Multipath richness a measure of MIMO capacity in an environment

Transcription

1 EUROEA COOERATIO I THE FIELD OF SCIETIFIC AD TECHICAL RESEARCH EURO-COST SOURCE: Aalborg Unversty, Denmark COST 73 TD 04) 57 Dusburg, Germany 004/Sep/0- ultpath rchness a measure of IO capacty n an envronment J. Bach Andersen, J.Ø.elsen Dept. of Communcaton Technology Aalborg Unversty els Jernes Vej 90 Aalborg, Denmark phone emal jba,jn@kom.aau.dk

2 ultpath rchness a measure of IO capacty n an envronment. Abstract J. Bach Andersen, J.Ø. elsen Department of Communcaton Technology Aalborg Unversty, Denmark The capacty of a IO system n an envronment depends on the envronment, the antennas and the SR. For a gven stuaton the normalsed egenvalues are determnng the capacty, and by expressng them as a cumulatve sum of ther log values, a measure of the ntrnsc multpath rchness s obtaned. For the case of waterfllng the capacty equals exactly the sum of the rchness and the log of water level. A set of ndoor measurements for a 6 by 3 element IO channel sounder s used to compare the rchness of the envronments. Defnton of rchness The capacty or spectral effcency of an envronment depends on the antenna structures at each end, knowledge of channel nformaton at the ends, correlatons between antennas and between paths, the dstrbuton of scatterers and most mportantly on the SR level. It s most easly expressed n the well-known formula for capacty [] C = log det I + HH ')) b/s/hz ) whch may be calculated for many dfferent cases once the channel matrx H has been measured. s the power normalsed to the nose power, the SR, and the number of transmt antennas. It s of nterest to fnd a smple relatonshp or curve, whch expresses the multpath rchness of the channel matrx wthout reference to the power or SR. Of course, one could use the capacty for a gven SR as such a measure, but that s just one number, whch does not contan any addtonal nformaton. As suggested n [] the EDOF effectve degrees of freedom) s essentally the slope of capacty versus SR at one value of SR and gves an ndcaton of the rank of the system. In [3] the relatve sum of the channel sngular values of the channel matrx s used. Varous measures are also dscussed n [4]. In ths formulaton the channel s frst supposed to be unknown at the transmtter, and the power dvded equally between the transmt antennas. For the moment we assume receve antennas, where <, whch means that the maxmum number of non-zero egenvalues equals. The number of sgnfcant egenvalues or sngular values determnes ths rchness, and n ths note we frst explore ths by lettng SR become so large that we can gnore the dentty matrx I n ). As t has been emphassed n [5] we can thnk of the egenvalues of HH as gans of the ndependent, orthogonal channels. Expressng the determnant through the egenvalues we can obtan a convenent measure for the multpath rchness, ndependent of SR, and f expressed n db also a convenent measure of the gans. The channel matrces are normalsed to have mean gan of 0 db), whch means that there s a constrant on the total gan, = )

3 Usng the arguments above to expand equaton ) we fnd C = log = log» log = log = 0.33 det I + = = + ) + ) = db)) + HH ' )) ) for log = log ) > ) 3) where the factor 0.33 stems from the transformaton from log to db. The rchness curve or vector) s now defned as the cumulatve sum of the log of the egenvalues k Rk) = log ) 4) = As wll be seen ths measure has a sgnfcant amount of nformaton concernng the multpath rchness, and apart from an easly calculated constant term dependng on the SR the capacty equals the rchness. It s the same rchness no matter whch end s the transmtter. The egenvalues are ordered n decreasng order. The fnal rchness and capacty) usng all egenvalues and suffcently hgh power equals R), but f t < 5) for a partcular value of =t then the egenvalues from t and above do not contrbute to the rchness, and we can use Rt) as the measure of rchness for that partcular value of SR. Waterfllng The formulaton becomes even more appealng when we consder the case, when the channels are known at the transmtter, so that waterfllng may be appled. The total power s now dstrbuted among the egenvalues nstead of among the antennas after the followng scheme, whch maxmses the capacty. The total power relatve to the nose power s denoted, and the powers allocated to the ndvdual channels or egenvalues are. The channels are flled up to a water level µ + = + =... = µ 6)

4 for the n channels wth postve powers. The -n channels are allocated zero power. Addng the equatons n 6) we fnd and the capacty n µ =? + 7) n n n n n n C = log + ) = log + µ - )) = log µ ) = log ) + nlog µ ) 8) so the capacty smply equals the sum of the rchness and the waterlevel expressed n log. wth only actve channels ncluded. It s nterestng to compare eqs. 8) and 3). In the case of unform dstrbuton of powers among antennas µ=/ compared wth µ=/n for large. Ths explans why there s only lttle dfference between the two cases for symmetrc systems where =n= for large, but a sgnfcant dfference when >>. Expermental results The followng cases are based on measurements taken n varous ndoor envronments at Aalborg Unversty, the measurement technques and envronments beng descrbed n [6]. The measurements are performed wth a 00 Hz sounder at 5.8 GHz. In ths secton we lmt the dscusson to a sngle frequency the narrowband rchness) and later look at the rchness as a functon of delay. The antennas are planar monopole arrays wth 6 or 3 elements at the ends, a 4 by 4 and a 4 by 8 array. In Fgure we show the rchness curves for the complete arrays wth 5 6*3) dfferent paths for one partcular poston of the two antennas, bur the dversty order s so hgh, that t s assumed to be representatve. The rchness s plotted as a curve, although t s only defned at the nteger values. The expermental envronments are ) a laboratory full of equpment lab), ) a rather empty basement room basement), 3) the lnk between two offces at two dfferent levels levels), and 4) a lnk between two offces at the same floor offce-to-offce). Apart from these the theoretcal uncorrelated Raylegh case and the theoretcal case of equal egenvalues called max. capacty) are shown. All sgnals below 30 db from the peak are gnored. The general shape of the rchness curve s parabolc, ndcatng an exponental decay of the egenvalues. The value of the frst egenvalue, the combned gan of the two antennas should be around 0*log0 v + ) [5] f uncorrelated, equal to 5.5 db or 0.33*5.5=5.3 n rchness. The general order of the rchness between the envronments for suffcent power s lab, offce-to-offce, basement, levels, and the dfferences n rchness are also the dfferences n capacty. ote that the turnng pont occurs where? = 0 db) and the maxmum rchness occurs for SR=, cf. equaton 5.

5 Fgure. Rchness for dfferent envronments for a 6*3 case, ncludng the d Raylegh case, and the maxmum capacty case wth equal egenvalues. There s a consderable dfference between the laboratory rchness full of equpment) and the levels case offces at two dfferent floors), n fact the dfference n capacty s 30 b/s/hz ndependent of SR as long as t s large. For =3, =6 and SR = 0 db, the lab capacty s 8 b/s/hz and levels capacty 5 b/s/hz. The rchness curves are ndependent of the envronments for up to 6 egenvalues of the total of 6. A more realstc case wth,)=4,4) s shown n fgure for the same envronments. The 4 antennas are a random subset of the 6 and 3 antennas of the complete antenna.

6 Fgure. The same envronments as n Fgure, but wth,)=4,4). The apparent rchness and thus the capacty are now much less, whch s natural wth the smaller number of elements. The levels case s now no longer the worst case, ndcatng that the capacty s a strong functon of the antenna structure as well as the envronment. It should be understood though that ths case s just one realsaton and not a mean of many realsatons. Roughly speakng the rchness for small egenvalues s vald for low SRs, about 0 db, the mddle range for around 0 db and hgh range wth all 4 egenvalues for SRs above 0 db. Dfferent envronments have the largest capacty for dfferent SRs. There s a cross-over such that the capacty s largest for ntermedate SRs for basement relatve to levels, whle t s the opposte for large SRs. The gan of the frst egenvalue s close to db 3*3.8) vald for a sngle large egenvalue for the levels and offce-to-offce case, whle the other cases have gans close to 0 db, whch s the true mean value for a Raylegh 4*4 case [5]. Ths s an ndcaton of small angular spreads for the levels and offce-to-offce case. In general the propagaton aspects of the dfferences wll be explored n future work. Rchness n the delay doman Snce the matrx may be expressed as a functon of delay t s possble to evaluate the rchness as a functon of delay at a fxed poston. Ths s not drectly related to capacty, but llumnates the more transent state of the mpulse. Fgure 3 a) and b) show the power delay profle, averaged over all antennas, and the rchness for the 6*3 case for levels as a functon of delay. At the peak of the mpulse the rchness s domnated by a few domnant egenvalues ndcatng that not all scatterers are actvated, and the rchness saturates at5 a later tme. ote that the maxmum values are

7 consderably hgher than the narrowband values n Fgure levels, b/s/hz). It s nterestng that the maxmum rchness occurs later than the arrval of maxmum power, but t s consstent wth the shape of the mpulse response, where the exponental decay corresponds to the scatter, whch presumably has both a temporal and angular spread. Fgure 3 a) power delay profle for levels and b) rchness versus delay

8 Concluson The successful applcaton of IO systems n ndoor envronments depends on a suffcently hgh rchness of multpath components. The capacty depends strongly on suffcent power avalable, but t s of nterest to defne the multpath rchness ndependent of the power level. Such a measure s the cumulatve sum of the log of the egenvalues of the channel matrx, approxmatng the capacty apart from an addtve term dependng on the antennas and the power, n the case of unknown channels at the transmtter. In the case of known channels waterfllng may be appled, and the capacty equals the rchness up to a certan egenvalue plus the water level. For the envronments measured all egenvalues are above a nose threshold, ndcatng full rank for all envronments. There s however consderable dfference between the envronments, the rchest beng a laboratory full of equpment. It s possble to assocate dfferent parts of the rchness curve wth varous power levels. For very small power levels the envronments are almost dentcal, utlsng only the frst 5 to 6 egenvalues. The rankng of the envronments s also dependent on the sze of the arrays. The rchness defnton seems to be a useful tool for comparng envronments. The multpath rchness may also be calculated as a functon of delay, and for the specfc envronments measured, the rchness peaks somewhat later then the man mpulse, whch has a low rchness. Acknowledgements The work has been supported by DoCoo Eurolabs, unch, Germany References [] G. Foschn and. J. Gans, On lmts of wreless communcatons n a fadng envronment when usng multple antennas, Wreless ersonal Communcatons, vol. 6, no. 3, pp , ar [] D-S Shu, G.J.Foschn,.J.Gans, J..Kahn, Fadng Correlaton and Its Effect on the Capacty of ultelement Antenna Systems IEEE Transactons on Communcatons, vol 48, no 3., arch 000, pp [3] J.W.Wallace,. A. Jensen, IO Capacty Varaton wth SR and ultpath Rchness from Full-wave Indoor FDTD Smulatons, Antennas and ropagaton Socety Internatonal Symposum,Volume:, -7 June 003, pp53-56 [4].C.F.Eggers, Dual drectonal channel formalsms and descrptons relevant for Tx-Rx dversty and IO, COST73 TD03) 044, Jan. 003 [5] J. Bach Andersen, Array Gan and Capacty for Known Random Channels wth ultple Element Arrays at Both Ends, IEEE Journal on Selected Areas n Communcatons, Vol. 8, o., ovember 000 [6] J. Ø. elsen, J. B. Andersen,. C. F. Eggers, G. F. edersen, K. Olesen, E. H. Sørensen, H. Suda, easurements of Indoor 6 3 Wdeband IO Channels at 5.8 GHz, IEEE Internatonal Symposum on Spread Spectrum Technques and Applcatons ISSSTA), Sydney, Australa, September 004