Ergodic and Outage Capacity of Narrowband MIMO Gaussian Channels

Transcription

1 Ergodic and Outage Capacity of Narrowband MIMO Gaussian Channels Yang Wen Liang Department of Electrical and Computer Engineering The University of British Columbia, Vancouver, British Columbia Abstract This paper presents exact results on the capacity of narrowband multiple-input-multiple-output MIMO Gaussian channels when perfect channel state information CSI is assumed at the receiver, but not always at the transmitter Expressions for the MIMO channel capacity over ergodic flat fading channel, and formulas for the variance of the instantaneous capacity are derived The cumulative distribution function CDF of the instantaneous channel capacity is also obtained using a Gaussian approximation This CDF specifies the probability of not achieving a certain level of capacity outage capacity of the non-ergodic MIMO channels I INTRODUCTION More recently, the application of multiple antennas at both the transmitter and the receiver configurations has attracted considerable interest both within academia and industry as a mean of providing significant capacity gains over conventional single antenna array based solutions [], [2], [3] This configuration is the so-called Multiple-input-multipleoutput MIMO system This novel approach offers a unique solution for increasing demand for high performance next generation wireless communications As demonstrated by Bell Laboratories [4], the spectral efficiencies is up to 42 bps/hz, a spectacular increase compared to currently achievable spectral efficiencies of 2-3 bps/hz in wireless mobile and LAN system Therefore, study and analysis the capacity of MIMO channels using information theory would provide a deeper understanding of the configuration, and realization of the fundamental capacity limit of this approach Channel capacity is defined as the maximum rate at which data can be transmitted at an arbitrary small error probability [5] The capacity of MIMO channels has been well studied for the iid Rayleigh fading scenario [2], [3], and an exact formula for the ergodic or mean channel capacity was obtained in [3] assuming the receiver has the perfect channel state information CSI Few years later, a closed-form expression for the iid Rayleigh MIMO capacity with equal power allocation to each of the transmit antennas was derived [6], [7] The capacity variance of this MIMO channel was then derived in [8], and [8] proposed that the MIMO channel capacity can be accurately approximated by a Gaussian random variable when the CSI is not available at the transmitter equal power allocation The implication of this Gaussian approximation is that only the capacity mean and variance are needed to get an accurate approximation to the capacity cumulative distribution function CDF It specifies the probability of not achieving a certain capacity, defined as the outage capacity probability It was shown by simulations that the Gaussian approximation technique is very accurate even for a small number of antenna elements, and with a wide range of signal-to-noise ratio SNR [8], [9] More recently, researchers are woring on calculation of MIMO channel capacity under different scenarios: Correlation fading [], Moment Generating Function MGF approach for independent Rician fading [] and so on The remainder of this paper is organized as follows The next section we review some relevant results from linear algebra and numerical analysis We present the MIMO system model in section III The capacity of MIMO systems with fixed and deterministic coefficients will be studied in section IV In section V, we discuss the capacity of MIMO systems with iid random channel coefficients for fast and bloc Rayleigh fading channel model In section VI, we discuss the outage capacity of MIMO systems for slow iid Rayleigh fading model In section VII some conclusions are presented II PRELIMINARIES We begin by establishing the notation and by reviewing the results from linear algebra and numerical analysis that we will employ For any matrix A, A denote the Hermitian transpose conjugate of A Recall from linear algebra that an n n matrix is Hermitian if A = A The Matrix is nonnegative definite if x Ax for any n complex vector x An n n matrix U is unitary, if U U = I n where I n is the n n identity matrix An eigenvector v of an n n matrix A corresponding to eigenvalue λ is a n vector of unit length such that va = λv for some complex number λ The vector space spanned by the eigenvectors of A corresponding to the non-zero eigenvalue has dimension r, where r is the ran of A The eigenvalues of a Hermitian matrix are real The eigenvalues of a nonnegative definite Hermitian matrix are nonnegative Given a n n Hermitian Matrix A, there exists a unitary matrix U and a real diagonal matrix D such that U AU = D The columns of U are an orthonormal basis of the n- dimensional complex space, C n, given by eigenvectors of A The diagonal elements of D are the eigenvalues λ i, i =, 2,, n of A

2 2 The trace of a product of two square matrices, A and B, is independent of the order of the multiplication That is trab = trba deti + AB = deti + BA, where the product of matrices A and B is a square matrix A n complex random vector x is said to be Gaussian if the 2n real random vector [ ˆx ] consisting of its Rx real and imaginary parts, ˆx =, is Gaussian The Ix expectation and covariance of ˆx are E[ˆx] R 2n and E[ˆx E[ˆx]ˆx E[ˆx] ] R 2n 2n respectively A complex Gaussian random vector x is circularly symmetric [3] if the covariance of the corresponding ˆx has the structure E[ˆx E[ˆx]ˆx E[ˆx] ] = [ ] RQ IQ 2 IQ RQ for some Hermitian non-negative definite matrix Q C n n III MIMO SYSTEM MODEL Let us consider a single point-to-point MIMO system with arrays of t transmit and r receive antennas We focus on a complex baseband linear system model describe in discrete time The system bloc diagram is shown in Figure The transmitted signals in each symbol period are denoted by a t complex vector x, where the x i refers to the transmitted signal from antenna i The total power of the complex transmitted signal x is constrained to P regardless of the number of transmit antennas That is E[x x] = tr E[xx ] = P The transmitted signal bandwidth is narrow enough, so its frequency response can be considered as flat The received signal y is given by y = Hx + n 2 Where H is a r t complex channel gain matrix The ijth entry, h ij, of the matrix H represents the channel fading coefficient gain from the jth transmit to the ith receive antenna The r vector n is the noise at the receiver We mae an assumption that components of n are statistically independent complex zero mean Gaussian random variables with independent and equal variance of real and imaginary parts The covariance matrix of n is given by E[nn ] = σ 2 I r 3 Where σ 2 is the identical noise power at each of the receive antennas We denote the average power at the output of each receive antenna by P r The average SNR at each receive antenna is defined as SNR = P r σ 2 4 We assume that the total power per receive antenna is equal to the total transmitted power In other worlds, signal attenuations and amplifications in the propagation process are ignored We also assume that H is perfectly nown to the receiver, but not Fig Space-time encoder x x 2 x t h r h t h r2 h 2 h h 22 Bloc diagram of MIMO system h rt h 2 h 2t y y 2 y r Space-time decoder always at the transmitter The channel matrix can be estimate at the receiver by transmitting a training sequence The estimated CSI can be communicated to the transmitter via a reliable feedbac channel IV MIMO CAPACITY FOR CHANNELS WITH FIXED COEFFICIENTS The channel matrix, H, is deterministic in this case By the singular value decomposition theorem [3], any r t matrix H can be written as H = UDV 5 where U and V are r r and t t unitary matrices respectively The columns of U are the eigenvector of HH and the columns of V are the eigenvectors of H H D is a r r diagonal matrix with nonnegative square roots of the eigenvalues of matrix HH in diagonal D ii, i =, 2,, r, are called the singular value of H, and denoted by λ i, i =, 2,, r By substituting 5 into 2, we get y = UDV x + n 6 Let ỹ = U y, x = V x, ñ = U n Thus, ỹ = D x + ñ 7 Note that U and V are unitary matrices, so ñ and x has the same distribution as n and x respectively [3] The covariance matrix for ñ is E[ññ ] = E [ U nn U ] = U E [ nn ] U = σ 2 I r 8 For the r t matrix H, the ran, r, is at most minr, t Equation 7 becomes { λi x i + ñ i i r ỹ i = 9 r + i r ñ i That is the rest of the components of ỹ if any do not depend on the transmitted signal As 9 indicates, the equivalent MIMO channel can be considered as consisting of r uncoupled parallel subchannels According to information theory [5], the optimum distribution of transmitted signals, x, are Gaussian for a Gaussian channel in the sense of achieving maximum mutual information channel capacity In other worlds, the original transmitted signals x should be Gaussian distributed Thus the overall channel capacity, denoted by C, is the sum of the subchannels capacities by the Shannon capacity formula [5] r C = ln + P ri σ 2 nats/s/hz

3 3 where P ri is the received signal power at the ith subchannel Depending on whether the CSI is nown at the transmitter, there are two cases, equal transmit power allocation and adaptive transmit power allocation A Equal Transmit Power Allocation In this case, the power allocated to subchannel i is given by P i = P/t, i =, 2,, t, and P ri is given by P ri = λ ip t Thus the channel capacity can be written as r C = ln + λ ip r tσ 2 = ln + λ ip tσ 2 nats/s/hz B Adaptive Transmit Power Allocation 2 For the case when the CSI is nown at the transmitter, the capacity given by 2 can be increased by allocating the transmit power to different antennas according to the waterfilling method [5] + P i = µ σ2, i =, 2,, r 3 λ i where µ is chosen to meet the power constraint so that r P i = P and a + denotes maxa, The received signal power at the ith subchannel can be written as P ri = λ i µ σ Thus the channel capacity can be written as r C = ln [ + λ iµ σ 2 + ] = = r r σ 2 [ ] + λi µ ln + σ 2 [ ] + λi µ ln nats/s/hz 5 σ 2 V CAPACITY OF MIMO FAST AND BLOCK RAYLEIGH FADING CHANNELS For Rayleigh fading, the entries in H are iid complex, zero mean Gaussian random variables with normalized unit magnitude variance The capacity of a random MIMO channel with power constraint tr E[xx ] = P can be expressed as { } C = E H max I x; y 6 px:tre[xx ]=P where E H denotes the expectation over all channel realizations and Ix; y represents the mutual information between x and y Equation 6 states that the capacity of the channel is defined as the maximum of the mutual information between the input, x, and the output, y, over all statistical distributions, px, on the input that satisfy the power constraint The realization, H, of H is assumed to be nown at the receiver, but not at the transmitter Thus the channel output is the pair y, H and equation 6 is equivalent to, from [3], C = max I x; y, H 7 px:tre[xx ]=P According to [3], the transmit signal x maximizes the differential entropy when x is a circularly symmetric complex Gaussian with covariance matrix Q given by E[xx ] The differential entropy of x is Hx = ln [detπeq] nats 8 The covariance matrix of the zero mean received signal y given by 2 with realization of H = H is E[yy ] = E [ Hx + nx H + n ] = E [ Hxx H ] + E [ nn ] = HQH + σ 2 I r 9 The mutual information between input and output of the MIMO system is then Ix; y, H = Ix; H + Ix; y H = Ix; y H = E H [Ix; y H = H] = E H [Hy H = H] E H [Hy x, H = H] { [ = E H ln det πehqh + σ 2 I r ]} Hn [ π r e r det ]} HQH + σ 2 I r = E H {ln π r e r σ 2r = E H {ln det I r + σ } 2 HQH 2 When the transmitter has no nowledge about the CSI, it is optimal to use equal power allocation to each transmit antenna [3] The transmit covariance matrix is then given by Q = P σ 2 t I t 2 Then the ergodic mean capacity of the system can be derived by substituting 2, 2 into 7 as C = E H {ln det I r + P } σ 2 t HH 22 = E H {ln det I t + P } σ 2 t H H 23 For the special case that H is deterministic, the capacity of the system is exactly the same as 2 derived from 22 for equal power allocation [3], [2] Not surprisingly, same expression can be obtained as 5 derived from 7 and 2 for adaptive power allocation as well [3] Let n = maxr, t and m = minr, t The random matrix HH for r < t, or H H for r t, has the Wishart distribution with parameters m, n [3] and the unordered eigenvalues have the joint density, from [3], pλ,, λ m = m!k m,n m i λ n m i e λ i λ i λ j 2 24 i<j

4 4 where K m,n is a normalizing factor Thus the distribution of one of the unordered eigenvalue, say λ, is the result of integrating out other random eigenvalues λ 2,, λ m Then the distribution of any one of the unorder eigenvalues is given by, from [3], pλ = m m =! [ L n m λ ] 2 λ n m e λ 25 + n m! where L n m λ is the associated Laguerre polynomial of order, and it is given by, from [3], [6], L n m λ = l + n m! l!n m + l!l! λl 26 l= Using equation 5 and determinant identity given in Section I, the mean capacity given by equations 22, 23 can be evaluated as C = E λ {ln det = E λ { m = me λ [ln I m + P ln + P } σ 2 t diagλ, λ,, λ m } 27 σ 2 t λ i ] + P σ 2 t λ 28 That is C = ln + P m! [ L n m λ ] 2 λ n m e λ σ 2 dλ t + n m! = 29 Dohler [6] derived a closed-form expression for 29 in terms of finite sums allowing for simple numerical implementation m! C = + d! = l = l 2 = A l, d A l2, d l +l 2 Z l +l 2 +d P σ 2 t 3 where d = n m, A l, d = + d!/[ l!d + l!l!], and Z i a = i j= i! i j!! a i j + [ i j = i j a j i e /a Ei ] a with Eix = x e t t dt, and Eix is a typical build-in function found in most of the simulation software, eg MATLAB For bloc fading channels, the channel capacity can be calculated by using the same expression as 22 as long as the channel is ergodic [3] for equal power allocation case In the adaptive transmit power allocation case, the instantaneous MIMO channel capacity is given by 5 The average capacity can be obtained by averaging over all realization of the channel matrix H for an ergodic channel From the simulation results given in [2], when the number of the transmit antennas is the same or lower than the number of receive antennas, there is almost no gain in adaptive power allocation However, when the number of transmit antennas is larger than the number of receive antennas, there is a significant gain achieved by water filling method According to the number of transmit and receive antennas, we give 3 different cases and their corresponding analytical results of the mean capacity in the following subsections Again, the assumption that the transmitter has no nowledge of the CSI is made A The Number of Receive Antenna is For a MIMO system with t transmit and one receive antenna on a fast Rayleigh fading channel The values of the capacity are shown in Figure 2 for various value of t and SNR As the number of transmit antennas increases, the capacity Channel Capacity nats/s/hz Fig The value of the capacity for r = vs Number of Tx Antennast SNR = 35dB SNR = 3dB SNR = 25dB SNR = 2dB SNR = 5dB SNR = db SNR = 5dB SNR = db Number of Tx Antennas t Channel capacity vs t for r = and various value of SNR approaches the asymptotic value, from [3], [2], lim C = ln + Pσ t 2 nats/s/hz 3 The system behaves as if the total power is transmitted over a single unfaded channel In other words, the effect of fading attenuates with increasing number of transmit antennas B The Number of Transmit Antenna is For a MIMO system with one transmit and r receive antennas on a fast Rayleigh fading channel The values of the capacity are shown in Figure 3 for various value of r and SNR As the number of receive antennas increases, the capacity approaches the asymptotic value, from [3], lim C = ln r + rp σ 2 nats/s/hz 32 Theoretically, the channel capacity increase logarithmically as r increasing C The Number of Receive Antenna Equals The Number of Transmit Antenna For a MIMO system with equal number of transmit and receive antennas on a fast Rayleigh fading channel The values

5 5 Channel Capacity nats/s/hz Fig 3 Channel Capacity nats/s/hz Fig The value of the capacity for t = vs Number of Rx Antennasr SNR = 35dB SNR = 3dB SNR = 25dB SNR = 2dB SNR = 5dB SNR = db SNR = 5dB SNR = db Number of Rx Antennas r Channel capacity vs r for t = and various value of SNR SNR = db SNR = 5dB SNR = db SNR = 5dB SNR = 2dB SNR = 25dB SNR = 3dB SNR = 35dB The value of the capacity for t = r vs Number of Rx Antennasr Number of Rx Antennas r Channel capacity vs r for r = t and various value of SNR of the capacity are shown in Figure 4 for various value of t and SNR As the number of antennas increases, the capacity approaches the asymptotic value, from [3], lim C = r t=r [ π 4 ln + Pσ ] ν 2 ν 4 dν nats/s/hz 33 Equation 33 can be lower bounded by, from [2], [ ] P lim C = r ln t=r σ 2 nats/s/hz 34 It shows that the capacity increase linearly with the number of antennas and logarithmically with the SNR VI CAPACITY OF MIMO SLOW RAYLEIGH FADING CHANNELS In this case, H is chosen randomly, according to a Rayleigh distribution at the beginning of transmission, and held fixed for all channel uses In other words, the channel is nonergodic In this regard, the capacity is a random variable, and no matter how small the rate threshold would be, there is always a non-zero probability that the channel is incapable of supporting arbitrarily low error rates [3] The capacity complementary cumulative distribution function ccdf is often used to characterize such channels [2], [3], [8] The ccdf defines the probability, denoted by P ccdf, that a specified capacity is provided The outage capacity probability, denoted by P out, is the probability of not achieving a threshold capacity, outage capacity, and it is defined as P out = inf p ln deti r + HQH < R th 35 Q:Q trq P where the infimum is taen over all positive semi-definite Q with the power constraint trq P, and the probability is taen over all realizations of the random matrix H P out, the cumulative distribution function CDF, of the outage probability, is related to ccdf by P out = P ccdf Smith [8] demonstrated that the narrowband Rayleigh MIMO channel capacity can be accurately approximated by Gaussian approximation, for the case when the receiver has the perfect CSI but the transmitter does not equal power allocation The implication of this result is that only mean and variance of the capacity are needed to get an accurate approximation of the outage capacity From 27, the instantaneous channel capacity is given by m C ins = ln + P σ 2 t λ i 36 where the notations follow exactly as before Recall that the mean channel capacity is given as 3 After some mathematical manipulations, the variance of C ins is obtained as, from [8], V arc ins = mv ar ln + P λ + mm Cov = m m j= { ln m [ σ 2 t + P λ σ 2t w 2 λpλdλ, ln + P λ σ 2 t i!j! i + n m!j + n m! ] } 2 λ n m e λ L n m i λln m j λwλdλ 37 where wλ = ln + P λ σ 2 t, and Ln m is defined as 26 The Gaussian approximated outage capacity probability P out with equal power allocation can now be obtained as P out = p C ins < R th R th E C ins = Q V ar Cins E C ins R th = Q V ar Cins 38 The Q-function is tail integral of a unit-gaussian pdf, and is defined as Q x = e z2 2 dz 39 2π x

6 6 The outage capacity probabilities are shown in Figure 5 for a MIMO channel with 4 transmit and 4 receive antennas for various value of SNR As Figure 5 shows, the variance of the 9 8 The outage capacity of MIMO channel with Tx = 2 and SNR = 5dB and various number of Rx The outage capacity of MIMO channel with Tx = Rx = 4 for various SNR dB 5dB 25dB P out = pc=<r th Rx = Rx = 4 P out = pc=<r th db db 2dB 3dB 3 2 Rx = 7 Rx = Rate Threshold in nats/s/hz Fig 5 The outage capacity of MIMO channel with Tx = Rx = 4 and for various SNR instantaneous capacity increased as the SNR increased for the same system The outage capacity probabilities are shown in Figure 6 for a MIMO channel with 2 receive antennas and various number of transmit antennas for SNR equals 5dB In contrast, P out = pc=<r th The outage capacity of MIMO channel with Rx = 2 and SNR = 5dB and various number of Tx Tx = Tx = 4 Tx = 7 Tx = Rate Threshold in nats/s/hz Fig 6 The outage capacity of MIMO channel with Rx = 2 and SNR=5dB for various number of transmit antennas the outage capacity probabilities are shown in Figure 7 for a MIMO channel with 2 transmit antennas and various number of receive antennas for SNR equals 5dB As Figure 6 and 7 shows, the variance of the instantaneous capacity decreased as the number of transmit antennas or receive antennas increased when other parameters are fixed VII CONCLUSION In this paper, we go through detail derivations of the capacity of MIMO channel under 3 major scenarios depending on the channel matrix H We focused in the case which the channel state information is nown at the receiver but not at the Rate Threshold in nats/s/hz Fig 7 The outage capacity of MIMO channel with Tx = 2 and SNR=5dB for various number of receive antennas transmitter Exact expressions of mean channel capacity, and the variance of the instantaneous channel capacity are obtained in the derivation As the result confirms that the use of MIMO system will greatly increase the achievable rate on Rayleigh fading channels ACKNOWLEDGMENT The author would lie to than Dr Lampe for his support REFERENCES [] J H Winters, On the capacity of radio communications with diversity in a Rayleigh fading environment, IEEE J Selected Area Commun, vol SAC-5, pp , Jun 987 [2] G J Foschini and M J Gans, On limits of wireless communication in a fading environment when using multiple antennas, Wireless Personal Communications, vol 6, pp 3-335, Mar 998 [3] I E Telatar, Capacity of multi-antenna Gaussian channels, Technical Report # BL TM, AT & T Bell Laboratories, 995 [4] G J Foschini Layered space-time architecture for wireless communication in a fading environment when using multiple antennas, Bell Labs Technical Journal, 2:4-59, Autumn 996 [5] R G Gallager, Information Theory and Reliable Communication New Yor: John Wiley & Sons, 968 [6] M Dohler, H Aghvami, A Closed Form Expression of MIMO capacity over Ergodic Narrowband Channels, IEEE Comm Letter, vol 8, Issue: 6, pp , June 24 [7] H Shin, J H Lee, Closed-form Formulas for Ergodic Capacity of MIMO Rayleigh Fading Channels, IEEE ICC, May 23, pp [8] P J Smith, M Shafi, On a Gaussian approximation to the capacity of wireless MIMO systems, IEEE ICC 22, New Yor, April 22 [9] M Kang, L Yang, M S Alouini, G Oien, How Accurate are the Gaussian and Gamma Approximations to the Outage Capacity of MIMO Channels?, 6th Baiona Worshop on Signal Processing in Communications, Baiona, Spain, September 8-, 23 [] C Chuah, D Tse, J M Kahn, R A Valenzuela, Capacity Scaling in MIMO Wireless System Under Correlated Fading, IEEE Trans Inform Theory, vol 48, pp , Mar 22 [] M Kang, M S Alouini, On the Capacity of MIMO Rician channels, Proc 4th Annual Allterton Conference on Communication, Control, and Computing Allerton 22, Monticello, IL, Oct 22, pp [2] B Vucetic, J Yuan, Space-time coding New Yor: John Wiley & Sons, 23 [3] M Dohler, H Aghvami, On the Approximation of MIMO Capacity, IEEE Letter Wireless Communications, July 23, submitted