research article micro-doppler ambiguity resolution based...

Research ArticleMicro-Doppler Ambiguity Resolution Based onShort-Time Compressed Sensing

Jing-bo Zhuang Zhen-miao Deng Yi-shan Ye Yi-xiong Zhang and Yan-yong Chen

School of Information Science and Engineering Xiamen University Xiamen 361005 China

Correspondence should be addressed to Zhen-miao Deng dzm ddbxmueducn

Received 4 May 2015 Revised 5 August 2015 Accepted 5 August 2015

Academic Editor Igor Djurovic

Copyright copy 2015 Jing-bo Zhuang 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

When using a long range radar (LRR) to track a target with micromotion the micro-Doppler embodied in the radar echoes maysuffer from ambiguity problem In this paper we propose a novelmethod based on compressed sensing (CS) to solvemicro-Dopplerambiguity According to the RIP requirement a sparse probing pulse train with its transmitting time random is designed Aftermatched filtering the slow-time echo signals of the micromotion target can be viewed as randomly sparse sampling of Dopplerspectrum Select several successive pulses to form a short-time window and the CS sensing matrix can be built according to thetime stamps of these pulses Then performing Orthogonal Matching Pursuit (OMP) the unambiguous micro-Doppler spectrumcan be obtained The proposed algorithm is verified using the echo signals generated according to the theoretical model and thesignals with micro-Doppler signature produced using the commercial electromagnetic simulation software FEKO

1 Introduction

Estimating and extracting micromotion information haveattracted much attention in recent years [1ndash6] Micromotioncan be defined as the mechanical vibration rotation or otherhigher order motion components excluding translationalmotion of a target and will produce a frequency modulationon the returned signal that generates sidebands about the tar-getrsquos Doppler frequency This is known as the micro-Dopplereffect This effect reflects the unique dynamic and structuralcharacteristics of the target which offers an approach for therecognition and identification of specific targets [7]

In Synthetic Aperture Radar (SAR)Inverse SyntheticAperture Radar (ISAR) imagery micro-Doppler effect willintroduce nonstationary phase modulation into returnedsignals which will significantly decrease the readability ofimages [5 8 9] In [8] an estimation method based on thediscrete fractional Fourier transform is proposed to estimatethe instantaneous vibration accelerations and frequencies Toseparate the rigid body and the micro-Doppler parts an L-statistics-based method for micro-Doppler effects removal isproposed in [9]

Most of the previous researches assume that the probingfrequency to a micromotion target is large enough and thus

there is no Doppler ambiguity andmicro-Doppler ambiguityHowever a long range radar usually works in low PRF whichcauses serious Doppler ambiguity and micro-Doppler ambi-guity In [10 11] a CS-based Doppler ambiguity resolutionmethod is proposed However thismethod cannot be appliedto resolve micro-Doppler ambiguities Therefore it is neces-sary to study how to extract the unambiguousmicro-Dopplertime-frequency spectrumwhen the PRF is low In [12] the CSis employed to remove undesirable cross terms in theWigner-Ville distribution of themicro-Doppler radar signature How-ever the micro-Doppler ambiguity problem is not discussed

To solve micro-Doppler ambiguity problem we proposea novel method based on CS in this paper According to theRIP requirement of the sensing matrix a sparse pulse trainwith random time stamps is designed based on the fixed-PRFpulsesThe echo signals after matched filtering can be viewedas randomly sparse sampling of themicro-Doppler spectrumTo reconstruct the micro-Doppler signature from the sparsesamples we propose a short-time-compressed-sensing time-frequency analysismethodA short-timewindow slides alongthe slow-time domain echo signals and the reconstructionalgorithm OMP is applied within this window to reconstructthe micro-Doppler spectrum Two kinds of echo signalsone generated according to the theoretical model and one

Hindawi Publishing CorporationJournal of Electrical and Computer EngineeringVolume 2015 Article ID 864508 7 pageshttpdxdoiorg1011552015864508

2 Journal of Electrical and Computer Engineering

produced by the commercial electromagnetic simulationsoftware FEKO are used to verify the proposed algorithm

2 Theory

21 Radar Echo Signal Model The transmitted chirp signalcan be modeled as

119904tran ( 119905119898) = rect(

119879119901

) exp(1198952120587(119891119888119905 +

12120574

2)) (1)

where rect(119906) = 1 |119906|le050 |119906|gt05 119891119888 and 120574 are the center frequency

and the chirp rate respectively 119879119901and are the pulse width

and fast time respectively 119905 = + 119905119898is the full time where

119905119898denotes the slow time The received echo signal 119904rec( 119905119898)

can be defined as

119904rec ( 119905119898)

= 119860rect( minus 2119877

119905119888

119879119901

) 119890119895120587120574(minus2119877

119905119888)

2119890minus119895(4120587119891

119888119888)119877119905

(2)

where 119860 is the backscattered field amplitude for the pointscatter 119888 is the speed of electromagnetic wave propagationand 119877

119905is the instantaneous distance between the radar and

the target Performingmatched filtering to the received signaland transforming the results into range-frequency domainyield

119903 (119891119903 119905119898) = 1198601015840rect(

119891119903

119861

) exp(minus1198954120587119888

(119891119903+119891119888) 119877119905) (3)

where 119861 is the signal bandwidth Ignoring the accelerationjerk of the target 119877

119905can be approximated by

119877119905asymp 1198770 + V119905119898 + 119903micro (119905119898) (4)

where 1198770 is the distance between the target and radar at time1199050 V is the radial velocity and 119903micro(119905119898) is the micromotionof the target Substituting (4) into (3) yields the slow-timedomain echo signal

119903 (119891119889 119905119898) = 11986010158401015840 exp [minus1198952120587119891

119889(119905119898) 119905119898] (5)

where11986010158401015840 = 1198601015840exp(2(119891

119903+119891119888)1198770119888) and 119891119889(119905119898) is the Doppler

frequency and can be written as

119891119889(119905119898) =

12120587

119889120593 (119905119898)

119889119905119898

=

2120582

119889119903 (119905119898)

119889119905119898

=

2120582

[V+119889119903micro (119905119898)

119889119905

] = 119891119887 119889

+119891119898 119889

(119905119898)

(6)

which consists of two parts that is the Doppler frequency119891119887 119889

caused by translational motion and the micro-Dopplerfrequency 119891

119898 119889(119905119898) caused by micromotion

22 Requirements of the PRF According to the Nyquist-Shannon sampling theorem for a band-limited basebandsignal the sampling rate must be greater than or equal to two

times the highest frequency of the signal For a pulsed radarthe Doppler frequency 119891

119889(119905119898) of the target is sampled with

the sampling rate as the PRF When the micro-Doppler fre-quency exceeds half of the PRF themicro-Doppler ambiguityphenomenon will occur Thus the PRF must be greater thanor equal to the highest micro-Doppler frequency shift

In a short-time interval 119891119889(119905119898) can be seen as a constant

If there are 119870 scatterers each with different micro-Dopplerfrequencies the echo signal can be rewritten as

119903 (119891119889 119905119898) =

119870

sum

119896=111986010158401015840 exp minus1198952120587119891

119889119896119905119898 (7)

If the PRF is smaller than max119896=1119870119891119889119896 the spectrum

aliases will occur in (7) According to the sub-Nyquistsampling theorem [13] or CS theory the sparse signal canbe sampled at a rate lower than the Nyquist The slow-timedomain radar echo signal can be viewed as the scalar sumof 119870 sinusoidal signals which is sparse in frequency domain[14]Thus the requirement for PRF can be loosened Next webriefly review the compressing sensing theory

23 Compressed Sensing The theory of compressed sensingshows that when the signal is sparse or compressible the sig-nal can be reconstructed accurately or approximately by gath-ering very few projective values of the signal Suppose 119909 (119909 isin

119877119873

) is a 119896-sparse (has 119896 nonzero values) or compressivesignal after orthogonalmapping projection and 119904 is the sparserepresentation of 119909 then a measurement matrix Φ whichis irrelevant with the orthogonal map Ψ could be built tomeasure 119909 linearly resulting in only119872 (119872 ≪ 119873) measuredvalues 119910 (119910 isin 119877

119872

)

119910 = Φ119909 (8)

The dimension of 119910 is less than that of 119909 so the equationhas infinity solutions By solving the optimal problem abovesignal 119909 can be approximately reconstructed 119909 is sparse inΨdomain that is

119909 = Ψ119904 (9)

Substituting (9) into (8) yields

119910 = Φ119909 = ΦΨ119904 = Θ119904 (10)

where Θ = ΦΨ is called sensing matrix with 119872 times 119873

dimension As long as Θ satisfies the Restricted IsometryProperty (RIP) the sparse signal 119904 could be recovered fromthe measured values 119910

Signal reconstruction is the process of recovering 119909 fromthe linear measured values 119910 The simplest method is to solvethe ℓ0 norm

min119909isin119877119873

1199090

st Φ119909 = 119910

(11)

It is optimal in theory but impractical in numerical computa-tion belonging to a nondeterministic polynomial (NP) hard

Journal of Electrical and Computer Engineering 3

problem Donoho and Elad proved that ℓ1 norm and ℓ0 normminimizations are equivalent if the solution is sufficientlysparse [15] ℓ1 norm

min119909isin119877119873

1199091

st Φ119909 = 119910

(12)

is an optimization problem and could be solved using linearprogramming

According to the description of Section 22 we knowthat the radar echo signals can be viewed as the scalar sumof sinusoidal signals and are sparse in frequency domainthus the compressed sensing can be utilized The linearmeasurement 119910 is a119872times 1 column vector which is randomlyextracted from themeasurements of (7) with the correspond-ing random sampling time 119905

119898 Since 119903(119891

119889 119905119898) is sparse in

frequency domain the Fourier basis is chosen as the basismatrixΨ with its element defined by

120595119894119896=

1radic119873

exp(minus1198952120587 119894119896119873

) 0 le 119894 119896 le 119873 (13)

Extracting119872 rows corresponding to the random transmittedtime 119905

119898from the identity matrix I

119873times119873yields the measure-

ment matrix Φ Thus the sensing matrix Θ (Θ = ΦΨ)

is obtained by randomly extracting 119872 row from the sparsematrixΨ and meets the RIP property

Generally the value of 119872 is related to the relevancy119906(Φ119873times119873

Ψ) between the sparse matrix Ψ and the measure-ment matrixΦ and satisfies

119872 ge 119862 sdot 1199062(Φ119873times119873

Ψ) sdot 119870 sdot log119873 (14)

where 119862 is a constant and 119906(Φ119873times119873

Ψ) is defined as [16]

119906 (Φ119873times119873

Ψ) = radic119873 sdot max1le119896119895le119873

10038161003816100381610038161003816(120593119896 120595119895)

10038161003816100381610038161003816 (15)

The classical recovery algorithms of compressed sensingare Basis Pursuit and Greedy Matching Pursuit BP algo-rithm is the global optimization algorithm which has severaladvantages including superresolution and stability but itis also accompanied with high computational complexityGreedy Matching Pursuit algorithm such as OrthogonalMatching Pursuit (OMP) is a local optimization algorithmand has the low computational complexity and high levelof localization accuracy The characteristics of OMP such aseasy implementation and fast speed make it a better choicethan BP algorithm [16]

3 Short-Time Compressed Sensing

Doppler frequency shifts generally have the time-varyingcharacteristic and should be analyzed via the joint time-frequency analysis technique However when the PRF islower than the Nyquist the traditional time-frequency anal-ysis methods are invalid According to the above theoreticalanalysis results and the time-variant properties of micro-Doppler we design a window-weighted compressed sensing

1 2 43 5 6 7 8 9 10

1 3 6 109

t

t

Transmitted time sequences with fixed-PRF and additional random

Sparse and random transmitted time sequences

7perturbation

middot middot middot


Figure 1 Illustration of radar dwell scheduling

method According to the RIP of the CS sensing matrixa sparse pulse train is randomly extracted from traditionalfixed repetition frequency pulses By adding a randomperturbation item to the transmitting time of each selectedpulse we can obtain a new transmitting time sequencewhich is equivalent to those extracted from a high PRF pulsetransmitting time set Then the corresponding measurementmatrix can be obtained according to the transmitting timesequence After matched filtering to the radar echoes ashort-time window slides along the slow-time domain echosignals and the CS method is applied within this windowto reconstruct the micro-Doppler spectrum Similar to theshort-time Fourier transform (STFT) we call this method asshort-time compressed sensing

The PRF of a LRR is usually low For a LRR workingin fixed PRF mode the slow-time domain echoes sufferfrom serious micro-Doppler ambiguity By modifying the119894th pulse transmitting time 119905

119894to 1199051015840

119894= 119905119894+ Δ119894 where Δ

119894is

a random perturbation we can obtain a new transmittingtime sequence Since Δ

119894is random the equivalent PRF for

1199051015840

119894is larger than that for 119905

119894 The unambiguous micro-Doppler

frequency range for 1199051015840119894will be larger than that for 119905

119894since

higher PRFmeanswider unambiguousmicro-Doppler rangeWhen the value of 119872 meets (14) the designed sensing

matrix can satisfy the RIP property and the slow-time echosignals can be reconstructed based on the sparse measure-ments Therefore we can reduce the number of transmittedpulses by randomly selecting 119872 pulses from the traditionalfixed-PRF transmitted time sequences and adding a randomperturbation to them The building of sparse and randomtransmitted time sequences is demonstrated in Figure 1

Since 119891119889(119905119898) is relevant to the slow time 119905

119898 there is no

applicable sensing matrix to reconstruct the micro-Dopplerspectrum via compressed sensing directly However if wemultiplied a short-time window to the slow-time echoes119891119889(119905119898) within this window can be seen as a constant and

then the sensing matrix can be generated Assuming thetime precision is Δ119905 the corresponding equivalent PRF is1198911015840

119904= 1Δ119905 and the Doppler unambiguous range for 1199051015840

119894is

represented as 1198911015840119904= (1Δ119905)119891

119904 Thus the unambiguous micro-

Doppler frequency range expands to (minus11989110158401199042 11989110158401199042)

The signal within the window is written as

119903 (119891119889 119905119898) = 119908 (119905

119899 119905119898)

119870

sum

119896=111986010158401015840 exp minus1198952120587119891

119889119896(119905119899) 119905119899 (16)

where the sliding short-time window is defined as119908(119905119899 119905119898) =

1 119905119898le119905119899le119905119898+119871minus1

0 else In the window the transmitted time sequenceis 119905119899isin 119905119898 119905119898+1 119905119898+119871minus1 The length of the window is


Generating fixed-PRF

Extracting randomly

Adding a random Designing sensingmatrix

Matched filtering toradar echo

Radar transmits Weighting with asliding window

Compressed sensingreconstruction

time sequence t998400i

time sequence ti

from ti yields sparse

perturbation to t998400i

according to t998400i

Figure 2 Flowchart of the proposed method

119871 that is the length of the observed signal is 119872 = 119871 Inorder to obtain more accurate Doppler estimate an adaptiverefinement algorithm of the sensing matrix proposed in [17]can be used to reconstruct the slow-time domain signal withhigher precision The column vector 120595

119899of the basis matrix

Ψ = [1205951 1205952 120595119873] is defined as 120595119899(119891) = exp(1198952120587119891119905)

and Ψ is an 119873 times 119873 matrix where 119873 = 1Δ119905 and Δ119891 =

(119891max minus 119891min)119873 According to 119905119899 extracting 119871 rows from the

basis matrixΨ119873times119873

yields the sensing matrixΘ119872times119873

Θ119872times119873

=

[

[

[

[

[

[

[

[

119890119895212058711989101199050

119890119895212058711989111199050

sdot sdot sdot 1198901198952120587119891119873minus11199050

119890119895212058711989101199051

119890119895212058711989111199051


d

11989011989521205871198910119905119871minus1

11989011989521205871198911119905119871minus1

sdot sdot sdot 1198901198952120587119891119873minus1119905119871minus1

]

]

]

]

]

]

]

]

(17)

With the classical recovery algorithms the unambiguousDoppler frequency 119891

119889119896(119905119898) can be reconstructed Sliding

the short-time window along the time axis and performingthe CS operation the unambiguous micro-Doppler time-frequency spectrum can be obtained

To satisfy the RIP the perturbation Δ119894= 120576 sdot 119875119877119868 where

120576 is a random variable uniformly distributed in (0 1) that is120576 sim 119880(0 1) In practical application there are two reasonscausing that the term Δ

119894could not be completely random

(1) the time sequence of radar is controlled by a referenceclock So the precision of Δ

119894is limited to the precision of

the clock (2) theoretically higher precision of Δ119894would

yield larger unambiguous Doppler frequency range throughrefining the basis matrix However as pointed out in [14]the relationship between the DFT matrix refining factorwhich is referred to the frequency redundancy factor andthe measurement number 119872 should satisfy (20) in [14] Inotherwords119872would increase if the refining factor increasestherefore the tradeoff between these two factors should becarefully considered Furthermore when the refining factorincreases the performance of recovery algorithmunder noiseenvironment will also degrade Thus we should carefullyselect proper Δ119905 to ensure a good performance

The process of the short-time compressed sensing onmicro-Doppler spectrum reconstruction is summarized asfollows

(i) A new transmitting time sequence is generated byextracting randomly from the fixed-PRF transmitting timesequenceThen a perturbation is added to each element of thenew transmitting time sequenceThe radar transmits probingpulses according to the scheduled time sequence Performingmatched filtering to the echo pulses the slow-time domainsignals are obtained

(ii) Slide a short-time window along the slow-timedomain signals and construct the corresponding sensingmatrixΨ according to the time stamps of the window

(iii) Performing the Orthogonal Matching Pursuit algo-rithm to the signals in the sliding window the instantaneousmicro-Doppler frequency is obtained

The flowchart for the implementation of the algorithm isshown in Figure 2

4 Simulation

In this section simulations are performed to verify the pro-posed method The proposed method can be applied to dif-ferent kinds of micromotion In the following twomicromo-tions that is spin and precession are simulated

Experiment 1 (one spin scatter case) The simulation parame-ters are shown in the list below One scatter case is as follows

Simulation Parameters

119891119888= 10GHz

119861 = 5MHz119891119903= 300Hz 119905 = 2 s

120596 = 2120587 rads1199031 = 06m

The corresponding slow-time echo signal for spin scattererscan be written as

119903 (119891119889 119905119898) =

1sum

119896=111986010158401015840 expminus1198952120587

2120596119896119903119896cos (120596

119896119905119898)

120582

119905119898 (18)

The simulation results for spin motion are shown in Figure 3


minus1000

100

f (H

z)

05 1 15 20t (s)

(a)

02 04 06 08 10t (s)

minus2000

200

f (H

z)

(b)

02 04 06 08 10t (s)

minus2000

200

f (H

z)

(c)

Figure 3 Micro-Doppler time-frequency spectrum ambigu-ous time-frequency spectrum via STFT (a) unambiguous time-frequency spectrum of adding-perturbation fixed-PRF via CS (b)unambiguous time-frequency spectrum of adding-perturbationsparse-PRF via CS (c)

According to the simulated parameters the maximummicro-Doppler frequency shift is plusmn250Hz Since the PRFis not large enough the micro-Doppler spectrum obtainedfrom the STFT is ambiguous Add a random perturbationΔ119894to each time stamp of the transmitting time sequence

1199051 1199052 119905119899 respectively The accuracy of the perturbationis Δ119905 = 1667119890 minus 4 and the corresponding PRF1015840 = 6000 Thelength of the time window is 119871 = 10 the number of mea-surements is119872 = 10 the length of the reconstructed signalis119873 = 6000 the sparse rate is 119870 = 2 the sensing matrixΘ isa 12 times 6000 matrix The unambiguous micro-Doppler time-frequency spectrum can be recovered using the CS recon-struction algorithm which is shown in (b) of Figure 3 Ran-domly selecting a portion of elements from 1199051 1199052 119905119899 andadding a random perturbation Δ

119894to each of the selected ele-

ments we can generate sparse probing pulses (c) of Figure 3shows the unambiguous micro-Doppler spectrum recon-structed from these sparse pulses with good performanceThe sparse rate that is the number of sparse probing pulsesdivided by the number of the original fixed-PRF pulses is 06

In order to investigate the effect of the number of mea-surements119872 on the reconstruction quality for a fixed sparserate we select different 119872 values and compare the recon-struction results which are shown in Figure 4We can see thatlarger119872 corresponds to better reconstruction performance

Experiment 2 (two spin scatterersrsquo case) Two spin scatterersare simulated and the simulation parameters are shown inthe list below To demonstrate the ambiguity resolving abilityof our method we change the rotating radius 1199031 = 06m ofExperiment 1 to 1199031 = 18m With these simulation parame-ters the micro-Doppler of this scatterer will be ambiguous

02 04 06 08 10t (s)

minus200

0

200

f (H

z)

(a)

02 04 06 08 10t (s)

minus300minus200minus100

0100200300

f (H

z)

(b)

Figure 4 CS reconstruction quality with different measurementnumber (a)119872 = 10 (b)119872 = 6

minus200

0

200

f (H

z)

05 1 15 20t (s)

(a)

minus5000

500

f (H

z)

05 1 15 20t (s)

(b)

minus5000

500

f (H

z)

05 1 15 20t (s)

(c)


twice which can be seen from (a) of Figure 5 Performingthe short-time compressed sensing the unambiguous micro-Doppler can be recovered successfully The sparse rate of (c)of Figure 5 is 075


119891119888= 10GHz


05 1 15 20t (s)

minus200

0

200

f (H

z)

(a)

05 1 15 20t (s)

minus2000

200

f (H

z)

(b)

05 1 15 20t (s)

minus2000

200

f (H

z)

(c)


119861 = 5MHz119891119903= 400Hz 119905 = 2 s

120596 = 2120587 rads1199031 = 18m 1199032 = 006m

Experiment 3 (two precession scatterersrsquo case) Two preces-sion scatterers are simulated and the simulation parametersare shown in the list below (19)The simulation results for pre-cession motion can be shown in Figure 6The correspondingslow-time echo signal for precession scatterers can be writtenas

119903 (119891119889 119905119898) =

2sum

119896=111986010158401015840

sdot expminus1198952120587[21205961119903119896cos (1205961119905119898)

120582

+

21205962119877119896cos (1205962119905)

120582

]

sdot 119905119898

(19)


119891119888= 10GHz

119861 = 5MHz119891119903= 400Hz 119905 = 2 s

1205961 = 2120587 rads 1205962 = 4120587 rads1199031 = 03m 1198771 = 015m1199032 = 003 1198772 = 0015

10 03 04 05 06 07 08 0901 02

t (s)


0100200300

f (H

z)

(a)

02 03 04 05 06 07 0801

t (s)

minus240minus180minus120minus60

060120180240300

f (H

z)

(b)

Figure 7 Micro-Doppler time-frequency spectrum ambigu-ous time-frequency spectrum via STFT (a) unambiguous time-frequency spectrum of adding-perturbation sparse-PRF via CS (b)

The simulation results for the precession case are similarto those for the spin case The sparse rate of (c) of Figure 6 is05 The simulation results in Figures 3ndash6 show that the pro-posed method can expand the unambiguous micro-Dopplerfrequency range

Experiment 4 In this experiment we use the electromagneticsimulation software FEKO to generate sparse-sampled slow-time echo signals and reconstruct the unambiguous micro-Doppler spectrum via CS A scatterer is rotating with aradius of 30 cm The rate of angular motion is 2120587 rads Thecarrier frequency is 10GHz The bandwidth is 500MHz Theazimuth and pitch angle are 0∘ and 90∘ respectively Thedynamic position of the scatterer at each sparse probing timestamp is calculated with MATLAB and then the correspond-ing slow-time echo is resolvedwith FEKOThePRF is 200Hz

According to the configuration the maximum micro-Doppler frequency is 119891max = 40120587 Since the PRF 119891

119903lt 2119891max

the micro-Doppler is ambiguous The ambiguous micro-Doppler spectrum with fixed-PRF probing pulses is shownin (a) of Figure 7 and the unambiguous micro-Doppler spec-trum with sparse-PRF probing pulses is shown in (b) of Fig-ure 7 It can be seen that our proposedmethod is also suitablefor the simulated echoes with FEKO

5 Conclusion

The proposed short-time compressed sensing method cansolve the micro-Doppler ambiguity problem The sensingmatrix built based on the sparse probing time stamps canmeet the requirement of RIP property Performing theOMP algorithm to the signals within the sliding short-time window the ambiguous micro-Doppler spectrum canbe reconstructed The proposed method can work in low


PRF circumstances and requires transmitting fewer probingpulses while at the same time it can achieve larger unambigu-ous micro-Doppler range

Conflict of Interests

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

Acknowledgments

The research was supported by the National High-Tech RampDProgram of China the Open-End Fund National Laboratoryof Automatic Target Recognition (ATR) the Open-End Fundof BITTT Key Laboratory of Space Object Measurement andthe National Natural Science Foundation of China (Grant no62101196)

References

[1] T Thayaparan L J Stankovic M Dakovic and V PopovicldquoMicro-Doppler parameter estimation from a fraction of theperiodrdquo IET Signal Processing vol 4 no 3 pp 201ndash212 2010

[2] K Li Y Liu K HuoW Jiang X Li and Z Zhuang ldquoEstimationofmicro-motion parameters based on cyclostationary analysisrdquoIET Signal Processing vol 4 no 3 pp 218ndash223 2010

[3] T Thayaparan K Suresh S Qian K VenkataramaniahS SivaSankaraSai and K S Sridharan ldquoMicro-doppleranalysis of a rotating target in synthetic apertureradarrdquo IET Signal Processing vol 4 no 3 Article IDISPECX000004000003000245000001 pp 245ndash255 2010

[4] T Thayaparan L Stankovic and I Djurovic ldquoMicro-Doppler-based target detection and feature extraction in indoor andoutdoor environmentsrdquo Journal of the Franklin Institute vol345 no 6 pp 700ndash722 2008

[5] Q Wang M Pepin A Wright et al ldquoReduction of vibration-induced artifacts in synthetic aperture radar imageryrdquo IEEETransactions on Geoscience and Remote Sensing vol 52 no 6pp 3063ndash3073 2014

[6] P Suresh T Thayaparan T Obulesu and K VenkataramaniahldquoExtracting micro-doppler radar signatures from rotating tar-gets using fourier-bessel transform and time-frequency analy-sisrdquo IEEE Transactions on Geoscience and Remote Sensing vol52 no 6 pp 3204ndash3210 2014

[7] L Liu D McLernon M Ghogho W Hu and J Huang ldquoBal-listic missile detection via micro-Doppler frequency estimationfrom radar returnrdquo Digital Signal Processing vol 22 no 1 pp87ndash95 2012

[8] S B Colegrove S J Davey and B Cheung ldquoSeparation of targetrigid body and micro-Doppler effects in ISAR imagingrdquo IEEETransactions on Aerospace and Electronic Systems vol 42 no 4pp 1496ndash1506 2006

[9] L Stankovic T Thayaparan M Dakovic and V Popovic-Bugarin ldquoMicro-doppler removal in the radar imaging analy-sisrdquo IEEE Transactions on Aerospace and Electronic Systems vol49 no 2 pp 1234ndash1250 2013

[10] Y H Quan L Zhang M D Xing and Z Bao ldquoVelocityambiguity resolving formoving target indication by compressedsensingrdquo Electronics Letters vol 47 no 22 pp 1249ndash1251 2011

[11] Y-X Zhang J-P Sun B-C Zhang and W Hong ldquoDopplerambiguity resolution based on compressive sensing theoryrdquoJournal of Electronics amp Information Technology vol 33 no 9pp 2103ndash2107 2011

[12] N Whitelonis and H Ling ldquoRadar signature analysis using ajoint time-frequency distribution based on compressed sens-ingrdquo IEEE Transactions on Antennas and Propagation vol 62no 2 pp 755ndash763 2014

[13] MMishali and Y C Eldar ldquoSub-nyquist samplingrdquo IEEE SignalProcessing Magazine vol 28 no 6 pp 98ndash124 2011

[14] M F Duarte and R G Baraniuk ldquoSpectral compressive sens-ingrdquoApplied and Computational Harmonic Analysis vol 35 no1 pp 111ndash129 2013

[15] D L Donoho and M Elad ldquoOptimally sparse representationin general (nonorthogonal) dictionaries via ℓ

1minimizationrdquo

Proceedings of the National Academy of Sciences of the UnitedStates of America vol 100 no 5 pp 2197ndash2202 2003

[16] J-H Wang Z-T Huang Y-Y Zhou et al ldquoGeneralized inco-herence principle in compressed sensingrdquo Signal Processing vol28 no 5 pp 675ndash679 2012

[17] L Hu Z Shi J Zhou and Q Fu ldquoCompressed sensing of com-plex sinusoids an approach based on dictionary refinementrdquoIEEE Transactions on Signal Processing vol 60 no 7 pp 3809ndash3822 2012

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



produced by the commercial electromagnetic simulationsoftware FEKO are used to verify the proposed algorithm

2 Theory

21 Radar Echo Signal Model The transmitted chirp signalcan be modeled as

119904tran ( 119905119898) = rect(

119879119901

) exp(1198952120587(119891119888119905 +

12120574

2)) (1)

where rect(119906) = 1 |119906|le050 |119906|gt05 119891119888 and 120574 are the center frequency

and the chirp rate respectively 119879119901and are the pulse width

and fast time respectively 119905 = + 119905119898is the full time where

119905119898denotes the slow time The received echo signal 119904rec( 119905119898)

can be defined as

119904rec ( 119905119898)

= 119860rect( minus 2119877

119905119888

119879119901

) 119890119895120587120574(minus2119877

119905119888)

2119890minus119895(4120587119891

119888119888)119877119905

(2)

where 119860 is the backscattered field amplitude for the pointscatter 119888 is the speed of electromagnetic wave propagationand 119877

119905is the instantaneous distance between the radar and

the target Performingmatched filtering to the received signaland transforming the results into range-frequency domainyield

119903 (119891119903 119905119898) = 1198601015840rect(

119891119903

119861

) exp(minus1198954120587119888

(119891119903+119891119888) 119877119905) (3)

where 119861 is the signal bandwidth Ignoring the accelerationjerk of the target 119877

119905can be approximated by

119877119905asymp 1198770 + V119905119898 + 119903micro (119905119898) (4)

where 1198770 is the distance between the target and radar at time1199050 V is the radial velocity and 119903micro(119905119898) is the micromotionof the target Substituting (4) into (3) yields the slow-timedomain echo signal

119903 (119891119889 119905119898) = 11986010158401015840 exp [minus1198952120587119891

119889(119905119898) 119905119898] (5)

where11986010158401015840 = 1198601015840exp(2(119891

119903+119891119888)1198770119888) and 119891119889(119905119898) is the Doppler

frequency and can be written as

119891119889(119905119898) =

12120587

119889120593 (119905119898)

119889119905119898

=

2120582

119889119903 (119905119898)

119889119905119898

=

2120582

[V+119889119903micro (119905119898)

119889119905

] = 119891119887 119889

+119891119898 119889

(119905119898)

(6)

which consists of two parts that is the Doppler frequency119891119887 119889

caused by translational motion and the micro-Dopplerfrequency 119891

119898 119889(119905119898) caused by micromotion

22 Requirements of the PRF According to the Nyquist-Shannon sampling theorem for a band-limited basebandsignal the sampling rate must be greater than or equal to two

times the highest frequency of the signal For a pulsed radarthe Doppler frequency 119891

119889(119905119898) of the target is sampled with

the sampling rate as the PRF When the micro-Doppler fre-quency exceeds half of the PRF themicro-Doppler ambiguityphenomenon will occur Thus the PRF must be greater thanor equal to the highest micro-Doppler frequency shift

In a short-time interval 119891119889(119905119898) can be seen as a constant

If there are 119870 scatterers each with different micro-Dopplerfrequencies the echo signal can be rewritten as

119903 (119891119889 119905119898) =

119870

sum

119896=111986010158401015840 exp minus1198952120587119891

119889119896119905119898 (7)

If the PRF is smaller than max119896=1119870119891119889119896 the spectrum

aliases will occur in (7) According to the sub-Nyquistsampling theorem [13] or CS theory the sparse signal canbe sampled at a rate lower than the Nyquist The slow-timedomain radar echo signal can be viewed as the scalar sumof 119870 sinusoidal signals which is sparse in frequency domain[14]Thus the requirement for PRF can be loosened Next webriefly review the compressing sensing theory

23 Compressed Sensing The theory of compressed sensingshows that when the signal is sparse or compressible the sig-nal can be reconstructed accurately or approximately by gath-ering very few projective values of the signal Suppose 119909 (119909 isin

119877119873

) is a 119896-sparse (has 119896 nonzero values) or compressivesignal after orthogonalmapping projection and 119904 is the sparserepresentation of 119909 then a measurement matrix Φ whichis irrelevant with the orthogonal map Ψ could be built tomeasure 119909 linearly resulting in only119872 (119872 ≪ 119873) measuredvalues 119910 (119910 isin 119877

119872

)

119910 = Φ119909 (8)

The dimension of 119910 is less than that of 119909 so the equationhas infinity solutions By solving the optimal problem abovesignal 119909 can be approximately reconstructed 119909 is sparse inΨdomain that is

119909 = Ψ119904 (9)

Substituting (9) into (8) yields

119910 = Φ119909 = ΦΨ119904 = Θ119904 (10)

where Θ = ΦΨ is called sensing matrix with 119872 times 119873

dimension As long as Θ satisfies the Restricted IsometryProperty (RIP) the sparse signal 119904 could be recovered fromthe measured values 119910

Signal reconstruction is the process of recovering 119909 fromthe linear measured values 119910 The simplest method is to solvethe ℓ0 norm

min119909isin119877119873

1199090

st Φ119909 = 119910

(11)

It is optimal in theory but impractical in numerical computa-tion belonging to a nondeterministic polynomial (NP) hard



min119909isin119877119873

1199091

st Φ119909 = 119910

(12)



119898 Since 119903(119891

119889 119905119898) is sparse in


120595119894119896=

1radic119873

exp(minus1198952120587 119894119896119873

) 0 le 119894 119896 le 119873 (13)








119872 ge 119862 sdot 1199062(Φ119873times119873

Ψ) sdot 119870 sdot log119873 (14)



119906 (Φ119873times119873


10038161003816100381610038161003816(120593119896 120595119895)

10038161003816100381610038161003816 (15)




1 2 43 5 6 7 8 9 10

1 3 6 109

t

t



7perturbation






119894to 1199051015840

119894= 119905119894+ Δ119894 where Δ

119894is



1199051015840




119894since




119898 there is no




119894is

represented as 1198911015840119904= (1Δ119905)119891




119903 (119891119889 119905119898) = 119908 (119905

119899 119905119898)

119870

sum

119896=111986010158401015840 exp minus1198952120587119891

119889119896(119905119899) 119905119899 (16)


1 119905119898le119905119899le119905119898+119871minus1




Extracting randomly






time sequence ti












Θ119872times119873

=

[

[

[

[

[

[

[

[

119890119895212058711989101199050

119890119895212058711989111199050


119890119895212058711989101199051

119890119895212058711989111199051


d

11989011989521205871198910119905119871minus1

11989011989521205871198911119905119871minus1


]

]

]

]

]

]

]

]

(17)
















4 Simulation




119891119888= 10GHz

119861 = 5MHz119891119903= 300Hz 119905 = 2 s

120596 = 2120587 rads1199031 = 06m


119903 (119891119889 119905119898) =

1sum

119896=111986010158401015840 expminus1198952120587

2120596119896119903119896cos (120596

119896119905119898)

120582

119905119898 (18)



minus1000

100

f (H

z)

05 1 15 20t (s)

(a)

02 04 06 08 10t (s)

minus2000

200

f (H

z)

(b)

02 04 06 08 10t (s)

minus2000

200

f (H

z)

(c)








02 04 06 08 10t (s)

minus200

0

200

f (H

z)

(a)

02 04 06 08 10t (s)


0100200300

f (H

z)

(b)


minus200

0

200

f (H

z)

05 1 15 20t (s)

(a)

minus5000

500

f (H

z)

05 1 15 20t (s)

(b)

minus5000

500

f (H

z)

05 1 15 20t (s)

(c)




119891119888= 10GHz


05 1 15 20t (s)

minus200

0

200

f (H

z)

(a)

05 1 15 20t (s)

minus2000

200

f (H

z)

(b)

05 1 15 20t (s)

minus2000

200

f (H

z)

(c)


119861 = 5MHz119891119903= 400Hz 119905 = 2 s

120596 = 2120587 rads1199031 = 18m 1199032 = 006m


119903 (119891119889 119905119898) =

2sum

119896=111986010158401015840


120582

+

21205962119877119896cos (1205962119905)

120582

]

sdot 119905119898

(19)


119891119888= 10GHz

119861 = 5MHz119891119903= 400Hz 119905 = 2 s

1205961 = 2120587 rads 1205962 = 4120587 rads1199031 = 03m 1198771 = 015m1199032 = 003 1198772 = 0015

10 03 04 05 06 07 08 0901 02

t (s)


0100200300

f (H

z)

(a)

02 03 04 05 06 07 0801

t (s)


060120180240300

f (H

z)

(b)





119903lt 2119891max


5 Conclusion






Acknowledgments


References
















1minimizationrdquo






RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











min119909isin119877119873

1199091

st Φ119909 = 119910

(12)



119898 Since 119903(119891

119889 119905119898) is sparse in


120595119894119896=

1radic119873

exp(minus1198952120587 119894119896119873

) 0 le 119894 119896 le 119873 (13)








119872 ge 119862 sdot 1199062(Φ119873times119873

Ψ) sdot 119870 sdot log119873 (14)



119906 (Φ119873times119873


10038161003816100381610038161003816(120593119896 120595119895)

10038161003816100381610038161003816 (15)




1 2 43 5 6 7 8 9 10

1 3 6 109

t

t



7perturbation






119894to 1199051015840

119894= 119905119894+ Δ119894 where Δ

119894is



1199051015840




119894since




119898 there is no




119894is

represented as 1198911015840119904= (1Δ119905)119891




119903 (119891119889 119905119898) = 119908 (119905

119899 119905119898)

119870

sum

119896=111986010158401015840 exp minus1198952120587119891

119889119896(119905119899) 119905119899 (16)


1 119905119898le119905119899le119905119898+119871minus1




Extracting randomly






time sequence ti












Θ119872times119873

=

[

[

[

[

[

[

[

[

119890119895212058711989101199050

119890119895212058711989111199050


119890119895212058711989101199051

119890119895212058711989111199051


d

11989011989521205871198910119905119871minus1

11989011989521205871198911119905119871minus1


]

]

]

]

]

]

]

]

(17)
















4 Simulation




119891119888= 10GHz

119861 = 5MHz119891119903= 300Hz 119905 = 2 s

120596 = 2120587 rads1199031 = 06m


119903 (119891119889 119905119898) =

1sum

119896=111986010158401015840 expminus1198952120587

2120596119896119903119896cos (120596

119896119905119898)

120582

119905119898 (18)



minus1000

100

f (H

z)

05 1 15 20t (s)

(a)

02 04 06 08 10t (s)

minus2000

200

f (H

z)

(b)

02 04 06 08 10t (s)

minus2000

200

f (H

z)

(c)








02 04 06 08 10t (s)

minus200

0

200

f (H

z)

(a)

02 04 06 08 10t (s)


0100200300

f (H

z)

(b)


minus200

0

200

f (H

z)

05 1 15 20t (s)

(a)

minus5000

500

f (H

z)

05 1 15 20t (s)

(b)

minus5000

500

f (H

z)

05 1 15 20t (s)

(c)




119891119888= 10GHz


05 1 15 20t (s)

minus200

0

200

f (H

z)

(a)

05 1 15 20t (s)

minus2000

200

f (H

z)

(b)

05 1 15 20t (s)

minus2000

200

f (H

z)

(c)


119861 = 5MHz119891119903= 400Hz 119905 = 2 s

120596 = 2120587 rads1199031 = 18m 1199032 = 006m


119903 (119891119889 119905119898) =

2sum

119896=111986010158401015840


120582

+

21205962119877119896cos (1205962119905)

120582

]

sdot 119905119898

(19)


119891119888= 10GHz

119861 = 5MHz119891119903= 400Hz 119905 = 2 s

1205961 = 2120587 rads 1205962 = 4120587 rads1199031 = 03m 1198771 = 015m1199032 = 003 1198772 = 0015

10 03 04 05 06 07 08 0901 02

t (s)


0100200300

f (H

z)

(a)

02 03 04 05 06 07 0801

t (s)


060120180240300

f (H

z)

(b)





119903lt 2119891max


5 Conclusion






Acknowledgments


References
















1minimizationrdquo






RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Extracting randomly






time sequence ti












Θ119872times119873

=

[

[

[

[

[

[

[

[

119890119895212058711989101199050

119890119895212058711989111199050


119890119895212058711989101199051

119890119895212058711989111199051


d

11989011989521205871198910119905119871minus1

11989011989521205871198911119905119871minus1


]

]

]

]

]

]

]

]

(17)
















4 Simulation




119891119888= 10GHz

119861 = 5MHz119891119903= 300Hz 119905 = 2 s

120596 = 2120587 rads1199031 = 06m


119903 (119891119889 119905119898) =

1sum

119896=111986010158401015840 expminus1198952120587

2120596119896119903119896cos (120596

119896119905119898)

120582

119905119898 (18)



minus1000

100

f (H

z)

05 1 15 20t (s)

(a)

02 04 06 08 10t (s)

minus2000

200

f (H

z)

(b)

02 04 06 08 10t (s)

minus2000

200

f (H

z)

(c)








02 04 06 08 10t (s)

minus200

0

200

f (H

z)

(a)

02 04 06 08 10t (s)


0100200300

f (H

z)

(b)


minus200

0

200

f (H

z)

05 1 15 20t (s)

(a)

minus5000

500

f (H

z)

05 1 15 20t (s)

(b)

minus5000

500

f (H

z)

05 1 15 20t (s)

(c)




119891119888= 10GHz


05 1 15 20t (s)

minus200

0

200

f (H

z)

(a)

05 1 15 20t (s)

minus2000

200

f (H

z)

(b)

05 1 15 20t (s)

minus2000

200

f (H

z)

(c)


119861 = 5MHz119891119903= 400Hz 119905 = 2 s

120596 = 2120587 rads1199031 = 18m 1199032 = 006m


119903 (119891119889 119905119898) =

2sum

119896=111986010158401015840


120582

+

21205962119877119896cos (1205962119905)

120582

]

sdot 119905119898

(19)


119891119888= 10GHz

119861 = 5MHz119891119903= 400Hz 119905 = 2 s

1205961 = 2120587 rads 1205962 = 4120587 rads1199031 = 03m 1198771 = 015m1199032 = 003 1198772 = 0015

10 03 04 05 06 07 08 0901 02

t (s)


0100200300

f (H

z)

(a)

02 03 04 05 06 07 0801

t (s)


060120180240300

f (H

z)

(b)





119903lt 2119891max


5 Conclusion






Acknowledgments


References
















1minimizationrdquo






RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus1000

100

f (H

z)

05 1 15 20t (s)

(a)

02 04 06 08 10t (s)

minus2000

200

f (H

z)

(b)

02 04 06 08 10t (s)

minus2000

200

f (H

z)

(c)








02 04 06 08 10t (s)

minus200

0

200

f (H

z)

(a)

02 04 06 08 10t (s)


0100200300

f (H

z)

(b)


minus200

0

200

f (H

z)

05 1 15 20t (s)

(a)

minus5000

500

f (H

z)

05 1 15 20t (s)

(b)

minus5000

500

f (H

z)

05 1 15 20t (s)

(c)




119891119888= 10GHz


05 1 15 20t (s)

minus200

0

200

f (H

z)

(a)

05 1 15 20t (s)

minus2000

200

f (H

z)

(b)

05 1 15 20t (s)

minus2000

200

f (H

z)

(c)


119861 = 5MHz119891119903= 400Hz 119905 = 2 s

120596 = 2120587 rads1199031 = 18m 1199032 = 006m


119903 (119891119889 119905119898) =

2sum

119896=111986010158401015840


120582

+

21205962119877119896cos (1205962119905)

120582

]

sdot 119905119898

(19)


119891119888= 10GHz

119861 = 5MHz119891119903= 400Hz 119905 = 2 s

1205961 = 2120587 rads 1205962 = 4120587 rads1199031 = 03m 1198771 = 015m1199032 = 003 1198772 = 0015

10 03 04 05 06 07 08 0901 02

t (s)


0100200300

f (H

z)

(a)

02 03 04 05 06 07 0801

t (s)


060120180240300

f (H

z)

(b)





119903lt 2119891max


5 Conclusion






Acknowledgments


References
















1minimizationrdquo






RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










05 1 15 20t (s)

minus200

0

200

f (H

z)

(a)

05 1 15 20t (s)

minus2000

200

f (H

z)

(b)

05 1 15 20t (s)

minus2000

200

f (H

z)

(c)


119861 = 5MHz119891119903= 400Hz 119905 = 2 s

120596 = 2120587 rads1199031 = 18m 1199032 = 006m


119903 (119891119889 119905119898) =

2sum

119896=111986010158401015840


120582

+

21205962119877119896cos (1205962119905)

120582

]

sdot 119905119898

(19)


119891119888= 10GHz

119861 = 5MHz119891119903= 400Hz 119905 = 2 s

1205961 = 2120587 rads 1205962 = 4120587 rads1199031 = 03m 1198771 = 015m1199032 = 003 1198772 = 0015

10 03 04 05 06 07 08 0901 02

t (s)


0100200300

f (H

z)

(a)

02 03 04 05 06 07 0801

t (s)


060120180240300

f (H

z)

(b)





119903lt 2119891max


5 Conclusion






Acknowledgments


References
















1minimizationrdquo






RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













Acknowledgments


References
















1minimizationrdquo






RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article micro-doppler ambiguity resolution based...

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