Direction of Arrival Estimation under Spread Spectrum Reference Signal Assisted Radio

Transcription

1 Communiations and Network, 013, 5, ublised Online November 013 (tt://wwwsirorg/journal/n) tt://dxdoiorg/10436/n Diretion of Arrival Estimation under Sread Setrum Referene Signal Assisted Radio eodoros N Kaifas Deartment of ysis, Aristotle University, essaloniki, reee tkaif@ysisautgr Reeived Setember 5, 013; revised Otober, 013; aeted Otober 30, 013 Coyrigt 013 eodoros N Kaifas is is an oen aess artile distributed under te Creative Commons Attribution Liense, wi ermits unrestrited use, distribution, and rerodution in any medium, rovided te original work is roerly ited ABSRAC In te resent work, a diretion of arrival estimator, under sread setrum referene of signal-assisted radio oerating in a Rayleig fading annel, is roosed e analysis, wi is alied to general reeiver antenna array onfigurations, demonstrates te ig erformane of te estimator wi is due to te double disreading (ode word and referene signal) e robability distribution funtion of te estimator is extrated and te system s robustness in regard to large number of interferers is demonstrated Keywords: DOA Estimator; Sread Setrum; Smart Antenna System 1 Introdution Under te ressure of te teleommuniations reuirements, in 4 and mu more in 5 [1], radios must be euied wit ig erformane systems tat ould auire and use all available information about te annel [] ere we fous on Diretion of Arrival (DOA), estimation e system is assumed to be imlemented in every arrier of te multiarrier sread setrum system [1] Bot sread setrum tenology and te information of te referene/ilot bits are emloyed Indeed, after te ode word disreading of te array s outut vetor, te ilot signal is used to erform te seond disreading e resulted vetor is used as te weigt vetor of te array e rodued ower in te far field is written in uadrati form and its average is omuted e maximum of te rodued beam is te DOAs estimate In te urrent work, te funtionality of te roosed system is addressed and its erformane is, bot analytially and numerially, verified In fat, after te robability Distribution Funtion (DF) of te estimator is extrated, its otential to oe wit very large number of interferers exibiting fine resolution is demonstrated e aer is organized as follows: First, in Setion, te signal and system models are given for an ulink senario of a sread setrum system en, in Setion 3, te double disreading funtion is studied and in Setion 4 te roer onvergene of te far-field is demonstrated e DF of te estimator is extrated in Setion 6 were its statistial roerties are resented Numerial evaluation of te roosed system s erformane follows and last, onlusions are stated Signal and System Model Consider an ulink senario in an asynronous sread setrum system wit mobile users Let 1 b b b i b N (1) be an N-bit seuene for te -t user wit te i-t bit eual to b i 1 Subsrit denotes te transose oeration Also, let 1 i () be an -i seuene for te -t user wit te i-t i eual to i 1 b is sread by us, te resetive transmitted signal an be reresented by a N 1 vetor: x b (3) were is te kroneker rodut Sine, te ulink oerates asynronously te ortogonal roerty of te ode vetors breaks down and teir elements sould be modeled as indeendent identially distributed (iid), binary RVs wit zero mean e same matematial modeling is given to te data bits from te different inoming signals Oen Aess

2 34 N KAIFAS o reresent te reeived signal one as to onsider a reeiver wit an antenna array So, we assume tat te array onsists of M elements and reeives signals from users tat arrive at te array from different diretions 1,,, e diretion resonse vetor assoiated wit te -t user is given by a e resetive signal an be given in te form of a M N matrix wit elements: y a x (4) is te reeived omlex signal amlitude e random variable (RV), as a Rayleig DF and te ase of is uniformly distributed in te domain (π, +π) At tis oint we make te usual and ratial assumtion tat te fading roess is onstant for a time eriod of a blok of Ν suessive bits e reeived base-band signal aross te reeiving antenna array an be exressed as: y x N (5) were N is an M N matrix wit elements reresenting satially and temorally wite omlex aussian noise wit zero mean and variane N 3 Double Disreading Funtion In order to ik out te signal of te -t user, a odemated filter ontaining te seified, and roerly time aligned, N seuene,, is alied to y erforming te first disreading funtion: r r z y a b a b N were we ave defined: (6) r ; r (7) It is easy to rove tat: 0;,, E r E r (8) us, using te Central Limit eorem, tat olds for big, te RV r, is a aussian random variable wit zero mean and variane eual to Also for te term N it is: 0; M E N E N I (9) were I M is te M M unit matrix e Central Limit eorem olds for tis ase also Now, assuming tat tere exist a referene signal, b, wi is igly orrelated wit b, we an erform te seond disreading forming te inner-rodut of z and b : w z b r r a b b r r a N b In (10), using b b, we ave defined: r b ; b N r (10) b b b b (11) e average value of te (8), (9) and (11), reads: w vetor, using Euations E w N a (1) e revious euation an be used to give te array s resonse vetor for te signal of interest (SOI) is an form te basis for te develoment of a robust DOA estimation algoritm Using (1), te angle of arrival SOI an be omuted using various teniues ere we use w to find te array s ower attern e DOA estimate is given by te angle of te attern s maximum Details are given next 4 Convergene of te rodued Average Far-Field attern Using w for te array s weigt vetor, te rodued ower attern is given by: a w w a (13) is attern an be farter imroved by using te average value: Ea w w a a Ew w a (14) So, we define te average ower attern by using te following: a Ra; R Ew w (15) Finally we ave: w w a a R E N b,, a a E r E r E N b (16) Euation (16) is rodued due to te fat tat te various terms of te summation are mutually unorrelated Also, it is easy to rove: b, ; E r N E N b I N (17) M Oen Aess

3 N KAIFAS 35 Using (17) and (18) in (16) and inserting te result in (15) gives: were N N (18) N a a a a (19) is te array s ower attern wen te maximum is towards te -t inoming signal o study te attern s onvergene we an write (18) as: N N N (0) Now it is obvious tat inreasing Ν, wi means, inreasing te roessing ain and inreasing te number of ilot bits, under te assumtion tat fading and DOAs remain onstant, (0) omes to te limit: (1) wi results tat DOA estimation gets more aurate is limit does not deend on te number of te inoming signals us, te metod an give te orret result assuming tat Ν an get an adeuate value In te following we study te statistial roerties of te roosed estimator 5 Statistial roerties of te DOA Estimator In te literature, tere are metods addressing te statistial roerties of DOA estimators We roeed as in [3] and we assume tat te signal to interferene lus noise ratio (SINR), is suffiiently ig so tat te zeros of ˆ are witin one Newton iteration ste of te true : and () denote te first and seond derivatives of te ower attern wit reset to alulated at Using (0), () beomes: N N By definition we ave tat: 0 (3) (4) Also, for te denominator of (3), aroximately (inreasing te N rodut), te next olds: (5) N Using (3)-(5) after rearranging te various terms, () beomes: N (6) e extration of te DF of te RV given in (6) is not an easy task sine te summation terms an take negative values [4] We start from (6) wi is written as: ; N sign e variable an take te following form: ; (7) (8) were are te terms wit sign 0 So, tere are ositive variables tat onstitute te RV and negative variables tat teir absolute values are summed to form te variable Now, te RV is distributed like: Oen Aess

4 36 N KAIFAS 1 f e u (9) E ; u is te unit ste funtion Euation (9) omes from te fat tat is Rayleig distributed wit te exeted value of te suare amlitude eual to e onditioning of on te angle of arrival of te desired user is exressed troug te After tis, it an be easily sown tat te RVs of (8) are distributed like: were e f u (30) 1 1 ; s s s 1 s Now let us define te variable: s (31) u (3) e random variable is distributed like: 1 u u f d f f u 0 e e 0 0 e RV λ is distributed like: e f u 1 ; ; (33) E (34) Let us now roeed to te RV ρ e DF of ρ is given by te following: f f f d (35) Before losing te setion devoted to te statistial roerties of te estimator a note is in order e DF of te estimator is onditioned on te DOAs of te signals not of interest and te SOI Neverteless, it an be easily roved tat te averaging over te DOAs still gives N and var N 1 6 Numerial erformane Evaluation In te urrent setion a verifiation of te extrated DF of te DOA estimator is given A omuter ode was written to simulate te reeiver and te signal models e statistial samling (Monte Carlo), simulation arameters are as follows: e signal to noise ratio, in te ase of a single reeiver, is te same for every user All users ave te same ower e average fading ower is set eual to 1 wile te fading is assumed to be Rayleig A Uniform Linear Array (ULA), aving seven omnidiretional elements wit λ/ interelement distane is used e number of interfering signals is set eual to 30 e DOA of te SOI is set eual to 30 e DOAs of te users are uniformly distributed between 60 e roessing gain is set eual to 16 e DF of te estimated DOA is given in Figure 1 e ontinuous urve is rodued from Euation (1) wile te istogram resents te Monte Carlo extrated DF after using 10 3 realizations of te satial signature e orizontal axis is te estimation angle in degrees 9-31, wile in te vertial axis, te absolute ourrene of te events is reresented For examle nearly 300 (87), times out of a total of 10 3 te estimator rodued an outome between 30 ± 005 If we araterize as aurate an estimate in te area 30 ± 05 ten te estimator rodues an aurate estimate aroximately 90% of te trials 7 Conlusions A double disreading based DOA estimator is roosed for a sread setrum referene of signal-assisted radio A losed form relation for te DF of te estimator is extrated to assess its erformane Instead of following te usual at, assuming DOA estimates orruted by simle additive wite noise, te realisti senario of Rayleig fading is addressed From te teoretial derivation, te estimator s ability to rovide aurate estimates, even in te ase of large Figure 1 e DF verifiation of te estimator Oen Aess

5 N KAIFAS 37 number of interferers, is demonstrated e roosed DOA metod is readily integrable in te Multiarrier sread setrum systems [1] REFERENCES [1] K Fazel and S Kaiser, Multi-Carrier and Sread Setrum Systems: From OFDM and MC-CDMA to LE and WiMAX, nd Edition, Jon Wiley & Sons, London, 008 [] L C odara, Smart Antennas, CRC ress, New York, 004 [3] F Li, Liu and R J Vaaro, erformane Analysis for DOA Estimation Algoritms: Unifiation, Simlifiation and Observation, IEEE ransations on Aerosae and Eletroni Systems, Vol 9, No 10, 1993, [4] M K Simon and M-S Alouini, On te Differene of wo Ci-suare Variates wit Aliation to Outage robability Comutation, IEEE ransations on Communiations, Vol 49, No 11, 001, Oen Aess