DIMENSIONALITY LOSS IN MIMO COMMUNICATION SYSTEMS

DIMENSIONALITY LOSS IN MIMO COMMUNICATION SYSTEMS Segey Loya, Amma Koui School of Infomation Technology and Engineeing (SITE) Univesity of Ottawa, 6 Louis Pasteu, Ottawa, Ontaio, Canada, KN 6N5 Email: segey.loya@ieee.og Depatment of Electical Engineeing Ecole de Technologie Supeieue, Note-Dame St. West Monteal (Quebec), H3C K3, Canada Abstact Taditional viewpoint is that the effect of coelation is to decease MIMO capacity. In this pape, the effect of coelation is studied fom a diffeent pespective: a concept of effective dimensionality (ED) of a MIMO system is intoduced and investigated using the coelation matix appoach. It is shown that the channel coelation esults in ED decease. Simple fomulas, which give an explicit dependence of ED on the channel coelation, ae given fo pactically-impotant cases. A compaison of the ED concept with the ecently-intoduced concept of effective degees of feedom is pesented. Key wods MIMO Channels, Capacity, Coelation. Intoduction Multiple-Input Multiple-Output (MIMO) communication achitectue has ecently emeged as a new paadigm fo efficient wieless communications in ich multipath envionments [,]. Using multi-element antenna aays (MEA) at both tansmitte and eceive, which effectively exploits the thid (spatial) dimension in addition to the time and fequency dimensions, this achitectue achieves channel capacities fa beyond those of taditional techniques. In uncoelated Rayleigh channels the MIMO capacity scales linealy as the numbe of antennas [,]. Howeve, thee ae seveal limitations to the pefomance of this achitectue in eal-wold conditions [,3-6]. One of the mao limitations is the coelation of individual subchannels, i.e. lins between one tansmitte and one eceive antennas, of the matix channel, which may esult in sevee degadation of MIMO pefomance [3-6]. One way to chaacteise this phenomenon is to conside the MIMO system dimensionality. When we have n n MIMO system (i.e. n tansmit and n eceive antennas) and n paallel independent sub-channels, we say that the system dimensionality is n. The MIMO channel capacity achieved unde these conditions (and also unde some easonable assumptions) is maximum. If some, o all, of the sub-channels ae coelated, the channel capacity deceases. Spatial dimensions, which ae used though n tansmit and n eceive antennas, povide smalle advantage in this case. Hence, we say that the system effective dimensionality (ED) deceases. We give a definition of the MIMO system effective dimensionality though a compaison of the MIMO capacity fo coelated and uncoelated channels, and investigate it numeically and analytically using the coelation matix appoach. We show that stong coelation between some subchannels esults in ED eduction. We also give a compaison of the effective dimensionality with the numbe of effective degees of feedom (EDOF) intoduced in [3]. By intoducing the ED concept, we ty to isolate and study the effect of coelation. In the ideal MIMO space, all the dimensions ae othogonal and the capacity achieves its maximum. Fo an actual MIMO space, some o all dimensions ae non-othogonal (coelated) and the capacity deceases. Fom a viewpoint of system pefomance, it is equivalent to going into a space of fewe dimensions. In fact, ED tells us how many of the actual numbe of tansmit/eceive banches ae effectively used. In a sense, the ED concept is simila to the powe efficiency concept, when lossless (ideal) and lossy (actual) systems ae compaed.. Effective Dimensionality of a MIMO System Unde some easonable assumptions, the channel capacity of n n MIMO system is []: + ( n) C = log det I+ρHH / bits/s/hz () whee n is the numbe of tansmit/eceive antennas, ρ is the signal-to-noise atio (SNR), I is n n identity matix, H is the nomalized channel matix, and + means tanspose conugate. The following nomalization of H is adopted in this pape [5]: + t ( HH ) = n, whee t means tace. Then ρ/n is pe-subchannel SNR. Anothe nomalization can also be used but ρ/n will have a diffeent meaning in this case. Fo simplicity, we futhe conside the case of equal eceived powes in evey eceive banch. In this case i h = and () simplifies to (, ) = log det ( +ρ / ) C n n R I R, ()

whee R is the nomalized channel coelation matix,, whose components ae = h h *, whee * i i i denotes complex conugate. In fact, i is the coelation coefficient of i-th and -th eceive banches. Stictly speaing, this definition is valid fo fixed channels only. Howeve, an expectation opeato can be employed fo stochastic channels. () emphasizes that the MIMO channel capacity C(R,n) is a function of the coelation matix R and of the numbe n of antennas. We define the MIMO effective dimensionality n e fom the following equation: = C R, n C I, ne, (3) whee C( I,n e ) is the MIMO capacity of uncoelated channel and is given by []: C( I, n ) = n log ( +ρ/ n ) (4) e e e Thus, the effective dimensionality is the numbe of dimensions of a MIMO system opeating ove an uncoelated channel, which has the same channel capacity as the actual system opeating ove the actual coelated channel (the signal-to-noise atio ρ being the same fo both cases). The ED shows how efficiently we use the actual eceive and tansmit banches and is, hence, a system pefomance paamete. In geneal, (3) is a tanscendental equation and cannot be solved analytically fo n e. Numeical methods can be applied to solve it. No convegence poblems ae anticipated since both sides of (3) ae monotonous functions of n. Fo some specific cases, analytical techniques can be used which allow us to gain some insight and to obtain simple analytical solutions fo pactically-impotant cases. A simila concept has been intoduced in [3] as the numbe of effective degees of feedom of a MIMO system: d δ EDOF = C( ρ) dδ δ= In some cases, the ED and the EDOF give vey close values while in some othe cases thei values ae vey diffeent. We give below a compaative analysis of the ED and the EDOF fo some impotant cases and discuss the implication of thei diffeences fo pactical system pefomance analysis. Simila paametes have also been consideed in [7]. 3. The Coelation Matix Appoach Let us now conside the case whee the coelation matix R has a bloc diagonal stuctue: whee n zeo matix, and (5) R R = (6) In I is the ( n ) ( n ) identity matix, is the R is a coelation sub-matix with non-zeo components. In this model, only eceive banches ae coelated; the est (n-) banches ae independent. In ode to demonstate how the ED concept wos, we adopt hee the exponential model fo R, which can appoximate some ealistic scenaios [5]: i, i R i = *, (7) i, i> [ ] whee is the (complex) coelation coefficient of adacent eceive banches. Using (6) and (7), C ( R, n) can be pesented in the following fom: whee and n ρ C( R, n) = log + det n R, (8) R is the following matix: i β, i R =, i i = (9) i β ( ) *, i > ρ ρ β= n + n A closed-fom expession fo the Appendix A, () det R is deived in n n n n α+ξ α ξ α α+ξ α ξ D = + + ξ () whee α= (β ), γ= ( β )., ξ= α 4 γ. Fo some pactically-impotant cases, a simplified expession may be obtained. Fo a system having a lage signal-to-noise atio ( ρ n >> ) and a lage numbe of antennas (n>>), afte some tansfomations which do not change the deteminant, we obtain (see Appendix A fo details): ( ) det R β () Appoximate solution of (3) then taes the simple fom: whee e ( ) n n σ (3) othe models of R (using, fo example, electomagnetic simulation) can also be used fo the pesent analysis

ρ ( n ) ρ ( + n ) log + σ= log (4) Fo =, one obtains σ= and, consequently, n e n +. Thus, the eduction in system effective dimensionality n due to the stong coelation of eceive banches is n = n ne. This is a physically easonable conclusion because we cannot tansmit infomation independently ove these banches but have to use them as only one banch. Fo =, σ= and n =, and the system has full dimensionality as it should be. The intepetation of this effect in tems of eigenvalue analysis is staightfowad. Fo lage coelation,, the eigenvalues of Appendix A): ( ) λ... λ <<, λ + R can be appoximated as (see ( ) ( ) + (5) Recall that the eigenvalues ae the vitual channel gains. Hence, we have one full-gain channel and (-) channels with educed gains (due to coelation). When =, λ = and λ =... =λ =, as expected, i.e. thee is only one channel out of, and the optimum stategy is to do the beamfoming and to tansmit only one bit steam ove channels. channels ae effectively collapsed into one. In the case of zeo coelation, =, one obtains λ =... =λ =, as expected, i.e. thee ae channels with full gains. It is inteesting to note that, despite of the fact that (5) has been deived unde the assumption of lage coelation,, it holds tue fo = as well. Fig. shows n as a function of fo diffeent values of computed using () and (3) and by numeical solution of (3) using (6) and (7). As one may see, () and (3) povide quite a good appoximation. An inteesting question, howeve, is how stong should the coelation be fo n. A detailed analysis (as well as an examination of Fig. ) gives the following ough estimation: n/ ρ (6) Fo the scenaio of Fig., one obtains:.995. Thus, fo almost all pactical cases (when ρ / n >> ) n will be smalle than. Let us now compae the ED and the EDOF concepts fo a pactically-impotant case of ρ n >> and n>>. A detail analysis shows that fo = and = the ED and the EDOF give appoximately the same pediction. But in between these two exteme cases thei values ae diffeent. One may wonde: why? Both paametes have vey simila physical meaning and both chaacteize the system pefomance fom the same viewpoint. Howeve, in the ED concept the eal system pefomance (channel capacity) is compaed to the ideal system pefomance, which opeates ove uncoelated paallel subchannels, fo the same total tansmitted powe that is distibuted between the effective dimensions only, not between the actual numbe of tansmittes. On the othe hand, EDOF is detemined though vaiation in SNR, i.e. in the tansmitted powe, Reduction in ED/EDOF 8 6 4 Reduction in ED (Eq. (3), (6) and (7)) Reduction in ED (Eq. (3) and (4)) Reduction in EDOF (Eq. (5), (6) and (7)) =5 =..4.6.8. Magnitude of coelation coefficient Figue. Reduction in ED/EDOF vesus the magnitude of coelation coefficient, n=, ρ=3 db. fo fixed channel coelation. The total tansmitted powe is always distibuted between the actual numbe of tansmittes. Both concepts can be used fo estimating MIMO system pefomance but fom diffeent viewpoints. In the EDOF concept, the effect of SNR is emphasized and, hence, it is moe elevant when one wants to now how the actual system pefomance vaies with tansmitted powe fo fixed coelation. On the contay, the effect of channel coelation is emphasized in the ED concept and, consequently, it is moe elevant when one wants to now how the system pefomance vaies with channel coelation fo fixed powe. In geneal, a definition of the numbe of dimensions of a MIMO system depends on a poblem consideed. It should be noted that the concept of effective dimensionality intoduced in this pape ae somewhat simila to the concept of effective divesity ode, which was successfully used in the analysis of divesity combining techniques ove coelated channels []. 4. Compaison to Othe Coelation Matix Models Let us now examine the effective dimensionality loss in coelated channels fo vaious coelation matix models. This allows one to find how sensitive the esults above ae with espect to coelation model vaiations.

Specifically, we examine the unifom model, the theediagonal model and the squaed exponential model. Fo the unifom model, R, i = =, Im i =, i, (7) [ ] { } assuming that is eal, the eigenvalues can be found explicitly in a closed fom, λ =... =λ =, λ = + ( ) (8) and the capacity can be expessed in a closed-fom as well [4]. Fo the ti-diagonal model,, i = R, = i i = *, i = + (9), othewise [ ] the eigenvalues can be expessed in the following closed fom [3], πi λ i = cos +, () Note that the model is physical only when π < cos +, () i.e. fo lage, < /. Othewise, as it can be seen fom (), some eigenvalues ae negative, which is not possible fo a physical coelation matix that is always positive definite. Fo the squaed-exponential model, which can be obtained based on some physical aguments [,], ( i), i R = *, i () ( i ), i > [ ] the eigenvalues ae not nown in a closed fom and numeical techniques should be used. Compaing all 4 models, one may note that the unifom model pesents a wost-case since all the banches ae equally coelated (in eal wold, coelation deceases fo widely-sepaated antennas, which is not accounted fo in the unifom model). The exponential model is moe physical since it accounts fo the coelation decease fo widely-sepaated antennas. The squaed exponential model, which is based on some physical aguments [,], also accounts fo the coelation decease fo distant antennas. In fact, it pedicts faste decease in coelation then the exponential model does. Finally, the thee-diagonal model accounts fo the coelation between neighboing antennas only assuming no coelation between distant antennas, which may be consideed as the best-case scenaio. The eason fo this model being nonphysical at > / is that it is not physically possible to have stong coelation between adacent antennas and no coelation fo the othes. Based on this easoning, one may expect that the capacities of the thee-diagonal, squaed exponential, exponential and unifom models coespondingly will satisfy to the following inequality: C3d Cse Ce Cu. Howeve, as a detailed numeical analysis demonstates (see Fig.), it is not that simple. The eason behind this is that the eigenvalues and the coelation matix enties ae elated in a vey complicated way. Fo example, almost all the enties of the squaedexponential model (except fo the diagonal ones) ae smalle than those of the exponential model, but some of the eigenvalues of the latte ae nonetheless lage than those of the fome. A detailed analysis shows that the unifom model capacity is always smalle than that of the exponential model and that the squaed exponential model capacity may be smalle o lage than that of the exponential model, depending on coelation and SNR values. It is inteesting to note that all the models (except fo the double-diagonal one, which is pathological at and above =.57, see ()) pedict almost the same behavio of the capacity vesus coelation. Hence, the esult seems to be of a geneal natue because it is quite modelindependent. Fig. 3 compaes loss in ED fo all 4 models. The esults ae again quite model-independent that suggest they may be of a geneal natue. Capacity, bit/s/hz 4 3 Unifom Exponential Squaed-exponential Tidiagonal..4.6.8. Figue. Capacity vesus coelation coefficient fo vaious models, n=5, ρ=3 db. Reduction in ED 5 4 3 Unifom Exponential Squaed-exponential Tidiagonal..4.6.8. Coelation coefficient Figue 3. Loss in ED vesus coelation coefficient fo vaious models, n=5, ρ=3 db.

Reduction in ED 4 Unifom Exponential Squaed-exponential 3 Tidiagonal..4.6.8. Coelation coefficient Figue 4. Loss in ED vesus coelation coefficient fo vaious models, n=5, ρ= db. As Fig. 4 suggests, thee is some vaiation due to the SNR. In paticula, the diffeence between the models slightly inceases as the SNR deceases. The squaed exponential model does povide bette pefomance and the unifom model povides the lowe bound on the pefomance (i.e. the wost-case scenaio) at ρ= db. The geneic tendency emains howeve the same. Oveall, as the SNR inceases all the cuves appoach close and close the unit step function at =. 5. Conclusions The effective dimensionality chaacteizes the MIMO system pefomance in a ealistic envionment, i.e., how efficiently we use multiple tansmit and eceive banches in a coelated channel. We have investigated it using the coelation matix appoach and have shown that fo stongly-coelated eceive banches the eduction in ED is appoximately -. Roughly speaing, ED is a facto in font of log(+snr) afte coelation has been taen into account. Its pupose is to compae the actual system pefomance to that of the ideal system, whose channel capacity is maximum, and to chaacteize in this way the effect of coelation. Fo example, when eceive banches ae completely coelated (=), this facto is (n-+) instead of n. The compaison of EDOF and ED shows that the fome gives a moe optimistic pediction of the system pefomance in a coelated channel. Refeences [] Foschini, G.J., Gans M.J.: On Limits of Wieless Communications in a Fading Envionment when Using Multiple Antennas, Wieless Pesonal Communications, vol. 6, No. 3, pp. 3-335, Mach 998 [] I.E. Telata, "Capacity of Multi-Antenna Gaussian Channels," AT&T Bell Lab. Intenal Tech. Memo., June 995 (Euopean Tans. Telecom., v., N.6, Dec.999) [3] Shiu, D.S., Foschini, G.J., Gans, M.J., Kahn, J.M., 'Fading Coelation and Its Effect on the Capacity of Multielement Antenna Systems,' IEEE Tans. on Communications, v. 48, N. 3, Ma., pp. 5-53. [4] S. Loya, G. Tsoulos, Estimating MIMO System Pefomance Using the Coelation Matix Appoach, IEEE Communication Lettes, v. 6, N., pp. 9-, Jan.. [5] S.L. Loya, Channel Capacity of MIMO Achitectue Using the Exponential Coelation Matix, IEEE Communication Lettes, v.5, N. 9, pp. 369 37, Sep. [6] S. Loya, A. Koui, New Compound Uppe Bound on MIMO Channel Capacity, IEEE Communications Lettes, v.6, N. 3, pp.96-98, Ma.. [7] Rayleigh, G.G., Gioffi, J.M.: "Spatio-Tempoal Coding fo Wieless Communications," IEEE Tans. Commun., v.46, N.3, pp. 357-366, 998. [8] S. Loya, A. Koui, Coelation and MIMO Communication Achitectue (Invited), 8th Intenational Symposium on Micowave and Optical Technology, Monteal, Canada, June 9-3, [9] M.A. Kon, T.M. Kon, Mathematical Handboo, McGaw Hill, New Yo, 968. [] M. Schwatz, W.R. Bennet, S. Stein, Communication Systems and Techniques, IEEE Pess, New Yo, 996. [] R.B. Etel at al, Oveview of Spatial Channel Models fo Antenna Aay Communication Systems, IEEE Pesonal Communications Magazine, Feb. 998, pp.-. [] T.S. Rappapot, Wieless Communications: Pinciples and Pactice, Pentice Hall, New Jesey,. [3] J.N. Piece, S. Stein, Multiple Divesity with Nonindependent Fading, IRE Poceed., Jan. 96, v. 48, pp. 89-4. [4] S.L. Loya, J.R. Mosig, Channel Capacity of N- Antenna BLAST Achitectue, Electonics Lettes, vol. 36, No.7, pp. 66-66, Ma.. 6. Apeendix A We use the following equivalent tansfomation of the deteminant: β* β*... β* β β*... β* det R = D = β β... β* = ( a) 3 β β β 3...

β *( β )... β β *( β )... ( b) = = = β β β... β β β 3... (β ) *( β )... β (β ) *( β )... β (β )...... (A) Whee we (a) fist multiply each ow (except fo the fist one) by * and substact it fom the pevious ow, and (b) multiply each colum by and substact it fom the pevious one (except fo the fist column). The last deteminant can be pesented in the following ecusive fom: whee D =αd γ D (Α) ( ), ( ). α= β γ= β. This is a diffeence equation [] and its solution can be pesented as: n n n n α+ξ α ξ α α+ξ α ξ D = + + ξ (Α3) whee ξ= α 4 γ. In high SNR mode, ρ / n >> β, this can be appoximated as, D ( ) α = (Α4) This appoximation is accuate povided that is not too close to. Moe accuate appoximation fo D can be deived as follows [5]. We stat with the second deteminant in (A) and note that, in high SNR egime, ρ / n >> β, the uppe diagonal elements (above the main diagonal) ae vey small and, hence, can be neglected. Then, D ( ) β... β β... = = β β β β... β β β 3... (A5) Using a simila appoach, we can now estimate the eigenvalues of R. We stat with the eigenvalue equation, R λ I = (A6) and, using the tansfomations simila to (A), tansfom the deteminant to the following fom:... γ* α γ... R λ I = γ* α... = (A7) α γ... λ whee α= λ +, γ= * λ. Fo, an appoximate solution of (A7) can be obtained as follows. Using (A3) and eeping the fist-ode tems in ξ, which is vey small when, one obtains α α α + λ = (-) smallest eigenvalues ae ( ) λ =... =λ = << + and the lagest eigenvalue is ( ) λ = ( ) + (Α8) (Α9) (Α) Note that fo =, λ = and λ =... =λ =, as expected. Detailed numeical analysis shows that the lagest and the smallest eigenvalues ae pedicted quite accuately, and that the pediction accuacy is wose fo the eigenvalues in between. Howeve, when computing the capacity these inaccuacies tend to compensate each othe so that the capacity is pedicted quite accuately.