target vibration estimation in sar based on phase-analysis ......an efficient vibration estimation...

RESEARCH Open Access

Target vibration estimation in SAR basedon phase-analysis methodWeijie Xia* and Linlin Huang

Abstract

An efficient vibration estimation method for synthetic aperture radar (SAR) systems based on phase-analysismethod is presented. Small vibrations, which consist of multiple frequency components and are usually ratherweak, introduce phase modulation in radar echoes. They contain important signatures of objects. Extracting andanalyzing the phase from radar echoes is one of the significant approaches to acquire vibration characteristics. Inorder to separate different targets in different range cells, the proposed method acts the range compression andrange migration correction on the radar echoes, followed by extracting the phase history of each target’s radarechoes. Since the phase history contains the time-varying ranges from the targets to airborne radar besides thevibration signal, we apply wavelet transform to the phase history to extract the vibration signal. Then the vibrationsignatures would be estimated quantitatively. The method is a more efficient and convenient approach to estimatethe vibration amplitudes and frequencies of one or more targets, whose vibration signal is a single-frequency ormulti-frequency signal. Moreover, it can be applied in the case where the airborne radar moves along the desirableor undesirable trajectory. Simulation results indicate that this method provides a larger range of estimable vibrationfrequency, higher estimation precision, and lower computation complexity.

Keywords: Linear frequency modulated (LFM) signal, Phase analysis, Vibration, Wavelet transform (WT)

1 IntroductionVibration is ubiquitous in the real world. Objects’ vibra-tion signatures bear vital information about the type ofthe objects [1]. Spaceborne synthetic aperture radar (SAR)can operate over thousands of kilometers, which is an ef-fective way to acquire vibration signatures of structuresplaced in remote locations. Small vibrations introducephase modulation in the returned SAR signals, which isoften referred to as the micro-Doppler effect [2]. Byanalyzing the micro-Doppler effect, we can identify andclassify the objects according to the vibration signa-tures. In addition, the vibration signatures can be usedto compensate the phase errors in SAR returns to elim-inate the image blur [3, 4]. Therefore, target vibrationsignatures estimation has great significance.Several methods have been presented in the literature

which deal with the problem of how to extract the vibra-tion signature with SAR systems. The phase-modulated

signals induced by vibration with Cohen’s class time-fre-quency method were analyzed, and the time-frequencysignatures and the SAR image of a vibrating target werestudied in [5–7]. In [8] and [9], the Doppler frequency ofthe rotating and vibrating targets was analyzed withsmoothed pseudo Wigner-Ville distribution. A novel vi-bration estimation method based on the discrete fractionalFourier transform (DFRFT) was presented in [10–12]. Themethod in [13] with sinusoidal frequency modulationFourier transform (SFMFT) was proposed to obtain thefrequency spectrum of vibration traces. In [14], the tar-get was processing with autofocus to estimate the phaseerror caused by vibration, and the vibration parameterswere calculated based on the phase error. Anotherextracting method of vibrating features based on slowtime envelope (STE) signatures was presented in [15].In [16] and [17], a micro-Doppler reconstruction methodusing azimuth time-frequency tracking of the phase his-tory was proposed.In this paper, a vibration estimation method using

SAR is presented based on phase analysis. In comparisonwith the previous works, our approach differs in two

* Correspondence: [email protected] Laboratory of Radar Imaging and Microwave Photonics (Nanjing Univ.Aeronaut. Astronaut.), Ministry of Education, College of Electronic andInformation Engineering, Nanjing University of Aeronautics and Astronautics,Nanjing 210016, China

EURASIP Journal on Advancesin Signal Processing

© 2016 The Author(s). Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, andreproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link tothe Creative Commons license, and indicate if changes were made.

Xia and Huang EURASIP Journal on Advances in Signal Processing (2016) 2016:94 DOI 10.1186/s13634-016-0390-7

http://crossmark.crossref.org/dialog/?doi=10.1186/s13634-016-0390-7&domain=pdfhttp://orcid.org/0000-0002-9616-8259mailto:[email protected]://creativecommons.org/licenses/by/4.0/

main ways. On the one hand, this method is applicableto the case where a single scatterer vibrating in multiplefrequencies and several scatterers distributed in differentrange cells are placed within a SAR scene. In addition, itis also suitable for the situation where the spaceborneradar moves along the undesirable trajectory. On theother hand, compared to methods which rely on theinterpretation of time-frequency representations, theproposed method provides a quantitative estimation ofthe vibration signatures by offering the history of theinstantaneous displacement and the spectrum of thevibrating object. The estimation results contain completeparameters, not only the vibration amplitudes and fre-quencies but also the initial phases included.The procedure of this method is as follows. It starts

with the conventional SAR processing procedure to ob-tain a nonstationary signal from the vibrating target.First, the returned SAR signals are demodulated. Second,range compression is acted on the demodulated SARechoes, followed by an application of range cell migra-tion correction to range lines. Hence, the signal from avibrating target is focused on a range line. Since smallvibrations modulate the phase in radar echoes, we ex-tract the phase history of each range line containingvibrating objects and also analyze the restrictive con-dition of phase ambiguity. Next, by performing thewavelet transform, the vibration displacement is ac-quired and vibration amplitudes and frequencies ofeach target are estimated quantitatively. The influenceof noises on the performance of the proposed approach isalso discussed. The method is capable of extracting vibra-tion signatures in the case where the airborne radar movesalong the desirable or undesirable trajectory. Finally, thesimulation is given and this method has been comparedwith other radar-based vibration estimation method. Theresults illustrate the superior of this method.The remainder of this paper is organized as follows. In

Section 2, the motion model is established and the math-ematical formulas of SAR returns containing vibration sig-natures are derived. In Section 3, the phase-analysisalgorithm including phase extraction and wavelet transformis introduced, followed by the discussion related to thesignal-to-noise ratio (SNR) in Section 4. Simulations areprovided and a comparison between the proposed methodand some of others is presented in Section 5. Section 6gives the conclusion.

2 Signal modelFigure 1 shows a 3-D SAR flight geometry. Aircraft withthe airborne radar sensor flies along the axis OX fromthe point A to B, whose instantaneous location is (x(η),y(η), h), where η is the azimuth time. The vibrating targetis located at the point C (0, y0, 0). The nominal line-of-sight distance from the target to the radar sensor is

r ηð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 ηð Þ þ y ηð Þ−y0ð Þ2 þ h2

qð1Þ

Let rv(η) denote the projection of the vibration displace-ment onto the line of sight (LOS) from the radar sensor tothe target. The instantaneous slant range becomes

R ηð Þ ¼ r ηð Þ þ rv ηð Þ ¼ r ηð Þ þ rv0 ηð Þ⋅ cosθ ηð Þ ð2Þwhere θ(η) is the angle between the instantaneous slantrange and the shortest distance from the target to theradar sensor and rv0 represents rv(η) for θ(η) = 0. Sincethe change of θ(η) is usually small, we obtain thatrv(η) ≈ rv0(η). Then R(η) can be approximated by

RðηÞ≈rðηÞ þ rv0ðηÞ ð3ÞMost vibration signals are usually referred to as a peri-

odic function, which can be expanded in Fourier series.Any vibration that generated in practice satisfies theDirichlet conditions, and therefore, the Fourier series ofvibration signal is convergent [18]. Consequently, thetraces of the vibration can be concluded into a series ofperiodic sinusoids. Here, we assume that the vibrationdisplacement projected onto LOS is modeled by

rv0 ηð Þ ¼Xi¼1

ai sin 2πf viηþ φið Þ ð4Þ

where ai is the projection of the vibration amplitudeonto LOS, fvi is the vibration frequency, and φi is the ini-tial phase of the vibration signal.Consider a side-looking strip-mode SAR, whose

transmitting waveform is a chirp signal, with carrierfrequency f0 and chirp rate Kr. Each returned SARpulse is demodulated by the transmitting signal delayedappropriately by the round-trip time to the center ofthe illuminated patch. A demodulated pulse of a singlepoint target can be written as [19]

A

B

C

h

Z

O

X

Y0y

r

Fig. 1 Three-dimensional SAR flight geometry

Xia and Huang EURASIP Journal on Advances in Signal Processing (2016) 2016:94 Page 2 of 11

s0 τ; ηð Þ ¼ A0wr τ− 2R ηð Þc� �

wa η−ηc� �

⋅ exp −j4πf 0R ηð Þc

� �exp jπKr τ−

2R ηð Þc

2( )

ð5Þ

where A0 is a complex constant, denoting the reflectivityof the target, c is the propagation speed of the pulse, τ isthe range time, ηc represents zero Doppler time, wr[•] isthe envelope of the pulse, usually approximated as arectangular window and wa[•] is the radiation pattern ofthe radar antenna. If there are more than one point targets,the SAR returns become

s0 τ; ηð Þ ¼XNn¼1

A0nwr τ−2Rn ηð Þ

c

� �wa η−ηc

� �

⋅ exp −j4πf 0Rn ηð Þc

� �exp jπKr τ−

2Rn ηð Þc

2( )

ð6Þ

According to (3), (4), and (6), we know that the smallrange perturbation of vibrating targets modulates thephase term in radar echo, which is referred to as themicro-Doppler effect. Figure 2 shows a reconstructedSAR image of a vibrating target. Due to the micro-Doppler effect, azimuth compression cannot focus thevibrating scatterer on the correct cross-range position,with some significant sidelobes nearby, which is com-monly called ghost targets [20]. Consequently, we canextract some information related to micro-motion byanalyzing the phase of the SAR returns.

3 Algorithm3.1 Phase extractionRange compression is applied to the SAR returned signalto separate the scatterers in range. By performing the

matching filtering in frequency domain, we get the resultof the range compression

src τ; ηð Þ ¼XNn¼1

A0npr τ−2Rn ηð Þ=c½ �wa η−ηc� �

⋅ exp −j4πf 0Rn ηð Þ=cf g

ð7Þ

where pr[•] is the a sinc function. Then the range migra-tion correction algorithm can be applied to (7) to correctfor range cell migration, the signal becomes

src0 τ; ηð Þ ¼

XNn¼1

srcn0 τ; ηð Þ

¼XNn¼1

A0npr τ−2R0n=c½ �wa η−ηc� �

⋅ exp −j4πf 0Rn ηð Þ=cf g

ð8Þ

where R0n is the distance from nth point target to themidaperture. Now, the range-compressed phase historyof one point target is in the same range cell. The phaseof the range line containing nth target can be solved by

ϕn ηð Þ ¼ arg srcnʼ τ; ηð Þ½ � τ¼2R0n=cj ð9Þi.e.,

ϕn ηð Þ ¼ −4πf 0Rn ηð Þ=c ð10ÞFigure 3 shows the range lines of two vibrating targets,

which can be obtained by applying range compressionand range migration correction to the SAR returns. Thetwo point targets are separated in range.Since the measured phase is limited only between − π

and π, the phase ambiguity occurs when some values ofthe real phase − 4πf0Rn(η)/c are over this range. The phasetotal shift between the measured phase and the real phaseis n times of 2π, where n is an integer to be determined.Fortunately, this can be corrected by comparing the

Range/bin

Cro

ss-r

ange

/bin

50 100 150 200 250 300

100

200

300

400

500

600

700

800

900

1000

1100

Fig. 2 SAR image of a vibrating point target

Range/bin

Azi

mut

h/sa

mpl

e

50 100 150 200 250 300

100

200

300

400

500

600

700

800

900

1000

1100

Fig. 3 Range lines of two vibrating targets


adjacent samples. Supposing the maximal real phasedifference between adjacent samples is below π, a so-lution of the phase revising function is

prðηkÞ ¼prðηk−1 Þ−2π; ϕnðηkÞ−ϕnðηk−1Þ > πprðηk−1Þ þ 2π; ϕnðηk−1Þ−ϕnðηkÞ > πprðηk−1Þ; jϕnðηkÞ−ϕnðηk−1Þj < π

8<:

ð11Þwhere pr(ηk) is the phase total shift between the realphase and measurements at kth azimuth time. Thus, therevised phase can be expressed as

ϕn0ðηÞ ¼ ϕnðηÞ þ prðηÞ ð12Þ

However, the discrete signal is subjected to a problem. Ifthe real phase difference between adjacent samples isgreater than π, we cannot acquire the correct real phaseby above phase-revising method. For example, in one case,if the adjacent samples’ real phase difference is greaterthan 2π, like point 2 and 3 in Fig. 4a, the phase shift be-tween the real phase and the measured phase of point 3 isalso larger than 2π (actually the phase shift equals 4πhere). By applying the above method, we only add 2π tothe measured phase (point 3') to obtain the revised phase(point 3 ''), which is not equal to the real phase (point 3).In another case, as shown in Fig. 4b, the adjacent samples’real phase, like point 5 and 6, whose difference is smallerthan 2π but larger than π, so that the phase difference be-tween their measured phase (point 5 ' and 6 ') is below π.With application of the above phase-revising method, the

revised phase (point 6 '') is equal to the measured phase(point 6 '), but not the real phase (point 6). In order toavoid the occurrence of the above, we need to increasepulse repetition frequency (PRF) to make sure that the realphase difference between adjacent samples is below π.Therefore, the limitation of PRF has to be addressed here.First, we consider the situation where the airborne

radar moves in a straight line at a constant speed, andsuppose that there is only one vibrating target whosevibration displacement along the direction of radar in-cidence is a sin(2πfvη). Let fprf denote PRF, and thesampling interval of slow time is

Δη ¼ 1=f prf ð13Þ

Since the maximal phase difference between adjacentsamples is always occurred where the slope of vibrationdisplacement is maximal, it can be expressed as

Δϕmax ¼4πf 0⋅2asinð2πf v⋅Δη=2Þ

c

¼ 8πf 0asinðπf v⋅ΔηÞc

ð14Þ

Due to Δϕmax must be constrained below π, i.e., 8πf0asin(πfv ⋅Δη)/c < π, the requirement of PRF can be solved via

f prf >πf v

arcsin c8af 0

� ð15ÞAccording to Nyquist sampling theorem, PRF must be

larger than 2fv. And for the vibration amplitude meets

a > 0, arcsin c8af 0

�should be satisfied as 0 < arcsin

c8af 0

�< π2.

Then, we take another situation into account. The air-borne radar does not simply move in a straight trajec-tory, while it flies with a velocity vY, which is a velocityalong axis OY, that leads to the slant range R(η) fluctuat-ing with azimuth time. The range of fluctuation is muchlarger than vibration amplitude so that we can only takefluctuation into consideration. In this case, the maximalphase difference between adjacent samples becomes

Δϕmax ¼4πf 0 vYj jmax⋅Δη

cð16Þ

According to Δϕmax < π and (13), we can get the limit-ing condition of PRF

f prf >4f 0 vYj jmax

cð17Þ

3.2 Wavelet transformCompared to Fourier transform, wavelet transform (WT)is a local transform in time and frequency domain, which

1(1')

2(2')

3'

3

1' 2'

3'

4(4')

5(5')

6'

6

4' 5' 6'

3

5

5

3

Real Phase

Measured Phase

RevisingPhase

RevisingPhase

Revised Phase

a

b

Fig. 4 Two special cases about phase revising. a The adjacentsamples’ real phase difference is greater than 2π. b The adjacentsamples’ real phase difference is smaller than 2π but larger than π


can be used to extract information from signal effectively.Wavelet transform makes a multi-scale analysis of func-tions or signals by expansion, translation operations, sothat some characteristics can be fully highlighted. It isknown as mathematical microscope.According to (10), the phase of the range line is

− 4πf0Rn(η)/c, which can be written in

ϕn ηð Þ ¼ −4πf 0 r ηð Þ þ rv0 ηð Þð Þ=c ð18ÞThis expression demonstrates that the phase of the

range line is the sum of two parts: − 4πf0 ⋅ r(η)/c, which islinear proportional to the nominal line-of-sight distancefrom the target to the radar sensor and − 4πf0 ⋅ rv0(η)/c.The former varies slowly with azimuth time referred to asa low-frequency signal. The latter induced by vibration isequivalent to a high-frequency signal. Here, we extract thelow-frequency signal from the phase of range line bywavelet transform, and the remaining high-frequency sig-nal is the vibration signal that we need. The algorithmchart is shown in Fig. 5.If we applied Fourier transform to (18) directly, vibra-

tion signal would be submerged in the low-frequencysignal in spectrogram, as shown in Fig. 6. This is becausethe changed slant distance is equivalent to a low-frequencysignal, whose amplitude is much larger than vibration amp-litude. In this way, it is not easy to distinguish the vibrationsignal and acquire its frequency from the spectrum.Therefore, we have to separate the vibration signal fromthe low-frequency signal by WT.

3.3 Frequency, amplitude, and initial phase estimationAfter applying the wavelet transform to the phase historyof range lines, the vibration signal is extracted, which canbe expressed in the following form

~rv0 ηð Þ ¼Xi¼1

~ai sin 2π~f viηþ ~φi

�

ð19Þ

By applying the Fourier transform directly to (19), thespectrum is obtained. It is convenient to get the esti-

mated frequencies ~f vi from the spectrogram.

The amplitudes Ãi in frequency domain are also easyto obtain. Since the amplitudes in time domain and infrequency domain have the relationship as (20):

~Ai ¼ T2 =dη⋅~ai ð20Þ

where T is the total azimuth time and dη is the sampleinterval in azimuth direction, the estimated amplitudesare acquired.In order to estimate its initial phase, we define the

basis function as follows:

rvref η; φ̂ið Þ ¼ ~ai sin 2π~f viηþ φ̂i

�

ð21Þ

where φ̂i∈ 0; 2π½ � . The projection of estimated vibrationsignal ~rv0 ηð Þ on the basis function can be expressed as

S φ̂ið Þ ¼ ~rv0 ηð Þ; rvref η; φ̂ið Þh i ð22Þ

The estimated initial phase ~φi is the φ̂i in [0, 2π] whichmakes S φ̂ið Þ achieve the maximum.

4 DiscussionsIn real-world applications, the performance of the pro-posed method is affected by the presence of noise. IfSAR returns are highly corrupted by noise, some errorsmay exist in the estimated vibration displacement. Thus,we are interested in finding the SNR threshold abovewhich the estimation result is acceptable.One metric to evaluate the estimation accuracy is the

normalized root-mean-square error (NRMSE) [21]. Itdenotes the normalized Euclidean distance between theestimated vibration signal r̂ v0 ηð Þ and the real vibrationsignal rv0(η). NRMSE can be calculated by

Multi-scaleanalysis by

WT

Remove high-frequency

component fromcoefficient ofeach scale

Reconstruct low-frequency signalby inverse WT

Subtract low-frequency

signal from thephase of range

line

The phase ofthe range line

Low-frequency

signal

High-frequencysignal (vibration

signal)

Fig. 5 The algorithm chart of separating signal by WT

-500 0 5000

50

100

150

Low-frequency signal

Vibration signal

Frequency/Hz

Am

plitu

de

Fig. 6 Spectrum obtained directly by applying FFT to the phase


NRMSE ¼ ~rv0 ηð Þ−rv0 ηð Þk k————————rv0 ηð Þk k ð23Þ

where ‖ • ‖ denotes the l2 norm operator.Adding a zero-mean complex-valued white Gaussian

noise to every SAR return, we compute NRMSEs toevaluate the performance of the proposed method in es-timating the vibration signal under different SNR levels.Here, we define the SNR as

SNR ¼ 10 lg σ2

σ2wð24Þ

where σ2 is the power of each return echo and σ2w is thevariance of the additive noise.Figure 7 shows an example of the vibration signal esti-

mated by the proposed method. The real vibration signalis assumed to be a sinusoidal wave with a frequency of20 Hz. Figure 7b shows when SNR = 10 dB, the noisehas a little influence on the estimated signal. When SNRdecreases to 0 dB, the estimated signal is severely cor-rupted by the noise.In order to assess the SNR threshold clearly, we car-

ried out a simulations for three vibration frequenciesunder different SNRs. The normalized root-mean-squareerrors are plotted in Fig. 8. The vibration frequencies are10, 20, and 30 Hz, respectively. When SNR is lower than−15 dB, the NRMSEs are relatively higher. And with thedecrease of SNR, the NRMSEs rise extremely fast. In thiscase, the estimated frequency may be correct, but itseems not reliable. When SNR is increased to −15 dB,the NRMSEs drop to an acceptable level. But the errorsplateau as the SNR increases. The results also imply thatwhen SNR is larger than −15 dB, the higher the vibra-tion frequency is, the lower the NRMSEs are.

5 Simulations5.1 Airborne radar along desirable trajectory5.1.1 Single point targetThe simulated SAR is an airborne side-looking strip-mode SAR working in the Ku band. Table 1 lists the keysystem parameters associated with the simulation. Thevibrating target with amplitude of 5 mm, a frequency of20 Hz, and initial phase of π rad is located at (0,7000,0).After applying range compression and range migrationcorrection to the simulated SAR returns, the range lineof the vibrating object was obtained. Figure 9a shows thetime-frequency representation and the real phase of thevibrating target’s range line. Its time-frequency represen-tation illustrates that the small vibration modulates thephase history of the radar echoes. The real phase was ac-quired by the method proposed in Section 3.1, whichconsists of two components: the variation with azimuthtime of the nominal line-of-sight distance from the tar-get to the radar sensor and the vibration signal, whichare referred to as the low-frequency signal and the high-frequency signal, respectively. Figure 9b shows the resultof the estimation. The first figure is the nominal line-of-

0 200 400 600 800 1000 1200-0.01

0

0.01

0 200 400 600 800 1000 1200-0.01

0

0.01

0 200 400 600 800 1000 1200-0.02

0

0.02

Azimuth time/s

Mag

nitu

deM

agni

tude

Mag

nitu

de

a

b

c

Fig. 7 Vibration signal estimated by phase-analysis method. a Realvibration signal. b Estimated signal with SNR = 10 dB. c Estimatedsignal with SNR = 0 dB

-30 -20 -10 0 10 200

0.2

0.4

0.6

0.8

1

1.2

1.4

SNR/dB

The

nor

mal

ized

roo

t mea

n-sq

uare

-err

or

Vibration frequency:f=10 HzVibration frequency:f=20 HzVibration frequency:f=30 Hz

Fig. 8 Normalized root-mean-square errors in estimating vibrationsignal for three frequencies under different SNRs usingphase-analysis-based estimator

Table 1 SAR system parameters used in the simulation

Parameter Quantity

Carrier frequency 15 GHz

Bandwidth 400 MHz

Pulse duration 0.5 μs

Range sampling frequency 420 MHz

Range resolution 0.375 m

Plane velocity 100 m/s

Length of the synthetic aperture 125 m

Pulse repetition frequency 880 Hz


sight distance varied with azimuth time, which is equivalentto a low-frequency signal separated from the phase of therange line by WT. Then subtracting this low-frequency sig-nal from the phase of the range line, we obtained the vibra-tion signal, as shown in the second figure. After applyingfast Fourier transform (FFT) to the estimated vibration sig-nal, the vibration frequency was acquired, as illustrated inthe last figure. The estimated amplitude and initial phase is4.9 mm and 3.0945 rad by the proposed method.Supposing that the vibration signal is a multi-frequency

sinusoidal signal, whose equation is given by

rv0 ηð Þ ¼ 0:003 sin 20πηþ 11π=9ð Þþ 0:006 sin 30πηþ πð Þþ 0:004 sin 40πηþ π=4ð Þ ð25Þ

Time-frequency representation and phase of the rangeline is shown in Fig. 9c. It is clear that a multi-frequencysignal modulates the phase history of SAR returned sig-nals. Figure 9d demonstrates the result of the estimation,from which we know that the estimated vibration signalcontains three frequencies. The estimated frequenciesare 9.917, 15.7, and 20.66 Hz, which are almost the sameas the real frequencies. The estimated amplitudes are

2.1, 5.7, and 3.8 mm correspondingly and the estimatedinitial phase is as Table 2.

5.1.2 Multiple point targets in different range cellsThen we considered the case where multiple pointtargets in different range cells are contained in thescene. Suppose that there are three vibrating targetsO, P, and Q, located at (0,6990,0), (0,7000,0), and(0,7005,0). The vibration amplitude, frequency, andinitial phase of O, P, and Q are shown in Table 3.The range lines of the three targets are shown inFig. 10a. Applying the proposed method to these rangelines respectively, we got the results of estimation forpoint O, P, and Q, as shown in Fig. 10b–d. The esti-mated vibration parameters are listed in Table 4, whichare almost the same as the corresponding parameters setin Table 3.

Time-frequency representation of the range line

Azimuth time/sample

Fre

quen

cy/H

z

200 400 600 800 1000-400

-200

0

200

400

0 200 400 600 800 1000 1200-0.4

-0.3

-0.2

-0.1

0Phase of the range line

Azimuth time/sample

Am

plitu

de o

f Pha

se

Time-frequency representation of the range line

Azimuth time/sample

Fre

quen

cy/H

z

200 400 600 800 1000-400

-200

0

200

400

0 200 400 600 800 1000 1200-0.4

-0.3

-0.2

-0.1

0Phase of the range line

Azimuth time/sample

Am

plitu

de o

f Pha

se

0 200 400 600 800 1000 1200-0.4

-0.2

0Low-frequency signal

Azimuth time/sample

0 200 400 600 800 1000 1200-0.01

0

0.01Vibration signal

Azimuth time/sample

-500 -400 -300 -200 -100 0 100 200 300 400 5000

2

4Spectrum of vibration signal

Frequency/Hz

X: 20.66Y: 2.713

0 200 400 600 800 1000 1200-0.2

-0.1


Azimuth time/sample

0 200 400 600 800 1000 1200-0.02

0


Azimuth time/sample

-200 -150 -100 -50 0 50 100 150 2000

2


Frequency/Hz

a b

c d

Fig. 9 The simulation results for one vibrating point target. a, b Time-frequency representation, the phase of the range line and the vibrationestimation results for a target whose vibration signal is a single-frequency sinusoidal. c, d Time-frequency representation, the phase of the rangeline and the vibration estimation results for a target whose vibration signal is a multi-frequency sinusoidal

Table 2 Comparison of real phase and estimated phase

Real phase/rad 11π/9 (3.8397) π (3.1416) π/4 (0.7854)

Estimated phase/rad 3.7856 3.3929 0.8011


5.2 Airborne radar along undesirable trajectoryAs mentioned in Section 3.1, if the aircraft flies with avelocity vY, the slant range R(η) would fluctuate with azi-muth time. Suppose that the aircraft’s track is a sinusoid,which can be written as

y ¼ sinðπ=62:5xÞ ð26Þ

The vibrating target with amplitude of 5 mm and afrequency of 10 Hz is located at (0,7000,0). Applying thesame method to the simulated SAR returns, we obtainedthe result as shown in Fig. 11.In fact, the aircraft’s fluctuation frequency is very low so

that it is also can be referred to as a low-frequency signalcompared to the vibration signal. Hence, the proposed

method still holds true in the case of the undesirabletrajectory.

5.3 Comparison with TF analysis method, DFRFT-basedmethod, and HAF-based methodAs described in Section 1, several methods about extract-ing the vibration signatures with SAR systems have beenproposed during the last few years. The time-frequency(TF) analysis method is an important tool to analyze themicro-Doppler effect. DFRFT is an effective method toestimate the time-varying accelerations, frequencies, anddisplacements associated with vibrating objects [10–12].Moreover, a time-frequency tracking algorithm based onthe high-order ambiguity function (HAF) is employed toreconstruct the instantaneous frequency law of the micro-Doppler [16, 17]. In this section, we compare them withphase-analysis method.

5.3.0.1 Time-frequency analysis method The time-frequency distributions to provide methods use time-frequency distributions to provide an analysis of themicro-Doppler effect. The time-frequency representationof the range line is shown in Fig. 12, which roughly re-veals a 4-Hz vibration component. However, this is only

Table 3 Vibration parameters of three targets

Targets O P Q

Amplitude/mm 3 6 4

Frequency/Hz 25 15 20

Initial phase/rad 2π/3 (2.0944) π (3.1416) π/4 (0.7854)

Range/bin

Azi

mut

h/sa

mpl

e

50 100 150 200 250 300

100

200

300

400

500

600

700

800

900

1000

1100

0 200 400 600 800 1000 1200-0.2

0

0.2Low-frequency signal

Azimuth time/sample

0 200 400 600 800 1000 1200-5

0

5x 10

-3 Vibration signal

Azimuth time/sample

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0


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a b

c d

O P

Q

Fig. 10 The simulation results for three vibrating point targets in different range cells. a Shows the range lines of the three targets. b–d Theresults of vibration estimation for target O, P, and Q


a qualitatively deduced observation. To obtain preciseestimations of the instantaneous displacement from thetime-frequency representation, further estimation proce-dures are required. In addition, the resolution of time-frequency distribution is closely related to the length ofthe time window, and we cannot get optimal time reso-lution and frequency resolution at the same time. If thevibration frequencies are higher, or there are multiple vi-bration components, the time-frequency representationwould not be clear enough to reveal all the vibrationcomponents. Whereas the proposed method provides dir-ect quantitative estimations of the vibration displacementand a larger range of the estimated vibration frequency.

5.3.0.2 DFRFT-based method The DFRFT-based methodapproximates the nonstationary signal by a chirp signalin a small time window. DFRFT is applied to estimatethe vibration acceleration. Then the history of the vibra-tion acceleration can be reconstructed and the vibrationfrequency can be obtained by calculating DFT of the ac-celeration. The proposed method extracts the phase ofthe global data directly and applies the WT to the phasehistory to obtain the vibration signal.A simulated example is provided to compare the capabil-

ity of the two methods on vibration signal reconstructionand spectrum estimation. We assumed that the point tar-get’s vibration signal is rv0(η) = 0.005 sin(8πη). The estima-tion results are shown in Fig. 13. The estimated frequencyof DFRFT-based method and phase-analysis method is

X: 0.0005687 Y: 17.42Index: 8725RGB: 0.813, 0, 0

Time/s

Fre

quen

cy/H

z

Time-Frequency Distribution

X: 0.2508 Y: 48.94Index: 5635RGB: 0.875, 1, 0.125

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

-400

-300

-200

-100

0

100

200

300

400

Fig. 12 The time-frequency representation of the range line (the actualvibration frequency is 4 Hz)

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6-4

-2

0

2

4x 10

-3

Azimuth time/s

Am

plitu

de

Vibration acceleration histroy

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0.1

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0.4

X: 4.355Y: 0.289

Frequency/Hz

Am

plitu

de

Spectrum of vibration signal

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0

0.01


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plitu

de

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X: 3.931Y: 0.7398


Frequency/Hz

Am

plitu

de

a

b

Fig. 13 The estimation results by two different methods. a Obtainedby DFRFT-based method. b Obtained by phase-analysis method

Table 4 Estimated vibration parameters of three targets

Targets O P Q

Amplitude/mm 2.6 5.4 3.8

Frequency/Hz 25.62 15.7 20.66

Initial phase/rad 2.0577 3.2358 0.8482

0 200 400 600 800 1000 1200-1

0


Azimuth time/sample

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0


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4

X: 9.253Y: 2.249


Frequency/Hz

Fig. 11 The estimation results in the case where airborne radarmoves along undesirable trajectory


4.355 and 3.931 Hz, and the computation time is 5.7754and 0.4108 s, respectively. It demonstrates that the phase-analysis method does an excellent reconstruction of smoothvibration displacement and provides more accurate estima-tion results. Furthermore, it also implies that the proposedmethod in this paper has lower computation complexity.

5.3.0.3 HAF-based method The HAF-based methodtracks the instantaneous frequency laws by dividing therange line of the vibrating target in nonoverlapping win-dows and locally approximating the phase with a poly-nomial expression. In each window, HAF algorithm isapplied, which is an iterative algorithm, to estimate thepolynomial coefficients. After the initial reconstruction,the instantaneous frequency laws must be filtered inorder to mitigate the possible gaps that may appear atthe transitions between estimation windows [17]. On theother hand, the phase-analysis method needs neither it-erative algorithm in each window nor filtering for esti-mated instantaneous frequency laws. Instead, it gives thehistory of the vibration displacement directly and thenthe vibration amplitudes, frequencies, and initial phasescan be estimated. Therefore, our proposed method mayhave relatively low computation complexity. Besides, ourmethod makes use of the global data so that the possibleerrors induced by transitions between the estimationwindows may be avoided.

6 ConclusionsIn this paper, a new method for radar-based vibrationestimation was presented, making the radar capable ofmore accurate vibration measurement. We obtain theSAR phase history by applying the range compressionand range migration correction to the SAR returns. Thenthe wavelet transform is acted on the phase history to ex-tract the vibration displacement. This method providesquantitative estimations of the vibration signature includ-ing vibration amplitudes, frequencies, and initial phases.Some simulations provided to demonstrate that it can esti-mate the vibration signature of one or several targets vi-brating in single or multiple frequencies successfully andit also can be used for the case where the aircraft with theradar sensor moves along desirable or undesirable trajec-tory. Note that in this paper, we assume that the aircraftflies at a constant speed. When the aircraft flies at thespeed varied with time, the fluctuation of the velocitywould generate a little interference to the phase history,but the vibration signal can be still extracted exactly.Therefore, the influence of the changed velocity can beneglected.However, this method has limitation that targets have

to be distributed in different range cells. We cannot ex-tract the vibration displacement of each target which isin the same range cell. In our future work, the case of

multiple point targets in the same range cell and themodels of real-world vibrating objects will be examinedcarefully, which will have great significance to objectrecognition, classification, and SAR imaging.

AcknowledgementsThis work was supported in part by the National Natural Science Foundationof China under Grant 61201366, in part by the Fundamental Research Fundsfor the Central Universities under Grant NS2016040, in part by the FundamentalResearch Funds for the Central Universities under Grant NJ20150020.

Competing interestsThe authors declare that they have no competing interests.

Received: 3 March 2016 Accepted: 25 August 2016

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AbstractIntroductionSignal modelAlgorithmPhase extractionWavelet transformFrequency, amplitude, and initial phase estimation

DiscussionsSimulationsAirborne radar along desirable trajectorySingle point targetMultiple point targets in different range cells

Airborne radar along undesirable trajectoryComparison with TF analysis method, DFRFT-based method, and HAF-based method

ConclusionsAcknowledgementsCompeting interestsReferences

target vibration estimation in sar based on phase-analysis ......an efficient vibration estimation...

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