because ofthe limitation of computational cost. In addition, comparedwith SRDN-𝛿, SRDN-NS provides a close performance; inother words, the denoising performance of SRDN-NS isapproaching the performance upper bound of SRDN.

4.4. Profiling Results. In this subsection, the profiling resultsby FFT after SRND-NS denoising are illustrated in Figure 4.The experimental parameters are the same as the experimentin Section 4.3. The original noisy HRRPs at different SNRsare shown in Figure 4(a), 10 dB in (I) and 1 dB in (II). Forcomparison, the profiling results withDWTDNare presentedin Figure 4(b), which still contain some noise componentsaround signal part, whereas SRDN-NS provides precise and

clear HRRPs in Figure 4(c). Notice that, in low SNR scenario,the denoised return by SRDN-NS may lose some weakenedsignal components or still contain a little strong noise (asshown in (II) of Figure 4(c)) due to the estimation error ofnoise level.

4.5. Real Data Profiling Results. We apply SRDN-NS todenoise the real return of AN-26 plane under differentnoise levels in this subsection. This dataset was previouslydescribed and utilized in [10]. The radar system parametersare the same as the simulated experimental parameters inTable 2. The profiling results by FFT after denoising byDWTDN and SRDN-NS are shown in Figure 5. The noisyHRRPs with different noise levels are shown in Figure 5(a),high SNR (I) and low SNR (II). Figure 5(b) shows that theHRRPs with DWTDN still contain some noise componentsaround the signal part, whereas, as shown in Figure 5(c),SRDN-NS provides much clearer HRRPs and preserves mostof the information.

5. Conclusions

This work has developed an effective denoising method,SRDN-NS, for high resolution range profiling at low SNR.Thedenoised return is recovered by solving a sparse represen-tation problem. Simulation and real data results demonstratethat SRDN-NS can greatly enhance SNRof radar return, lead-ing to a high-qualityHRRP. In this work, we only consider theinfluence of the zero-mean Gaussian white noise; however,there are many other complicated noise environments, suchas ground clutter and sea clutter, that the radar system may


Nor

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(II)

−60

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−20

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0

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0 50 100 150 200 250

(b)

0 50 100 150 200 250

(c)

Figure 5: Profiling results of AN-26 real return: (a) the noisyHRRPs at high SNR (I) and low SNR (II), respectively; (b)HRRPswithDWTDN;(c) HRRPs with SRDN-NS. All HRRPs are normalized by maximum component absolute mode and on the same scale.

face, so in the future we will continue our study on thesecomplicated environments.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgment

This work was supported by the National Science Foundationof China under Grant no. 61201311.

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Research Article Sparse Representation Denoising for Radar …downloads.hindawi.com/journals/ijap/2014/875895.pdf · 2019. 7. 31. · Research Article Sparse Representation Denoising