ON THE CAPACITY OF THE MIMO CHANNEL - A TUTORIAL INTRODUCTION - Bengt Holter

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1 ON HE CAPACIY OF HE MIMO CHANNEL - A UORIAL INRODUCION - Begt Holter Norwegia Uiversity of Sciece ad echology Departmet of elecommuicatios O.S.Bragstads plass B, N-7491 rodheim, Norway bholter@tele.tu.o ABSRAC Multiple-iput multiple-output MIMO) systems are today regarded as oe of the most promisig research areas of wireless commuicatios. his is due to the fact that a MIMO chael ca offer a sigificat capacity gai over a traditioal sigle-iput sigle-output SISO) chael. I this paper, a tutorial itroductio o the chael capacity of a MIMO chael will be give. 1. INRODUCION he icrease i spectral efficiecy offered by MIMO systems is based o the utilizatio of space or atea) diversity at both the trasmitter ad the receiver. Due to the utilizatio of space diversity, MIMO systems are also referred to as multiple-elemet atea systems MEAs). With a MIMO system, the data stream from a sigle user is demultiplexed ito separate sub-streams. he umber equals the umber of trasmit ateas. Each sub-stream is the ecoded ito chael symbols. It is commo to impose the same data rate o all trasmitters, but adaptive modulatio rate ca also be utilized o each of the sub-streams [1]. he sigals are received by R receive ateas. With this trasmissio scheme, there is a liear icrease i spectral efficiecy compared to a logarithmic icrease i more traditioal systems utilizig receivediversity or o diversity. he high spectral efficiecies attaied by a MIMO system are eabled by the fact that i a rich scatterig eviromet, the sigals from each idividual trasmitter appear highly ucorrelated at each of the receive ateas. Whe the sigals are coveyed through ucorrelated chaels betwee the trasmitter ad receiver, the sigals correspodig to each of the idividual trasmit ateas have attaied differet spatial sigatures. he receiver ca use these differeces i spatial sigature to simultaeously ad at the same frequecy separate the sigals that origiated from differet trasmit ateas.. CHANNEL CAPACIY At the iput of a commuicatio system, discrete source symbols are mapped ito a sequece of chael symbols. he chael symbols are the trasmitted/coveyed through a wireless chael that by ature is radom. I additio, radom oise is added to the chael symbols. I geeral, it is possible that two differet iput sequeces may give rise to the same output sequece, causig differet iput sequeces to be cofusable at the output. o avoid this situatio, a o-cofusable subset of iput sequeces must be chose so that with a high probability, there is oly oe iput sequece causig a particular output. It is the possible to recostruct all the iput sequeces at the output with egligible probability of error. A measure of how much iformatio that ca be trasmitted ad received with a egligible probability of error is called the chael capacity. o determie this measure of chael potetial, assume that a chael ecoder receives a source symbol every s secod. With a optimal source code, the average code legth of all source symbols is equal to the etropy rate of the source. If S represets the set of all source symbols ad the etropy rate of the source is writte as HS), the chael ecoder will receive o average HS) s iformatio bits per secod. 1 Assume that a chael codeword leaves the chael ecoder every c secod. I order to be able to trasmit all the iformatio from the source, there must be R = HS) c 1) s 1 he etropy rate is a fuctio of the statistical distributio of the source S. If the source S represet a discrete memoryless radom variable, the etropy rate of the source is equal to the source etropy, ad is defied as HS) = E[ log b p S )]. It is commo to use b = ad the etropy is the expressed i iformatio bits per source symbol.

2 iformatio bits per chael symbol. he umber R is called the iformatio rate of the chael ecoder. he maximum iformatio rate that ca be used causig egligible probability of errors at the output is called the capacity of the chael. By trasmittig iformatio with rate R, the chael is used every c secods. he chael capacity is the measured i bits per chael use. Assumig that the chael has badwidth W, the iput ad output ca be represeted by samples take s = 1 W secods apart. With a bad-limited chael, the capacity is measured i iformatio bits per secod. It is commo to represet the chael capacity withi a uit badwith of the chael. he chael capacity is the measured i bits/s/hz. It is desirable to desig trasmissio schemes that exploit the chael capacity as much as possible. Represetig the iput ad output of a memoryless wireless chael with the radom variables X ad Y respectively, the chael capacity is defied as [] C =maxix; Y ), ) px) where IX; Y ) represetsthemutual iformatio betwee X ad Y. Eq.) states that the mutual iformatio is maximized with respect to all possible trasmitter statistical distributios px). Mutual iformatio is a measure of the amout of iformatio that oe radom variable cotais about aother variable. he mutual iformatio betwee X ad Y ca also be writte as IX; Y )=HY ) HY X), 3) where HY X) represets the coditioal etropy betwee the radom variables X ad Y. he etropy of a radom variable ca be described as a measure of the amout of iformatio required o average to describe the radom variable. It ca also described as a measure of the ucertaity of the radom variable. Due to 3), mutual iformatio ca be described as the reductio i the ucertaity of oe radom variable due to the kowledge of the other. Note that the mutual iformatio betwee X ad Y depeds o the properties of the chael through a chael matrix H) ad the properties of X through the probability distributio of X). he chael matrix H used i the represetatio of the iput/output relatios of a MIMO chael is defied i the ext sectio. 3. SYSEM MODEL It is commo to represet the iput/output relatios of a arrowbad, sigle-user MIMO lik by the complex basebad vector otatio y = Hx +, 4) where x is the 1) trasmit vector, y is the R 1) receive vector, H is the R ) chael matrix, ad is the R 1) additive white Gaussia oise AWGN) vector at a give istat i time. hroughout the paper, it is assumed that the chael matrix is radom ad that the receiver has perfect chael kowledge. It is also assumed that the chael is memoryless, i.e., for each use of the chael a idepedet realizatio of H is draw. his meas that the capacity ca be computed as the maximum of the mutual iformatio as defied i ). he results are also valid whe H is geerated by a ergodic process because as log as the receiver observes the H process, oly the first order statistics are eeded to determie the chael capacity [3]. A geeral etry of the chael matrix is deoted by h ij }. his represets the complex gai of the chael betwee the jth trasmitter ad the ith receiver. With a MIMO system cosistig of trasmit ateas ad R receive ateas, the chael matrix is writte as h 11 h 1 h 1 h H = , 5) h R1 h R where h ij = α + jβ 6) = α + β e j arcta β α 7) = h ij e jφij. 8) I a rich scatterig eviromet with o lie-of-sight LOS), the chael gais h ij are usually Rayleigh distributed. If α ad β are idepedet ad ormal distributed radom variables, the h ij is a Rayleigh distributed radom variable. 4. SISO CHANNEL CAPACIY he ergodic mea) capacity of a radom chael with = R = 1 ad a average trasmit power costrait P ca be expressed as [] } C = E H max IX; Y ), 9) px):p P where P is the average power of a sigle chael codeword trasmitted over the chael ad E H deotes the expectatio over all chael realizatios. Compared to the defiitio i ), the capacity of the chael is ow defied as the maximum of the mutual iformatio betwee the iput ad the output over all statistical distributios o the iput that satisfy the power costrait. If each chael symbol at

3 the trasmitter is deoted by s, the average power costrait ca be expressed as P = E [ s ] P. 10) Usig 9), the ergodic mea) capacity of a SISO system = R = 1) with a radom complex chael gai h 11 is give by [4] C = E H log 1+ρ h11 )}, 11) where ρ is the average sigal-to-oise SNR) ratio at the receiver brach. If h 11 is Rayleigh, h 11 follows a chi-squared distributio with two degrees of freedom [5]. Eq.11) ca the be writte as [4] C = E H log 1+ρ χ )}, 1) where χ is a chi-square distributed radom variable with two degrees of freedom. Figure 1 shows the Capacity [bit/s/hz] SISO capacity SNR [db] Figure 1: Ergodic capacity a Rayleigh fadig SISO chael dotted lie) compared to the Shao capacity of a SISO chael solid lie). Shao capacity for a gaussia chael solid lie) ad the capacity of a Rayleigh fadig chael dotted lie) accordig to 1). he Rayleigh fadig curve preseted i Figure 1 equals the result i [6] capacity of a Rayleigh fadig chael with optimal power ad rate adaptio at the trasmitter uder the assumptio of perfect chael estimatio ad retur chael free of errors ad delay). 5. MIMO CHANNEL CAPACIY he capacity of a radom MIMO chael with power costrait P cabeexpressedas } C = E H max Ix; y), 13) px):trφ) P where Φ = Exx } is the covariace matrix of the trasmit sigal vector x. he total trasmit power is limited to P, irrespective of the umber of trasmit ateas. By usig 4) ad the relatioship betwee mutual iformatio ad etropy, 13) ca be expaded as follows for a give H Ix; y) = hy) hy x) 14) = hy) hhx + x) 15) = hy) h x) 16) = hy) h), 17) where h ) i this case deotes the differetial etropy of a cotiuous radom variable. It is assumed that the trasmit vector x ad the oise vector are idepedet. Eq. 17) is maximized whe y is gaussia, sice the ormal distributio maximizes the etropy for a give variace []. he differetial etropy of a real gaussia vector y R with zero mea ad covariace matrix K is equal to 1 log πe) det K). For a complex gaussia vector y C, the differetial etropy is less tha or equal to log detπek), with equality if ad oly if y is a circularly symmetric complex Gaussia with Eyy } = K [3]. Assumig the optimal gaussia distributio for the trasmit vector x, the covariace matrix of the received complex vector y is give by E yy } = E Hx + )Hx + ) } 18) = E Hxx H } + E } 19) = HΦH + K 0) = K d + K. 1) he superscript d ad deotes respectively the desired part ad the oise part of 1). he maximum mutual iformatio of a radom MIMO chael is the give by I = hy) h) [ = log det πe K d + K ))] log πek )] [ = log det K d + K )] log K )] K = log d + K ) K ) 1)] )] = log K d K ) 1 + I R )] = log HΦH K ) 1 + I R. Whe the trasmitter has o kowledge about the chael, it is optimal to use a uiform power distributio [3]. he trasmit covariace matrix is the give by Φ = P I. It is also commo to assume he superscript deotes Hermitia traspose

4 ucorrelated oise i each receiver brach described by the covariace matrix K = σ I R. he ergodic mea) capacity for a complex AWGN MIMO chael cathebeexpressedas[3,4] C = E H log I R + his ca also be writte as C = E H log P σ HH I R + ρ HH )]}. ) )]}, 3) where ρ = P σ is the average sigal-to-oise SNR) ratio at each receiver brach. By the law of large umbers, the term 1 HH I R as gets large ad R is fixed. hus the capacity i the limit of large is C = E H R log 1 + ρ)}. 4) Further aalysis of the MIMO chael capacity give i 3) is possible by diagoalizig the product matrix HH either by eigevalue decompositio or sigular value decompositio. By usig eigevalue decompositio, the matrix product is writte as HH = EΛE, 5) where E is the eigevector matrix with orthoormal colums ad Λ is a diagoal matrix with the eigevalues o the mai diagoal. Usig this otatio, 3) ca be writte as: C = E H log I R + ρ )]} EΛE. 6) he matrix product HH ca also be described by usig sigular value decompositio o the chael matrix H writte as H = UΣV, 7) where U ad V are uitary matrices of left ad right sigular vectors respectively, ad Σ is a diagoal matrix with sigular values o the mai diagoal. All elemets o the diagoal are zero except for the first k elemets. he umber of o-zero sigular values k equals the rak of the chael matrix. Usig 7) i 3), the MIMO chael capacity ca be writte as C = E H log I R + ρ )]} UΣΣ U. 8) After diagoalizig the product matrix HH,thecapacity formulas of the MIMO chael ow icludes uitary ad diagoal matrices oly. It is the easier to see that the total capacity of a MIMO chael is made up by the sum of parallel AWGN SISO subchaels. he umber of parallel subchaels is determied by the rak of the chael matrix. I geeral, the rak of the chael matrix is give by rakh) =k mi, R }. 9) Usig 9) together with the fact that the determiat of a uitary matrix is equal to 1, 6) ad 8) ca be expressed respectively as k C = E H log 1+ ρ ) } λ i 30) k = E H log 1+ ρ ) } σi. 31) I 30), λ i are the eigevalues of the diagoal matrix Λ ad i ad 31), σi are the squared sigular values of the diagoal matrix Σ. he maximum capacity of a MIMO chael is reached i the urealistic situatio whe each of the trasmitted sigals is received by the same set of R ateas without iterferece. It ca also be described as if each trasmitted sigal where received by a separate set receive ateas, givig a total umber of R receivig ateas. With optimal combiig at the receiver ad receive diversity oly = 1), the chael capacity cabeexpressedas[4] )} C = E H log 1+ρ χ R, 3) where χ R is a chi-distributed radom variable with R degrees of freedom. If there are trasmit ateas ad optimal combiig betwee R ateas at the receiver, the capacity ca be writte as 3 [4] [ C = E H log 1+ ρ ]} χ R. 33) Eq.33) represet the upper boud of a Rayleigh fadig MIMO chael. I Figure, the Shao capacity of a SISO chael is compared to the upper boud of 33) with = R = 6. Eve though this boud o the MIMO chael represet a special case, Figure clearly shows the potetial of the MIMO techology. Whe the chael is kow at the trasmitter, the maximum capacity of a MIMO chael ca be achieved by usig the water-fillig priciple [] o the trasmit covariace matrix. he capacity is the give by k ) } ρ C = E H log 1+ɛ i λ i 34) k ) } ρ = E H log 1+ɛ i σi, 35) 3 Assumig the artificial case of o iterferece betwee the received sigals.

5 Capacity [bit/s/hz] Ergodic capacity of a MIMO fadig chael SNR [db] Figure : he Shao capacity of a SISO chael dotted lie) compared to the ergodic capacity of a Rayleigh fadig MIMO chael solid lie) with = R =6. where ɛ i is a scalar, represetig the portio of the available trasmit power goig ito the ith subchael. he power costrait at the trasmitter ca be expressed as ɛ i. Clearly, with a reduced umber of o-zero sigular values i 31) ad 35), the capacity of the MIMO chael will be reduced because of a rak deficiet chael matrix. his is the situatio whe the sigals arrivig at the receivers are correlated. Eve though a high chael rak is ecessary to obtai high spectral efficiecy o a MIMO chael, low correlatio is ot a guaratee of high capacity [7]. I [8], the existece of pi-hole chaels is demostrated. Such chaels exhibit low fadig correlatio betwee the ateas at both the receiver ad trasmitter side, but the chaels still have poor rak properties ad hece low capacity. 6. ANENNA SELECION he MIMO chael capacity has so far bee optimized based o the assumptio that all trasmit ad receive ateas are used at the same time. Recetly, several authors have preseted papers o MIMO systems with either trasmit or receive atea selectio. As see earlier i this paper, the capacity of the MIMO chael is reduced with a rak deficiet chael matrix. A rak deficiet chael matrix meas that some colums i the chael matrix are liearly depedet. Whe they are liearly depedet, they ca be expressed as a liear combiatio of the other colums i the matrix. he iformatio withi these colums is the i some way redudat ad is ot cotributig to the capacity of the chael. he idea of trasmit atea selectio is to improve the capacity by ot usig the trasmit ateas that correspod to the liearly depedet colums, but istead redistributig the power amog the other ateas. Sice the total umber of parallel subchaels i the sums of eq.31) ad 35) is equal to the rak of the chael matrix, the optimal choice is to distribute the trasmit power o a subset of k trasmit ateas that maximizes the chael capacity. It is show i [9] that the optimal choice of k trasmit ateas that maximizes the chael capacity results i a chael matrix that is full rak. I [10], a computatioally efficiet, ear-optimal search techique for the optimal subset based o classical waterpourig is described. I [11], the capacity of MIMO systems with receive atea selectio is aalyzed. With such a reducedcomplexity MIMO scheme, a selectio of the L best ateas of the available R ateas at the reciever is used. his has the advatage that oly L receiver chais are required compared to R i the full-complexity scheme. I [11], it is demostrated through Mote Carlo simulatios that for =3ad R =8,thecapacity of the reduced-complexity scheme is 0bits/s/Hz compared to 3bits/s/Hz of a full-complexity scheme. 7. OUAGE CAPACIY I this paper, the ergodic mea) capacity has bee used as a measure for the spectral efficiecy of the MIMO chael. he capacity uder chael ergodicity where i 9) ad 13) defied as the average of the maximal value of the mutual iformatio betwee the trasmitted ad the received sigal, where the maximizatio was carried out with respect to all possible trasmitter statistical distributios. Aother measure of chael capacity that is frequetly used is outage capacity. With outage capacity, the chael capacity is associated to a outage probability. Capacity is treated as a radom variable which depeds o the chael istataeous respose ad remais costat durig the trasmissio of a fiite-legth coded block of iformatio. If the chael capacity falls below the outage capacity, there is o possibility that the trasmitted block of iformatio ca be decoded with o errors, whichever codig scheme is employed. he probability that the capacity is less tha the outage capacity deoted by C outage is q. his ca be expressed i mathematical terms by Prob C C outage } = q. 36) I this case, 36) represets a upper boud due to fact that there is a fiite probability q that the chael capacity is less tha the outage capacity. It ca also be writte as a lower boud, represetig the case where there is a fiite probability 1 q) thatthe

6 chael capacity is higher tha C outage, i.e., Prob C >C outage } =1 q. 37) 8. SUMMARY I this paper, a tutorial itroductio o the capacity of the MIMO chael has bee give. he use of multiple ateas o both the trasmitter ad receiver side of a commuicatio lik have show to greatly improve the spectral efficiecy of both fixed ad wireless systems. he are may research papers published o MIMO systems, reflectig the perceptio that MIMO techology is see as oe of the most promisig research areas of radio commuicatio today. [10] S. Sadhu, R. U. Nabar, D. A. Gore, A. Paulraj, Near-optimal selectio of trasmit ateas for a MIMO chael based o shao capacity, Coferece Record of the 34th Asilomar Coferece o Sigals, Systems ad Computers, 1: , 000. [11] A. F. Molisch, M. Z. Wi, J. H. Witers, A. Paulraj, Capacity of MIMO systems with atea selectio, I Proc. IEEE Iteratioal Coferece o Commuicatios ICC), : , REFERENCES [1] S.Catreux,P.F.Driesse,L.J.Greestei, Attaiable throughput of a iterferece-limited multiple-iput multiple-output MIMO) cellular system, IEEE rasactios o Commuicatios, 498): , aug 001. [].M.Cover,J.A.homas, Elemets of Iformatio theory, Joh Wiley & Sos, Ic., [3] I. elatar, Capacity of multi-atea gaussia chaels, A& echical Memoradum, ju [4] G. J. Foschii, M. J. Gas, O limits of wireless commuicatios i a fadig eviromet whe usig multiple ateas, Wireless Persoal Commuicatios, 6: , aug [5] J. G. Proakis, Digital Commuicatios, McGraw-Hill, Ic., [6] M-S. Alouii, A. Goldsmith, Capacity of akagami multipath fadig chaels, Proc. Vehicular echology Coferece, 1:358 36, [7] D. Chizhik, G. J. Foschii, R. A. Valezuela, Capacities of multi-elemet trasmit ad receive ateas: Correlatios ad keyholes, Electroic Letters, 3613): , ju 000. [8]D.Gesbert,H.Bölcskei, D. Gore, A. Paulraj, MIMO wireless chaels: capacity ad performace predictio, Global elecommuicatios Coferece GLOBECOM 00), : , 000. [9] D. A. Gore, R. U. Nabar, A. Paulraj, Selectig a optimal set of trasmit ateas for a low rak matrix chael, Proc. ICASSP 000, 498): , aug 000.