Closed-fom Fomulas fo Egodic Capacity of MIMO Rayleigh Fading Channels Hyundong Shin and Jae Hong Lee School of Electical Engineeing Seoul National Univesity Shillim-dong, Gwanak-gu, Seoul 151-742, Koea Fax: +82 2 88 8224 Email: shd71@snu.ac.k Abstact We pesent a new closed-fom fomula fo the egodic capacity of multiple-input multiple-output MIMO wieless channels. Assuming independent and identically distibuted i.i.d. Rayleigh flat-fading between antenna pais and equal powe allocation to each of the tansmit antennas, the channel capacity is expessed in closed fom as finite sums of the exponential integals which ae the special cases of the complementay incomplete gamma function. Using the well-known asymptotic behavio of the MIMO capacity, we also give a simple appoximate expession fo the channel capacity. Numeical esults show that the appoximation is quite accuate fo the entie ange of aveage signal-to-noise atios. I. INTRODUCTION In eseach aeas on wieless communications, multipleinput multiple-output MIMO systems equipped with multiple antennas at both tansmit and eceive ends have ecently dawn consideable attention in esponse to the inceasing equiements on data ate and quality in adio links [1] [4]. Multipath signal popagation has long been thought as an impaiment limiting the system capacity and eliable communication in wieless channels, togethe with the constaint of powe and bandwidth. In MIMO systems, this multipath popagation due to the ich scatteing in wieless channels is used to impove achievable data ate and link quality and it has been shown that the use of multiple antennas at both the tansmitte and the eceive significantly inceases the infomation-theoetic capacity fa beyond that of single-antenna systems [1], [2]. As the numbe of antennas at both the tansmitte and the eceive gets lage, the capacity inceases linealy with the minimum of the numbe of tansmit and eceive antennas fo a given fixed signal-to-noise atio SNR [1] [3], even if the fades between antenna pais ae coelated [4]. In this pape, we extend the analysis in [1] to obtain closedfom expessions fo the capacity of i.i.d. MIMO Rayleigh flat-fading channels. In addition, using the asymptotic ate of the capacity gowth fo MIMO channels, we pesent vey accuate appoximation fomulas fo the channel capacity. This wok was suppoted in pat by the National Reseach Laboatoy NRL Pogam and the Bain Koea 21 Poject of Koea. II. CAPACITY OF MIMO CHANNELS In this section, we biefly eview the capacity fomula fo MIMO channels. Conside a point-to-point communication link with t tansmit antennas and eceive antennas. Thoughout the pape we efe to α = mint, } and β = maxt, } and estict ou analysis to the fequency-flat fading case. We assume that the channel is pefectly known to the eceive while unknown to the tansmitte. When the tansmitted signal vecto is composed of statistically independent equal powe components each with a ciculaly symmetic complex Gaussian distibution, the channel capacity unde aveage tansmit powe constaint is given by [1], [2] whee C = log 2 det I + t HH bits/s/hz 1 I identity matix; H t andom channel matix; aveage SNR at each eceive antenna; tanspose conjugate of a matix. The enty H ij of H is the complex channel gain between tansmit antenna j and eceive antenna i, which is modeled as an i.i.d. complex Gaussian andom vaiable with zeo mean and unit vaiance, i.e., H ij CN, 1. The egodic capacity of the andom MIMO channel, which is the Shannon capacity obtained by assuming it is possible to code ove many independent channel ealizations, is evaluated by aveaging with espect to the andom matix channel as follows [1]: [ C t, = E log 2 det I + t HH ]. 2 Fo some specific models of the channel matix, C t, can be evaluated by statistical simulations, howeve these numeical matix calculations may be vey lengthy, especially when the numbe of antennas is vey lage. The andom matix HH fo < t o H H fo t has the cental Wishat distibution with paametes α and β, and its andom eigenvalues ae of geat inteest in multivaiate statistics see [9] and efeences theein. Using the singula -783-782-4/3/$17. 23 IEEE 2996
value decomposition theoem and the geneal esults fom the andom matix theoy, Telata [1] deived the analytic fomula fo the egodic capacity of MIMO channels in 2 as follows: C t, = α log 2 1 + λ/t p λ λ dλ 3 whee p λ λ is the distibution of a andomly selected eigenvalue of the Wishat matix with paametes α and β. The distibution p λ λ is given by [1] p λ λ = 1 α k= k!λ β α e λ 2 L β α k λ}, λ 4 k + β α! whee L m k x is the Laguee polynomial of ode k defined as [11, eq. 8.97.1] L m k x = ex x m d k e x k! dx k x k+m} k k + m x = 1 i i k i i!. 5 i= whee n k = n!/k!n k! is the binomial coefficient. Using the limit theoem on the distibution of the eigenvalues of lage dimensional andom matices [1], the asymptotic behavio of the MIMO capacity 2 has been epoted in [1] and [4]. If t and tend to infinity in such a way that β/α tends to a limit τ 1, wehave lim t, β/α τ whee C t, α = 1 2π bτ aτ log 2 1 + νx bτ xx aτ dx x = τ log 2 1+ν Jν, τ log + log 2 1+ντ Jν, τ 2 e J ν, τ 6 ν aτ = τ 1 2, bτ = τ +1 2 if t ν = /τ othewise J x, z 1 4 x z +1 2 +1 x 2 z 1 +1} 2. Note that the closed-fom expession 6 was pesented in the context of the capacity analysis fo code-division multipleaccess CDMA systems with andom speading sequences [6], [7]. In [6], 6 was indiectly obtained by using the fact that the pefect successive cancele with minimum meansquae-eo MMSE pefilteing achieves asymptotically the same capacity as the maximum-likelihood decode. On the ode hand, Rapajic and Popescu [7] deived the closed-fom expession diectly fom the integal and poved the esults in [6]. Fo t = = n, 6 educes to 1 C n,n lim n n = 2 log 2 1+ 4 +1 log 2e 4 4 +1 1 2 2. 7 III. CLOSED-FORM FORMULAS FOR THE CAPACITY We now deive a closed-fom expession fo the capacity of MIMO channels. Theoem 1: The egodic capacity in bits/s/hz of an i.i.d. MIMO Rayleigh fading channel with t tansmit antennas and eceive antennas unde aveage tansmit powe constaint and equal powe allocation is given by C t, = e t/ log 2 e 2k 2l k l k 2l k= l= i= β α+i 2l +2β 2α 2l i 1 i 2l!β α + i! 2 2k i l!i!β α + l! t } 8 whee E n z is the exponential integal of ode n defined by [11, p. xxxv] E n z = 1 e zx x n dx, n =, 1,, Rez} >. 9 Poof: Fom 5 and the identity of [11, eq. 8.976.3] [L m k x] 2 = Γk + m + i 2 2k k! k l= 2l! 2k 2l l!γm + l +1 L2m 2l 2x 1 whee Γz is the gamma function, the distibution p λ λ can be ewitten as p λ λ = 1 α out. = 1 α k= k= l= k!λ β α e λ [ β α + k! k L β α k λ 2l! 2k 2l 2 2k l!β α + l! ] 2 λ β α e λ L 2β 2α 2l 2λ = 1 k 2l 1 i 2l!λ β α+i e λ α 2 2k i l!i!β α + l! k= l= i= } 2k 2l 2β 2α +2l, λ. 11 k l 2l i 1 Thee exists a mino typo in [4, eq. 9] the tem -2 in 8 was missed 2997
Substituting 11 into 3, the capacity 3 is witten as C t, = log 2 e k 2l k= l= i= 1 i 2l! 2k 2l 2 2k i l!i!β α + l! } 2l +2β 2α ln1 + λ/tλ β α+i e λ dλ. 2l i 12 To evaluate the integal in 12, we use the following esult fom [5, Appendix B]: I n µ= =n 1!e µ ln1 + xx e µx dx, µ>,n=1, 2, n j=1 Γ n + j, µ µ j 13 whee Γa, z is the complementay incomplete gamma function defined by [11, eq. 8.35.2]. Accoding to 13, the integal in 12 is evaluated as ln1+λ/tλ β α+i e λ dλ =β α+i! β α+i+1 e t/ j=1 β α+i+1 t Γ β + α i 1+j, t/ t/ j. 14 The exponential integal E n z of 9 is the special case of the complementay incomplete gamma function, i.e., Fom 14 and 15, we have E n z =z Γ1 n, z. 15 ln1 + λ/tλ β α+i e λ dλ β α+i =β α + i!e t/ t. 16 Inseting 16 into 12, the channel capacity C t, bits/s/hz can be expessed in closed fom as 8. Example 1: Conside t = = n. Fom 8 with α = β = n, the capacity of a MIMO channel with n antennas at both the tansmitte and the eceive is given by C n,n = e n/ log 2 e k 2l 1 i k= l= i= 2k 2l k l 2l i 2 2k i 2l l in i } n. 17 Fig. 1. Nomalized channel capacity fo MIMO, MISO, and SIMO channels vesus the numbe of antennas n at =1dB. Example 2: Conside t = n and =1, i.e., multiple-input single-output MISO channels. Fom 8 with α = 1 and β = n, the capacity of a MISO channel with n tansmit antennas is given by C n,1 = e n/ log 2 e n. 18 Example 3: Conside t =1and = n, i.e., single-input multiple-output SIMO channels. Fom 8 with α =1and β = n, the capacity of a SIMO channel with n eceive antennas is given by C 1,n = e 1/ log 2 e 1. 19 Note that inceasing the numbe of eceive antennas fom n 1 to n in the SIMO channel yields an additional capacity advantage equal to e 1/ log 2 ee n 1/ bits/s/hz and 19 is in ageement with the fomely known esult of the exact capacity fomula fo Rayleigh fading channels with eceive divesity if applying the identity 15 to [5, eq. 4]. Fig. 1 shows C t, /n vesus the numbe of antennas n at =1 db fo the following five cases: a t = n and =2n; bt =2n and = n; ct = = n; dt =1 and = n SIMO; e t = n and =1MISO. If the numbe of antennas is inceased at both the tansmitte and the eceive, i.e., the cases a, b, and c, C t, /n, whee n = mint, }, conveges vey fast to its asymptotic value accoding to 6 as n inceases. The asymptotic values fo a, b, and c ae 4.114, 3.1253, and 2.7233 bits/s/hz, espectively. On the ode hand, fo SIMO and MISO channels with n eceive o tansmit antennas, i.e., the cases d and e, espectively, C 1,n /n and C n,1 /n decease with n and appoach to zeo as n. This 2998
Fig. 2. Channel capacity and its appoximation fo MIMO channels with t = = n vesus aveage SNR. Fig. 3. Channel capacity and its appoximation fo MIMO channels with t = n and = η n vesus aveage SNR. implies that if the numbe of antennas is inceased only at the tansmitte o the eceive, the asymptotic ate of the capacity gowth is equal to zeo. Also, we see that due to the total tansmitted powe constaint, the ate of the capacity gowth of the MISO channel appoaches to zeo faste than that of the SIMO channel as the numbe of antennas inceases. IV. APPROXIMATE FORMULAS FOR THE CAPACITY As obseved fom Fig. 1, it is obvious that the capacity of MIMO channels is vey well appoximated by a linea function of mint, } with a ate accoding to the asymptotic value 6 fo a given aveage SNR. Theefoe, the following appoximations ae staightfowad. The capacity of a MIMO channel with n antennas at both the tansmitte and the eceive in 17 is simply appoximated by C a,a C n,n C 1,1 + lim a a = e 1/ log 2 ee 1 1/+ 2 log 2 1+ 4 +1 } log 2e 2 4 +1 1 2. 2 4 Fo t<, an appoximate fomula fo the egodic capacity is given by C t, C a,b C 1, /t +t 1 lim a,b a b/a /t = /t 1 1 e 1/ log 2 e +t 1 + log 2 [ t log 2 1+ J } 1+ t J, t, } t log 2e J, ] 21 t whee z epesents the smallest intege geate than o equal to z. Similaly, when t >, the egodic capacity is appoximated by C a,b C t, C t/,1 + 1 lim a,b a a/b t/ t/ 1 t/ = e t/ / log 2 e [ t + 1 log 2 1+ t J t, t } + log 2 1+ J t, t } t log 2 e J t, t ]. 22 2999
[4] C.-N. Chuah, D. Tse, J. M. Kahn, and R. A. Valenzuela, Capacity scaling in MIMO wieless systems unde coelated fading, IEEE Tans. Infom. Theoy, vol. 48, no. 3, pp. 637 65, Ma. 22. [5] M.-S. Alouini and A. J. Goldsmith, Capacity of Rayleigh fading channels unde diffeent adaptive tansmission and divesity-combining techniques, IEEE Veh. Technol., vol. 48, no. 4, pp. 1165 1181, July 1999. [6] S. Vedu and S. Shamai, Spectal efficiency of CDMA with andom speading, IEEE Tans. Infom. Theoy, vol. 45, no. 2, pp. 622 64, Ma. 1999. [7] P. B. Rapajic and D. Popescu, Infomation capacity of a andom signatue multiple-input multiple-output channel, IEEE Tans. Commun., vol. 48, no. 8, pp. 1245 1248, Aug. 2. [8] T. M. Cove and J. A. Thomas, Elements of Infomation Theoy. New Yok: Wiley, 1991. [9] A. Edelman, Eigenvalues and Condition Numbes of Random Matices. Ph.D. thesis, M.I.T., Cambidge, MA, May 1989. [1] J. W. Silvestein, Stong convegence of the empiical distibution of eigenvalues of lage dimensional andom matices, J. Multivaiate Anal., vol. 55, pp. 331 339, 1995. [11] I. S. Gadshteyn and I. M. Ryzhik, Table of Integals, Seies, and Poducts. San Diego, CA: Academic, 6th ed., 2. Fig. 4. Channel capacity and its appoximation fo MIMO channels with = n and t = η n vesus aveage SNR. To assess the accuacy of the appoximate expessions of the capacity fo MIMO channels, we compae the appoximations with the exact calculations in Figs. 2 4. Fig. 2 shows the exact capacity of MIMO channels with t = = n and its appoximation fom 2 vesus aveage SNR fo vaious n. Fig. 3 shows the exact capacity of MIMO channels with t = n and = η n and its appoximation fom 21 vesus aveage SNR fo n =2, 6, 1, and η =1.5 and 3. Fig. 4 shows the exact channel capacity fo = n and t = η n and its appoximation fom 22 vesus aveage SNR fo the same values of n and η as those in Fig. 3. Fom these plots we can see that the appoximations closely match the exact channel capacity 8 fo the entie ange of aveage SNR and get moe accuate if the numbe of tansmit antennas is a multiple of that of eceive antenna and vice vesa. These closed-fom appoximations can theefoe be safely used to pedict the capacity of MIMO channels. V. CONCLUSIONS In this pape we obtained a closed-fom expession and an accuate appoximation fo the egodic capacity of i.i.d. MIMO Rayleigh flat-fading channels unde aveage tansmit powe constaint and equal powe allocation. By using these expessions, we can easily pedict the capacity pefomance of MIMO channels without any numeical integations o statistical simulations with numeical matix calculations. REFERENCES [1] I. E. Telata, Capacity of multi-antenna Gaussian channels, Euopean Tans. Telecommun., vol. 1, no. 6, pp. 586 595, Nov./Dec. 1999. [2] G. J. Foschini and M. J. Gans, On limits of wieless communications in a fading envionment when using multiple antennas, Wieless Pesonal Commun., vol. 6, no. 3, pp. 311 335, Ma. 1998. [3] G. G. Raleigh and J. M. Cioffi, Spatio-tempoal coding fo wieless communications, IEEE Tans. Commun., vol. 46, no. 3, pp. 357 366, Ma. 1998. 3