low-altitude and slow-speed small target detection based

Research ArticleLow-Altitude and Slow-Speed Small Target Detection Based onSpectrum Zoom Processing

Xuwang Zhang1 Songtao Lu 2 Jinping Sun 1 andWei Shangguan3

1School of Electronic and Information Engineering Beihang University Beijing 100191 China2Department of Electrical and Computer Engineering Iowa State University Ames IA 50011 USA3National Laboratory of Radar Signal Processing Xidian University Xirsquoan 710071 China

Correspondence should be addressed to Jinping Sun sunjinpingbuaaeducn

Received 2 December 2017 Accepted 4 April 2018 Published 10 May 2018

Academic Editor Wanquan Liu

Copyright copy 2018 Xuwang Zhang et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper proposes a spectrum zoom processing based target detection algorithm for detecting the weak echo of low-altitude andslow-speed small (LSS) targets in heavy ground clutter environments which can be used to retrofit the existing radar systemsWith the existing range-Doppler frequency images the proposed method firstly concatenates the data from the same Dopplerfrequency slot of different images and then applies the spectrum zoom processing After performing the clutter suppression thetarget detection can be finally implementedThrough the theoretical analysis and real data verification it is shown that the proposedalgorithm can obtain a preferable spectrum zoom result and improve the signal-to-clutter ratio (SCR)with a very low computationalload

1 Introduction

The existing radar systems are mainly designed to detectthe high-speed military targets such as the fighter planemissile and armed helicopter These targets usually fly ata high altitude with a high speed so that it is very easyto separate the target echo and groundsea clutter in thefrequency domain In this case the target detection pro-cess of radar systems can be simply described as followsfirstly perform the coherent accumulation and then detectthe targets with some traditional constant false alarm rate(CFAR) detection algorithms [1ndash4] In practice the coherentaccumulation pulse number is small such that the detectionresult can be updated quickly with a low computationalcomplexity

In recent years there are various types of aircraft suchas the paraglider light helicopter and rotorcraft emergingquickly in the market These aircraft usually fly under 1000meters with a speed lower than 200 kmh and a radar crosssection (RCS) smaller than 2m2 Therefore these aircraft aremostly called the low-altitude and slow-speed small (LSS)targets and they have the common characteristics simple

manipulation easy accessibility and excellent concealmentNowadays the LSS target detection and tracking has becomean important research direction in the field of radar signalprocessing andhas a great significance in protecting the safetyof major events and maintaining the order of airport flightsFor a LSS target the small RCS signifies that the target echois very weak and the low flying height usually results inthe mixture of the target echo and heavy ground clutter Inthe time domain the ground clutter has a noticeable maskeffect on the target echo Meanwhile the Doppler frequencyof ground clutter is very close to that of target echo whenthe target speed is slow Therefore it is also difficult toseparate the target echo and ground clutter in the frequencydomain All these factors make the LSS target detection verychallenging Unfortunately the existing radar systems cannotbe competent for this problem Consequently it is necessaryto retrofit the existing radar systems such that they can obtainthe LSS target detection ability but also keep the high-speedtarget detection ability

There have been already many studies in the literaturesfocusing on the problem of detecting LSS targets underthe background of sea clutter For example a short-time

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 4146212 10 pageshttpsdoiorg10115520184146212

2 Mathematical Problems in Engineering

fractional Fourier transform based detection algorithm wasproposed in [5] by investigating the micro-Doppler effectof the targets at the sea surface In order to improve thedetection performance an adaptive waveform was designeddynamically in [6] where the expectation-maximizationalgorithm is used to estimate the time-varying parametersof the compound-Gaussian sea clutter For the maneuveringtarget detection two algorithms which apply the adaptiveChirplet decomposition and spectral subtraction respec-tively were proposed in [7] A time-frequency iterationdecomposition method was proposed in [8] by focusing onthe nonstationarity of scattered echo from the slow movingweak target at the sea surface Meanwhile a time-frequencymethod was also applied to detect the small accelerating tar-get in the background of sea clutter [9] In addition Fourier-Bessel transformwas combined with time-frequency analysisto decompose the nonstationary echo of the maneuveringtarget into multiple components [10] Besides the methodof time-frequency analysis applied in maneuvering targetdetection problems the effectiveness of track-before-detect(TBD) method in suppressing the sea clutter was assessedwith the real data [11] Furthermore a polynomial fittingbased signal phase training structure was studied in [12] forthe LSS target detection in the sea clutter

In fact it is also very significant to study the LSS targetdetection problem under the background of the groundclutter especially in the complex urban environment Withthe fast development of the technology and reduction of theprice of aircraft a large number of paragliders and rotorcraftsappear in the peoplersquos daily lifeThese cheap aircraftmainly flyon the ground and the illegal manipulation on these aircraftcan result in serious security issues However there are fewliteratures focusing on the slow moving target detection inthe ground clutter [13ndash17] A space-time adaptive processingmethod was proposed to detect the ground moving targetwith range migration (RM) [14] In order to reduce theamount of echo data and achieve a wide observation swatha parametric sparse representation method was used for themotion parameter estimation of the ground target in [15]In particular some spectrum zoom algorithms provide anew path for the LSS target detection For example thechirp-Z transform (CZT) which can be used to obtain anarbitrary frequency resolution was introduced in [18] Basedon the CZT an interlaced CZT was proposed in [19] and thistransform can produce a spectrum with denser frequencysamples at any required place Furthermore the warped dis-crete Fourier transform which can produce inhomogeneousfrequency samples was studied in [20 21] These algorithmsmay have some value in the LSS target detection butthe computational complexity is very high More seriouslythe above methods do not consider the characteristics ofexisting radar systems and the convenience of retrofittingprocess

A new detection algorithm based on spectrum zoomprocessing is proposed in this paper according to thecharacteristics of LSS targets By retrofitting the existingradar systems with some simple operations the proposedalgorithm can make the radar systems obtain an excellentLSS target detection ability Meanwhile the high-speed target

detection ability is still retained With the available range-Doppler frequency image in the existing radar systems theproposed algorithm firstly concatenates the data from thesame low Doppler frequency slot of different images andthen performs the spectrumzoomprocessing on the obtaineddata Finally the clutter suppression and target detection areperformed on the spectrum zoom result Spectrum zoomprocessing is the key step which can effectively separate thetarget echo and ground clutter in the frequency domain andsignificantly improve the signal-to-clutter ratio (SCR)

This paper is organized as follows In Section 2 thebasic theory of spectrum zoom processing is introduced InSection 3 the spectrum zoom processing based LSS targetdetection algorithm is proposed In Section 4 the perfor-mance of the proposed algorithm is theoretically analyzed interms of the SCR improvement and computational load InSection 5 we verify the theoretical analysis in Section 4 withsome real data Finally conclusions are drawn in Section 6

2 Basic Theory of Spectrum Zoom Processing

21 Signal Model Let 119904(119905) 0 le 119905 lt 119872119873119879119904 be a continuoustime complex signal with a finite length where119872 and119873 arepositive integers and 119879119904 gt 0 Dividing 119904(119905) into 119872 signals1199040(119905) 1199041(119905) 119904119872minus1(119905) with the length of119873119879119904 we can get119904 (119905) = 1199040 (119905) + 1199041 (119905) + sdot sdot sdot + 119904119872minus1 (119905) (1)

where119904119898 (119905) = 119904 (119905) 119898119873119879119904 le 119905 lt (119898 + 1)1198731198791199040 other 119898 = 0 1 119872 minus 1 (2)

Shifting the signal 119904119898(119905) to 0 le 119905 lt 119873119879119904 along the time axiswe have 119909119898 (119905) = 119904119898 (119905 + 119898119873119879119904) (3)

Combining (1) with (3) we can obtain119904 (119905) = 1199090 (119905) + 1199091 (119905 minus 119873119879119904) + sdot sdot sdot+ 119909119872minus1 (119905 minus (119872 minus 1)119873119879119904) (4)

After sampling the continuous time signal 119904(119905) and 119909119898(119905)with a period of 119879119904 respectively the resulting discrete timesequences are119904 [119899] = 119904 (119899119879119904) 119899 = 0 1 119872119873 minus 1119909119898 [119899] = 119909119898 (119899119879119904) 119899 = 0 1 119873 minus 1 (5)

Obviously 119904[119899] is just the arrangement of 1199090[119899] 1199091[119899] 119909119872minus1[119899] with the sequential orderAccording to [22] the Fourier transform (FT) of 119904(119905) can

be written as 119878 (120596) = intinfinminusinfin

119904 (119905) 119890minusj120596119905d119905 (6)

Mathematical Problems in Engineering 3

Meanwhile the discrete time Fourier transform (DTFT) of119904[119899] can be written as119878 (Ω) = 119872119873minus1sum119899=0

119904 [119899] 119890minusjΩ119899 (7)

and the discrete Fourier transform (DFT) of 119904[119899] can bewritten as 119878 [119896] = 119872119873minus1sum

119899=0

119904 [119899]119882119896119899119872119873 (8)

where119882119872119873 = 119890minusj2120587(119872119873) and 119896 = 0 1 119872119873 minus 1Obviously the sequence 119878[119896] is just the sample of 119878(Ω) at

the frequency 119891 = 119896Δ119891 that is119878 [119896] = 119878 (Ω)|119891=119896Δ119891 (9)

where Δ119891 = 119865119904(119872119873) Assume that the whole energy of 119904(119905)is concentrated in119891 isin [0 119865119904]Therefore the sampling processof 119904(119905) obeys Nyquistrsquos law and 119878(Ω) is the extension of 119878(120596)with the period of 119891 = 119865119904 In this case 119878[119896] is also the sampleof 119878(120596) at 119891 = 119896Δ119891 that is119878 [119896] = 119878 (Ω)|119891=119896Δ119891 = 119878 (120596)|119891=119896Δ119891 (10)

Similar to (6)ndash(10) the FT of 119909119898(119905) can be written as119883119898 (120596) = intinfinminusinfin

119909119898 (119905) 119890minusj120596119905d119905 (11)

Meanwhile the DTFT and DFT of 119909119898[119899] can respectively bewritten as 119883119898 (Ω) = 119873minus1sum

119899=0

119909119898 [119899] 119890minusjΩ119899 (12)

119883119898 [119896] = 119873minus1sum119899=0

119909119898 [119899]119882119896119899119873 (13)

where 119898 = 0 1 119872 minus 1 119896 = 0 1 119873 minus 1 and 119882119873 =119890minusj2120587119873 We also have119883119898 [119896] = 119883119898 (Ω)1003816100381610038161003816119891=119896Δ119891 = 119883119898 (120596)1003816100381610038161003816119891=119896Δ119891 (14)

where Δ119891 = 119865119904119873In addition taking the FT on both sides of (4) we arrive

at 119878 (120596) = 1198830 (120596) + 1198831 (120596) 119890minusj120596119873119879119904 + sdot sdot sdot+ 119883119872minus1 (120596) 119890minusj120596(119872minus1)119873119879119904= 119872minus1sum119898=0

119883119898 (120596) 119890minusj120596119898119873119879119904 (15)

which shows the relationship between 119904(119905) and 1199090(119905) 1199091(119905) 119909119872minus1(119905) in the frequency domain

22 Spectrum Zoom Processing The sequence 119878[119896] 119896 = 0 1 119872119873minus1 shows the whole spectrum information of 119904(119905) in[0 119865119904] However only partial spectrum information of 119904(119905) isnecessary in some applications Consider a specific problemAssume that the sequence 119883119898[119896] 119896 = 0 1 119873 minus 1 isknownThen divide the frequency range [0 119865119904] into119873 bandswith equal lengths In this case how to obtain the spectruminformation of 119904(119905) at the 119901th band that is 119878[119901119872] 119878[119901119872 +1] 119878[119901119872 +119872 minus 1] 119901 isin 0 1 119873 minus 1

An obvious method is as follows firstly calculate thesequence 119878[119896] 119896 = 0 1 119872119873 minus 1 according to (8) andthen select out 119878[119901119872] 119878[119901119872 + 1] 119878[119901119872 +119872minus 1] Herewe introduce a fast method for obtaining 119878[119901119872] 119878[119901119872 +1] 119878[119901119872 + 119872 minus 1] with 119883119898[119896] 119896 = 0 1 119873 minus 1 asfollows This method will be applied to retrofit the existingradar systems in Section 3

For simplicity let1198910 ≜ (119901119872 + 119902) 119865119904119872119873 1205960 ≜ 21205871198910 = 2120587 (119901119872 + 119902) 119865119904119872119873 (16)

where 119901 = 0 1 119873 minus 1 and 119902 = 0 1 119872 minus 1 Accordingto (10) we have119878 [119901119872 + 119902] = 119878 (120596)|119891=1198910 = 119878 (120596)|120596=1205960 (17)

Substituting (15) into (17) we can get119878 [119901119872 + 119902] = 119872minus1sum119898=0

119883119898 (1205960) 119890minusj1205960119898119873119879119904= 119872minus1sum119898=0

119883119898 (1205960) 119890minusj(2120587(119901119872+119902)119865119904119872119873)119898119873119879119904= 119872minus1sum119898=0

119883119898 (1205960) 119890minusj2120587119901119898119890minusj2120587119902119898119872= 119872minus1sum119898=0

119883119898 (1205960)119882119902119898119872 (18)

where 119882119872 = 119890minusj2120587119872 It can be seen that 119878[119901119872 + 119902] isjust the DFT of 1198830(1205960) 1198831(1205960) 119883119872minus1(1205960) Therefore itis very easy to obtain 119878[119901119872 + 119902] by applying the sequence1198830(1205960) 1198831(1205960) 119883119872minus1(1205960)

Since 1205960 includes the variable 119902 we need to calculate anew sequence 1198830(1205960) 1198831(1205960) 119883119872minus1(1205960) for a given 119902which can result in a high computational load Instead wecan approximate 1205960 as follows1205960 asymp 2120587119901119872119865119904119872119873 = 2120587119901119865119904119873 ≜ 0 (19)

Equation (18) can be approximated as119878 [119901119872 + 119902] asymp 119872minus1sum119898=0

119883119898 (0)119882119902119898119872 (20)


According to (14) we can get119883119898 [119901] = 119883119898 (120596)1003816100381610038161003816120596=2120587119901119865119904119873 = 119883119898 (0) (21)

Substituting (21) into (20) we can get119878 [119901119872 + 119902] asymp 119872minus1sum119898=0

119883119898 [119901]119882119902119898119872 (22)

Hence 119878[119901119872 + 119902] can be approximated as the DFT of1198830[119901] 1198831[119901] 119883119872minus1[119901] Since the variable 119902 is not con-tained in the sequence1198830[119901] 1198831[119901] 119883119872minus1[119901] for a given119901 only the same sequence 1198830[119901] 1198831[119901] 119883119872minus1[119901] needsto be calculated when 119902 takes all the 0 1 119872 minus 1 Thisway can significantly reduce the computational load In thecase when 119883119898[119896] 119896 = 0 1 119873 minus 1 is known a fast wayfor obtaining the spectrum information of 119904(119905) at the 119901thband (ie 119878[119901119872] 119878[119901119872+ 1] 119878[119901119872+119872minus1]) is to selectout the sequence1198830[119901] 1198831[119901] 119883119872minus1[119901] from the knowninformation and then perform the DFT

According to (10) and (14) 119883119898[119896] 119896 = 0 1 119873 minus 1is the sample of 119883119898(Ω) with the period of 119865119904119873 whereas119878[119896] 119896 = 0 1 119872119873 minus 1 is the sample of 119878(Ω) with theperiod of 119865119904(119872119873) From (22) we know that the frequencyspectrum sample 119878[119896] with a short sampling period can beobtained from the frequency spectrum sample 119883119898[119896] witha relative longer sampling period by DFT which is the so-called spectrum zoom processing According to the aboveanalysis we can summarize the spectrum zoom processingas follows (i) sample the continuous time complex signal 119904(119905)whose frequency spectrum locates in [0 119865119904] with a periodof 119879119904 and obtain 119872 sequences 119909119898[119899] 119899 = 0 1 119873 minus1 with the length 119873 (ii) write the DFT of 119909119898[119899] 119899 =0 1 119873 minus 1 as 119883119898[119896] 119896 = 0 1 119873 minus 1 (iii) foran arbitrary 119901 isin 0 1 119873 minus 1 the DFT of sequence1198830[119901] 1198831[119901] 119883119872minus1[119901] can be approximately regarded asthe spectrum sampling result of 119904(119905) in [119901119865119904119873 (119901 + 1)119865119904119873)with a period of 119865119904(119872119873) or can be equivalently regardedas the DFT of the sequence 119904[119899] 119899 = 0 1 119872119873 minus 1where 119904[119899] is the sequential arrangement of the119872 sequences1199090[119899] 1199091[119899] 119909119872minus1[119899] 119899 = 0 1 119873 minus 1 in the fre-quency range [119901119865119904119873 (119901 + 1)119865119904119873) (ie 119878[119901119872] 119878[119901119872 +1] 119878[119901119872 + 119872 minus 1]) It can be seen that the spectrumzoom processing provides a simple and fast approach forobtaining the refined spectrum from the coarse spectrumwhich is very significant in the applications where only therefined spectrum on a partial band is necessary

3 Spectrum Zoom Processing Used forthe LSS Target Detection

The existing radar systems are mainly designed for high-speed targets Considering the facts that the instantaneousposition of the high-speed target varies quickly and theDoppler frequency of target echo varies in a board range thedetector usually takes a small coherent accumulation pulsenumber Such an operation can obtain a high frame rate toupdate the target state quickly with a low computational com-plexity Meanwhile the resulting range-Doppler frequency

image has a large spectrum interval However as a largenumber of LSS targets appear in recent years the existingradar systems cannot perform an effective detection on thesetargets The main reasons are as follows (1) the echo of LSStarget is very weak and more coherent pulses are necessaryto accumulate the target energy (2) the Doppler frequencyof LSS target echo is very close to that of the ground clutterand the spectrum should be refined enough to separate themTherefore the existing radar systems should be retrofittedproperly

Based on the spectrum zoom processing introducedin Section 2 a simple and feasible scheme is designed forretrofitting the existing radar systems in this section Thisscheme applies the available range-Doppler frequency imagein the existing radar systems to obtain the refined spectruminformation of the observation data in the low Dopplerfrequency band which can effectively separate the target echoand ground clutter in the frequency domain and improve theaccumulation effect of the target energy Next this retrofittingscheme will be introduced in detail

31 Whole Retrofitting Scheme A typical model of the pulseDoppler radar observation data is shown in Figure 1 Thehorizontal axis represents the fast time dimension containing119871 range gates in total The vertical axis represents theslow time dimension containing 119872119873 pulses in total Eachsequential119873 pulses are regarded as a frame of the observationdata and it contains 119872 frames of the observation data inFigure 1 Write the 119899th pulse as z119899 = (1199111198991 1199111198992 119911119899119871)where 119911119899119897 represents the sample of the 119897th range gate in the119899th pulse 119899 = 1 2 119872119873 119897 = 1 2 119871 Define 119873 times 119871matrix

Z119898 =(z(119898minus1)119873+1z(119898minus1)119873+2

z119898119873

) (23)

which represents the119898th frame of the observation data119898 =1 2 119872Figure 2 shows thewhole scheme for retrofitting the exist-

ing radar systems by the spectrum zoomprocessing Only theoperations after the pulse compression are considered Thetop dashed box contains the original detection process of theexisting radar systems which can effectively detect the high-speed target The lower dashed box contains the concreteretrofitting operations on the existing radar systems It can beseen that all the original processing operations are retainedand only some simple operations including data selectionDFT and detection are added in this retrofitting scheme

32 Original Detection Process The detection process inthe existing radar systems are mainly used to detect thehigh-speed target Firstly perform the coherent accumula-tion Specifically perform the DFT on each frame of theobservation data along the slow time axis In fact this isusually realized by the fast Fourier transform (FFT) After thecoherent accumulation we can obtain 119872 frames of 119873 times 119871


Fast Time

Slow

Tim

e

N

N

N

M F

ram

es

L Range Gates

z1z2z3

zN

zN+1zN+2

zN+3

z2N

z(Mminus1)N+1z(Mminus1)N+2z(Mminus1)N+3

zMN

Figure 1 Typical model of the pulse Doppler radar observation data

Track

Frame 1

Frame M

Frame 2 DFT

DFT

DFT

Detect

Detect

Detect

DFT

DFT Detect

Range-timeimage

Range-Dopplerfrequency image

High-speed target High-speed targetdetection result track

Orig

inal

sign

al p

roce

ssing

Retro

fittin

g sch

eme

Stick

Single Dopplerfrequency slot data



Slow-speed targetdetection result

DFT

+QH

Dop

pler

freq

uenc

y slo

t

0D

oppl

er fr

eque

ncy

slot

minusQH

Dop

pler

freq

uenc

y slo

t

bullbull

bull

bullbullbull

bullbullbull

Figure 2 Whole scheme for retrofitting the existing radar systems by the spectrum zoom processing

range-Doppler frequency images The 119898th frame of range-Doppler frequency image can be written as

X119898 = (x1198981 x1198982 x119898119871) (24)

where

x119898119897 = (119909(119898minus1)119873+1119897 119909(119898minus1)119873+2119897 119909119898119873119897)T (25)

Then perform the target detection on each frame ofrange-Doppler frequency image The Doppler frequency ofthe high-speed target echo is much higher than that of theground clutter Hence it is very easy to separate them in thefrequency domain In general a simple CFAR algorithm canfind out the potential high-speed target Finally when thedetection results from the multiple frames of range-Doppler


frequency images are obtained we can jointly analyze theseresults to improve the detection performance and estimatethe target track

33 Concrete Retrofitting Operations To obtain the LSStarget detection ability for the radar systems some properretrofitting operations are necessary Here the spectrumzoom processing is mainly used to realize the two points (1)separate the LSS target echo and the ground clutter in thefrequency domain (2) improve the SCR

Before introducing the concrete retrofitting operationssome necessary analysis will be made as follows The velocityof most LSS targets is no more than Vmax = 200 kmhThis prior information can be applied to reduce the targetdetection range According to [23] the Doppler frequency oftarget echo can be expressed as119891119889 = 2V120582 (26)

where V denotes the radial velocity and 120582 denotes thewavelength of radar transmission signal Write the pulserepetition frequency of the observation data in Figure 1 as 119865119904Then the interval of two adjacent Doppler frequency slots inthe range-Doppler frequency image of existing radar systemsis Δ119891 = 119865119904119873 According to (26) the difference of radialvelocities corresponding to two adjacent Doppler frequencyslots is ΔV = 120582Δ1198912 and the radial velocity correspondingto each Doppler frequency slot can respectively be writtenas minus(1198732)ΔV minus(1198732 minus 1)ΔV (1198732 minus 1)ΔV Hence theDoppler frequency slot index corresponding to the maximalvelocity Vmax of LSS targets is119876119867 = round(VmaxΔV ) (27)

where the function round(119909) represents the rounding of 119909In the existing radar systems since ΔV is often large weknow that 119876119867 is small Generally 119876119867 le 5 The Dopplerfrequencies and radial velocities in the 119896 = minus119876119867 minus119876119867 +1 119876119867 Doppler frequency slots are very low so theseDoppler frequency slots are called the lowDoppler frequency(LDF) area Meanwhile the rest of Doppler frequency slotsare called the high Doppler frequency (HDF) area

To obtain the enough elevating force most LSS targetshave a minimal velocity denoted as Vmin In general Vminis comparable to ΔV Therefore we can believe that thedistribution range of LSS target echo is the whole LDFarea The ground clutter consists of the echoes of groundbuildings and trees Generally the ground clutter mainlylocates near the 119896 = 0Doppler frequency slot As a result theecho of LSS target is very close to the ground clutter in theLDF area In some cases both of themmay locate in the sameDoppler frequency slot This makes it very difficult to detectthe LSS target and separating the echo of LSS target and theground clutter is the key of this procedure Next the concreteretrofitting operations will be introduced

Firstly take out the data in the LDF area Specifically takeout the data in the 119896 = 119876119867 Doppler frequency slot from the

available119872 frames of range-Doppler frequency images of theexisting radar system and concatenate the data together toform a119872times119871matrixMeanwhile perform the same operationon the data in the 119896 = minus119876119867 minus119876119867 + 1 119876119867 minus 1 Dopplerfrequency slots respectively In this way we can obtain 2119876119867+1matrices with a dimension of119872times 119871

Then respectively perform the DFT on the 2119876119867 + 1matrices along the vertical axis such that 2119876119867 + 1 new119872times119871matrices are obtained These new matrices are still range-Doppler frequency images where the horizontal axis repre-sents the range gate and the vertical axis represents the refinedDoppler frequency slot The range of Doppler frequency ofthe 119896th image is [(minus12+119896)Δ119891 (12+119896)Δ119891] 119896 = minus119876119867 minus119876119867+1 119876119867 Concatenate the 2119876119867+1 frames of range-Dopplerfrequency images to form a frame of (2119876119867 + 1)119872 times 119871 range-Doppler frequency image The range of Doppler frequency is[minus(12 + 119876119867)Δ119891 (12 + 119876119867)Δ119891] the interval of two adjacentDoppler frequency slots is Δ119891119872 and the correspondingvelocity interval is ΔV119872 in the (2119876119867 + 1)119872 times 119871 range-Doppler frequency image Comparedwith the range-Dopplerfrequency image in the existing radar system the spectrumzoom degree improves 119872 times in the new range-Dopplerfrequency image obtained by the second DFT In fact thismeans that the separation degree of the ground clutter andthe echo of LSS target improve119872 times In the new (2119876119867 +1)119872 times 119871 range-Doppler frequency image the ground clutteris still near the zero Doppler frequency whereas the echoof LSS target is away from the zero Doppler frequencyHence the echo of LSS target and the ground clutter areobviously separated in the frequency domain In additionthe DFT is essentially a coherent accumulation process Anadditional benefit of the second DFT is improving the SCRabout 119872 times enhancing the ability of detecting the LSStarget

In the (2119876119867+1)119872times119871 range-Doppler frequency image theDoppler frequency slot index corresponding to the minimalvelocity Vmin of LSS targets is119876119871 = round( VminΔV119872) = round(119872VminΔV ) (28)

Hence there is no echo of the LSS target in the Doppler fre-quency slotswhose indexes areminus(119876119871minus1) minus(119876119871minus2) 119876119871minus1whereas the ground clutter mainly locates in these Dopplerfrequency slots To suppress the ground clutter it is straight-forward to eliminate these Doppler frequency slots from the(2119876119867 + 1)119872 times 119871 range-Doppler frequency image Finallyperform the target detection on the rest of Doppler frequencyslots with the common detection algorithms and a LSS targetdetection result is obtained In the spectrum zoom basedretrofitting scheme we can obtain a LSS target detectionresult whenever 119872 frames of observation data are receivedSuch a detection output rate can meet the requirement ofthe LSS target detection because the position of LSS targetvaries very slowly Whenmultiple sets of detection results areobtained we can jointly analyze these results to estimate thetrack of LSS target

The spectrum zoom based retrofitting scheme retainsall the original processing operations of the existing radar


systems and makes full use of the available range-Dopplerfrequency images The concrete retrofitting operations onlyincludes data selection DFT and threshold detection whichare very easy to implement with hardware Overall thespectrum zoom based target detection algorithm can makethe radar system obtain a good ability of LSS target detectionwhile the original ability of high-speed target detection canstill be retained

4 Performance Analysis

This section analyzes the performance of the spectrum zoombased LSS target detection algorithm from the SCR improve-ment and computational load Two traditional detectionalgorithms are selected as the reference algorithms Thefirst reference algorithm is implemented as follows take 119872successive frames of the observation data (ie119872119873 pulses) asa whole and then perform theDFT along the time dimensionto obtain a119872119873 times 119871 range-Doppler frequency image finallyperform the target detection This algorithm only performsthe DFT once Hence it can be called the one DFT basedalgorithm The other reference algorithm is realized as fol-lows take 119872 successive frames of the observation data as awhole and then perform the CZT along the time dimensionto obtain a refined range-Doppler frequency image where thecorresponding velocity range is [minusVmax Vmax] finally performthe target detection This algorithm is called the CZT basedalgorithm

The original intention of this paper is to retrofit the exist-ing radar system to obtain the ability of LSS target detectionHence the performance analysis is performed in this specificapplication scene Firstly consider the performance of theone DFT based algorithm the CZT based algorithm and thespectrum zoom based algorithm in the SCR improvementThe one DFT based algorithm considers the119872 frames of theobservation data as a whole and performs the coherent accu-mulation In theory this algorithm can improve the SCRwith119872119873 times The range-Doppler frequency image obtained bythe CZT based algorithm is just the area corresponding to thevelocity range [minusVmax Vmax] of the range-Doppler frequencyimage obtained by the one DFT based algorithm Hencethe CZT based algorithm can also improve the SCR with119872119873 times The spectrum zoom based algorithm containsa two-stage coherent accumulation process The first stagewhich is an original operation in the existing radar systemcan improve the SCR with 119873 times The second stage isintroduced in retrofitting the radar systemwith the spectrumzoom processing The SCR is improved less than 119872 timesslightly in the second stage because of the approximationin (22) Therefore the SCR improvement capacity of thespectrum zoom based algorithm is less than 119872119873 timesslightly

Next consider the performance of the three algorithms incomputational load In order to simplify the analysis assumethat 119872 is 2 to the 1205811th power and 119873 is 2 to the 1205812th powerwhere 1205811 and 1205812 are integers In addition assume all theDFT is implemented by FFT Take the 119872 frames of theobservation data as an example for performing the targetdetection

In the coherent accumulation process according to [22]the number of complex additions and the number of complexmultiplications required in the one DFT based algorithm are1198871 = 119871 times (119872119873) log2 (119872119873) 1198872 = 119871 times (1198721198732 ) log2 (119872119873) (29)

In the CZT based algorithm the sampling number ofthe time-domain signals is 119872119873 in each range gate and thesampling number of the Doppler frequencies is119867 = round( 2VmaxΔV119872) (30)

The length of DFT is Γ = 2119872119873 in the CZT process because119872 and 119873 are 2 to the integer powers According to [18 19]the number of complex additions and the number of complexmultiplications required in the CZT based algorithm are1198881 = 2119871Γ log2Γ1198882 = 119871 (Γ log2Γ + 119867 +119872119873 + Γ) (31)

The spectrum zoom based algorithm is performed on thebasis of the existing119873 times 119871 range-Doppler frequency imagesAccording to (27) the spectrum zoom processing only needsto be implemented on the 119896 = minus119876119867 minus119876119867+1 119876119867Dopplerfrequency slots Therefore the number of complex additionsand the number of complex multiplications required in thespectrum zoom based algorithm are1198891 = 119871 (2119876119867 + 1) times119872 log21198721198892 = 119871 (2119876119867 + 1) times (1198722 ) log2119872 (32)

Assume that the computational load required by a com-plex multiplication and a complex addition is equal Also weassume that119872 and119873 are much larger than 1 For simplicitytake119872 = 119873 In this case the computational load of the oneDFT based algorithm in the coherent accumulation processis 119887 = 1198871 + 1198872 = 31198711198722log2119872 (33)

The computational load of the CZT based algorithm is119888 = 1198881 + 1198882 = 119871 (3Γ log2Γ + 119867 +119872119873 + Γ)= 91198711198722 + 121198711198722log2119872 asymp 121198711198722log2119872 (34)

The computational load of the spectrum zoom based algo-rithm is 119889 = 1198891 + 1198892 = 15 (2119876119867 + 1) 119871119872 log2119872asymp 3119876119867119871119872 log2119872 (35)

It can be seen from comparing (33) with (35) that thecomputational load of the spectrum zoom based algorithmis only 119876119867119872 of that of the one DFT based algorithm


According to the analysis in Section 3 119876119867 is usually verysmall Hence the computational load of the spectrum zoombased algorithm is much lower than that of the one DFTbased algorithm By comparing (34) with (35) we can findthat the computational load of the spectrum zoom basedalgorithm is also much lower than that of the CZT basedalgorithm The main reason is because the spectrum zoombased algorithm can make full use of the available range-Doppler frequency images in the existing radar systemTherefore the proposed spectrum zoom based algorithm ismore suitable for retrofitting the existing radar system

The range-Doppler frequency images obtained by theabove three algorithms have the same spectrum zoomdegreeand every algorithm only needs to perform the targetdetection on the area corresponding to the velocity range[minusVmax Vmax] Therefore the total numbers of the detectedcells for the three algorithms are equal This means that thethree algorithms require the same computational load in thetarget detection process

In summary the SCR improvement capacity of the spec-trum zoom based algorithm is similar to that of the one DFTbased algorithm and the CZT based algorithm Howeverthe spectrum zoom based algorithm requires a much lowercomputational load because this algorithm canmake full useof the available coarse range-Doppler frequency images

5 Simulation Results with Real Data

To test and verify the effectiveness of the spectrum zoombased detection algorithm we choose a real data from theground-based radarThe real data contains 3200 frames of theobservation data in total and each frame of the observationdata contains 119873 = 64 pulses The pulse repetition period is119879119901 = 18 120583s and the sampling rate is 119891119904 = 160MHz There are119873119903 = 300 range gates in total and the length of a range gateis Δ = 09375m There are two small targets flying at a lowaltitude in the observation area One is a slow target markedas Target 1 and the other is a fast target marked as Target 2Target 1 moves away from the radar with a constant velocityV1 = 435ms It started in the 173rd range gate at the initialtime and stopped in the 190th range gate at the terminaltime Meanwhile Target 2 moves away from the radar with aconstant velocity V2 = 5013ms It started in the 47th rangegate at the initial time and stopped in the 244th range gate atthe terminal time

In order to detect the high-speed target the existing radarsystems separately process each frame of the observationdata Figure 3 shows the coherent accumulation result ofthe first frame of the observation data in the existing radarsystem It can be seen that both the echo of Target 1 and theground clutter locate near the 119896 = 0 Doppler frequency slotWe cannot distinguish the echo of Target 1 and the groundclutter according to their spectrumcharacteristics in Figure 3The echo of Target 2 locates in the 119896 = minus5Doppler frequencyslot Meanwhile the HDF area is free of the ground clutterwhich is consistent with the analysis in Section 3

Figure 4 shows the refined range-Doppler frequencyimages obtained from the first 119872 = 32 frames of theobservation data for the one DFT based algorithm the

Table 1 Running time for different algorithms

Running time (s)One DFT based algorithm 3623CZT based algorithm 8067Spectrum zoom based algorithm 612

30

25

20

0

5

10

15

0

100

200

300

Am

plitu

de

Doppler Frequency Slot

Rang

e Gat

e

Ground Clutter

Target 1Target 2

3020100minus10minus20minus30

Figure 3 Range-Doppler frequency image obtained by the existingradar system

CZT based algorithm and the proposed spectrum zoombased algorithm Figure 4(a) is the result of the one DFTbased algorithm Figure 4(b) is the result of the CZT basedalgorithm and Figure 4(c) is the result of the spectrum zoombased algorithm In order to show the echo of Target 1 andthe ground clutter clearly only the 64 Doppler frequencyslots near the zero Doppler frequency are plotted in eachimage It can be observed that the range-Doppler frequencyimages obtained by the three algorithms are very similarThe spectrum zoom degree is very high and the echo ofTarget 1 and the ground clutter are well separated in thethree range-Doppler frequency images The heavy groundclutter mainly locates near the 119896 = 0 Doppler frequencyslot whereas the echo of Target 1 locates in the 119896 = minus13Doppler frequency slot and the 173rd range gate ComparingFigures 4(a) and 4(b) with Figure 4(c) we can find that thespectrum zoom based algorithm has similar performancewith the classic algorithms in separating the echo of LSStarget and the ground clutterThe amplitudes of Target 11015840 echoare respectively 209 209 and 203 in Figures 4(a) 4(b) and4(c)This implies that the performance of the spectrum zoombased algorithm is very close to the classic algorithms in theenergy accumulation

The running time of processing the first 119872 = 32 framesof the observation data by the one DFT based algorithmthe CZT based algorithm and the spectrum zoom basedalgorithm is shown in Table 1 The simulation is performedon a personal computer and programmed with MATLABIt can be seen that the running time of the spectrum zoombased algorithm is only about 16 of that of the oneDFT based


200

150

100

50

0

20

0

minus200

100

200

300

Am

plitu

de

Doppler Frequency SlotRange Gate

Ground Clutter

Target 1

(a)

200

150

100

50

0

20

0

minus200

100

200

300

Am

plitu

de


Ground ClutterTarget 1

(b)

200

150

100

50

0

20

0

minus200

100200

300

Am

plitu

de



(c)

Figure 4 Refined range-Doppler frequency images for different algorithms (a) one DFT based algorithm (b) CZT based algorithm (c)spectrum zoom based algorithm

algorithm and 113 of that of the CZT based algorithm Thisillustrates that the spectrum zoom based algorithm requiresa much lower computational load than the one DFT basedalgorithm and the CZT based algorithm

According to the results shown in Figures 3 and 4 andTable 1 we can conclude that the proposed spectrum zoombased detection algorithm has similar performance with theone DFT based algorithm and the CZT based algorithm inrefining spectrum improving SCR and detecting targets butthe computational load is much lower than the existing twomethods

6 Conclusions

The existing radar systems which are mainly designed forthe high-speed targets cannot perform an effective detectionon the LSS targets whose number is growing fast in recentyears For this problem a spectrum zoom processing basedtarget detection algorithm is proposed in this paper to retrofitthe existing radar systems This algorithm can make full useof the available range-Doppler frequency images obtainedin the existing radar systems and the implementation of

this algorithm is easy to follow Firstly concatenate the datafrom the same Doppler frequency slot of different imagesand then perform the spectrum zoom processing Finallythe clutter suppression and target detection are performedThe proposed algorithm can obtain the same spectrumzoom degree with the one DFT based algorithm and theCZT based algorithm but it only requires a much lowercomputational load More importantly only a few retrofittingoperations are needed to retrofit the existing radar systemswith the proposed algorithm Therefore the spectrum zoomprocessing based target detection algorithm is more suitablefor retrofitting the existing radar systems

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant no 61471019)


References

[1] M Skolnik G Linde and K Meads ldquoSenrad An advancedwideband air-surveillance radarrdquo IEEE Transactions on Aero-space and Electronic Systems vol 37 no 4 pp 1163ndash1175 2001

[2] J M Loomis ldquoArmy Radar Requirements for the 21st Centuryrdquoin Proceedings of the 2007 IEEE Radar Conference pp 1ndash6Waltham MA USA April 2007

[3] X Zhang J Sun Y Zhang S Lu and C Liu ldquoH-PMHT track-before-detect processing with DP-based track initiation andterminationrdquo IET Signal Processing vol 10 no 9 pp 1118ndash11252016

[4] X Zhang J Sun J Fu and S Lu ldquoFast implementationmethod of permutation test with valid strategyrdquo Journal ofSignal Processing vol 31 no 10 pp 1233ndash1239 2015 (Chinese)

[5] X Chen J Guan Z Bao and Y He ldquoDetection and extractionof target with micromotion in spiky sea clutter via short-timefractional fourier transformrdquo IEEE Transactions on Geoscienceand Remote Sensing vol 52 no 2 pp 1002ndash1018 2014

[6] S P Sira D Cochran A Papandreou-Suppappola et alldquoAdaptive waveform design for improved detection of low-RCStargets in heavy sea clutterrdquo IEEE Journal of Selected Topics inSignal Processing vol 1 no 1 pp 56ndash66 2007

[7] Y Zhang S Qian and TThayaparan ldquoDetection of a manoeu-vring air target in strong sea clutter via joint time-frequencyrepresentationrdquo IET Signal Processing vol 2 no 3 pp 216ndash2222008

[8] L Zuo M Li X Zhang Y Wang and Y Wu ldquoAn efficientmethod for detecting slow-moving weak targets in sea clutterbased on time-frequency iteration decompositionrdquo IEEE Trans-actions on Geoscience and Remote Sensing vol 51 no 6 pp3659ndash3672 2013

[9] A Yasotharan and TThayaparan ldquoTime-frequencymethod fordetecting an accelerating target in sea clutterrdquo IEEETransactionsonAerospace andElectronic Systems vol 42 no 4 pp 1289ndash13102006

[10] P Suresh T Thayaparan and K Venkataramaniah ldquoFourier-Bessel transform and time-frequency-based approach fordetecting manoeuvring air target in sea-clutterrdquo IET RadarSonar amp Navigation vol 9 no 5 pp 481ndash491 2015

[11] A Aprile E Grossi M Lops and L Venturino ldquoTrack-before-detect for sea clutter rejection Tests with real datardquo IEEETransactions on Aerospace and Electronic Systems vol 52 no3 pp 1035ndash1045 2016

[12] L Hao and Z Lefeng ldquoLow velocity small radar target detectionin maritime environmentrdquo in Proceedings of the 2010 2ndInternational Conference on Signal Processing Systems (ICSPS)pp V3-385ndashV3-388 Dalian China July 2010

[13] J D Park and J F Doherty ldquoTrackDetection of LowObservableTargets Using a Motion Modelrdquo IEEE Access vol 3 pp 1408ndash1415 2015

[14] Z Zhou S Zhigang and W Renbiao ldquoMethod for detectingground moving target with range migrationrdquo in Proceedings ofthe IET International Radar Conference 2009 pp 141-141 GuilinChina

[15] F-F Gu Q Zhang Y-C Chen W-J Huo and J-C Ni ldquoPara-metric Sparse Representation Method for Motion ParameterEstimation of Ground Moving Targetrdquo IEEE Sensors Journalvol 16 no 21 pp 7646ndash7652 2016

[16] J Sun Q Li X Zhang and W Sun ldquoAn Efficient Implemen-tation of Track-Oriented Multiple Hypothesis Tracker Using

Graphical Model ApproachesrdquoMathematical Problems in Engi-neering vol 2017 Article ID 8061561 2017

[17] X Zhang J Sun S Lu and GWang ldquoNon-parametric detectorin nonhomogeneous clutter environments with knowledge-aided permutation testrdquo IET Radar Sonar amp Navigation vol 10no 7 pp 1310ndash1318 2016

[18] L R Rabiner R W Schafer and C M Rader ldquoThe chirp z-transform algorithm and its applicationrdquo Bell System TechnicalJournal vol 48 no 5 pp 1249ndash1292 1969

[19] I Sarkar and A T Fam ldquoThe interlaced chirp Z transformrdquoSignal Processing vol 86 no 9 pp 2221ndash2232 2006

[20] A Makur and S K Mitra ldquoWarped discrete-Fourier transformTheory and applicationsrdquo IEEE Transactions on Circuits andSystems I Fundamental Theory and Applications vol 48 no 9pp 1086ndash1093 2001

[21] S Franz S K Mitra J C Schmidt and G Doblinger ldquoWarpeddiscrete Fourier transform A new concept in digital signalprocessingrdquo in EEE International Conference on AcousticsSpeech and Signal Processing (ICASSP) vol 2 pp 1205ndash1208Orlando USA May 2002

[22] A V Oppenheim R W Schafer and J R Buck Discrete-TimeSignal Processing Prentice-Hall USA 1999

[23] M A Richards Fundamentals of Radar Signal ProcessingMcGraw-Hill USA 2005

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fractional Fourier transform based detection algorithm wasproposed in [5] by investigating the micro-Doppler effectof the targets at the sea surface In order to improve thedetection performance an adaptive waveform was designeddynamically in [6] where the expectation-maximizationalgorithm is used to estimate the time-varying parametersof the compound-Gaussian sea clutter For the maneuveringtarget detection two algorithms which apply the adaptiveChirplet decomposition and spectral subtraction respec-tively were proposed in [7] A time-frequency iterationdecomposition method was proposed in [8] by focusing onthe nonstationarity of scattered echo from the slow movingweak target at the sea surface Meanwhile a time-frequencymethod was also applied to detect the small accelerating tar-get in the background of sea clutter [9] In addition Fourier-Bessel transformwas combined with time-frequency analysisto decompose the nonstationary echo of the maneuveringtarget into multiple components [10] Besides the methodof time-frequency analysis applied in maneuvering targetdetection problems the effectiveness of track-before-detect(TBD) method in suppressing the sea clutter was assessedwith the real data [11] Furthermore a polynomial fittingbased signal phase training structure was studied in [12] forthe LSS target detection in the sea clutter

In fact it is also very significant to study the LSS targetdetection problem under the background of the groundclutter especially in the complex urban environment Withthe fast development of the technology and reduction of theprice of aircraft a large number of paragliders and rotorcraftsappear in the peoplersquos daily lifeThese cheap aircraftmainly flyon the ground and the illegal manipulation on these aircraftcan result in serious security issues However there are fewliteratures focusing on the slow moving target detection inthe ground clutter [13ndash17] A space-time adaptive processingmethod was proposed to detect the ground moving targetwith range migration (RM) [14] In order to reduce theamount of echo data and achieve a wide observation swatha parametric sparse representation method was used for themotion parameter estimation of the ground target in [15]In particular some spectrum zoom algorithms provide anew path for the LSS target detection For example thechirp-Z transform (CZT) which can be used to obtain anarbitrary frequency resolution was introduced in [18] Basedon the CZT an interlaced CZT was proposed in [19] and thistransform can produce a spectrum with denser frequencysamples at any required place Furthermore the warped dis-crete Fourier transform which can produce inhomogeneousfrequency samples was studied in [20 21] These algorithmsmay have some value in the LSS target detection butthe computational complexity is very high More seriouslythe above methods do not consider the characteristics ofexisting radar systems and the convenience of retrofittingprocess

A new detection algorithm based on spectrum zoomprocessing is proposed in this paper according to thecharacteristics of LSS targets By retrofitting the existingradar systems with some simple operations the proposedalgorithm can make the radar systems obtain an excellentLSS target detection ability Meanwhile the high-speed target

detection ability is still retained With the available range-Doppler frequency image in the existing radar systems theproposed algorithm firstly concatenates the data from thesame low Doppler frequency slot of different images andthen performs the spectrumzoomprocessing on the obtaineddata Finally the clutter suppression and target detection areperformed on the spectrum zoom result Spectrum zoomprocessing is the key step which can effectively separate thetarget echo and ground clutter in the frequency domain andsignificantly improve the signal-to-clutter ratio (SCR)

This paper is organized as follows In Section 2 thebasic theory of spectrum zoom processing is introduced InSection 3 the spectrum zoom processing based LSS targetdetection algorithm is proposed In Section 4 the perfor-mance of the proposed algorithm is theoretically analyzed interms of the SCR improvement and computational load InSection 5 we verify the theoretical analysis in Section 4 withsome real data Finally conclusions are drawn in Section 6

2 Basic Theory of Spectrum Zoom Processing

21 Signal Model Let 119904(119905) 0 le 119905 lt 119872119873119879119904 be a continuoustime complex signal with a finite length where119872 and119873 arepositive integers and 119879119904 gt 0 Dividing 119904(119905) into 119872 signals1199040(119905) 1199041(119905) 119904119872minus1(119905) with the length of119873119879119904 we can get119904 (119905) = 1199040 (119905) + 1199041 (119905) + sdot sdot sdot + 119904119872minus1 (119905) (1)

where119904119898 (119905) = 119904 (119905) 119898119873119879119904 le 119905 lt (119898 + 1)1198731198791199040 other 119898 = 0 1 119872 minus 1 (2)

Shifting the signal 119904119898(119905) to 0 le 119905 lt 119873119879119904 along the time axiswe have 119909119898 (119905) = 119904119898 (119905 + 119898119873119879119904) (3)

Combining (1) with (3) we can obtain119904 (119905) = 1199090 (119905) + 1199091 (119905 minus 119873119879119904) + sdot sdot sdot+ 119909119872minus1 (119905 minus (119872 minus 1)119873119879119904) (4)

After sampling the continuous time signal 119904(119905) and 119909119898(119905)with a period of 119879119904 respectively the resulting discrete timesequences are119904 [119899] = 119904 (119899119879119904) 119899 = 0 1 119872119873 minus 1119909119898 [119899] = 119909119898 (119899119879119904) 119899 = 0 1 119873 minus 1 (5)

Obviously 119904[119899] is just the arrangement of 1199090[119899] 1199091[119899] 119909119872minus1[119899] with the sequential orderAccording to [22] the Fourier transform (FT) of 119904(119905) can

be written as 119878 (120596) = intinfinminusinfin

119904 (119905) 119890minusj120596119905d119905 (6)



119904 [119899] 119890minusjΩ119899 (7)


119899=0

119904 [119899]119882119896119899119872119873 (8)





119909119898 (119905) 119890minusj120596119905d119905 (11)


119899=0

119909119898 [119899] 119890minusjΩ119899 (12)

119883119898 [119896] = 119873minus1sum119899=0

119909119898 [119899]119882119896119899119873 (13)




119883119898 (120596) 119890minusj120596119898119873119879119904 (15)







119883119898 (1205960) 119890minusj1205960119898119873119879119904= 119872minus1sum119898=0

119883119898 (1205960) 119890minusj(2120587(119901119872+119902)119865119904119872119873)119898119873119879119904= 119872minus1sum119898=0


119883119898 (1205960)119882119902119898119872 (18)




119883119898 (0)119882119902119898119872 (20)




119883119898 [119901]119882119902119898119872 (22)








Z119898 =(z(119898minus1)119873+1z(119898minus1)119873+2

z119898119873

) (23)





Fast Time

Slow

Tim

e

N

N

N

M F

ram

es

L Range Gates

z1z2z3

zN

zN+1zN+2

zN+3

z2N


zMN


Track

Frame 1

Frame M

Frame 2 DFT

DFT

DFT

Detect

Detect

Detect

DFT

DFT Detect

Range-timeimage



Orig

inal

sign

al p

roce

ssing

Retro

fittin

g sch

eme

Stick





DFT

+QH

Dop

pler

freq

uenc

y slo

t

0D

oppl

er fr

eque

ncy

slot

minusQH

Dop

pler

freq

uenc

y slo

t

bullbull

bull

bullbullbull

bullbullbull



X119898 = (x1198981 x1198982 x119898119871) (24)

where

x119898119897 = (119909(119898minus1)119873+1119897 119909(119898minus1)119873+2119897 119909119898119873119897)T (25)







































30

25

20

0

5

10

15

0

100

200

300

Am

plitu

de


Rang

e Gat

e

Ground Clutter

Target 1Target 2






200

150

100

50

0

20

0

minus200

100

200

300

Am

plitu

de


Ground Clutter

Target 1

(a)

200

150

100

50

0

20

0

minus200

100

200

300

Am

plitu

de



(b)

200

150

100

50

0

20

0

minus200

100200

300

Am

plitu

de



(c)




6 Conclusions





Acknowledgments



References
































Journal of












Journal of







Volume 2018






Volume 2018










119904 [119899] 119890minusjΩ119899 (7)


119899=0

119904 [119899]119882119896119899119872119873 (8)





119909119898 (119905) 119890minusj120596119905d119905 (11)


119899=0

119909119898 [119899] 119890minusjΩ119899 (12)

119883119898 [119896] = 119873minus1sum119899=0

119909119898 [119899]119882119896119899119873 (13)




119883119898 (120596) 119890minusj120596119898119873119879119904 (15)







119883119898 (1205960) 119890minusj1205960119898119873119879119904= 119872minus1sum119898=0

119883119898 (1205960) 119890minusj(2120587(119901119872+119902)119865119904119872119873)119898119873119879119904= 119872minus1sum119898=0


119883119898 (1205960)119882119902119898119872 (18)




119883119898 (0)119882119902119898119872 (20)




119883119898 [119901]119882119902119898119872 (22)








Z119898 =(z(119898minus1)119873+1z(119898minus1)119873+2

z119898119873

) (23)





Fast Time

Slow

Tim

e

N

N

N

M F

ram

es

L Range Gates

z1z2z3

zN

zN+1zN+2

zN+3

z2N


zMN


Track

Frame 1

Frame M

Frame 2 DFT

DFT

DFT

Detect

Detect

Detect

DFT

DFT Detect

Range-timeimage



Orig

inal

sign

al p

roce

ssing

Retro

fittin

g sch

eme

Stick





DFT

+QH

Dop

pler

freq

uenc

y slo

t

0D

oppl

er fr

eque

ncy

slot

minusQH

Dop

pler

freq

uenc

y slo

t

bullbull

bull

bullbullbull

bullbullbull



X119898 = (x1198981 x1198982 x119898119871) (24)

where

x119898119897 = (119909(119898minus1)119873+1119897 119909(119898minus1)119873+2119897 119909119898119873119897)T (25)







































30

25

20

0

5

10

15

0

100

200

300

Am

plitu

de


Rang

e Gat

e

Ground Clutter

Target 1Target 2






200

150

100

50

0

20

0

minus200

100

200

300

Am

plitu

de


Ground Clutter

Target 1

(a)

200

150

100

50

0

20

0

minus200

100

200

300

Am

plitu

de



(b)

200

150

100

50

0

20

0

minus200

100200

300

Am

plitu

de



(c)




6 Conclusions





Acknowledgments



References
































Journal of












Journal of







Volume 2018






Volume 2018











119883119898 [119901]119882119902119898119872 (22)








Z119898 =(z(119898minus1)119873+1z(119898minus1)119873+2

z119898119873

) (23)





Fast Time

Slow

Tim

e

N

N

N

M F

ram

es

L Range Gates

z1z2z3

zN

zN+1zN+2

zN+3

z2N


zMN


Track

Frame 1

Frame M

Frame 2 DFT

DFT

DFT

Detect

Detect

Detect

DFT

DFT Detect

Range-timeimage



Orig

inal

sign

al p

roce

ssing

Retro

fittin

g sch

eme

Stick





DFT

+QH

Dop

pler

freq

uenc

y slo

t

0D

oppl

er fr

eque

ncy

slot

minusQH

Dop

pler

freq

uenc

y slo

t

bullbull

bull

bullbullbull

bullbullbull



X119898 = (x1198981 x1198982 x119898119871) (24)

where

x119898119897 = (119909(119898minus1)119873+1119897 119909(119898minus1)119873+2119897 119909119898119873119897)T (25)







































30

25

20

0

5

10

15

0

100

200

300

Am

plitu

de


Rang

e Gat

e

Ground Clutter

Target 1Target 2






200

150

100

50

0

20

0

minus200

100

200

300

Am

plitu

de


Ground Clutter

Target 1

(a)

200

150

100

50

0

20

0

minus200

100

200

300

Am

plitu

de



(b)

200

150

100

50

0

20

0

minus200

100200

300

Am

plitu

de



(c)




6 Conclusions





Acknowledgments



References
































Journal of












Journal of







Volume 2018






Volume 2018









Fast Time

Slow

Tim

e

N

N

N

M F

ram

es

L Range Gates

z1z2z3

zN

zN+1zN+2

zN+3

z2N


zMN


Track

Frame 1

Frame M

Frame 2 DFT

DFT

DFT

Detect

Detect

Detect

DFT

DFT Detect

Range-timeimage



Orig

inal

sign

al p

roce

ssing

Retro

fittin

g sch

eme

Stick





DFT

+QH

Dop

pler

freq

uenc

y slo

t

0D

oppl

er fr

eque

ncy

slot

minusQH

Dop

pler

freq

uenc

y slo

t

bullbull

bull

bullbullbull

bullbullbull



X119898 = (x1198981 x1198982 x119898119871) (24)

where

x119898119897 = (119909(119898minus1)119873+1119897 119909(119898minus1)119873+2119897 119909119898119873119897)T (25)







































30

25

20

0

5

10

15

0

100

200

300

Am

plitu

de


Rang

e Gat

e

Ground Clutter

Target 1Target 2






200

150

100

50

0

20

0

minus200

100

200

300

Am

plitu

de


Ground Clutter

Target 1

(a)

200

150

100

50

0

20

0

minus200

100

200

300

Am

plitu

de



(b)

200

150

100

50

0

20

0

minus200

100200

300

Am

plitu

de



(c)




6 Conclusions





Acknowledgments



References
































Journal of












Journal of







Volume 2018






Volume 2018













































30

25

20

0

5

10

15

0

100

200

300

Am

plitu

de


Rang

e Gat

e

Ground Clutter

Target 1Target 2






200

150

100

50

0

20

0

minus200

100

200

300

Am

plitu

de


Ground Clutter

Target 1

(a)

200

150

100

50

0

20

0

minus200

100

200

300

Am

plitu

de



(b)

200

150

100

50

0

20

0

minus200

100200

300

Am

plitu

de



(c)




6 Conclusions





Acknowledgments



References
































Journal of












Journal of







Volume 2018






Volume 2018









200

150

100

50

0

20

0

minus200

100

200

300

Am

plitu

de


Ground Clutter

Target 1

(a)

200

150

100

50

0

20

0

minus200

100

200

300

Am

plitu

de



(b)

200

150

100

50

0

20

0

minus200

100200

300

Am

plitu

de



(c)




6 Conclusions





Acknowledgments



References
































Journal of












Journal of







Volume 2018






Volume 2018









References
































Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








low-altitude and slow-speed small target detection based

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