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Millimeter-Wave Circular Synthetic Aperture RadarImaging

Shahrokh Hamidi∗ and Safieddin Safavi-Naeini∗

Abstract—In this paper, we present a high resolution mi-crowave imaging technique using a compact and low cost singlechannel Frequency Modulated Continuous Wave (FMCW) radarbased on Circular Synthetic Aperture Radar (CSAR) technique.We develop an algorithm to reconstruct the image from the rawdata and analyse different aspects of the system analytically.Furthermore, we discuss the differences between the proposedsystems in the literature and the one presented in this work.

Finally, we apply the proposed approach to the experimentaldata collected from a single channel FMCW radar operating at79 GHz and present the results.

Index Terms—Circular Synthetic Aperture Radar (CSAR),FMCW radar, high resolution imaging

I. INTRODUCTION

Synthetic Aperture Radar (SAR) imaging is a well-knowntechnique to produce high resolution radar images [1], [2].In the range direction the idea of pulse compression is usedto achieve high resolution. In the azimuth direction, it is therelative motion of the radar with resect to the target that givesrise to higher resolution. When the synthetic aperture is createdcircularly, the system is called Circular Synthetic ApertureRadar (CSAR). Previously, CSAR imaging has been addressedin the literature such as the airborne CSAR systems in [2]–[5]in which the circular aperture is created around the scene tobe imaged. In fact, the circular synthetic aperture encircles thescene. The Geostationary CSAR system operates in the sameway [6]. The ground based CSAR systems, a.k.a., ArcSAR,have also been studied extensively [7]–[11]. The ground basedCSAR systems are different from the airborne and spaceborneCSAR systems in a way that the area to be imaged can beoutside of the circular synthetic aperture which provides aunique opportunity to monitor the 360o surroundings of theCSAR system using a small circular aperture. Of course, weshould mention that the Unmanned Aerial Vehicle (UAV) andHelicopter based CSAR systems, while stationary with respectto the ground, can also operate in the same mode as the groundbased CSAR systems meaning that the scene to be imaged canbe outside of the synthetic circular aperture exactly like theconfiguration we consider in this work.

In [11], the authors have addressed the previous work donein the field of CSAR imaging presented in [7]–[9] and havedeveloped a ground based CSAR system which is the sameas the model we develop in this paper. They have created

Shahrokh Hamidi is with the Faculty of Electrical and Computer Engineer-ing, University of Waterloo, 200 University Ave W, Waterloo, ON., Canada,N2L 3G1. e-mail: Shahrokh.Hamidi@uwaterloo.ca.

Safieddin Safavi-Naeini is with the Faculty of Electrical and ComputerEngineering, University of Waterloo, 200 University Ave W, Waterloo, ON.,Canada, N2L 3G1. e-mail: safavi@uwaterloo.ca.

a circular aperture at X band. The scene to be imaged isoutside of the synthetic aperture. Their CSAR system operatesin two different modes which they have referred to as, scanmode and spot mode. The scan mode is referred to the casethat the angle between the antenna axis and the circular trackremains the same while the radar is collecting data from thescene to be imaged. In the spot mode, however, the antennais spinning on its axis while it is moving along the track todirect the mainlobe on the target at all times during the datagathering process. For the image reconstruction they have usedthe Range Doppler algorithm.

In [12]–[14], sub-aperture processing, Range Dopplermethod as well as ω − k algorithms have been used for theCSAR image reconstruction. In [15], a helicopter based CSARimaging system has been developed to create high resolutionimages from the ground while the helicopter is stationary.

In this paper, we specifically focus on CSAR imaging inwhich the scene to be imaged is located outside of the aperture.We further consider the case in which the angle between themain axis of the antenna and the circular synthetic apertureremains fixed during the data gathering process. This allowsus to cover the entire 360o surroundings of the radar in onefull rotation. As we mentioned before, our CSAR systemis similar to that of [11]. However, the algorithm that wedevelop to reconstruct the image is different. In [11] the RangeDoppler algorithm has been used for image reconstruction.First, they perform range compression and then they have usedthe Taylor expansion of the radial distance between the radarand the target up to the second order to perform azimuthcompression. The Range Doppler algorithm, however, canperform well only for narrow swath width in narrow beammode. Furthermore, for radars with high range resolution, suchas the one we consider in this work, the Range Cell Migration(RCM) phenomenon [1], if not compensated for, can degradethe quality of the image considerably. The RCM effect refersto the case that the energy of a point target appears in differentrange cells per different azimuth lines. When we are dealingwith small range cells (high range resolution) and gatheringdata with a wide beam over a large synthetic aperture, whichis the case we consider in this paper, the RCM is unavoidable.In [11], the authors have not addressed the RCM phenomenon.We, on the other hand, develop a joint range-azimuth algorithmcapable of forming a 2D image without considering anyapproximation. Our proposed algorithm is an exact solutionfor the CSAR imaging and it can operate for large swath widthin wide beam mode. We further describe the angular samplespacing analytically which [11] has not addressed.

In [12]–[14], the authors have used the sub-aperture pro-cessing, the Range Doppler as well as the ω − k algorithms

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Fig. 1. The geometry of the set-up. The radar track is a circle with radius r.

for CSAR image processing. Similar to the Range Doppleralgorithm, the sub-aperture processing and the ω−K algorithmalso suffer from the same issues we mentioned before. Toovercome the previously mentioned problems with the RangeDoppler algorithm, the sub-aperture processing and the ω− kmethod, in [15], a technique based on the z chirp transformhas been developed. However, the z chirp transform basedtechnique proposed in [15], is performed in 2D frequency do-main, namely, the range frequency and the azimuth frequencydomain which as a result the dependency of the RCM to therange is ignored [1].

We should also mention that, despite the fact that we haveconducted indoor experiments in this paper, the developedalgorithm in this work can be used for outdoor ground basedas well as UAV and Helicopter based CSAR imaging systemswithout any modification.

The structure of the paper is as follows. In section II, wedescribe the system model and formulate the problem. Insection III, we present an algorithm for image reconstruction.In section IV, we talk about the limits on the sample distancingin the angular domain. Section V discusses the resolutionlimit. Finally, section VI has been dedicated to applying theproposed algorithm to the experimental data gathered from asingle channel FMCW radar operating at 79 GHz followed byconcluding remarks.

II. MODEL DESCRIPTION

In this section, we develop the system model. Fig. 1 showsthe geometry of the model. The radar track is a circle withradius r on which the radar is moving with constant angularvelocity. The radar transmits a Linear Frequency Modulated(LFM) signal, a.k.a., chirp signal, toward the target. Thereflected signal from the target is received at the location of thereceiver and after being mixed with a copy of the transmittedsignal, the result is expressed as

s(R) = [s(1,R), s(2,R), · · · , s(N,R)]T , (1)

where s(i) is given as

s(i,R) = σ(R)ej2π[fc −

b

2+

b

N(i− 1)]

2Rc , (2)

in which b is the chirp bandwidth, N is the number of timesamples, fc and c are the center frequency and the speedof light, respectively. Also, i ∈ {1, 2, · · · , N} and σ(R) is

the complex valued reflection coefficient of a point reflectorlocated at R. The parameter R is the distance of the targetfrom the radar.

The transmitter and receiver of the FMCW radar are locatedat distance d from each other. Therefore, we are dealing witha Bistatic radar. However, if d�

√4α c

fcR, we can then use

the Monostatic approximation in which R is measured withrespect to the midpoint of the transmitter and receiver [16].The Monostatic approximation holds for the FMCW radar thatwe use in this paper. Therefore, our analysis will be based onMonostatic FMCW radar.

From Fig. 1, parameter R for a point reflector located at(Rt, θ̃t) is described as

R =

√Rt

2 + r2 − 2rRt cos(θ − θ̃t). (3)

Using (3), we can rewrite (1) for a point target located at(Rt, θ̃t) as

s(k, θm) = σ(Rt, θ̃t)×

ejk

√Rt

2 + r2 − 2rRt cos(θm − θ̃t), (4)

where the ith element of the vector k is given as ki =

4π[fc − b

2 + bN (i− 1)]

cfor i ∈ {1, 2, · · · ,N}. The subscript

index m refers to the mth angle over which the data has beencollected.

III. IMAGE RECONSTRUCTION BASED ONBACKPROJECTION METHOD

To set the stage for image reconstruction process, we definethe following N ×M matrix

A(k, R̂l, θ̂l,N,Θ) =

[a1(k, R̂l, θ̂l),a2(k, R̂l, θ̂l), . . . ,aM (k, R̂l, θ̂l)], (5)

where am(k, R̂l, θ̂l) is a N × 1 vector which is given as

am(k, R̂l, θ̂l) = ejk

√R̂2l + r2 − 2rR̂l cos(θm − θ̂l). More-

over, in (5), N = {1, 2, · · · , N} and Θ = {θ1, θ2, · · · , θM} isthe range of angles over which the data is collected.

Furthermore, we define the following N ×M matrix

S(k,Θ) = [s(k, θ1), s(k, θ2), · · · , s(k, θM )], (6)

where s(k, θm) is given in (4).The next step is defining the Region Of Interest (ROI) for

the problem which has been shown in Fig. 2. Finally, weexpress the image reconstruction algorithm as

I(R̂l, θ̂l) =tr[(S(k,Θ)⊙

A†(k, R̂l, θ̂l,N,Θ))

∗ (1M×1 ∗ 1TN×1)], (7)

in which I(R̂l, θ̂l) is the intensity for a hypothetical targetlocated at (R̂l, θ̂l) in the ROI. In (7), tr stands for the traceof matrix, (.)

† and (.)T represent the complex conjugate

and the transpose operators, respectively. Moreover,⊙

is theHadamard matrix product (element-wise product), ∗ stands forthe matrix product and 1m×1 is a m×1 column vector with allits elements equal to 1. Also, the matrices A(k, R̂l, θ̂l,N,Θ)and S(k,Θ) are given in (5) and (6), respectively.

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Radar

(R̂, θ̂)

Fig. 2. The figure shows the ROI which is composed of range of cells forhypothetical targets.

IV. ANGULAR SAMPLE SPACING

In this section, we discuss the distance between samplestaken in angular direction.

For Rt � r, we can use paraxial approximation and rewritethe phase term for the signal given in (4) as

φ(θ) = k[Rt + r cos(θ − θ̃t)]. (8)

Using x = r cos(θ) and y = r sin(θ) for the location of theradar, we can describe (8) as

φ(x, y) = k[Rt − x cos(θ̃t)− y sin(θ̃t)]. (9)

Therefore, we can describe the wavenumber of the signal inx direction as

kx =∂[k(Rt − x cos(θ̃t)− y sin(θ̃t))]

∂x,

= k cos(θ̃t). (10)

As a result, the spatial distance between samples in x directionis expressed as

∆x ≤2π

2[(kx)max − (kx)min],

=c

2 cos(θ̃t)[fmax − fmin]. (11)

Using | ∆x |= r sin(θ)∆θ, we can write (11) as

∆θ ≤c

r sin(2θ̃t)[fmax − fmin]. (12)

From (12), we infer that to satisfy the Nyquist rate in the θdirection the maximum sample spacing is given as

∆θ ≤c

r[fmax − fmin]. (13)

V. RESOLUTION ANALYSIS

In this section, we analyse the resolution of the system inboth range and angular domains. The range resolution dependson the bandwidth of the transmitted signal and it is given as

δR =c

2b. (14)

Regarding the angular resolution, we know that for a linearSAR system the angular resolution is described as δa = λ

2Ls

in which Ls is the length of the synthetic aperture and λ = cfc

represents the wavelength of the signal [1], [2]. In the case ofCSAR imaging, the length of the synthetic aperture is r×θ3dBwhere θ3dB is the 3 dB beamwidth of the antenna. Therefore,the angular resolution for the CSAR system is expressed as

δa =λ

2r × θ3dB. (15)

In fact, the angular resolution depends on the exposure timefor a point target and from (15), we can see that the angularresolution for CSAR system depends on r and θ3dB since acombination of both parameters will determine the exposuretime.

VI. EXPERIMENTAL RESULTS

In this section, we apply the algorithm that we have devel-oped to the experimental data. Fig. 3 shows the single channelFMCW radar from MediaTek company with on-chip antennasthat we have used for our experiments. Fig. 4 illustrates theantenna pattern in azimuth plane. As can be seen from Fig. 4,the 3 dB beamwidth is around 100o. The center frequency

The radar chip

Fig. 3. The single channel FMCW radar from MediaTek company.

of the radar is 79 GHz and the bandwidth has been set to3.49 GHz which results in δR = 4.3 cm resolution in therange direction. The chirp time is 68.8 µs. We have set themaximum range of the radar to 5.5 m. We have 128 samplesper each chirp which means N = 128.

We have chosen r = 13 cm for the radius of the circle overwhich the radar is rotating. As a result, for a target that isfully exposed by the 3 dB beamwidth of the antenna we obtainδa = 0.24o angular resolution. The choice of r = 13 cm is notunique. However, it is important to be aware of the dependencyof the angular resolution, the angular sample distancing and

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the size of the synthetic aperture on the parameter r as wehave explained them clearly in the paper.

The range of the angles over which we have taken thedata, is 180o. Based on (13), the angular step size should be∆θ ≤ 37.82o. In our experiments, we have set the angularstep size to 0.2o which as a result, the number of samplestaken in angular direction is M = 900. The experimental set-

θo

Fig. 4. The antenna pattern in the azimuth plane. The 3dB beamwidth andmaximum gain at boresight are approximately 100o and 5dB, respectively.

up is shown in Fig. 5. A 20 dbsm along with two 10 dbsmcorner reflectors have been chosen as targets. To create theROI shown in Fig. 2, we have chosen 0 ≤ R̂ ≤ 4 m and0o ≤ θ̂ ≤ 90o. Fig. 6 shows the radar mounted on the robot

Corner reflectors The radar mounted on the robot

Fig. 5. The experimental set-up composed of the single channel FMCW radarmounted on the robot as well as a 20 dbsm and two 10dbsm corner reflectors.

in close-up. The robot rotates the radar in a circular pattern.The distance from the center of the robot to the radar isr = 13 cm. Fig. 7 shows the corner reflectors located infront of the radar. Fig. 8 shows the corner reflectors and theirdistances from one another as well as from the center of thecoordinate system. Finally, the reconstructed image based on(7) has been presented in Fig. 9. We have conducted anotherexperiment only using two 10 dbsm corner reflectors. The

The robot The FMCW radar

Fig. 6. The single channel FMCW radar mounted on the robot.

10 dbsm corner reflectors The 20 dbsm corner reflector

Fig. 7. A 20 dbsm along with two 10 dbsm corner reflectors.

horizontal distance between them is 24 cm. The radial distancefrom the center point of the two corner reflectors to the originof the coordinate system is 4.73 m. Fig. 10 illustrates thelocation of the corner reflectors with respect to each other andalso with respect to the radar. To create the ROI shown inFig. 2, we have chosen 4 m ≤ R̂ ≤ 6 m. The reconstructedimage for the experimental set-up described in Fig. 10 hasbeen shown in Fig. 11.

Fig. 8. The position of a 20 dbsm as well as two 10 dbsm corner reflectorsin front of the radar used to collect the experimental data based on the set-upshown in Fig. 5.

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y[m]

x[m]

Fig. 9. The reconstructed image based on the experimental data gathered bythe set-up shown in Fig. 5 and using the method given in (7). The colorbaris in dB scale.

Fig. 10. The position of the two 10 dbsm corner reflectors in front of theradar.

VII. CONCLUSION

In this paper, we developed a high resolution imaging tech-nique based on a compact and low cost single channel FMCWradar by utilizing circular motion. The imaging method allows3600 coverage and the synthetic aperture is created over asmall circle. We analysed the model at length and at theend applied the proposed technique to the experimental datagathered from a single channel FMCW radar operating at79 GHz and discussed the results.

VIII. ACKNOWLEDGMENT

The authors would like to express their sincere gratitude toMediaTek company.

REFERENCES

[1] I. Cumming and F. Wong, Digital Processing of Synthetic ApertureRadar Data: Algorithms and Implementation. Artech House, 2005.

[2] M. Soumekh, Synthetic aperture radar signal processing with MATLABalgorithms. John Wiley, 1999.

[3] M. Soumekh, “Reconnaissance with slant plane circular SAR imaging,”IEEE Transactions on Image Processing, vol. 5, no. 8, pp. 1252–1265,1996.

[4] Y. Li and D. Zhu, “The geometric-distortion correction algorithm forcircular-scanning SAR imaging,” IEEE Geoscience and Remote SensingLetters, vol. 7, no. 2, pp. 376–380, 2010.

y[m]

x[m]

Fig. 11. The reconstructed image of two 10dbsm corner reflectors based onthe experimental data gathered by the set-up shown in Fig. 10 and using themethod given in (7).

[5] Y. Lin, W. Hong, W. Tan, Y. Wang, and Y. Wu, “Interferometric circularSAR method for three-dimensional imaging,” IEEE Geoscience andRemote Sensing Letters, vol. 8, no. 6, pp. 1026–1030, 2011.

[6] L. Kou, X. Wang, M. Xiang, and M. Zhu, “Interferometric estimationof three-dimensional surface deformation using geosynchronous circularSAR,” IEEE Transactions on Aerospace and Electronic Systems, vol. 48,no. 2, pp. 1619–1635, 2012.

[7] A. Broquetas, R. De Porrata, L. Sagues, X. Fabregas, and L. Jofre,“Circular synthetic aperture radar C-SAR system for ground-basedapplications,” Electronics Letters, vol. 33, no. 11, pp. 988–989, 1997.

[8] H. Rudolf, D. Leva, D. Tarchi, and A. J. Sieber, “A parallelogram shapedarm for improving circular SARs,” vol. 1, pp. 553–555, 1999.

[9] M. Bara, L. Sagues, F. Paniagua, A. Broquetas, and X. Fabregas, “High-speed focusing algorithm for circular synthetic aperture radar C-SAR,”Electronics Letters, vol. 36, no. 9, pp. 828–830, 2000.

[10] D. S. Garmatyuk and R. M. Narayanan, “Ultra-wideband continuous-wave random noise arc-SAR,” IEEE Transactions on Geoscience andRemote Sensing, vol. 40, no. 12, pp. 2543–2552, 2002.

[11] H. Lee, J. Lee, K. Kim, N. Sung, and S. Cho, “Development of a truck-mounted arc-scanning synthetic aperture radar,” IEEE Transactions onGeoscience and Remote Sensing, vol. 52, no. 5, pp. 2773–2779, 2014.

[12] B. Sun, Y. Zhou, J. Chen, , and C. Li, “Operation mode of circular tracescanning sar for wide observation,” vol. 30, no. 12, pp. 2805–2808, 2008.

[13] D. Li, G.-S. Liao, Q. Xu, and W. Wang, “High resolution imagingalgorithm for helicopter-borne rotating synthetic aperture radar,” vol. 35,no. 7, pp. 1389–1395, 2013.

[14] Y. Liao, M. Xing, L. Zhang, and Z. Bao, “A novel modified omega-k algorithm for circular trajectory scanning sar imaging using seriesreversion,” vol. 64, 2013.

[15] W. Li, G. Liao, S. Zhu, and J. Xu, “A novel helicopter-borne rosarimaging algorithm based on the azimuth chirp z transform,” IEEEGeoscience and Remote Sensing Letters, vol. 16, no. 2, pp. 226–230,2019.

[16] J. H. G. Ender and J. Klare, “System architectures and algorithms forradar imaging by MIMO-SAR,” pp. 1–6, 2009.

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Shahrokh Hamidi was born in Sanandaj, Kurdistan,Iran, in 1983. He received his B.Sc., M.Sc. and Ph.D.degrees all in Electrical and Computer Engineering.His current research areas focus on statistical signalprocessing, image processing, wireless communi-cation, machine learning, optimization and arrayprocessing.

SAFEDDIN SAFAVI-NAEINI was born in Gach-saran, Iran, in 1951. He received the B.Sc. degree inelectrical engineering from the University of Tehran,Tehran, Iran, in 1974, and the M.Sc. and Ph.D.degrees in electrical engineering from the Universityof Illinois, Urbana Champaign, in 1975 and 1979,respectively. He joined the Department of Electricaland Computer Engineering, University of Tehran, asan Assistant Professor, in 1980, where he became anAssociate Professor, in 1988. In 1996, he joined theDepartment of Electrical and Computer Engineering,

University of Waterloo, ON, Canada, where he is currently a Full Professorand the RIM/NSERC Industrial Research Chair of intelligent radio/antennaand photonics. He is also the Director of a newly established Center forIntelligent Antenna and Radio System (CIARS). He has published over 80journal papers and 200 conference papers in international conferences. Hisresearch activities deal with RF/microwave technologies, smart integratedantennas and radio systems, mmW/THz integrated technologies, nano-EMand photonics, EM in health sciences and pharmaceutical engineering, an-tenna, wireless communications and sensor systems and networks, new EMmaterials, bio-electro-magnetics, and computational methods. He has ledseveral international collaborative research programs with research institutesin Germany, Finland, Japan, China, Sweden, and USA.