EURASIP Journal on Applied Signal Processing 2002:3, 316–329 c 2002 Hindawi Publishing Corporation
Rayleigh Fading Multi-Antenna Channels
Alex Grant Institute for Telecommunications Research, University of South Australia, Mawson Lakes Boulevard, Mawson Lakes, SA 5095, Australia Email: [email protected]
Received 29 May 2001
Information theoretic properties of flat fading channels with multiple antennas are investigated. Perfect channel knowledge at the receiver is assumed. Expressions for maximum information rates and outage probabilities are derived. The advantages of transmitter channel knowledge are determined and a critical threshold is found beyond which such channel knowledge gains very little. Asymptotic expressions for the error exponent are found. For the case of transmit diversity closed form expressions for the error exponent and cutoff rate are given. The use of orthogonal modulating signals is shown to be asymptotically optimal in terms of information rate. Keywords and phrases: space-time channels, transmit diversity, information theory, error exponents.
1. INTRODUCTION receivers such as the decorrelator and MMSE filter [10]; (b) mitigation of fading effects by averaging over the spatial Wireless access to data networks such as the Internet is ex- properties of the fading process. This is a dual of interleav- pected to be an area of rapid growth for mobile communi- ing techniques which average over the temporal properties cations. High user densities will require very high speed low of the fading process; (c) increased link margins by simply delay links in order to support emerging Internet applica- collecting more of the transmitted energy at the receiver. tions such as voice and video. Even in the low mobility in- Recently, it has been realized that coordinated use of di- door environment, the deleterious effects of fading and the versity can be achieved through the use of space-time codes. need for very low transmit power combine to cause prob- Rather than relying solely on array processing of uncoded lems for radio transmissions. Regardless of advanced coding transmissions, forward error correction codes which add re- techniquessuchasturbo-codes[1], channel capacity remains dundancy in both the temporal and spatial domains are de- an unmovable barrier (without yet considering inefficien- signed specifically for channels with multiple transmit and cies introduced by higher layer network overheads). Without receive antennas. There are currently two main approaches changing the channel itself not much can be done. Fortu- to realizing the capacity potential of these channels: coordi- nately, increasing the number of antennas at both the base nated space-time codes and layered space-time codes. and mobile stations accomplishes exactly that, resulting in Coordinated space-time block codes [11, 12, 13]andtrel- channels with higher capacity. Such systems can theoretically lis codes [14, 15, 16, 17, 18] are designed for coordinated use increase capacity by up to a factor equaling the number of in space and time. The data is encoded using multidimen- transmit and receive antennas in the array [2, 3, 4, 5, 6]. sional codes that span the transmit array. Such codes are effi- Spatial diversity has been proposed for support of cient for small arrays, and can achieve within 3 dB of the 90% very high rate data users within third generation wide- outage capacity rate calculated in [3]. The other approach band Code-Division Multiple Access (CDMA) systems such uses layered space-time codes [19, 20, 21], where the chan- as cdma2000 [7]. Using multiple antennas, these systems nel is decomposed into parallel single-input, single-output achieve gains in link quality and therefore capacity. Clas- channels. sically, diversity has been exploited through the use of ei- Increasing the number of antennas and using space-time ther beam-steering (for antenna arrays with correlated el- codes involves both a physical and computational complex- ements), or through diversity combining (for independent ity/performance trade-off. Simple schemes such as multi- antenna arrays) [8, 9]. Use of these array processing tech- carrier transmit diversity or array processing techniques, niques can achieve any combination of the following: (a) re- such as maximum ratio combining may be easily imple- duction of multiple access interference through the “nulling” mented. How is their performance limited, when compared of strong interferers. Such techniques are complementary to to powerful space-time codes? What is the advantage of chan- (and share the mathematical formulations of) multiple-user nel knowledge at the transmitter? In this paper, we attempt Rayleigh Fading Multi-Antenna Channels 317 to answer some of these questions from an information the- matrix. For a random variable X,E[X]andvarX are its ex- oretic point of view. We do not consider the design of space- pectation and variance. time codes. The paper is organized in the following way. The space- 2. THE SPACE-TIME CHANNEL time channel model, along with the Rayleigh fading statistical model are introduced in Section 2. Consider a point-to-point communication link with t trans- Under the assumption of fast Rayleigh fading and per- mit antennas and r receive antennas. Throughout the paper fect channel knowledge at the receiver, various channel ca- we refer to m = min{r, t} and n = max{r, t}. We let the y j ≤ j ≤ r pacity computations are carried out in Section 3.1.Inpartic- match-filtered received signal k, at antenna 1 at ular, we find simple bounds on capacity for the case where time k be given by the transmitter has no channel knowledge. These bounds are used to further investigate the rate of growth of capacity with t L−1 y j = xi f i,j nj , the number of antennas. An asymptotic (in the number of k k−l k,l + k (1) users) expression for the multiple access space-time channel i=1 l=0 capacity is shown to be independent of the number of trans- xi ≤ i ≤ t where k is the signal transmitted from antenna 1 mit antennas, indicating that for large numbers of users mul- k f i,j j l tiuser diversity dominates transmit diversity. We then pro- at time ; k,l is the response at antenna at time to an i k nj vide bounds on the outage capacity for slow fading Rayleigh impulse applied to antenna at time ;and k is a discrete- channels. These bounds provide additional support to a con- time white Gaussian noise process, independent over time j j j j n n = σ2 n n = j = j jecture of Telatar concerning the optimal power distribution. and antennas, with E[ k ¯k] and E[ k ¯k ] 0for In Section 3.2, we consider perfect channel knowledge and/or k = k. The integer L>0 is the (maximum) delay at the transmitter. Asymptotic expressions for capacity are spread of the fading channel. given, and the advantage of using channel knowledge at the The information theoretic analyses presented so far in the transmitter is quantified. We find a transmit power thresh- literature have concentrated on the flat fading case, we like- old beyond which transmitter channel knowledge yields only wise restrict our present scope. For flat fading, the impulse a marginal capacity advantage. We also consider a subop- f i,j response sequences k become scalars. We simplify (1)ob- timal approach, sometimes referred to as downlink beam- taining a linear algebraic model [2]. At each symbol interval, forming, in which the transmitter uses only the most pow- the received vector y ∈ Cr depends on the transmitted vector erful eigenchannel. Expressions for capacity and outage ca- x ∈ Ct according to pacity are given for this case. In Section 4, we give asymptotic expressions for a ran- y = Hx + n. (2) dom coding error exponent. In Section 5, we consider trans- mit diversity (using only a single receive antenna). In particu- Element yj is the matched-filter output from antenna j, lar, we derive a closed form expression for the error exponent while xi is the signal transmitted from antenna i. The ma- ff r×t and show that the cut-o rate increases logarithmically with trix H ∈ C has as elements Hji, which is the complex the number of antennas. We also show that asymptotically, gain between transmit antenna i and receive antenna j.The the use of orthogonal carriers for each transmit antenna is vector n contains i.i.d. circularly symmetric Gaussian noise ∗ optimal. samples [22, page 134], E[nn ] = σ2Ir . We place the follow- Throughout, we make extensive use of results from the ing power constraint on the transmitted signal (independent theory of random matrices. Of particular interest are Wishart of t), matrices, which model certain statistics of the Rayleigh chan- nel. We make use of both the large systems approach, now fa- E [x∗x] ≤ P. (3) miliar in the analysis of multiple-input multiple-output sys- tems, as well as using results for finite dimension matrices. We denote the signal-to-noise ratio (SNR) as γ = P/σ 2. We collect in Appendix A several definitions and known re- sults for reference. Several lemmas used throughout the pa- Definition 1 (Rayleigh channel). Randomly select the en- per are also proved in Appendix A. Finally, in Appendix B we tries of H, independently of x and n as follows. Let Hji, give for reference definitions and some results concerning hy- i = 1, 2,...,t, j = 1, 2,...,r be i.i.d., zero mean Gaussian pergeometric functions of matrix argument, which are useful with independent real and imaginary parts, each with vari- in certain matrix variate distribution calculations. ance 1/2. We use the following notations. The vector x ∈ Cn is a column vector with complex entries xi, i = 1, 2,...,n. The complex entries of H are independent with uni- m×n Likewise A ∈ C is a matrix with complex entries Aij, formly distributed phase and Rayleigh distributed magni- i = 1,...,m, j = 1,...,n. The superscripts , ∗ and −1de- tudes, modelling the scenario in which the antenna separa- note, respectively, transposition, Hermitian adjoint, and ma- tion is large enough to ensure independent fading. A ma- trix inverse. By A > 0 we indicate that A is positive definite. trix H selected according to Definition 1 is matrix normal Likewise A > X means A − X > 0, det A is the determinant, distributed (i.e., H ∼ ᏺ(r, t) according to Definition A.1). ∗ tr A the trace and etr A = exp(tr A), and In is the n×n identity The matrix product HH will be of great interest and has 318 EURASIP Journal on Applied Signal Processing the Wishart distribution, HH∗ ∼ ᐃ(r, t)byDefinition A.2. concept of forming an average capacity is somewhat less Appendix A describes some useful properties of Wishart ma- straightforward, since the transmitter has the additional op- trices that we will use throughout the paper. Appendix B tion of optimizing the power allocation over time as well as summarizes some results concerning hypergeometric func- over the eigenvalues (while maintaining the required average tions of matrix argument that are required for certain calcu- power restriction). We do not consider average capacities for lations concerning Wishart matrices. this case. The channel matrix may either be fixed, or time vary- ing. In the time varying case, we can further distinguish 3.1. Channel unknown at transmitter between fast variations (compared to the data transmission Now concentrate on the case in which only the receiver per- rate), which allow us to compute average information rates, fectly knows the time varying Rayleigh channel matrix H. and slowly varying channels, for which outage probability is The capacity of the fast time varying Rayleigh channel was more appropriate. We assume throughout that the channel found by Telatar [2]. We rewrite this capacity using an alter- matrix is known perfectly at the receiver. We consider both native form in the following theorem. the case where the channel is unknown at the transmitter and the case where it is known perfectly. The theory for the Theorem 1. At each symbol interval, let H be selected ac- case in which the channel matrix is unknown at both ter- cording to Definition 1, and assume that H is known to the minals is still in development. It is known that the capacity receiver, but not at the transmitter. Let m = min{r, t} and gains to be achieved depend upon the coherence time of the n = max{r, t}. Then the capacity of the channel (2) is given channel [23, 24, 25]. Special space-time codes have also been by designed to operate in the absence of channel state informa- γ tion [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36]. C γ, t, r = ∗ ( ) E log det I + t HH ∞ (m − 1)! γ = x e−xxn−m (7) 3. CAPACITIES n − log 1+ t ( 1)! 0 In this section, we try to gain some insight into the scaling × Lm−1 x 2 − Lm x Lm−2 x dx, of capacity with the number of transmit and receive anten- n−m+1( ) n−m+1( ) n−m+1( ) nas. We are particularly interested in quantifying the capacity Lα k increase available through the availability of channel knowl- where k is the generalized Laguerre polynomial of order [38]. edge at the transmitter. We assume throughout that the re- Capacity is achieved for x circularly symmetric zero mean com- ∗ = P/t ceiver has perfect channel knowledge. plex Gaussian vector with E[xx ] It . Consider the channel (2), with average transmit power Proof. The expression follows from [2,Theorem2],us- constraint (3). Restrict xi, i = 1, 2,...,tto be i.i.d. zero mean Gaussian variables, each with variance P/t. Then the maxi- ing [39, equation (8.974.1)]. mum transmission rate is given by [2]as Although straightforward to numerically compute, (7) γ does not give much information about the scaling of capac- I γ, r, t, = ∗ . r t ( H) log2 det Ir + t HH bit/channel use (4) ity with and . Using limiting arguments, it is by now well known, and often quoted that in the case of r = t = k that This is simply an achievable rate, rather than the capacity, C(γ, k, k) is linear in k. We now give convenient upper and since no optimization over the source distribution is per- lower bounds to (7). formed. Such optimization would require knowledge of H at the transmitter. In the case of fast Rayleigh fading according Theorem 2. The capacity (7) is upper and lower bounded ac- to Definition 1, the channel capacity is given by C(γ, r, t) = cording to E[I(γ, r, t, H)] and is achieved for the i.i.d. Gaussian input γ t distribution just described. n−m C(γ, t, r) ≤ m log +logm! + log Lm − , (8) Given perfect channel knowledge at the transmitter, t γ channel capacity takes a different form, and is achieved by γ m−1 water filling on the parallel orthogonal channels defined C γ, t, r ≥ m ψ n − i , ( ) log t + ( ) (9) by the singular vectors of H,see,forexample,[37,Theo- i=0 rem 8.5.1]. Let λ1,...,λm be the m nonzero eigenvalues of HH∗. Then the capacity is given parametrically [2]by where ψ is Euler’s digamma function [39, equation (8.36)]. Proof. Inequality (8) follows directly from Jensen’s inequality CWF = log µλi, (5) −1 and Theorem A.4. The lower bound follows from det(I+A) ≥ i:λi ≤µ det A for Hermitian A and Lemma A.2. −1 P = µ − λi . (6) −1 i:λi ≤µ It is clear from the method of proof that the upper and lower bounds (8)and(9)aretightasγ →∞. In fact, ex- Note that in the case of transmitter channel knowledge, the panding the digamma as ψ(z) = log z − (2z)−1 − θ(12z)−2,for Rayleigh Fading Multi-Antenna Channels 319 some 0 <θ<1, we may further approximate (9)by 3.1.1 Multiple users Now consider a K user multiple access channel, in which the γ n m m − , channel from each user to the common receiver is an in- log t +log ! + log m c (10) dependent space-time channel. We suppose each user has t transmit antennas and that there are r receive antennas. At where c ≈ 0.577 is the Euler-Mascheroni constant [40, Sec- k = , ,...,K each symbol interval, user 1 2 transmits a vector tion 3.1.9.4]. Now t (k) x[k] ∈ C by sending Pk/t xi over antenna i = 1, 2,...,t ∗ such that E[x[k] x[k]] ≤ 1, corresponding to a transmit r Pk ∈ C n−m t n power constraint .Thereceivedvectory ,observed√ lim Lm − = , (11) γ→∞ γ m at the output of the r receive antennas is y = H Px + n, where x = (x[1], x[2],...,x[K]) and H ∈ Cr×Kt contains the ffi ≤ k ≤ K hence we expect polynomial tightness of the upper and lower fading coe cients. For a given integer 1 , the ele- H bounds in γ. ment j,kt+i−1 is the complex gain between transmit antenna i k j Kt×Kt The lower bound is however poor when simultaneously of user and receive antenna .The diagonal matrix = P ,P ,...,P /t ∈ Cr γ t and t ≈ r.Thisproblemmaybesomewhatoffset by P diag( 1It 2It K It) .Thevectorn contains ∗ = σ2 noting that C is monotonic in both n and m, and hence the i.i.d. zero-mean Gaussian noise samples, E[nn ] I.We lower bound near n = m may be improved by using smaller assume that H is known at the receiver, but not at the trans- values of either n or m. mitter. It is therefore not unreasonable to use (8) as an approx- Passing to the band-limited case via the usual arguments, imation for the purposes of determining the rate of growth suppose that the total bandwidth available for transmission KW W KW of capacity with t and r, since for sufficient SNR, the upper is ,where is some fixed quantity ( should be less and lower bounds will display the same gross behavior. Three than the coherence bandwidth of the channel for the flat fad- cases are of interest: (a) fix t and let r →∞,(b)fixr and let ing assumption to be realistic). Then the sum capacity of this t →∞, and (c) fix t/r and let both t,r →∞. channelisgivenby Considering case (a), we hold t constant and send r large ∗ HPH to see that C(P, H) = KW log det Ir + bit/s. (15) KWσ2
γ r! γ If H is independently selected at each symbol interval accord- C(γ, t, r) ≈ t log +log −→ t log + t log r (12) t (r − t)! t ing to Definition 1, the capacity of the resulting channel is found by taking the expectation of (15) with respect to the r / r − t
R −1 number of transmit antennas comes at the price of decreas- Pout(R, Q) ≤ Pr(X Pout(R, P) = inf Pout(R, W). (17) ter have perfect channel knowledge. We are interested in two Q:Q>0,tr Q≤P cases, firstly we compute a large systems limit for the water- filling capacity C and establish the optimal power profile. We consider H chosen once for all time according to WF Secondly, we consider a simple suboptimum case, in which our Rayleigh channel definition. Now the distribution transmitter selection diversity is used, that is, the transmit- of log det(I + W) is unknown. Apart from the mean ter simply transmits using all its power along the maximum (Theorem A.4), the moments of this quantity do not have eigenvector. In this second case, we compute the large sys- a simple expression (although in limiting cases they are tems capacity and the outage probability. Without loss of known [41]). The following theorem demonstrates the type generality, we let σ2 = 1, that is, γ = P. All of the results in of difficulty one encounters. this section will be obtained as large systems limits and rely m, n →∞ Theorem 4. Theoutageprobabilityisupperbounded on the convergence of the Wishart spectrum as in such a way that m/n → β ≤ 1. sR Pout(R, P) ≤ inf 2F0 (s,n; Q) e , (18) Given a particular eigenvalue distribution, the capacity s>0 Q>0,tr Q≤P may be found using (5). It is however a well-known fact that for a wide class of random matrices, the normalized eigen- where mFn (·; ·; ·) is a hypergeometric function of matrix argu- value distribution converges to a nonrandom limit as the ma- ment as defined in Appendix B. trices are taken large. We use this fact to determine the large systems capacity limit. Proof. Now W HQH∗ ∼ ᐃ(m, n, Q)and Theorem 6. Let m ≤ n approach infinity such that m/n → P (R, Q) = Pr log det(I + W) 1 . m CWF(Pcrit(β),β) 1 025 5 1.02 4 20 dB 1.015 3 15 dB /C WF (nats) C 1.01 /m 10 dB 2 WF C 5dB 1.005 1 0dB 1 0.20.40.60.81 β 00.20.40.60.81 β Figure 2: Relative capacity gain from water-filling along the locus P = P Figure 1: Asymptotic water-filling capacity. of crit. 2.75 Proof. Under the assumption µ−1 ≤ a(β), the limit (22)be- . comes 2 5 2.25 C b(β) WF −→ log µ + log λdF(λ) m /m 2 a(β) (25) WF 1 − β 1 C 1.75 = log µ + log − 1, β 1 − β 1.5 where the first line is due to integrating over the entire sup- 1.25 port of λ and the second line is from [42, Corollary 5.1]. It remains to find µ.Nowfrom(23), −10 0 10 20 P,dB b β λ − a β b β − λ 1 ( ) ( ) ( ) Figure 3: Relative capacity gain from water-filling at β = 1. P −→ µ − dλ. (26) πβ λ2 2 a(β) of transmitter channel knowledge may not cost too much. This integral is of the form [39, equation (2.267.2)] and after For P>P , the capacity advantage for using transmitter straightforward calculation is found to be 1/(1 − β). crit channel knowledge is at most about 2%. P Pcrit (left of dot-dashed line), or for large SNR that there is only a 2 C γ −→ γm β small advantage in CWF/m to be obtained from water-filling. ( ) log 1+ 1+ a.s. (27) This is investigated further in Figure 2, where the ratio of the water-filling capacity to the corresponding limiting value of This indicates that by transmitting along the maximum C is shown for P = Pcrit(β). This loss will be even less for eigenvector, performance superior to the unfaded AWGN higher power levels (to the left of the dot-dashed line on channel is obtained. Note however that the capacity increase Figure 1). The relative difference, upper bounded by approxi- is only logarithmic with min{t,r}, as compared to the lin- mately 1.022 at β ≈ 0.1 is preserved through the scaling by m. ear capacity increase available through the use of all transmit This result indicates that in certain scenarios that the absence antennas, even in the absence of channel knowledge at the 322 EURASIP Journal on Applied Signal Processing transmitter. For finite systems, the following theorem may which is well known for the normal distribution to be given be used to compute outage probabilities for the selection di- by versity approach. s2σ2 R M(s) = exp µs + . (34) Theorem 8. Let α = (e − 1)/P, then 2 Γm (m) nm Taking the supremum on both sides of (33) and using (34) Pout(R, P) = α 1F1 (n; n + m; −αI) . (28) Γm (n + m) gives the result. P R, P = Λ α Proof. From Theorem A.5, out( ) ( ). µ(ρ) may be calculated using [2, equation (13)], or may be lower bounded using [48, Theorem 3]. 4. ERROR EXPONENTS In addition to maximum information transmission rates, it 5. TRANSMIT DIVERSITY is interesting to consider the error exponent [37]forspace- In this section, we consider transmit diversity, r = 1, t>1. time channels. The error exponent gives some indication of From (4), it is easy to see that how difficult it may be to achieve a certain bit error rate. In this section, we derive a limiting expression for the error ex- γ ponent. Telatar [2] gives the following lower bound to the C = E log 1+ X , (35) 2t error exponent. where X is a chi-squared random variable with 2t degrees of Lemma 1. The probability of error averaged over all randomly freedom.2 As t increases, C → log(1+γ) (since X → 2t by the selected (N,NR) block codes is bounded law of large numbers), which is as if all the power were trans- mitted over a single, nonfaded link. By increasing the num- Pe ≤ exp − N max E0(ρ) − ρR , (29) ff 0≤ρ≤1 ber of antennas, the e ects of the fading may be completely removed. In practice, only a small number of transmit anten- where nas (typically two or three for a wide range of γ) are required (see [2, Figure 3]). −ρ γ ∗ Increasing the number of transmit antennas however has E (ρ) = − log E det Ir + HH . (30) 0 t(1 + ρ) the desirable effect of increasing the error exponent, as we now show. Theorem 10. Let r = 1, then Reliability functions for fading channels with correlated t = multiple antennas have also been considered in [45]for 1, t ρ t t ρ see also [46]. E ρ = − (1 + ) Ψ t,t − ρ (1 + ) , 0( ) log γ +1 ; γ The following theorem uses the theory of random deter- minants to give a limiting expression for the error exponent. (36) Theorem 9. Let t/r be fixed. The following limiting relation where Ψ is the confluent hypergeometric function [39, Sec- holds. tion 9.2]. Furthermore, letting CG(γ) = log(1+γ) be the Gaus- sian channel capacity with SNR γ we have the following limit- σ2 ρ ρ2 ( ) ing expression for the error exponent: sup E0(ρ) = µ(ρ)ρ − , (31) t→∞ 2 γ γ γ ∗ C − R, ≤ R ≤ C − , where, for convenience A(ρ) = I + γ/(t(1 + ρ))HH and G if 0 G 2 2 2(γ +2) lim Er = t→∞ γ γ ρC − ρR(ρ), otherwise, µ(ρ) = lim E [log det A] = lim C ,t,r , G 1+ρ t→∞ t→∞ 1+ρ (32) (37) σ2(ρ) = lim var log det A. t→∞ where for the second case, CG(γ/2) − (γ/2(γ +2)) 7 0.4 30 dB 6 5 25 dB 0.3 20 dB Er 4 AWGN 0.2 (nats) Diversity 0 3 15 dB R 2 10 dB 0.1 1 5dB 0dB 0 0.10.20.30.40.50.60.7 246810 R (nats) t Figure 4: Error exponent, γ = 0dB. Figure 5: Cut-off rate versus number of antennas. Proof. From (30), we have Proof. Equation (41) is obtained from (39) via the identity Ψ α, α z = ezΓ − α, z ( ; ) (1 ). γ −ρ E0(ρ) = − log E 1+ X , (39) For t>1 the cut-off rate may be lower bounded as fol- 2t(1 + ρ) lows: X t where is a chi-squared random variable with 2 degrees of γ 1 Ψ R ≥ log 1+ 1 − . (43) freedom. Using an integral representation for [39,equa- 0 2 t tion (9.211.4)], we get (36) after some algebraic manipula- tions. This bound is tight as t →∞or γ →∞.HenceR0 increases t X → t As increases, 2 in probability we have at least as the logarithm of t/(t − 1). Figure 5 shows R0 (solid t γ line) and the lower bound (43) (dashed) plotted versus for lim E (ρ) = ρC , (40) various SNR. t→∞ 0 G 1+ρ 5.1. Orthogonal transmit diversity which, after some manipulations (along the lines of [37, Sec- tion 7.4]) proves (37). It is interesting to consider the use of mutually orthogo- nal transmit waveforms for each transmit antenna,3 as pro- Figure 4 compares the error exponents for the additive posed for cdma2000 [7, Section 3.2.1.1.5]. We continue to white Gaussian noise channel [37, equation (7.4.33)] and the consider transmit diversity only (r = 1), and assume code- limiting t →∞transmit diversity channel with Rayleigh fad- books with i.i.d. circularly symmetric complex Gaussian let- ing (37). The signal-to-noise ratio is zero dB. Further numer- ters. This means that for a given channel, transmission rates ical investigations have shown that the gap in the error expo- are upper bounded by I as given by (4). If the channel is ran- nents only disappears as γ → 0. It is interesting to note that dom, according to Definition 1, the transmission rate is up- t →∞ C → C γ although results in G( ) there is an inherent per bounded by C as given by (7). complexity penalty to be paid (at least in terms of the error Examples of orthogonal waveforms include (a) time di- exponent). vision, in which the scalar input is successively transmit- Closed form determination of Er for finite t using (36) ted from each antenna at disjoint time intervals; (b) fre- seems difficult, although numerical methods can be used. We quency division (multi-carrier transmit diversity [7, Sec- ff R = E therefore concentrate on the cut-o rate, 0 0(1), which tion 3.2.1.1.5.1]), in which the signal for each antenna gives the error exponent up the critical rate, and a lower is modulated by an orthogonal frequency carrier; and (c) bound beyond that. Substituting ρ = 1 into the expressions code division (direct-sequence transmit diversity [7, Sec- of Theorem 10, we obtain the following corollary. tion 3.2.1.1.5.2]), where each antenna is modulated by (for example) an orthogonal Walsh code. ff Corollary 2. The cut-o rate is given by It is easy to see (e.g., through application of the data pro- γ 2t 2t cessing theorem [49]) that restriction to orthogonal carriers R (t,γ) = t log − − log Γ 1 − t, , (41) can only decrease the maximum possible mutual informa- 0 2t γ γ tion, when compared to the use of unconstrained space-time ∞ α−1 −t codes. In this section, we show that use of a sufficiently large where Γ(α, z) = z x e dt is the incomplete gamma func- tion. Furthermore, γ 3What counts is that the waveforms are orthogonal at the receiver, for lim R (t,γ) = log 1+ . (42) t→∞ 0 2 example, flat fading and no Doppler spread. 324 EURASIP Journal on Applied Signal Processing number of transmit antennas makes this penalty insignifi- (8.214.1)] that cant. This may be interesting since use of orthogonal carriers ∞ xk should result in simple receiver structures [11, 12]. Narula et x = −x ,x<. Ei( ) c+log( )+ k · k 0 (47) al. [6] have found the average mutual information (assuming k=1 ! a Gaussian codebook) in the case of orthogonal signaling. Using this identity, and performing some simple rearrange- Theorem 11. Let H be a fixed matrix, known at the receiver. ments, we can write Then the average mutual information under the assumption of ∞ − /γ k orthogonal transmit waveforms is given by 1/γ 1/γ ( 1 ) E [I⊥] − e log(γ) = −e c+ . (48) k · k! t 2 k=1 1 Hi P I⊥ = log 1+ . (44) t 2 σ2 γ →∞ i=1 Taking limits as on both sides yields the desired result, since 1/γ → 0, forcing the exponential term to unity, and Furthermore, for this channel I⊥ ≤ I with equality if and only each element inside the sum to zero. if |Hi| = |Hj | for all i, j = 1, 2,...,t. Hence for any given channel, the use of either multi- 6. CONCLUSION carrier transmit diversity, or orthogonal direct-sequence In this paper, we have given several calculations regarding the transmit diversity, while convenient from an implementa- capacity and error exponents of multiple antenna Gaussian tion point of view, incurs a loss, when compared to space- channels. We used simple bounds to determine the scaling time coding. Since this is true of any given channel, the aver- of capacity with the number of antennas. In particular, we ages over random channel selection (e.g., Definition 1) also found that if the number of transmit and receive antennas I ≤ C obey E[ ⊥] E[ ]. The following result gives an expression are increased proportionately, a linear increase is obtained, I for E[ ⊥] and shows its limiting behavior for large signal-to- and the rate of this increase was determined. We further dis- noise ratios. covered that asymptotically, there is no capacity advantage for the number of transmit antennas to exceed the number Theorem 12. Let H be randomly selected every symbol interval of receive antennas. In the case that the number of transmit according to Definition 1,andletH be known at the receiver. antennas is held constant, we found that capacity increases Then the maximum transmission rate, under the assumption of 1/γ logarithmically with the number of receive antennas. In the orthogonal transmit diversity is given by E[I⊥] = −e Ei(−1/γ) ∞ reverse situation, holding the number of receive antennas x = e−x/x dx nats/channel use, where Ei( ) −x is the exponential fixed, increasing the number of transmit antennas results in integral function. Furthermore, convergence to a fixed capacity. A simple calculation showed that for multiple-access space-time channels that capacity is lim log 1+γ − E [I⊥] = c, (45) γ→∞ dominated by multiuser diversity, rather than space diversity. For large numbers of users, there is no total capacity advan- where c ≈ 0.577 is the Euler-Mascheroni constant. E[I⊥] → tage in using multiple antennas. We also gave bounds on the log(1 + γ) in the sense that their ratio tends to unity. outage probability for nonergodic channels and discovered that a constant power allocation over some subset of anten- r = t C → It is easy to verify that for 1 and increasing , nas minimized one of these bounds. γ log(1 + ). Hence even though a rate penalty is incurred for Large systems limits were calculated for the capacity in using orthogonal transmit diversity, this penalty disappears the case of perfect channel knowledge at the transmitter. For as the signal-to-noise ratio and the number of antennas is power levels above a certain critical threshold, this limit can increased. be easily calculated in closed form. We found numerically Proof. From (44), Definition 1, and the linearity of the ex- that for power levels greater than this same critical threshold that there is very little to be gained by the water-filling ap- pectation operator, we have that E[I⊥] = E[log(1 + γX)], where X is an exponentially distributed4 random vari- proach, although for smaller powers we found that capacity able [22]. Hence may be increased significantly. We also considered briefly the use of selection diversity at the transmitter and found that ∞ this simple strategy outperforms a single antenna link with I = γx e−x dx E [ ⊥] log(1 + ) no fading. 0 (46) The remainder of the paper was devoted to calculation of = −e1/γ − 1 , Ei γ error exponents. In particular, we found a limiting expres- sion for E0(ρ), and a closed form expression for the case of according to [39, equation (4.337(2))], which is the first part transmit diversity (r = 1). We further showed that for r = 1 of the theorem. Continuing, we note from [39,equation the cut-off rate increases logarithmically with the number of transmit antennas. Although capacity is near to maximum for a small number of antennas (typically 2 or 3), use of ad- 4 χ2 ditional antennas improves performance through increase of Or equivalently, a 2 variable. Rayleigh Fading Multi-Antenna Channels 325 the error exponent. We have also shown that use of a suf- The following theorem gives the distribution of a complex ficiently large number of transmit antennas offsets the rate Wishart matrix. The proof of this theorem follows [53,Theo- penalty suffered due to orthogonal transmissions from each rem 3.2.3], using however the complex hypergeometric func- antenna, such as those described for cdma2000. tions, as outlined in Appendix B. Theorem A.2 (Wishart distribution). Let W ∼ ᐃ(m, n). APPENDICES Then A. WISHART MATRICES Γm (m) n P(W < X) = det X 1F1 (n; n + m; −X) . (A.4) Wishart matrices are of interest in statistics and signal pro- Γm (n + m) cessing. We now review some facts concerning Wishart ma- trices, and derive some results concerning expectations of Theorem A.3 (Ordered eigenvalue distribution). Let W ∼ certain determinantal forms and further investigate the dis- ᐃ(m, n) with ordered eigenvalues λ1 ≥ λ2 ≥···≥λm. Then tribution and asymptotics of the largest eigenvalue. Books m m− providing an introduction to matrix variate distributions π ( 1) p λ1,λ2,...,λm = are [50, 51, 52, 53]. These texts concentrate on real variables. Γm (n) Γm (m) Although the corresponding complex cases that we require m m n−m 2 are easily developed, they are noticeably absent from these × exp λi λi λi − λj . references. i=1 i=1 i In fact Theorem A.1 may be used as the definition of the m−1 Wishart matrix, from which the statement of Definition A.2 E [log det W] = ψ(n − i), i=0 may be proved as a theorem. The ordering of definition and (A.8) theorem however become less important in the light of [53, m−1 = ψ n − i , Theorem 3.3.3], where it is proved that a matrix W with var log det W ( ) i= the Wishart distribution may always be factored W = XX∗, 0 where X ∼ ᏺ(m, n). It is useful however to remember that where ψ(x) = Γ(x)/Γ(x) is Euler’s digamma function [39, there are other matrix forms besides Definition A.2 that lead equation (8.36)]. to the Wishart distribution. We can also consider Wishart matrices with arbitrary co- The following lemma is from [56, Section 37]. variance structure. For A ∈ Cm×m nonsingular and Σ = AA∗, the product W = AXX∗A∗ is Wishart ᐃ(m, n, Σ)withdensity Lemma A.3. For α ∈{1, 2,...,n},letA(α) be the principal ∈ Cn×n submatrix [57, page 17] formed from A by deleting n −1 − n−m p(W) = det Σ Γm (n) etr(−Σ 1W)detW . (A.3) rows and columns not indexed by the elements of α.Thenfor 326 EURASIP Journal on Applied Signal Processing any n × n matrix A function Λ(x) = Pr(λ r k> κ = k ,k ,...,k p i −a A general formula for the coefficients of zonal polyno- 1F0 (a; X) = det(I − X) . (B.7) mials has not been found. A recurrence for the coefficients may be found in [51]. An easily proved useful identity is The power series representation given below is due to Con- k Cκ (αX) = α Cκ (X). stantine [68]. The following constant occurs frequently in integrals p × p concerning zonal polynomials. Note that this differs from Theorem B.1. Let X be a Hermitian matrix. 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