Analysis of miro-doppler signatures V.C. Chen, F. Li, S.-S. Ho and H. Wehsler Abstrat: Mehanial vibration or rotation of a target or strutures on the target may indue additional frequeny modulations on the returned radar signal whih generate sidebands about the target s Doppler frequeny, alled the miro-doppler effet. Miro-Doppler signatures enable some properties of the target to be determined. In the paper, the miro-doppler effet in radar is introdued and the mathematis of miro-doppler signatures is developed. Computer simulations are onduted and miro-doppler features in the joint time frequeny domain are exploited. 1 Introdution Radar transmits a signal to a target, interats with the target, and returns bak to the radar. The hange in the properties of the returned signal ontains harateristis of interest of the target. When the transmitted signal of a oherent radar system hits moving targets, the arrier frequeny of the signal will be shifted, known as the Doppler effet. The Doppler frequeny shift reflets the veloity of the moving target. Mehanial vibration or rotation of a target, or strutures on the target, may indue additional frequeny modulations on the returned radar signal, whih generate sidebands about the target s Doppler frequeny, alled the miro-doppler effet [1, ]. Miro-Doppler signatures enable us to determine some properties of the target. The miro-doppler effet was originally introdued in oherent laser radar systems. In a oherent system, the phase of a signal returned from a target is sensitive to the variation in range. In many ases, a target or strutures on the target may have vibrations or rotations in addition to target translation, suh as a rotor on a heliopter or a rotating radar antenna on a ship. Motion dynamis of the rotating rotor or antenna will produe frequeny modulation on the baksattered signals and indue additional Doppler variations to the translation Doppler shift. From the eletromagneti point of view, when a target has vibration, rotation or other nonuniform motions, the radar baksattering is subjet to modulations that onstitute features in the signature [3, 4]. Miro-Doppler an be regarded as a unique signature of the target and provides additional information that is omplementary to existing methods. To exploit these unique miro-doppler features, traditional analysis, suh as the Fourier transform, or the sliding window or short time Fourier transform, may not possess the neessary resolution for extrating IEE Proeedings online no. 0030743 doi: 10.1049/ip-rsn:0030743 Paper first reeived 0th Deember 00 and in revised form 0th June 003 V.C. Chen is with the Radar Division, Naval Researh Laboratory, Code 5311, 4555 Overlook Ave. SW, Washington, DC 0375, USA F. Li, S.-S. Ho and H. Wehsler are with the Department of Computer Siene, George Mason University, Fairfax, VA 030, USA these features. Therefore, high-resolution time frequeny analysis is neessary for extrating the time-varying Doppler signature []. Mathematis of miro-doppler effet Mathematis of the miro-doppler effet an be derived by introduing vibration or rotation to onventional Doppler analysis. A target an be represented as a set of point satterers. The point sattering model may simplify the analysis while preserving the miro-doppler effet. As shown in Fig. 1, the radar is stationary and loated at the origin Q of the radar o-ordinate system (U, V, W). The target is desribed in the attahed loal o-ordinate system (x, y, z) and has translation and rotation with respet to the radar o-ordinates. For the purpose of mathematial analysis, a referene o-ordinate system (X, Y, Z) is introdued, whih has the same translation as the target loal o-ordinates (x, y, z) but has no rotation with respet to the radar o-ordinates (U, V, W). Thus, the referene o-ordinate system shares the same origin O with the target loal o-ordinates and is assumed to be at a distane R 0 from the radar. Assume that the azimuth and elevation angle of the target in the radar o-ordinates (U, V, W) are a and b, respetively, and the unit vetor of the radar line of sight (LOS) diretion is defined by n ¼ R 0 =kr 0 k¼ðos os ; sin os ; sin Þ T ð1þ where kk represents the Eulidean norm. Suppose the target has a translation veloity v with respet to the radar and an angular rotation veloity v; whih an be represented in the referene o-ordinate system as v ¼ðo X ; o Y ; o Z Þ T : Thus, a point satterer P, whih is loated at r 0 ¼ðX 0 ; Y 0 ; Z 0 Þ T ; at time t ¼ 0 will move to P 0 at time t. The movement an be onsidered as, first, a translation from P to P 00 with veloity v; or OO 0 ¼ vt; and then, a rotation from P 00 to P 0 with an angular veloity v: The rotation from P 00 to P 0 an be desribed by a rotation matrix Rot( ) [5, 6]. At time t, the loation of P 0 an be alulated as r ¼ O 0 P 0 ¼ RotðtÞ O 0 P 00 ¼ RotðtÞr 0 ðþ and the range vetor from the radar to the satterer at P 0 beomes IEE Pro.-Radar Sonar Navig., Vol. 150, No. 4, August 003 71
Fig. 1 Geometry of a radar and a target with translation and rotations QP 0 ¼ QO þ OO 0 þ O 0 P 0 ¼ R 0 þ vt þ r ¼ R 0 þ vt þ RotðtÞr 0 Thus, the salar range is R t ¼ RðtÞ ¼kR 0 þ vt þ RotðtÞr 0 k ð3þ ð4þ If the radar transmits a sinusoidal waveform with a arrier frequeny f, then the baseband of the returned signal from the point satterer is a funtion of R t n sðtþ¼ðx;y;zþexp jf R t o ¼ðx;y;zÞexpfjFðR t Þg ð5þ where ðx;y;zþ is the refletivity funtion of the point satterer P desribed in the target loal o-ordinates (x, y, z), is the propagation speed of the eletromagneti wave and the phase of the baseband signal is FðR t Þ¼f R t ð6þ By taking the time derivative of the phase, the Doppler frequeny shift indued by the target s motion an be obtained where dfðr t Þ f D ¼ 1 dt ¼ f 1 d R t ¼ f d dt R t dt ½ðR 0 þvtþrotðtþr 0 Þ T ðr 0 þvtþrotðtþr 0 ÞŠ ¼ f vþ d T dt ðrotðtþr 0Þ n P ð7þ n P ¼ R 0 þvtþrotðtþr 0 kr 0 þvtþrotðtþr 0 k is the diretion unit vetor from the radar to the point satterer at P 0. The angular rotation veloity vetor v ¼ðo X ; o Y ; o Z Þ T defined in the referene o-ordinate system rotates along the unit rotation vetor v 0 ¼ v=kvk with a salar angular veloity O ¼kvk: Assuming the rotational motion at eah time interval an be onsidered to be infinitesimal, the rotation matrix an be written in terms of the matrix ^v as 7 where RotðtÞ ¼ expf ^vtg 3 0 o Z o Y ^v ¼ 4 o Z 0 o X 5 o Y o X 0 ð8þ ð9þ is alled the skew symmetri matrix assoiated with v ¼ ðo X ; o Y ; o Z Þ T ; whih is the linear transformation that omputes the ross produt of the vetor v with any other vetor, as desribed in the Appendix. Thus, the Doppler frequeny shift in (7) beomes f D ¼ f v þ d T dt ðe ^vt r 0 Þ n P ¼ f ðv þ ^ve ^vt r 0 Þ T n P ¼ f ðv þ ^vrþ T n P f ðv þ ^v rþ T n ð10þ where, beause kr 0 kkvt þ RotðtÞ rk; the diretion unit vetor n P an be approximated by n ¼ R 0 =kr 0 kn P Therefore, the Doppler frequeny shift is approximately f D ¼ f ½v þ v rš radial ð11þ where the first term is the Doppler shift due to the translation and the seond term is the mathematial expression of the miro-doppler f mirodoppler ¼ f ½v rš radial ð1þ 3 Time frequeny analysis of miro-doppler signatures A ommon method to analyse a time domain signal is transforming it from the time domain to the frequeny domain by using the Fourier transform. The frequeny domain shows the magnitude of different frequenies ontained in the signal over the overall time period the signal is analysed. When the radar returned signal from a vibrating or rotating target is viewed in the frequeny domain, its miro-doppler shifts an be seen by their deviation from the entre frequeny of the radar returns. Frequeny-domain signatures provide information about frequeny modulations generated by the vibration or rotation. Although the frequeny spetrum may indiate IEE Pro.-Radar Sonar Navig., Vol. 150, No. 4, August 003
the presene of miro-doppler shifts and possibly the relative amount of displaement toward eah side, beause of the lak of time information, it is not easy to tell the vibration or rotation rate from the frequeny spetrum alone. Therefore, the time frequeny analysis that provides time-dependent frequeny information is more useful and is omplementary to the existing time-domain or frequenydomain methods. To analyse the time-varying frequeny harateristis of the miro-doppler, the radar returned signal should be analysed in the joint time frequeny domain by applying high-resolution time frequeny transforms. From the joint time frequeny domain signature, the frequeny and the period of vibration or rotation an be found [, 7]. The diretion of movement of the target at a speifi time may also be found by examining the time data and the sign of the miro-doppler shift aused by the movement of the target. Time frequeny transforms inlude linear transforms, suh as the short-time Fourier transform (STFT) and bilinear transforms, suh as the Wigner Ville distribution (WVD). With a time-limited window funtion, the resolution of the STFT is determined by the window size. A larger window has higher frequeny resolution but poor time resolution. The bilinear WVD has better harateristis of the time-varying spetrum than any linear transform. However, it suffers the problem of ross-term interferene, i.e. the WVD of the sum of two signals is not the sum of their WVDs [8]. To redue the ross-term interferene, the kernel-filtered WVD an be used to preserve the useful properties of the time frequeny transform with a slightly redued time frequeny resolution and a largely redued ross-term interferene. The WVD with a linear lowpass filter are haraterised as the Cohen s lass. In our miro-doppler signature study, the smoothed pseudo Wigner Ville distribution is used to redue the ross-term interferene and ahieve higher resolution [9]. 4 Simulation study of miro-doppler signatures In this Setion, we present examples of vibrations and rotations that an indue miro-doppler effets. Based on the mathematial analysis, we an alulate theoretial results of miro-doppler signatures. Simulation study is used to verify the theoretial results. In the simulation, the point satterer model [10] is used for modelling targets beause it is simple ompared to the EM predition ode simulation and it is easy to observe the effet of vibration or rotations and separately study individual movements. 4.1 Miro-Doppler signature of a vibrating point satterer The geometry of the radar and a vibrating point-satterer is illustrated in Fig.. The vibration entre O is stationary with azimuth angle a and elevation angle b with respet to the radar. The point-satterer is vibrating at a vibration frequeny f v with maximum amplitude D v : The azimuth and elevation angle of the vibration diretion desribed in the referene o-ordinates (X, Y, Z) is P and P ; respetively. Beause of the vibration, the point-satterer P, whih is initially at time t ¼ 0 loated at ðx 0 ; Y 0 ; Z 0 Þ T in (X, Y, Z), will at time t move to 3 3 X X 0 6 7 6 7 4 Y 5 ¼ rðtþ n V þ 4 Y 0 5 Z ð13þ where n V ¼½os P os P ; sin P os P ; sin P Š T is the unit vetor of the vibration diretion. Therefore, beause of the vibration, the veloity of the satterer P beomes Z 0 Fig. Geometry of radar and vibrating point satterer; and time frequeny miro-doppler signatures a Geometry of a radar and a vibrating point satterer b Time frequeny miro-doppler signature alulated by (15) Time frequeny miro-doppler signature by simulation IEE Pro.-Radar Sonar Navig., Vol. 150, No. 4, August 003 73
d dt rðtþ ¼ d dt rðtþ n V ¼ D v f v osð f v tþ ðos P os P ; sin P os P ; sin P Þ T ð14þ From (7) and using RotðtÞr 0 ¼ r and n P n; the miro- Doppler shift indued by the vibration is f mirodoppler ¼ f d T dt rðtþ n ¼ 4f f vd v osð f v tþn V n ð15þ whih is a sinusoidal funtion of time osillating at the vibration frequeny. Assume the radar operates at f ¼ 10 GHz and a pointsatterer is vibrating about a entre point at (U 0 ¼ 1000 m; V 0 ¼ 5000 m; W 0 ¼ 5000 m). Thus, the unit vetor from the radar to the vibration entre is n ¼ðU 0 ; V 0 ; W 0 Þ T =ðu 0 þ V 0 þ W 0 Þ 1= If the amplitude and frequeny of the vibration is D v ¼ 0:01 m and f v ¼ Hz; and the azimuth and elevation angle of the vibration diretion are P ¼ 08 and P ¼ 108; respetively, the theoretial result of the miro-doppler signature alulated from (15) is shown in Fig. b. In our simulation study, the pulse radar with a pulse repetition frequeny (PRF) of 000 is assumed and a total of 048 pulses are used to generate the miro-doppler signature of the vibrating point-satterer. The simulation result is shown in Fig. and is idential to the theoretial analysis. 4. Miro-Doppler signature of a rotating target The geometry of the radar and a target having threedimensional rotations is depited in Fig. 3. The radar o-ordinate system is (U, V, W), the target loal o-ordinate system (x, y, z) and the referene o-ordinate system (X, Y, Z) is parallel to the radar o-ordinates (U, V, W) and loated at the origin of the target loal o-ordinates. The azimuth and elevation angle of the target in the radar o-ordinates (U, V, W) is a and b, respetively. Beause of the target s rotation, any point on the target desribed in the loal o-ordinate system (x, y, z) will move to a new position in the referene o-ordinate system (X, Y, Z). The new position an be alulated from its initial position vetor multiplied by an initial rotation matrix Rot Init determined by Euler angles (f, u, ) [6]. In the target loal o-ordinate system (x, y, z), when a target rotates about its axes x, y and z with the angular veloity v ¼ðo x ; o y ; o z Þ T ; a point-satterer P at r 0 ¼ ðx 0 ; y 0 ; z 0 Þ T in the loal o-ordinates will move to a new loation in the referene o-ordinates (X, Y, Z) desribed by Rot Init r 0 : The unit vetor of the rotation is defined by v 0 ¼ðo 0 x; o 0 y; o 0 zþ T ¼ Rot Init v ð16þ kvk To ompute the 3-D rotation matrix Rot(t) in (8), the Rodrigues rotation formula [6] RðtÞ ¼expð ^v tþ ¼I þ ^v 0 sin Ot þ ^v 0 ð1 os OtÞ ð17þ is an effiient method, where I is the identity matrix, the salar angular veloity O ¼kvk and ^v 0 is the skew symmetri matrix assoiated with v 0 ¼ðo 0 x; o 0 y; o 0 zþ T 0 o 0 ^v 0 z o 0 3 y ¼ 4 o 0 z 0 o 0 5 x o 0 y o 0 x 0 ð18þ Therefore, in the referene o-ordinate system (X, Y, Z), at time t the satterer P will move from its initial loation to a new loation r ¼ Rot t Rot Init r 0 : Aording to (1), the miro-doppler frequeny shift indued by the rotation is approximately f mirodoppler ¼ f ½Ov0 rš radial ¼ f O ^v 0 T r n ¼ f O ^v 0 T Rot t Rot Init r 0 n ¼ f O ^v 0 sin Ot ^v 0 3 os Ot þ ^v 0 ði þ ^v 0 Þ T Rot Init r 0 n ð19þ Fig. 3 Geometry of a radar and a ubi target with eight satterers 74 IEE Pro.-Radar Sonar Navig., Vol. 150, No. 4, August 003
rotations is shown in Fig. 4b, whih is idential to the theoretial result. From the miro-doppler signature, the period of the rotation period an be alulated as T ¼ =kvk ¼ 1:1547 s. We an see that the miro-doppler signature in the time frequeny domain is a sinusoid with initial phase and amplitude that depends on the initial positions of the satterer and the initial Euler angle (f, u, ). 5 Summary We have shown that the mehanial vibrations or rotations of a target, or strutures on the target, an indue additional frequeny modulation on radar returned signals and generate the miro-doppler effet. We derived mathematial formulas for miro-doppler, and also simulated miro- Doppler signatures of targets undergoing vibrations or rotations. The simulation results onfirmed that the mathematial analysis is valid. 6 Aknowledgments This work was supported in part by the US Offie of Naval Researh and the Missile Defense Ageny. 7 Referenes Fig. 4 Time frequeny miro-doppler signatures a Calulated from (8) b By simulation Beause the skew symmetri matrix ^v 0 is defined by the unit vetor of the rotation v 0 ; then ^v 0 3 ¼ ^v 0 and the rotation-indued miro-doppler frequeny beomes f mirodoppler ¼ f O ^v 0 ð ^v 0 sin Ot þ I os OtÞRot Init r 0 radial ð0þ Assume the radar arrier frequeny and the initial loation of the target entre is the same as desribed in Setion 4.1. The target is assumed to be a ube that onsists of eight point-satterers as illustrated in Fig. 3. The initial Euler angles are ( ¼ 458; ¼ 458; ¼ 458). If the target rotates along the x, y and z axes with an angular veloity v ¼½; ; Š T rad/s and initial positions of eight satterers in the target o-ordinate system are P 1 ¼ðx ¼ 0:5 m; y ¼ 0:5 m; z ¼ 0:5 mþ P ¼ðx ¼ 0:5 m; y ¼ 0:5 m; z ¼ 0:5 mþ P 3 ¼ðx ¼ 0:5 m; y ¼ 0:5 m; z ¼ 0:5 mþ P 4 ¼ðx ¼ 0:5 m; y ¼ 0:5 m; z ¼ 0:5 mþ P 5 ¼ðx ¼ 0:5 m; y ¼ 0:5 m; z ¼ 0:5 mþ P 6 ¼ðx ¼ 0:5 m; y ¼ 0:5 m; z ¼ 0:5 mþ P 7 ¼ðx ¼ 0:5 m; y ¼ 0:5 m; z ¼ 0:5 mþ P 8 ¼ðx ¼ 0:5 m; y ¼ 0:5 m; z ¼ 0:5 mþ Then, the miro-doppler frequeny shift an be alulated from (0) and is shown in Fig. 4a. With a PRF of 000 and 048 pulses transmitted within about 1.04 s. of dwell time, the simulated result of the miro-doppler indued by the 1 Zediker, M.S., Rie, R.R., and Hollister, J.H.: Method for extending range and sensitivity of a fiber opti miro-doppler ladar system and apparatus therefor, US Patent no. 6,847,817, 8 De. 1998 Chen, V.C., and Ling, H.: Time-frequeny transforms for radar imaging and signal analysis (Arteh House, 00) 3 Kleinman, R., and Mak, R.B.: Sattering by linearly vibrating objets, IEEE Trans. Antennas Propag., 1979, 7, pp. 344 35 4 Gray, J.E.: The Doppler spetrum of aelerating objets. Pro. IEEE Int. Radar Conf., Arlington, VA, USA, 1990, pp. 385 390 5 Chen, V.C., and Mieli, W.: Time-varying spetral analysis for radar imaging of maneuvering targets, IEE Pro. Radar Sonar Navig., 1998, 145, (5), pp. 6 68 6 Murray, R.M., Li, Z., and Sastry, S.S.: A mathematial introdution to roboti manipulation (CRC Press, Boa Raton, FL, USA, 1994) 7 Chen, V.C.: Analysis of radar miro-doppler signature with timefrequeny transform. Pro. 10th IEEE Workshop on Statistial signal and array proessing, Poono Manor, PA, USA, August 000, pp. 463 466 8 Cohen, L.: Time-frequeny analysis (Prentie Hall, Englewood Cliffs, NJ, 1995) 9 Auger, F., Flandrin, P., Gonalves, P., and Lemoine, O.: Timefrequeny toolbox for use with MATLAB, 1996 10 Chen, V.C., and Mieli, W.: Simulation of ISAR imaging of moving targets, IEE Pro., Radar Sonar Navig., 001, 148, (3), pp. 160 166 8 Appendix The ross-produt of a vetor a ¼ða x ; a y ; a z Þ and a vetor b ¼ðb x ; b y ; b z Þ is 3 a y b z a z b y a b ¼ 4 a z b x a x b z 5 a x b y a y b x 3 0 a z a y ¼ 4 a z 0 a x 5 b 3 x 4 b y 5 ¼ ^ab a y a x 0 b z ð1þ where 3 0 a z a y ^a ¼ 4 a z 0 a x 5 a y a x 0 ðþ is alled the skew symmetri matrix and ^a ¼ ð^aþ T ð3þ A rotation matrix that belongs to the speial orthogonal 3-D IEE Pro.-Radar Sonar Navig., Vol. 150, No. 4, August 003 75
rotation matrix group R 33 is denoted by frot R 33 jrot T Rot ¼ I; detðrotþ ¼þ1g ð4þ By taking a derivative of the onstraint RotðtÞRot T ðtþ ¼I with respet to time t, we have _RotðtÞRot T ðtþ ¼ ½_RotðtÞRot T ðtþš T ð5þ This means that the matrix _RotðtÞRot T ðtþ R 33 is a skew symmetri matrix. Therefore, we an find a rotation vetor v ¼ðo X ; o Y ; o Z Þ suh that the assoiated skew symmetri matrix ^v ¼ _RotðtÞRot T ðtþ ð6þ thus _RotðtÞ ¼ ^o RotðtÞ ð7þ By solving this linear ordinary differential equation (7), we obtain RotðtÞ ¼expf ^vtgrotð0þ Assuming Rotð0Þ ¼I for the initial ondition, we have RotðtÞ ¼expf ^vtg ð8þ The matrix is a 3-D rotation matrix that rotates about the axis v by kvkt rad. 76 IEE Pro.-Radar Sonar Navig., Vol. 150, No. 4, August 003