PAPER 2-Dimensional Accurate Imaging with UWB Radar Using Indoor Multipath Echoes for a Target in Shadow Regions

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1 2366 PAPER 2-Dmensonal Accurate Imagng wth UWB Radar Usng Indoor Multpath Echoes for a Target n Shadow Regons Shuhe FUJITA a), Student Member, Takuya SAKAMOTO, Member, and Toru SATO, Fellow SUMMARY UWB (Ultra Wde-Band) pulse radar s promsng for survellance systems because t has an outstandng hgh range-resoluton. To realze an accurate UWB radar magng system, we propose a new approach that employs multpath echoes from a target n an ndoor envronment. Usng multpath echoes, the proposed system can accurately estmate mages, even for targets n a shadow regon where the targets are out of sght of the antenna. We apply a smple nterferometry technque usng the multple mrror mage antennas generated by multpath propagaton. We fnd that ths smple method also produces many undesred false mage ponts. To tackle ths ssue, we also propose an effectve false mage reducton algorthm to obtan a clear mage. Numercal smulatons verfy that most of the false mage ponts are removed and the target shape s accurately estmated. key words: UWB pulse radar, multpath scatterng waves, shadow regon magng, nterferometry, Tme-Reversal magng 1. Introducton Survellance systems are ndspensable for mantanng a safe socety, and preventng crmes and terror attacks. Most of the current securty systems use cameras due to ther lowcost and hgh resoluton capabltes. A system usng radowaves s another canddate for ths purpose because t has the potental to avod some of the substantal lmtatons of camera-based systems. It has been reported that radowaves enable the detecton of targets n hdden places where cameras cannot work. Exstng communcaton nfrastructure lke WLAN statons has been employed for postonng purposes [1], [2]. Although these methods are capable of estmatng target locatons, the resoluton s not good enough to estmate the shape of the target. To obtan ths nformaton, UWB (Ultra Wde-Band) pulse radar s promsng for survellance magng because of ts hgh range resoluton. To acheve hgh cross-range resoluton, most of the conventonal algorthms [3] [5] ncludng the SEABED method [6], [7] use array antennas. These costly and large-scale systems are not realstc for commercal survellance systems. Another approach usng the moton of targets [8] has been developed to obtan an mage usng only 3 antennas. Furthermore, a new approach, the TR (Tme-Reversal) method [9], [10] usng multpath echoes makes t possble to calculate an mage usng only a sngle antenna [11], [12]. Assumng a pont-lke target, ths Manuscrpt receved June 9, Manuscrpt revsed February 14, The authors are wth the Department of Communcatons and Computer Engneerng, Graduate School of Informatcs, Kyoto Unversty, Kyoto-sh, Japan. a) E-mal: fujta-syuhe@denden18.mbox.meda.kyoto-u.ac.jp DOI: /transcom.E94.B.2366 method numercally back-propagates the receved sgnal n a known-shaped room to focus on an mage at the target locaton. The method cannot estmate a target shape although t gves an accurate target locaton. In ths paper, we propose a new magng method for UWB radar wth only a sngle antenna that combnes the deas of the TR and the nterferometry method [13] to accurately estmate a target shape n a multpath envronment. Ths method also has a partcular advantage over conventonal camera-based systems because t enables the magng of a target n an area that s out of sght. The proposed method pcks up multpath echoes from the receved sgnals to estmate the target shape. The procedure gves a correct target shape wth many undesrable false mages caused by the ambguty of the propagaton paths correspondng to multpath echoes n the receved sgnals. To elmnate these false mages, we also propose an effectve false mage reducton algorthm to obtan a clear mage. Frst, we explan the procedure of the proposed magng method, followed by some numercal smulaton results to show the performance of the proposed method compared to conventonal methods. 2. System Model For smplcty, we deal wth a 2-dmensonal problem n ths paper. It s assumed that a radar system s nstalled on a mechancal scanner n a room as n Fg. 1. We assume that the room s confgured as a known polygonal shape. A target (a human) s located at an unknown poston n the room. The room has a blnd area blocked by walls. The areas where drect waves cannot be receved from the antenna are called shadow regons n ths paper. Fgure 2 shows an example of the shadow regon blocked by a wall n a room. The Fg. 1 Overvew of survellance system usng UWB pulse radar. Copyrght c 2011 The Insttute of Electroncs, Informaton and Communcaton Engneers

2 FUJITA et al.: 2-DIMENSIONAL ACCURATE IMAGING WITH UWB RADAR USING INDOOR MULTIPATH ECHOES FOR A TARGET IN SHADOW REGIONS 2367 Fg. 2 System model A wth a shadow regon. Fg. 3 Mrror mage antennas for model A. shadow regon n the system s defned as the ntersecton of the shadow regons for all the antenna locatons, whch means that the shadow regon s the area not vsble from any locaton along the antenna-scannng lne. The regon panted the darkest gray n Fg. 2 represents the shadow regon. Fgure 3 shows a model of the system, where an antenna and a target are located n an L-shaped room made of PEC (Perfect Electrc Conductor). We defne real space as the space where the target and the antenna are located. We express the real space wth the parameters r = (x,y). The target s modeled as a smple-shaped PEC object n the shadow regon. The reflecton coeffcents of the walls and the target are set to 1.0, whch s the deal case wth the maxmum echo ntensty. The antenna s used for both transmttng and recevng, and s scanned along a straght lne. The -th antenna locaton s expressed as x = Δx + x 0,whereΔx s the nterval of the antenna locaton. An antenna s scanned along a lne y = y 0 n the x drecton. Rased-cosne-shaped UWB pulses, wth a roll-off factor of 1.0, a center frequency of 79 GHz and a bandwdth of 1.4 GHz, are transmtted and echoes are receved by the same antenna. The antenna s assumed to have an deally unform beam pattern wth a beamwdth of 180 and wth the manlobe n the drecton of the y axs. Note that the deal 180 unform beam pattern for the entre bandwdth of 1.4 GHz cannot be realzed n practce. The purpose of ths paper s to evaluate the performance of radar magng methods n deal cases, whch corresponds to the achevable performance lmt. It s an mportant future task to nvestgate the magng qualty under actual condtons wth a non-unform antenna pattern, waveform dstorton, and antenna couplng effect. We defne s (X, Y) as the receved sgnal at the antenna locaton (x,y) = (X,y 0 ), where we defne Y wth tme t and the speed of the radowave c as Y = ct. The ray tracng method [14], [15] s used to calculate the receved sgnal. Ray tracng solves the propagaton of waves by repeatedly generatng narrow beams through the medum. In our smplfed model, the waveforms change only ther ampltudes and delays, wth the sgnal s (Y) receved at r = (X,y 0 ) calculated as s (Y) = L A n w(y cτ n ), (1) n=1 where L s the number of paths, A n and τ n represent the ampltude and relatve delay of the n-th component and w(y)s the reference waveform. The drect echoes from walls are subtracted from s (X, Y). A flter matched wth the transmtted waveform s appled to the raw sgnal s (X, Y) to obtan a fltered sgnal s(x, Y). The magnary space expressed wth (X, Y) s called a data space. For convenence, we ntroduce mrror mage antennas that are located at symmetrcal postons wth respect to the room walls as n Fg. 3. Each multpath wave can be modeled wth an magnary echo from the correspondng mrror mage antenna. The j-th mrror mage antenna for the -th antenna poston s located at a ( j) = (x ( j),y ( j) ) = (Δx ( j) + x ( j) j) 0,y( 0 )( = 0,, M; j = 0,, N), where M s the number of antenna locatons, N s the number of mrror mage antennas, and Δx ( j) s the nterval of the j-th mrror mage antenna s locaton. The ndexes j > 0 are assgnedto the mrror mage antennas from left to rght, and from top to bottom n ascendng order. In the case of j = 0, t represents an actual antenna locaton. 3. Conventonal Methods 3.1 SEABED Method A hgh-speed UWB radar magng method, SEABED [6] has been proposed based on a smplfed target model. Ths method utlzes a reversble transform BST (Boundary Scatterng Transform) between the pont of real space (x,y)and the pont of data space (X, Y ), whch s extracted by the output of the matched flter s(x, Y), where Y = Y/2. The IBST (Inverse BST) s expressed as

3 2368 Fg. 6 Receved sgnals after applyng the matched flter. Fg. 4 System model B wthout shadow regons. Fg. 7 Image estmated by the TR method. Fg. 5 Image estmated by the SEABED method. { x = X Y dy /dx, (2) y = Y 1 (dy /dx) 2, (3) where dy /dx 1 holds. Although t s reported that the SEABED can obtan a hgh-qualty mage, ths method can be appled only to measurements usng only drect echoes receved wthout any multpath sgnals as n Fg. 4. Fgure 5 shows the estmated mage by applyng the IBST to the system model shown n Fg. 4. Here, a crcular target wth a radus of 0.5 m located at ( 3.0m, 4.0m) s assumed. We also set the other parameters (x 0,y 0 ) = (0.1m, 1.0m),Δx = 0.1m,and M = 38. In Fg. 5, although SEABED provdes an accurate target shape, we can obtan only a small part of the target shape because we use only drect echoes from the target wthout any multpath echoes. 3.2 Tme-Reversal Imagng Method The TR (Tme-Reversal) method s another approach for UWB radar magng that s lkely to be applcable even to multpath echoes [12]. In the TR method, extended to shadow regon magng, the mage I(r) s formulated as N N I(r)= H(p, q, r) (X, s r a (p) + r a (q) ) 2, (4) p=0 q=0 where H(p, q, r) s the functon defned as 1 (r Π p Π q ), H(p, q, r) = 0 (r Π p Π q ). In Eq. (5), Π p s the shadow regon from the locaton of p-th antenna. Equaton (4) ndcates that the mage I(r) sproduced by summng all the sgnals usng dfferent antenna pars after compensatng for the tme delay. The functon H(p, q, r) prevents the summaton from ncludng contradctory components that propagate through the PEC walls. We apply ths method to the system model shown n Fg. 3. Here, the observaton tme s 0 t 150 nsec, correspondng to a range of 45 m. We assume N = 6, whch means that the mage s produced wth multpath echoes wth the number of reflectons equal to or less than 3. Fgure 6 shows the receved sgnals s(x, Y) and Fg. 7 shows the mage obtaned by the TR method. The mage s (5)

4 FUJITA et al.: 2-DIMENSIONAL ACCURATE IMAGING WITH UWB RADAR USING INDOOR MULTIPATH ECHOES FOR A TARGET IN SHADOW REGIONS 2369 normalzed by the maxmum value of I(r).Asshown n ths fgure, although the TR method can estmate the locaton of the target, t cannot estmate an accurate shape thereof. 4. Proposed Method 4.1 Extracton of Range Pont Pars Ths secton descrbes the proposed magng algorthm to obtan hgh-resoluton mages n an ndoor envronment. Frst, n ths subsecton, we explan the procedure of the ntal data processng of the receved sgnals. (X, Y,k )s defned as the range pont that s extracted from the peak ponts of s(x, Y) as δ s(x, Y) δ Y 0, (6) (X,Y)=(X,Y,k )= s(x, Y,k ) ρ max s(x, Y), (7) Fg. 8 Schematc of extractng pars of range ponts. where X s the -th actual locaton of an antenna and Y,k s the k-th peak of the sgnal receved at (X,y 0 ). The parameter ρ 0 s emprcally determned. The k-th peak pont for X s extracted by fndng the local maxmum ponts wth the quas-newton method as Y,k = arg max s(x, Y) 2, (8) Y wth the ntal value Y = mδy s satsfyng s(x, (m 1)ΔY s < s(x, mδy s ) > s(x, (m + 1)ΔY s ). (9) ΔY s s the samplng nterval of Y. The sampled data of the sgnal s(x, Y) s nterpolated usng the snc functon n ths search. Moreover, we pck up pars of adjacent range ponts satsfyng the condton: Y,u Y +1,v T 0, (10) Fg. 9 Extracted pars of range ponts. where T 0 s the length of the transmtted pulse. The schematc of ths procedure s llustrated n Fg. 8. We set T 0 = 0.2 m for our system wth the bandwdth of B w = 1.4 GHz. T 0 s calculated as T 0 = c/b w. The black dots connected wth sold lnes n Fg. 9 show the pars of range ponts extracted by the procedure descrbed above. 4.2 Interferometry Imagng n an Indoor Envronment In ths subsecton, we descrbe the proposed magng method usng the range ponts extracted n the prevous subsecton. Note that the receved echoes nclude both monostatc and bstatc radar echoes. If the transmttng and recevng propagaton paths are dentcal as n the left-hand sde fgure of Fg. 10, ths s nterpreted as a monostatc radar sgnal wth a sngle actual/magnary antenna. In contrast, other reflected echoes propagate along a path dfferent from the transmttng propagaton; ths echo corresponds to a bstatc radar arrangement as n the rght-hand sde fgure of Fg. 10. Fg. 10 Two knds of propagaton paths. The nterferometry method [13] s employed for magng usng the extracted pars of range ponts. Interferometry s a commonly used technque for DOA (Drecton-Of- Arrval) estmaton usng the phase dfference between multple echoes receved wth dfferent antennas. By extendng ths prncple, the target shape s provded by solvng the ntersecton ponts of the followng couple of ellpses: r a (p) + r a (q) = Y,u, (11) r a (p) + r a (q) = Y +1,v. (12) Ths can be used for estmatng the DOA by measurng the dfference between the delays of multple echoes receved by dfferent antennas. The schematc of ths nterferometry

5 2370 Fg. 11 Schematc of bstatc nterferometry. Fg. 13 Schematc of false mage reducton process. Fg. 12 Estmated mage ponts wthout false mage reducton process. Fg. 14 Estmated mage ponts wth false mage reducton process. method s llustrated n Fg. 11, where an ellpse wth foc a (1) and a (2), and another ellpse wth foc a (1) +1 and a(2) +1 are used to calculate the target locaton. In the case of p = q,the soluton s gven by the ntersecton of a couple of crcles, not ellpses. The actual numercal procedure of calculatng the ntersectons of crcles or ellpses are explaned n Appendces A and B. We apply these methods to all possble combnatons of pars of range ponts and antennas to obtan an estmated mage. Fnally, f the estmated pont falls outsde the room, the pont s removed. The mage estmated by ths method s shown n Fg. 12, where a broken lne and black dots represent the actual target shape and the estmated mage. Although ths mage s a correct estmate of the crcular target, t also has many false mage ponts because t contans ncorrect combnatons of a range of ponts and antennas. Ths s because ncorrect pars of antennas are used to produce the mage. 4.3 False Image Reducton Method The problem s that we cannot know whch echo corresponds to whch antenna pars at ths stage. Frst, we calculate a rough mage usng the conventonal TR method to estmate the approxmate locaton of targets as n Fg. 7. We estmate the maxmum pont r max from the mage n Fg. 7. Next, we pck up consstent combnatons of range ponts that satsfy the relatonshp between the actual/magnary antenna scannng drecton and the estmated range values. We only use antenna pars based on the nclnaton of the lnes connectng the range ponts. In ths process, echoes are dvded nto two groups A and B as n Fg. 13. Each antenna par s classfed as a member of one of these groups A or B. Applyng the nterferometry method, we add the followng condton: ( Y,u Y +1,v r max a (p) + r max a (q) r max a (p) +1 + r max a (q) ) +1, (13) ( Y,u < Y +1,v r max a (p) + r max a (q) < r max a (p) + r max a (q) ). (14) Addtonally, we assume that true mage ponts exst wthn the vcnty of the pont r max as r r max < μ. The target shape estmated by the proposed method s shown n Fg. 14, where the whte square symbol represents the estmated target locaton r max,andwherewesetμ = 0.5m +1 +1

6 FUJITA et al.: 2-DIMENSIONAL ACCURATE IMAGING WITH UWB RADAR USING INDOOR MULTIPATH ECHOES FOR A TARGET IN SHADOW REGIONS 2371 and r max = ( 2.40 m, 4.10 m). In ths fgure, most of the false mages are removed and the true target shape s accurately estmated. The RMS error of the estmated shape s 0.47 mm. In our numercal smulatons, we assume a vacant room wthout any furnture. However, the proposed method can stll be appled even f there are some tems of furnture n the room. Ths s because the method can produce mages f t can employ the magnary mrror antennas correspondng to the propagaton paths ncludng the effect of furnture. As for the applcablty of the proposed method to a movng target, the performance depends on the scannng speed of the antenna. If the antenna scannng s much faster than the target moton, the proposed method can produce snap-shot mages. If an antenna array s employed nstead of mechancal scannng, the method can easly be appled to a movng target. 4.4 Parameter Optmzaton The parameters ρ and μ n our proposed algorthm have a sgnfcant effect on magng performance. In ths subsecton, we nvestgate the estmated range and accuracy of mages for varyng values of parameters ρ and μ. Fgure 15 and Fg. 16 show the estmated range and the estmaton RMS error for the parameter ρ and μ, where the estmated range s the percentage of the estmated area of the target surface over the total surface area. The results reveal that there s a trade-off relatonshp between the estmated range and accuracy. To ensure an estmated range of more than 7%, the parameterpar(ρ, μ) needs to be set to (0.3, 0.4), (0.4, 0.5) or (0.5, 0.5) as shown n Fg. 15. Moreover, to obtan the mnmum RMS error under the prevous condton, the parameter par (ρ, μ) ssetto(0.5, 0.5) as shown n Fg. 16. Hereafter, we use these values for our numercal smulatons. Note that we assume the system model A n calculatng these values. However, the optmum parameters depend on the target sze, shape and locaton, as well as the room shape. 5. Performance Evaluaton of the Proposed Method 5.1 Nose Tolerance We show the magng accuracy of the proposed method wth nosy data assumng the same scenaro as n the prevous secton. To produce a nosy sgnal numercally, whte Gaussan nose s added to the raw sgnals s (X, Y). We defne S/N as the rato of the peak nstantaneous sgnal power to the averaged nose power after applyng the matched flter. The RMS error of the estmated shape usng the proposed method s shown n Fg. 17. Ths fgure shows that the RMS error s relatvely small, less than 40 mm for the S/N db. Moreover, we have confrmed that the TR method cannot estmate an accurate target locaton n the case of S/N db, leadng to poor performance of the proposed method. The mage estmated n nosy envronments s shown n Fg. 18 for S/N = db. In Fg. 18, although there are napproprate false ponts, most of the estmated ponts are located on the target surface, gvng an accurate mage estmaton. 5.2 Performance Evaluaton wth Other Models Fg. 15 Estmated range for the parameters ρ and μ. Ths subsecton dscusses the performance of the proposed method wth dfferent models. Frst, we apply the proposed method to the system model B shown n Fg. 4. The target mage obtaned s shown n Fg. 19. Ths result verfes Fg. 16 RMS error for the parameters ρ and μ. Fg. 17 RMS error of the proposed method vs. S/N.

7 2372 Fg. 18 Estmated mage ponts for S/N=30.50 db. Fg. 21 Image estmated for the system model C. Fg. 19 Estmated mage ponts for a target n a lne-of-sght area. Fg. 22 Estmated mage ponts for an ellptcal target. Fg. 20 System model C for a hallway. that the regon contanng mages s extended compared wth Fg. 5 because the proposed method uses not only the drect echo but also multpath echoes for magng. The estmaton RMS error s 1.78 mm. Next, we show the performance of the proposed method assumng the envronment shown n Fg. 20, modelng a corner of a hallway. The estmated mage s shown n Fg. 21. In ths fgure, a dfferent part of the target boundary s accurately estmated. The estmaton RMS error s mm. Fnally, we apply the proposed method to an ellpt- cal PEC target n the model A shown n Fg. 3. Fgure 22 shows an example of the estmated mage for the target wth an nclnaton angle of 120, where part of the target shape s correctly estmated wth an estmaton RMS error of 0.02 mm. We have also appled the proposed method to the same ellptcal targets wth varous nclnaton angles for 60 ψ 120 and confrmed the average estmaton RMS error to be 2.57 mm. The accuracy of estmaton depends on many factors ncludng the target shape and locaton, and the room shape. Ths s because the nterference effect between echoes from dfferent paths s hghly senstve to these factors. In addton, the optmum parameters ρ and μ depend on assumptons of the room s shape, the wall s reflectvty and the target locaton. As such, t s dffcult to evaluate the magng accuracy n a general context. 6. Conclusons Ths paper proposed a sngle antenna magng method for targets n shadow regons. Frst, we establshed that the conventonal method does not provde suffcent resoluton of a target n a shadow regon. To obtan a hgh-resoluton mage, we proposed the magng method usng the prnc-

8 FUJITA et al.: 2-DIMENSIONAL ACCURATE IMAGING WITH UWB RADAR USING INDOOR MULTIPATH ECHOES FOR A TARGET IN SHADOW REGIONS 2373 ple of nterferometry and appled ths method to each of the mrror mage antennas. Addtonally, we proposed a false mage reducton algorthm usng an approxmate target locaton obtaned by the TR method and the nclnaton of the estmated pars of echoes n the data space. In ths process, most of the false mage ponts were removed and the target boundary was estmated accurately. We also nvestgated the performance of the proposed method for nosy data and clarfed that an accurate mage s obtaned when the S/N s hgher than about 25 db based on the results of the computer smulatons. Moreover, we nvestgated the performance of the proposed method n dfferent system models, confrmng the effectveness of the proposed method n a varety of stuatons. Note that the performance of the proposed method evaluated n the paper s based on a few partcular models. Therefore, the performance depends on multple factors ncludng the reflecton coeffcents of the walls and target, and the parameters ρ and μ. The proposed method assumes a system model wth a sngle target, and the algorthm s assumed to know ths model. To apply the method to multple targets, t s expected that some modfcatons would be requred to the method. For example, the peak detecton process of the TR method needs to be adequately modfed to extract multple peaks. Ths expanson of the method to multple targets s an mportant future task. References [1] S. Ikeda, H. Tsuj, and T. Ohtsuk, Indoor event detecton wth egenvector spannng sgnal subspace for home or offce securty, IEICE Trans. Commun., vol.e92-b, no.7, pp , July [2] K. Pahlavan, F.O. Akgul, M. Hedar, A. Hatam, J.M. Elwell, and R.D. Tngley, Indoor geolocaton n the absence of drect path, IEEE Wreless Commun., vol.13, no.6, pp.50 58, [3] C. Le, T. Dogaru, L. Nguyen, and M.R. Ressler, Ultra wdeband (UWB) radar magng of buldng nteror: Measurements and predctons, IEEE Trans. Geosc. Remote Sens., vol.47, no.5, pp , [4] X. Zhuge, T.G. Savelyev, A.G. Yarovoy, and L.P. Lgthart, UWB array-based radar magng usng modfed Krchhoff mgraton, 2008 IEEE Internatonal Conference on Ultra-WdeBand (ICUWB2008), vol.3, pp , 2008 [5] W.C. Khor, M.E. Balkowsk, A. Abbosh, N. Seman, and S. Crozer, An ultra wdeband mcrowave magng system for breast cancer detecton, IEICE Trans. Commun., vol.e90-b, no.9, pp , Sept [6] T. Sakamoto, A fast algorthm for 3-D magng wth UWB pulse radar systems, IEICE Trans. Commun., vol.e90-b, no.3, pp , March [7] S. Kdera, Y. Kan, T. Sakamoto, and T. Sato, A fast and hghresoluton 3-D magng algorthm wth lnear array antennas for UWB pulse radars, IEICE Trans. Commun., vol.e91-b, no.8, pp , Aug [8] T. Sakamoto, Y. Matsuk, and T. Sato, A novel UWB radar 2-D magng method wth a small number of antennas for targets wth arbtrary shapes and moton, 2009 IEEE Internatonal Conference on Ultra-WdeBand (ICUWB2009), pp.9 11, [9] E.A. Marengo and F.K. Gruber, Subspace-based localzaton and nverse scatterng of multple scatterng pont targets, EURASIP J. Appl. Sgnal Process., vol.2007, pp , [10] Y. Jn and J.M.E. Moura, Tme-reversal detecton usng antenna arrays, IEEE Trans. Sgnal Process., vol.57, pp , [11] T. Sakamoto and T. Sato, Tme-reversal UWB magng wth a sngle antenna n mult-path envronments, 3rd European Conference on Antennas and Propagaton (EuCAP) 2009, pp.23 37, [12] T. Sakamoto and T. Sato, A method of estmatng a room shape wth a sngle antenna n a multpath envronment, 4th European Conference on Antennas and Propagaton (EuCAP) 2010, pp.12 16, [13] D. Massonet and K.L. Fegl, Radar nterferometry and ts applcatons to changes n Earth s surface, Rev. Geophys., vol.36, no.4, pp , [14] M.F. Iskander and Z. Yun, Propagaton predcton models for wreless communcaton systems, IEEE Trans. Mcrow. Theory Tech., vol.50, pp , [15] M.C. Lawton and J.P. McGeehan, The applcaton of a determnstc ray launchng algorthm for the predcton of rado channel characterstcs n small-cell envronments, IEEE Trans. Veh. Technol., vol.43, pp , Appendx A: Monostatc Interferometry Method The ntersecton pont of two crcles Eqs. (11) and (12) for p = q s analytcally solved by ( ) r = a (p) cos(α ± β) + Y,m, (A 1) sn(α ± β) where α and β are represented as α = tan 1 y (p) y (p) +1 x (p) x (p) +1 (x (p) β=cos 1 x (p) 2Y,m Appendx B:, (A 2) +1 )2 +(y (p) y (p) (x (p) x (p) +1 )2 +(y (p) +1 )2 +Y 2,m Y2 +1,n y (p) +1 )2 Bstatc Interferometry Method. (A 3) The ntersecton pont of two ellpses s solved numercally. The ellpse defned by Eq. (11) s expressed wth the parameter θ n the followng equaton: r(θ) = R(γ) ( a cos θ b sn θ ) + a(p) + a (q) 2 (0 θ 2π), (A 4) where R( ) represents the rotaton matrx: ( ) cos φ sn φ R(φ) =, (A 5) sn φ cos φ and γ s a constant number gven by γ = tan 1 y (p) y (q) x (p) x (q). (A 6) The parameters a and b denote the long and short axes of the ellpse n Eq. (11), gven by a = Y,m /2, (A 7)

9 2374 b = Y 2,m (x(p) x (q) ) 2 (y (p) y (q) ) / 2 2. (A 8) When the pont r(θ) corresponds to the ntersecton pont, the pont satsfes Eq. (12). To use ths prncple for calculaton, the followng functon F(θ) s ntroduced: F(θ) = r(θ) a (p) + r(θ) a (q) Y +1,n. (A 9) +1 We fnd θ by numercal calculaton, where the θ satsfes F(θ ) = 0. The soluton s gven as r(θ ). +1 Shuhe Fujta receved the B.E. degree from Kyoto Unversty n He s currently studyng for an M.I. degree at the Graduate School of Informatcs, Kyoto Unversty. Hs current research nterest s n sgnal processng for UWB pulse radars. Takuya Sakamoto receved hs B.E. degree from Kyoto Unversty n 2000, and M.I. and Ph.D. degrees from the Graduate School of Informatcs, Kyoto Unversty n 2002 and He s an assstant professor n the Department of Communcatons and Computer Engneerng, Graduate School of Informatcs, Kyoto Unversty. Hs current research nterest s n UWB radar sgnal processng. He s a member of the IEEJ and the IEEE. Toru Sato receved hs B.E., M.E., and Ph.D. degrees n Electrcal Engneerng from Kyoto Unversty, Kyoto, Japan n 1976, 1978, and He has been wth Kyoto Unversty snce 1983 and s currently a Professor n the Department of Communcatons and Computer Engneerng, Graduate School of Informatcs. Hs major research nterests nclude system desgn and sgnal processng aspects of UWB radars, atmospherc radars, radar remote sensng of the atmosphere, and radar observaton of space debrs. He s a member of the Insttute of Electrcal and Electroncs Engneers, the Socety of Geomagnetsm and Earth, Planetary and Space Scences, the Japan Socety for Aeronautcal and Space Scences, and Amercan Meteorologcal Socety.