Convex Optimization-Based Rotation Parameter Estimation Using Micro-Doppler

Transcription

1 Journal of Electrical Engineering 4 (6) doi:.765/8-/6.4. D DAVID PUBLISHING Convex Optimization-Baed Rotation Parameter Etimation Uing Micro-Doppler Kyungwoo Yoo, Joohwan Chun, Seungoh Yoo and Chungho Ryu. Department of Electrical Engineering, Korea Advanced Intitute of Science and echnology, Daejeon, Republic of Korea. 8 th R&D Intitute, Agency for Defene Development, Republic of Korea Abtract: We preent a novel algorithm that can determine rotation-related parameter of a target uing FMCW (frequency modulated continuou wave) radar, not utilizing inertia information of the target. More pecifically, the propoed algorithm etimate the angular velocity vector of a target a a function of time, a well a the ditance of cattering point in the wing tip from the rotation axi, jut by analyzing Doppler pectrogram obtained from three or more radar. he obtained parameter value will be ueful to claify target uch a hotile warhead or miile for real-time operation, or to analyze the trajectory of target under tet for the intrumentation radar operation. he propoed algorithm i baed on the convex optimization to obtain the rotation-related parameter. he performance of the propoed algorithm i aeed through Monte Carlo imulation. Etimation performance of the propoed algorithm depend on the target and radar geometry and improve a the number of iteration of the convex optimization tep increae. Key word: Micro-Doppler, FMCW radar, SF (hort-time Fourier tranform), convex optimization, rotation parameter.. Introduction o find the characteritic of a target with a radar, we tranmit probing ignal to the target, and then meaure the time-delay and Doppler frequency hift from the echo ignal back from the target, to etimate the poition and velocity of the target. hi i the conventional ue of the radar. Recently, however, radar are alo being ued to analyze the micro-motion of a target uing micro-doppler information extracted from the echo ignal. he relationhip between the micro-motion and micro-doppler frequency ha been derived analytically [-] for each micro-motion uch a rotation, preceion and nutation. In the application ide, micro-doppler information obtained from the Doppler pectrogram ha been ued to claify human activitie or target [-4] uing SVM (upport vector machine)-baed algorithm. Furthermore, micro-motion parameter have been etimated utilizing the inertia of the target [5-6]. Correponding author: Kyungwoo Yoo, mater degree, reearch field: radar ignal proceing. In thi paper, we analyze the relationhip between the rotation of the wing or rotor and the micro-doppler frequency obtained from Doppler pectrogram, and propoe a rotation parameter etimation algorithm. hree or more monotatic radar are needed and the maximum Doppler frequency and angular velocity are extracted from the Doppler pectrogram of each radar. he propoed algorithm i baed on the convex optimization to obtain the rotation-related parameter. hi paper i organized a follow. FMCW radar ignal modeling and a flow of generating Doppler pectrogram are decribed in Section II. he propoed algorithm i given in Section III. Performance of the propoed algorithm i demontrated in Section IV through imulation, and concluion i made in Section V.. Doppler Spectrogram Generation Doppler pectrogram extracted from the range-doppler map i widely ued to analyze the ditribution of Doppler frequency a a function of time for the motion analyi or activity claification. In thi

2 58 Convex Optimization-Baed Rotation Parameter Etimation Uing Micro-Doppler ection, we briefly review FMCW radar ignal and the procedure for Doppler pectrogram generation.. FMCW Radar Signal Modeling In the cae of an FMCW radar, tranmitted ignal m, t and received ignal r m, t cattered from the target may be modeled a Eq. () [7]: m, texp j fct t, r m, texp j fct t t t t PRI, t mpri t, m,..., M () where m, fc, t, and PRI are repectively, index of the pule, the center frequency, low-time, chirp rate and PRI (pule repetition interval). he time delay t i aumed to be lowly changing, i.e., there i no range cell migration during the coherence proceing interval (CPI), M PRI, where M i the number of coherently integrated pule. Note that t can be written a t r t / c r vrt/ c for a monotatic radar. Alo rt, r and c denote, repectively, the range to the target a a function of time, initial range and peed of light. Radial velocity i denoted with v r. Now the deramped ignal will be m, t m, t m, t d r exp j fc t t exp j t t (). Range and Doppler Etimation In Eq. (), t i the beat frequency f b, which i proportional to the range to the target. Note that exp / j fc t t may be conidered a a contant. herefore Eq. () may be written a d m, t Aexp j fb t, and f b can be found by applying the Fourier tranform with repect to t. he range can be found from f b by r t cfb /. Doppler frequency can be obtained by applying the Fourier tranform to d m, t with repect to m,..., M for contant t ' where m, t ' i the maximum, i.e., at the target poition. Becaue we aume that there i no range cell migration, t ' may be conidered identical for all m. Now the Fourier tranform of d m, t ' become: M j km F d m, t' d m, t' exp m M () v Since f r d fc, the radial velocity v c r can be found from f d.. SF (Short-ime Fourier ranform) If the target ha micro motion uch a rotation, vibration, then it Doppler frequency alo change in a hort time []. Hence, to obtain the micro motion information, the SF need to be applied for M SF PRI where t ' i contant. Since the reolution of Doppler frequency i proportional to / M SF PRI, frequency reolution of SF i much poorer than that of F. o compenate thi, we pad zero at the end of d m, t ', m,..., M when the SF i applied. An example of the Doppler pectrogram i given in Fig.. hi figure correpond to the ignal obtained from a radar located at,, for a miile located at,,. he miilei aumed to have a ingle catterer at the edge of a 5cm-length wing, and rotate with the angular velocity of rad / Hz with repect to,,.. Rotation Parameter Etimation We hall decribe the propoed convex optimization-baed rotation parameter etimation algorithm uing the Doppler pectrogram and the relationhip between Doppler frequency and object dynamic.. Rotation Parameter Firt, the poition of the target i aumed to be known, for example by uing the trilateration with three radar. Let u conider two coordinate ytem; d

3 Convex Optimization-Baed Rotation Parameter Etimation Uing Micro-Doppler 59 Fig. Fig. Radar Doppler pectrogram.,, x x r X he reference frame and the local frame. the reference coordinate frame and the local coordinate frame that have the ame origin which i at the center of the gravity of the target. We may regard the reference frame a the navigation frame X, Y, Z which i completely determined when the target poition i known. he local frame i the body frame x, yz, which i attached on the target, and therefore, change with repect to the heading of target. See Fig. [8]. Z y z,, z r Y Referring Fig., note that the target angular velocity vector i,, z r and the catterer located at,, x rotate around,, zr in the local coordinate frame. hee two vector can be rewritten in reference coordinate uing coordinate tranform: Xr, Yr, Zr Rrot,, zr (4) r X, Y, Z Rrot x,, where R i the direction coine matrix (rotation rot matrix) that repreent the attitude of the local frame with repect to the reference frame [8]. Let the angular velocity vector and catterer point vector in reference frame be and r, repectively. It will be convenient to decompoe and r into their norm and r, and the unit vector / and r / r. he magnitude will be etimated from the Doppler pectrogram, and the unit vector / and r / r are etimated uing convex optimization. Before applying the convex optimization, we need to initialize r / r appropriately. Detailed decription of the algorithm i given in ubection..

4 6 Convex Optimization-Baed Rotation Parameter Etimation Uing Micro-Doppler. Convex Optimization-Baed Rotation Parameter Etimation We hall explain the detailed procedure in four tep. () Calculate from the Doppler pectrogram A hown in Fig., Doppler variation appear in Doppler pectrogram. Since we are intereted in the micro-motion only, the variation of micro Doppler i inuoidal. From the Doppler pectrogram, D time-frequency ignal can be obtained uing the morphological technique [5]. A a reult, if we extract the frequency component of D time-frequency ignal, we can etimate. hi well-known procedure i omitted in thi paper. () Set the initial value r he magnitude of catterer point vector r differ depending on the ize or type of the miile. In our imulation tudy, the initial value of r i et a 5 cm or cm. () Calculate / and etimation error baed on convex optimization Now the maximum Doppler frequency obtained from Doppler pectrogram i ued to calculate / and it etimation error. Conider the ith radar. See Fig.. Since the ith Doppler frequency f di, i proportional to the ith radial velocity v ri, between the ith radar and the target, we can write the ith Doppler frequency a: fc fc fdi, vri, xi r (5) c c where x i i the unit vector directed from the center of gravity of the target to the ith radar. he approximation preent in Eq. (5) i due to that fact that the ditance between the center of the target and the catterer point i much cloer than the ditance between the center of the target and the radar. A can be een in Eq. (5), when the and r are determined, note that the inner product of the x i and r hould be maximized to maximize f di. o, obtain the maximum inner product value, we apply convex optimization. In thi paper, while changing /, r / r i etimated to maximize f di, and etimation error compared to meaured maximum Doppler frequency i calculated. At thi tep, to etimate / and r / r efficiently, etting initial / cloe to true / i very important. Initial / can be computed for three radar a: x co x / f x x f x x f d, / co / / d, co / d, x / fd, / x / fd, x / f d, (6) Eq. (6) ue the relationhip between radial velocity and it Doppler frequency, where i i the angle between x i and /. Since the initial / calculated from Eq. (6) i near to the true /, / i modified by adding mall amount of x, y, z to the initial /. In our imulation, x, y, z are et to.5. / ' / x y z (7) / x y z Now, we can formulate a convex optimization problem [9] for each radar: fc min di,, min i i r f r r x r i c fc min rxi / ri / ri r i c ubject to / r / r = i i (8) Note that the maximization of f, di, r ha been changed to the minimization of fdi,, r in Eq. (8). We ue predetermined value for, r, / a explained before. herefore, f, di, r depend only on r i / r. i We define the etimation error a: where f f (9) N N et mea d,max,i d,max,i i mea fd,max,i et mea f d,max,i and d,max,i f are repectively, the etimated

5 Convex Optimization-Baed Rotation Parameter Etimation Uing Micro-Doppler 6 Z z X / RX (,, ) x X / RX (,,) ar (,, ) X / RX (,, ) Y Fig. Poition of radar and a target in the reference frame. able arget characteritic. Variable Angular velocity vector,, Norm of catterer point vector X y Scenario,,, 5, [cm] and the meaured Doppler frequency at the ith radar. he error i calculated whenever / ' change and conider / ' which ha the mallet a final etimation angular unit vector for r. he convex optimization problem can be olved with any widely available package. (4) Change r and calculate etimation error for each r In the previou tep, a quite accurate angular unit vector can be obtained. In thi tep, r i altered baed on thi vector. A varying / ', the unit catterer point vector i etimated by comparing for variou r candidate. If i almot the ame for varying r, then reetimate / ' for r which ha the mallet. Since r / r varie quite rapidly compared to other rotation parameter, etimating r / r during very hort time i meaningle. So, in thi paper, r / r i ued a a convex optimization variable only. 4. Simulation Reult 4. Simulation Environment hree monotatic FMCW radar are located a hown in Fig.. hey are aumed to be operated at GHz center frequency, with 75 MHz bandwidth,.5 GHz ampling frequency, 5.6 PRI and 6 coherent pule for Doppler frequency etimation. able how the true target characteritic for two cenario. 4. Simulation Reult Firt, r i etimated at an interval of cm. A hown in Fig. 4, for both cenario, the relative error become the minimum at true r. Note that there i etimation error even for noie-free environment. hi i due to the reolution limitation of x, y, z ued at / etimation. If x, y, z become maller or convex optimization i run enough, the etimation error will alo become maller. Fig. 5 how the RMSE (root mean quared error) of etimation for different iteration number of convex optimization. Iteration number i counted when all the for all / ' are compared. hu, at the next iteration, / ' which ha the mallet i et to the initial /. A can be een in Fig. 5, the RMSE of become maller, a the iteration number increae. hee reult alo demontrate that difference of the geometry affect the convergence

6 6 Convex Optimization-Baed Rotation Parameter Etimation Uing Micro-Doppler relative error 5 4 norm(r) = cm norm(r) = 5cm norm(r) = cm Scenario relative error r [m] Scenario 5 norm(r) = cm 4 norm(r) = 5cm norm(r) = cm r [m] Fig. 4 Plot of veru r when true r i, 5, cm for cenario,. error of [rad/] error of [rad/] RMSE of [rad/] Scenario X-axi - Y-axi Z-axi iteration number X-axi Scenario 5 Y-axi Z-axi iteration number iteration number Fig. 5 Etimation error and RMSE of veru iteration number for cenario and. Scenario Scenario

7 Convex Optimization-Baed Rotation Parameter Etimation Uing Micro-Doppler RMSE [rad/] poition error [m] Fig. 6 RMSE of veru poition error for cenario RMSE [rad/] X-axi. Y-axi Z-axi - - Meaured Doppler frequency error [Hz] Fig. 7 RMSE of with repect to frequency error for cenario.

8 64 Convex Optimization-Baed Rotation Parameter Etimation Uing Micro-Doppler peed of the propoed technique. Fig. 6 and 7 depict the effect of meaurement error in cenario. Fig. 6 i for the poition error, and Fig. 7 i for the Doppler frequency error. he poition error i et to [m], and the frequency error to [Hz]. he number of iteration for Monte Carlo trial i. Since the ditance between a radar to the target i, m, the correponding poition error level may be conidered to be mall. However, in the cae of frequency error, note that they are relatively large. 5. Concluion We preent a convex optimization-baed rotation parameter etimation algorithm uing micro-doppler without knowing inertial characteritic. A the number of iteration increae, the etimation performance of the propoed algorithm improve. Alo, the convergence rate of the propoed algorithm i related to the geometry of radar and target. Additionally, the performance of the propoed technique i related to the poition and Doppler frequency meaurement accuracy. Acknowledgement hi reearch wa upported by a grant from Algorithm development of multi-target poition etimation uing RF array enor (4--) funded by Defene Buine Agency and Agency for Defene Development (ADD). Reference [] Gao, H., Xie, L., Wen, S., and Kuang, Y.. Micro-Doppler Signature Extraction from Ballitic arget with Micro-motion. IEEE ran. Aerop. Electron. Syt. 46 (4): [] Clemente, C., and Soraghan, J. J.. Vibrating arget Micro-Doppler Signature in Bitatic SAR with a Fixed Receiver. IEEE ran. Geoci. Remote Sen. 5 (8): 9-7. [] Kim, Y., and Ling, H. 9. Human Activity Claification Baed on Micro-Doppler Signature Uing a Support Vector Machine. IEEE ran. Geoci. Remote Sen. 47 (5): 8-7. [4] Kim, Y., Ha, S., and Kwon, J. 5. Human Detection Uing Doppler Radar Baed on Phyical Characteritic of arget. IEEE Geoci. Remote Sen. Lett. (): [5] Lei, P., Sun, J., Wang, J., and Hong, W.. Micromotion Parameter Etimation of Free Rigid arget Baed on Radar Micro-Doppler. IEEE ran. Geoci. Remote Sen. 5 (): [6] Wang, J., Lei, P., Sun, J., and Hong, W.. Spectral Characteritic of Mixed Micro-Doppler imefrequency Data Sequence in Micro-motion and Inertialparameter Etimation of Radar arget. IE Radar Sonar Navig. 8 (4): [7] Zhang, J., Jin,., Qiu, L., and Zhou, Z. 5. Automatic Human Micro-Doppler Signature Separation by Hough ranform. Ninth International Sympoium on Multipectral Image Proceing and Pattern Recognition (MIPPR5), International Society for Optic and Photonic. [8] itterton, D. H., and Weton, J. L Strapdown Inertial Navigation echnology. Stevenage, U.K.: Peregrinu. [9] Boyd, S., and Vandenberghe, L. 4. Convex Optimization. Cambridge, U.K.: Cambridge Univ. Pre.