77 GHz ACC Radar Simulation Platform

Transcription

1 77 GHz ACC Rada Simulation Platfom Camilla Känfelt, Alain Péden, Ali Bazzi, Ghayath El Haj Shhadé, Mohamad Abbas, Thiey Chonavel and Fantz Bodeeau Abstact The development of a system simulation platfom fo Adaptive Cuise Contol (ACC) ada woking at 77 GHz is pesented. The simulation platfom allows us to test diffeent ada achitectues, modulation fomats and detection algoithms as well as to simulate diffeent scenaios, which impoves the decision-making befoe and duing the hadwae development. the modulation bandwidth, R the ange and T denotes the total duation of the up and down amps. 2 v f f m R fb = 4 ± m (1) c T c Fequency Tansmitted signal ; f +f m (t) Index Tems Automotive Rada, Adaptive Cuise Contol, Modulation Fomat, LFM, FMCW f + f m f +f m (t 2 ) f d Received signal ; f +f m (t)±f d T I. INTRODUCTION O impove and add functionality to ACC adas, a system simulato that takes into account the ada s font-end pefomance at 77 GHz as well as the signal modulation and signal pocessing at base-band fequencies, is a must. We pesented such a ada simulation platfom fo Fequency Shift Keying (FSK) modulation fomat in 27, [1]. Geneally FSK based adas have excellent speed esolution, but the distance esolution is not vey good. This makes it difficult to disciminate between two tagets with identical elative speed, and futhe, tagets with the same speed as the ada, i.e. a zeo elative speed, ae not detected at all. This is why a Linea Fequency Modulation (LFM) fomat that allows the handling of moe complex scenaios, [2], has been chosen as the modulation fomat fo ou wok, and a second-geneation simulation platfom using LFM was ecently pesented in [3]. This pape epots on some impovements of the latte of these simulation platfoms: noise has been intoduced in the font-end components, new modulation fomats fo bette beat fequency association at multiple-taget-scenaios have been added and finally a MATLAB scipt has been developed to enhance the signal teatment using taget identification algoithms. II. LFM WAVEFORMS One basic LFM wavefom is the Fequency Modulated Continuous Wave (FMCW), as pesented in Fig. 1, see also [4]-[6]. When using this modulation fomat the taget detection is obtained fom the beat fequencies pesented fo each taget on the up and down amp espectively. The beat fequency obtained on the up amp is denoted by f b+ and on the down amp it is denoted f b-. The two beat fequencies can be calculated accoding to eq. (1), whee v is the elative speed of the taget, f the caie fequency, c the speed of light, f m f +f m (t 1) f t 1 T /2 f b+ f b- Fig. 1. FMCW modulation wavefom. t 2 τ Time Once having detected the beat fequencies the elated taget s data ae found by the following manipulations: by adding f b+ to f b- and solving fo v the elative speed is obtained and by subtacting f b- fom f b+ and solving fo R the ange is obtained. This is summaized in the following equations: c v ( + = fb+ fb ) (2) 4 f R T c ( f f ) = b+ b 8 f m In this pape a double FMCW modulation fomat (Fequency Modulated Continuous Wave with two up and down amps) is used. This wavefom is inspied by the one poposed in [6]. The two tiangles do not have exactly the same size. Thus, thee is a small diffeence of the amp (o chip) inclination between the fist and the second tiangle. This ensues, as was shown in [6], that the sequence ode of the tagets beat fequencies emains the same fo the up and down amp espectively. The coect association between the beat fequencies of the diffeent amps is thus guaanteed, which is othewise a poblem fo a single tiangle FMCW. When using this double FMCW modulation fomat, fou beat fequencies ae associated with each taget, one on each amp, and the associated equations fo calculating the tagets elative speed and ange ae given in eq. (4) and (5). Hee f b+ denotes the beat fequency on the fist up amp, f b- the one on the fist down amp, f b+ the one on the second up amp and finally f b- the one on the second down amp. T is the total duation of the wave fom (5.12 ms) and θ is the duation of (3)

2 > ITST 29, # 6 < 2 the fist tiangle (2.695 ms). The caie fequency, f, used is 76.5 GHz and the modulation bandwidth, f m, is 6 MHz. c v = ( fb+ ' + fb ' + fb+ '' + fb '' ) (4) 8 f c R = (( f f ) + ( f f )) b+ ' 8 f m b ' 1 + θ T b+ '' 1 θ b '' III. THE RADAR SIMULATION PLATFORM In ode to obtain a usable simulation platfom, the complete font-end, which consists of the micowave monolithic integated cicuits (MMIC), the popagation channel, the antenna and the tagets, has to be modeled coectly, including pecise noise chaacteistics. We have chosen to develop the MMIC models based on vendos specifications. The othe sub-cicuits ae epesented by thei scatteing paametes found by electomagnetic simulations. The antenna is descibed by its gain, and the popagation channel by a time delay and attenuation elated to the distance. Fou types of tagets, diffeentiated by thei Rada Coss Section (RCS), ae modeled (pedestian, motocycle, ca, tuck). The ada achitectue is shown in Fig. 2. The yellow boxes indicate the paametes on tansmission and the violet boxes the eception. The I and Q data yield the sign of the beat fequency and theeby the tagets elative speed. The Rada platfom is developed in the Advanced Design System softwae fom Agilent, with its built-in Envelope simulato and base-band Data Flow contolle that handle the flow of the digital signal pocessing. The Envelope simulato is launched fom the Data Flow contolle that also applies the modulation to the Voltage Contolled Oscillato (VCO) and collects the base-band data. Fig. 2. The geneal ada achitectue and simulation platfom, whee the Data Flow simulation tool contols the Envelope simulation of the RF platfom. Table 1 pesents the device data used in this simulation platfom. A. MMIC modeling The complete set of MMICs intended to be used in this ada ae commecially available fom the UMS foundy. The elated specifications ae found in [7]-[1]. (5) TABLE 1. DATA OF THE RADAR FRONT-END COMPONENTS Component Paamete Value VCO Fequency GHz+f m (MMIC) Output powe 5 dbm Phase Noise at 1kHz at 1 khz at 1 MHz -75 dbc/hz -1 dbc/hz -123 dbc/hz Multiplie-bysix-amplifie Multiplication facto 6 Output powe 14.5 dbm (MMIC) Noise figue 8 db Low Noise Gain 15 db Amplifie Noise figue 4.5 db (MMIC) Mixe Convesion loss 7.5 db (MMIC) Noise figue at 1kHz at 1 khz 34 db 28 db at 1 khz at 2 khz 21 db 17 db Couples Losses 3.2 db (Duoid) Isolation 4 db Tansition Losses.25 db (antenna) Antenna Maximal gain (TX) 27 db Suface effective aea (RX) m² Noise tempeatue ~29 K Popagation Losses (pe unit aea) 1log 1 (4πR²) dbsm path Delay τ=r/c s Tagets Dopple fequency 2v f /c Hz RCS Pedestian -1 dbsm Motocycle 7 dbsm Ca min 1log 1 (R)+5 dbsm 2 dbsm Tuck 2log min 1 (R)+5 dbsm 45 dbsm 1) Voltage Contol Oscillato (VCO) The VCO, [7], is modeled by a VCO block and a phase noise modulato block. The oscillation fequency is set to GHz, the output powe is set to 5 dbm (14 dbm is also possible but not necessay since the following MMIC cicuit includes a Medium Powe Amplifie), and the tuning fequency is set to 1 MHz/V. The paametes in the phase noise block ae chosen as to esult in a phase noise that is as close as possible to the values listed in the specification, see Table 1. The esulting phase noise cuve is pesented in Fig. 3. Phase Noise (dbc/hz) PN1 PN1 PN1 1 1E1 1E2 1E3 1E4 1E5 1E6 1E7 1E8 PN1 noisefeq= 1.kHz PhaseNoise=-79.9 PN1 noisefeq= 1.kHz PhaseNoise= PN1 noisefeq= 1.MHz PhaseNoise= Fequency (Hz) Fig. 3 The VCO phase noise (dbc/hz) as a function of fequency (Hz). 2) Multiplie by Six-Medium Powe Amplifie (X6MPA) The X6MPA, [8], is modeled by a fequency multiplying block, an amplifie block and a noise voltage souce connected to an AM modulato.

3 > ITST 29, # 6 < 3 In the multiplying block, the gain of the unwanted hamonics is set to -6 db and the gain of the wanted 6 th hamonic is set to db. In the amplifie block the fowad tansmission coefficient (o gain) is set to 9 db (which will esult in a 14 dbm output powe). The gain compession paametes ae adjusted to achieve an output powe vesus input powe plot afte hamonic balance simulations as depicted in Fig. 4. Pout (dbm) Pin (dbm) Fig. 4. The MPA powe gain obtained afte the hamonic balance simulation. The phase noise incease, caused by the fequency multiplie, can be calculated as 2 log 1 (N) whee N is the multiplication facto. Thus the incease due to the multiplie by six is 15.6 db which is seen when compaing the phase noise of the output of the VCO with that of the output of the multiplie block, see Fig. 5. Phase Noise (dbc/hz) E1 1E2 1E3 1E4 1E5 1E6 1E7 Fequency (Hz) PN dep Delta=15.56 Delta Mode ON Fig. 5. The phase noise due to the multiplie by six afte simulation follows the theoetically calculated value, as expected. VCO phase noise (o), phase noise afte the multiplie by six (-). As an amplifie has a multiplication facto of 1 (i.e. only the fundamental signal and no hamonics on the output) the phase noise incease due to the MPA should be 2 log 1 (1)= and as a esult the phase noise diffeence fom the VCO output to the MPA output should be 15.6 db. Howeve, plotting the phase noise at the output of the MPA and again compaing this value to the phase noise at the VCO output, we see that hee the diffeence is only 12.6 db, which should indicate that the MPA has actually deceased the phase noise in the cicuit. PhaseNoise (dbc/hz) E1 1E2 1E3 1E4 1E5 1E6 1E7 Fequency (Hz) pn dep Delta=12.6 Delta Mode ON Fig. 6. The phase noise at the output of the MPA as compaed to the output of the VCO. The diffeence is less than the theoetically calculated value and is due to the MPA satuation. VCO phase noise (o), phase noise afte the X6MPA (-). This phenomenon which occus in the amplifies satuated egion is descibed and explained in [11]. In the ada fontend application, the MPA woks in the satuated egion. 3) Low Noise Amplifie (LNA) The LNA, [9], is modeled with an amplifie block with the fowad tansmission coefficient set to 15 db and the noise figue set to 4.5 db. The gain compession paametes wee adjusted fo the powe gain plot to fit the specification, see Fig. 7. Pout (dbm) Pin (dbm) Fig. 7. The LNA output powe vesus input powe. 4) Mixe The mixe used is a dual channel mixe, [1], with a typical convesion loss of 7.5 db at the system fequency. The low fequency noise figue of this component vaies with the fequency due to the 1/f noise geneation, see Table 1. The mixe is modeled by a mixe block with the noise figue set to 17 db and the convesion gain to -7.5 db. The 1/f noise is then geneated in a noise voltage souce block and then added to the output signal. In fact, the noise voltage paamete is given by an equation (V_Noise=k/(min_IFfequency b ) with k and b adjusted to obtain the coect 1/f noise pofile. It was found that k should be and b=.45 fo the best fit. The paamete min_iffequency is chosen as the lowest down conveted beat fequency fo each amp accoding to the tagets distances and elative speeds, which thus sets the associated noise fo that amp. This means that the noise level is not the same on the fou amps and the signal to noise atio (SNR) has to be monitoed on each amp sepaately. Since thee is no senso block within ADS that can be used to, within the mixe model duing simulation, detemine which is the lowest down conveted fequency, the model has to be povided with this infomation. Thus, fo each simulation case, a set of case specific distance/elative speed data is given as input paametes to the ADS model. Fom these paametes, the minimum beat fequency on each amp is calculated (using Vaiable Equations within the ADS model) and used as the min_iffequency to povide the amp and case specific 1/f noise. B. Modeling of passive elements 1) Passive elements on the mm-wave cad The passive elements on the mm-wave cad ae modeled in two ways; one using the ADS povided micostip blocks such as powe splitte fo the powe divide placed afte the X6MPA output, a 4-pot S-paamete block with ideal coefficients fo the ciculato (which is in fact ealized as a 3 db/9 db hybid). In this vesion no tansmission lines ae

4 > ITST 29, # 6 < 4 used since they do not add any functionality to the platfom evaluation. In the second vesion the whole mm-wave cad is modeled using the ADS Momentum simulato tool, and thus a lookalike cicuit is intoduced in the platfom. The difficulty with this moe ealistic appoach is that the Momentum cad model has to be optimized to povide an excellent LO isolation in ode to avoid the geneation of a lage DC signal on the mixe output. In eality a DC component in the output spectum is pobable, howeve this is eliminated in the filte befoe the Fouie tansfomation takes place and it could theefoe be omitted fom. The 3 db/9 hybid Momentum simulation data ae pesented in Fig. 8. Amplitude balance (db) Isolation (db) Fequency (GHz) a) Fequency (GHz) b) Fig. 8. The momentum simulation esults fo the couple cicuit on 127 µm thick, Roges RT 588, Duoid: a) Amplitude (o) and phase balance ( ) vesus fequency, b) Isolation vesus fequency. 2) Micowave line to waveguide tansition To inteconnect the mm-wave cad to the antenna a micostip to waveguide tansition is used. This tansition is modeled using the Ansoft HFSS electomagnetic simulato tool, see Fig. 9. The micostip line ends in a patch pobe which excites a wave guide of WR1 size. A metal shot cicuit is placed at a distance of λ g /4 ove the patch pobe, which will seve as an open cicuit in the feed plane enabling maximum electical field stength. Fig. 9. HFSS model of the micostip to waveguide tansition. The fowad tansmission loss of this tansition is.24 db at the caie fequency and bette than.5 db fom 7 to Phase balance ( ) 9 GHz. The eflection coefficients ae bette than -12 db fom 7 to 9 GHz with a minimum of -23 db at the design fequency. 3) Antenna The antenna is subject to detailed investigation thoughout this poject, since the choice of antenna concept has a geat impact on the oveall cost of the ada system. Howeve, since the antenna development has been unning in paallel with the simulation platfom wok, a simple antenna model has been used until the final choice is made and the actual antenna gain and adiation patten can be enteed in the platfom simulato. The antenna is designed fo a naow lobe width which is swept ove the oad, thus the taget s angula position is given by the sweep angle. This time dependent sweep behavio is not taken in concen in the simulation platfom, but a fowad pointing antenna is modeled by an output and an input amplifie block. The output amplifie has a fowad tansmission coefficient set to 27 db wheeas fo the input amplifie it is calculated by the effective aea as G R λ²/(4 π). Hee G R denotes the eception gain which is 27 db, and λ is the fee space wavelength at the caie fequency. 4) Popagation channel The popagation channel is epesented by an attenuato block and a time delay block in each diection (i.e. tansmission and eflection mode). Fo simplicity, only the Line of Sight (LOS) popagation path is consideed, since the oad eflected popagation path delives a much weake eflected signal which is shifted in fequency less than the fequency esolution of the simulation platfom pemits to detect. Moeove, the fact that some tagets might be concealed by the othes is not taken into account. 5) Tagets Fou tagets ae used in this study: the pedestian, the motocycle, the ca and the tuck. The model paametes ae the ada coss section (RCS) and the Dopple fequency shift associated to taget s elative speed. A few papes pesenting esults fom measuements of the ada coss section at 76 GHz exist, e.g. [12]-[16]. In [12], measuements ae pefomed fom 5 to 94 GHz fo seveal test vehicles in each goup: motocycles, cas and minivans, while vaying the diection of the vehicles fom -1 to +1 azimuth. These esults show a clea diffeence in the RCS depending on the fequency and azimuth. In [14] the RCS fo a pedestian at 76 GHz and a 5 m distance is found to be -8 dbsm. Howeve, it is noted that this value vaies a lot depending on the clothing of the pedestian. In [15] the vaiation of the RCS of a ca at 76 GHz is measued fo two distances, 5 m and 25 m, and found to be about 2 and 16 dbsm espectively. This shows that the RCS fo lage tagets vaies with distance in the nea field ange. Consequently, a fix RCS value can only be used at a distance sufficiently lage fo the whole taget to be illuminated by the lobe. When the taget is close to the ada, the ada lobe only illuminates a smalle pat of the vehicle and theefoe the RCS must be vaied

5 > ITST 29, # 6 < 5 accodingly. Othe esults extacted fom ay tacing simulations and measuement validations have also been poposed in [16] whee an azimuth dependent RCS is achieved. Hence, fom all this, we can conclude that the RCS fo diffeent tagets depend on the fequency, azimuth angle, distance, etc., and to be able to model the ada s behavio a set of RCS fo the tagets must be found. In this wok, the two smalle tagets ae modeled with a fix RCS value: 1dBsm and 7dBsm fo the pedestian and the motocycle espectively. Fo the two lage tagets, a modeling appoach, taking into account the RCS dependence of the distance, while fixing the caie fequency at 76.5 GHz and the azimuth at, has been taken. To find an empiical fomula to model the ca s and tuck s RCS, we have pefomed measuements using an antenna with a 5 lobe angle, an azimuth angle to the taget of and the distance, R, was then vaied. The fomulas obtained ae pesented in equation (6) and (7) below: 1 log1 ( R) + 5 RCS ca = min (dbsm) (6) 2 2 log1 ( R) + 5 RCS tuck = min (dbsm) (7) 45 The RCS fo the fou tagets ae epesented in Fig. 1 below. Also included ae the efeenced RCS values, [12]- [15], given at 76 GHz fo compaison. The values deviate fom each othes, pobably because of the angula beam width of the antenna not given in these efeences. Fig. 1. Rada coss section used thoughout the simulations: ca accoding to eq. (6), tuck accoding to eq. (7), motocycle 7 dbsm and pedestian -1 dbsm. Othe published data, [12]-[15]. IV. SIMULATION EXAMPLE Fig. 11. a shows the double FMCW modulation wave fom used in the following simulation example whee the tagets ae a tuck [15 m, +8 km/h, RCS=28.5 dbsm], a pedestian [15 m, +1 km/h, RCS=-1 dbsm], and a motocycle [15m, +1 km/h, RCS=7 dbsm]. Fig. 11.b shows the acquied downconveted signal fom this simulation. Since we have thee tagets and consequently thee beat fequencies on each amp, we should expect to detect the sum of these thee beat fequencies in Fig. 11. b, while we seem to obseve only a pue sinusoid wave. This is due to the fact that, fo this specific simulation case, thee is a vey stong powe diffeence between the thee eflected signals. The time obsevation only eveals the beat fequency of the most poweful eflected signal (which comes fom the tuck at 15 m). Nevetheless, all beat fequencies with a powe geate than the noise level will be detected by means of Fouie analysis. VCO contol (V) IFsignal (V) Time (ms) a) Time (ms) b) Fig. 11. a) Double FMCW modulation fomat. b) The etuned downconveted signal s I component, fo the simulation example, aound the fist maximum modulation fequency. A. Signal Pocessing Once the IF I and Q signals ae acquied, the signal pocessing is to be caied out. Fo an appopiate signal pocessing, seveal basic functionalities ae equied: 1. The signal spectum fo each amp has to be calculated. 2. All peaks in the spectum above a given noise level have to be detected and saved in a table. Hee a taget that etuns a beat fequency with an SNR above 15 db is consideed as detectable. 3. False tagets have to be eliminated and the emaining peaks in each spectum have to be associated coectly. 4. Once the association is done, the sign of each beat fequency has to be detemined fo coect distance and elative speed calculation. 5. The tagets types can then be classified fom thei estimated RCS. 1) Signal pocessing in ADS In ADS, built-in spectum analyzes ae used to captue the fequency spectum of each amp, as shown in Fig. 12. Unfotunately, once the specta ae available, the ADS data display offes vey limited possibilities to automatically detect the peaks that each above the noise floo and thee is no loop function available (which is a must fo the association of peaks). Hence, the peak-detection-association and taget-

6 > ITST 29, # 6 < 6 identification algoithm is almost impossible to implement. Theefoe, the time-vaiant base-band components (I and Q) of the eceived signal (as in Fig. 11.b) ae expoted fo futhe pocessing in MATLAB. Powe (dbm) Tuck Pedestian Upslope Fequency (khz) Motocycle Fig. 12. Fequency spectum of the fist up amp of the simulation example, fom the ADS data display. 2) Impoved Signal Pocessing using MATLAB Using MATLAB we pefom the following opeations to obtain the taget infomation fom the ADS ada simulation: 1. The impoted I and Q components ae combined to a complex signal, which is then divided into pieces, with lengths equal to the time of each modulation amp. 2. A Hanning window is applied to educe fluctuations in the specta. 3. Fouie tansfomation with zeo-padding is applied to each signal piece to enhance the estimation of the beat fequencies 4. The powe is calculated and specta ae plotted. 5. The noise level of each spectum is calculated ove the actual bandwidth (1/amp duation). 6. The peaks locations ae achieved by detection thesholding, which yields fequency intevals whee they lie. Within these intevals, the local maximum is found and identified as a beat fequency, if the SNR is bette than 15 db. 7. The association of peaks found in the specta is made by compaison of thei sequence ode and powe. Hence, false alams (andom noise peaks) can be eliminated. 8. The elated tagets data ae calculated and the tagets ae classified. Fig. 13. The spectum fo the fist up amp, obtained using the MATLAB signal pocessing. In Fig. 13 the fist up amp spectum fom the MATLAB signal pocessing is pesented (c.f. the coesponding specta in Fig. 11.b deliveed by ADS). Hee, the beat fequencies occu to be negative, which is in fact coect as can be veified by enteing the tagets data in eq. (1), while setting T equal to θ, which is the duation of the fist tiangle. So, using the complex signal in the FFT, we eceive diectly the coect sign infomation fo the beat fequencies. Fig. 14 shows that the MATLAB scipt has found the tagets, and thei paametes ae calculated coectly. The SNR fo the thee tagets ae well above the detection limit of 15 db, and consequently the thee tagets ae detected. Fig. 14. Results pesented afte pocessing the simulation data with MATLAB fo the example in Fig. 11.b. and Fig. 13. V. CONCLUSIONS We have developed a full system ada simulato using ADS simulations combined with a MATLAB scipt fo efficient signal pocessing. Diffeent achitectues, modulation fomats and detection algoithms can theefoe be studied befoe the hadwae is ealized. REFERENCES [1] G. Piel, A. Péden, and P. Mallejac, Vitual platfom fo the design and the analysis of the pefomance of an adaptive cuise contol ada, ITST 5 th Intenational Confeence on ITS Telecommunications, Best, Fance, June 25. [2] H. Rohling, and M.-M. Meinecke Wavefom design pinciples fo automotive ada systems, CIE Intenational Confeence on Rada, Beijing, China, Oct. 21. [3] C. Känfelt, A. Péden, G. El Haj Shhadé, M. Abbas and F. Bodeeau, Plate-fome de simulation globale pou ada ACC à 77 GHz, CoGIST9, St Quay Potieux, Fance, June 29, pp [4] H. Ruse, and V. Mágoi, Sweep lineaization of a micowave FMCW dopple senso by an ultasonic efeence, Poceedings of the 1997 IEEE Intenational Fequency Contol Symposium, pp [5] M.-M. Meinecke, H. Hemann, Combination of LFMCW and FSK modulation pinciples fo automotive ada systems, Geman Rada Symposium GRS2, Belin, Oct , 2 [6] S. Miyahaa, New algoithm fo multiple object detection in FM-CW ada, SAE Technical Pape Seies [7] CHV227 ev 3Jan9 (homepage visited 25 august 29). [8] CHU3377 ev 23Ma7 (homepage visited the 25 august 29). [9] CHA177a ev 21Oct8 (homepage visited the 25 august 29). [1] CHM2378a ev 2Dec6 (homepage visited the 25 august 29). [11] A. Geens and Y. Rolain, Noise behavio of an amplifie in compession, Digest-sping of 55th ARFTG Confeence, Vol. 37, June 2, pp.1-8. [12] H. Suzuki, Rada coss section of automobiles fo millimete wave band, JARI eseach Jounal, Vol. 22, No. 1, pp , 2. [13] N. Yamada, Thee-dimensional high esolution measuement of ada coss section fo ca in 76 GHz band, R&D eview of Toyota Cental R&D Labs., Vol. 36, No. 2 (21. 6). [14] N. Yamada, Rada coss section fo pedestian in 76 GHz, Poceedings of 25 Euopean Micowave Confeence, Vol. 2, 4-6 Oct. 24, pp [15] R. Schneide, H.-L. Blöche, K. M. Stohm, KOKON Automotive high fequency technology at 77/79 GHz, Poceedings of the 4th Euopean Rada Confeence, Octobe 27, Munich, Gemany, pp [16] K. Schule, D. Becke, and W. Wiesbeck, Extaction of vetical scatteing centes of vehicles by ay-tacing simulations, IEEE Tansactions on Antennas and Popagation, Vol. 56, No. 11, Novembe 28, pp