Hindawi Publishing Corporation EURASIP Journal on Wireless Communications and Networking Volume 2006, Article ID 45831, Pages 1–8 DOI 10.1155/WCN/2006/45831 Asymptotic Analysis in MIMO MRT/MRC Systems Quan Zhou and Huaiyu Dai Department of Electrical and Computer Engineering, NC State University, Raleigh, NC 27695-7511, USA Received 11 January 2006; Revised 20 July 2006; Accepted 16 August 2006 Recommended for Publication by Zhiqiang Liu Through the analysis of the probability density function of the largest squared singular value of a complex Gaussian matrix at the origin and tail, we obtain two asymptotic results related to the multi-input multi-output (MIMO) maximum-ratio- transmission/maximum-ratio-combining (MRT/MRC) systems. One is the asymptotic error performance (in terms of SNR) in a single-user system, and the other is the asymptotic system capacity (in terms of the number of users) in the multiuser scenario when multiuser diversity is exploited. Similar results are also obtained for two other MIMO diversity schemes, space-time block coding and selection combining. Our results reveal a simple connection with system parameters, providing good insights for the design of MIMO diversity systems. Copyright © 2006 Q. Zhou and H. Dai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. INTRODUCTION Multi-input multi-output (MIMO) systems can be exploited for spatial multiplexing or diversity gains. For a M IMO di- versity system, appropriate diversity combining techniques are employed at the transmit and receive end to effectively transform the MIMO channel into an equivalent single- input single-output (SISO) one, with increased robustness. Depending on whether the channel state information (CSI) is required at the transmitter, MIMO diversity schemes can be divided into two categories: open-loop and closed-loop. Among the former is the scheme that employs well-known space-time block coding at the transmitter and maximum ratio combining at the receiver, coined as STBC/MRC. As certain feedback often exists in a wireless network (e.g., in use scheduling discussed below), closed-loop schemes are also of great interest. This category includes simple selec- tion combining on both ends (SC/SC), joint maximum ratio transmission and maximum ratio combining (MRT/MRC), and various hybrid selection combining schemes in be- tween. For diversity usage, MRT/MRC systems provide the op- timal performance reference [1–5], but its analysis is also more involved than others (see relevant distribution func- tions in Section 2), w hich will be the focus of this paper. With the assumption that the receive beamforming vector is matched to the transmit one with unit modulus for all entries, the average output signal-to-noise ratio (SNR) of an MRT/MRC system is upper and lower bounded in [1], based on which the average symbol error rate (SER) and di- versity order for a BPSK system are approximately derived. With the restricting assumptions in [1] removed, it is known that (for white Gaussian noise) the optimal transmit and re- ceive beamformer are given by the principal right and left singular vector of the channel matrix H, respectively; and the MIMO channel is transformed into a SISO link with equiva- lent channel gain σ max , the largest singular value of H.For Rayleigh fading channels, the distribution of σ 2 max ,already derived in [6], is revisited in [2] and expressed in an alterna- tive form—a linear combination of Gamma functions. Based on this expression, the exact system SER is derived for gen- eral modulation schemes in [2]. The distribution of σ 2 max for Ricean fading is obtained in [4]. Unfortunately, results in [2] and [4] do not easily lead one to an insightful understanding of the impact of the system parameters, including the num- ber of transmit and receive antennas M and N,onperfor- mance. For example, in [2], the authors make two observa- tions on MIMO MRT/MRC systems through simulation re- sults: one is that when M +N keeps fixed, the antennas distri- bution with |M −N| minimized will provide the lowest SER, while the other is that when M × N is fixed, a distribution with the largest M+N gives the best performance. But the au- thors do not provide a rigorous justification for both obser- vations. Some similar observations are also made in [4]. In a 2 EURASIP Journal on Wireless Communications and Networking multiuser wireless network, there is another form of diversity called multiuser diversity, which reflects the fact of indepen- dent fluctuations of different users’ channels [7]. Multiuser diversity can be exploited to increase the system throughput, through intentionally transmitting to the user(s) with good channels at each instant (opportunistic scheduling). There exist some work on the joint spatial diversity and multiuser diversity systems. In particular, the system capacity analysis for Rayleigh fading channels is given in [8], and in [9]for more genera l Nakagami fading channels. While these results are accurate, simpler expressions are desired that can clearly reveal the interaction between these two forms of diversity. Aiming at obtaining succinct and insightful performance evaluation for MIMO MRT/MRC systems (more general MIMO diversity systems), we take a different approach in this paper by conducting asymptotic analysis. Asymptotic analysis is widely used in various areas of communications and networking. Besides mathematical tractability, asymp- totic analysis also helps reveal some fundamental relation- ship of key system parameters, which may be concealed in the finite case by random fluctuations and other transient prop- erties of channel matrices. This paper comprises two sub- topics: error performance in the single-user scenario and ca- pacity scaling law in the multiuser scenario. While presenting complementary aspects of MIMO MRT/MRC systems, these two are threaded together through a common theme, the in- vestigation of the approximate behavior of the distribution of σ 2 max at the extremes, with the former at the origin and the latter at the tail. The main contributions of this paper are summarized below. (1) By studying the behavior of the distribution function of σ 2 max at the origin, we obtain the asymptotic average SER (in terms of SNR) for MIMO MRT/MRC systems. As appli- cations we verify the two observations made in [2]. (2) By studying the behavior of the distribution function of σ 2 max at the tail, we obtain the asymptotic system capac- ity (in terms of the number of users) for MIMO MRT/MRC systems when multiuser diversity is exploited. (3) Similar analysis is also carried out for two other repre- sentative MIMO diversity schemes: STBC/MRC and SC/SC. Comparison among them enables better understanding of MIMO diversity and the interaction between spatial diversity and multiuser diversity. This paper is organized as follows. In Section 2,wegive our model for MIMO MRT/MRC systems. Then we pro- vide our asymptotic analysis for the average SER and sys- tem capacity in Sections 3 and 4, respectively, together with some numerical results for illustration purpose. Conclusion is given in Section 5. 2. SYSTEM MODEL We assume a narrowband MIMO MRT/MRC system with M transmit antennas and N receive antennas, modeled as y = Hx + n = Hw t u + n,(1) where y ∈ C N×1 is the received vector, H ∈ C N×M is the channel matrix, w t ∈ C M×1 is a unit-norm transmit weight vector, chosen as the principal right singular vector corre- sponding to the largest singular value σ max of H, u is the transmitted symbol with power P T ,andn ∈ C N×1 is a zero- mean circularly symmetric complex Gaussian noise vector with variance σ 2 n /2 per real dimension. We define γ t = P T /σ 2 n as the average transmit SNR. For illustration purpose, inde- pendent and identically distributed Rayleigh fading is con- sidered for H, but our analysis can be readily extended to other fading scenarios when appropriate distributions are available. When multiple MIMO users are involved, their channels are assumed independent. At the receiver side a weight vector w r ∈ C N×1 is applied on y to obtain a deci- sion statistic for u, chosen as the principal left singular vector of H here. Other diversity schemes can be equivalently repre- sented with w t and w r appropriately defined. The cumulative distribution function (CDF) of γ = σ 2 max is given by [6] F MRT/MRC γ (x) = Ψ c (x) Π s k =1 Γ(t − k +1)Γ(s −k +1) , x ∈ (0,+∞), (2) where s = min(M, N), t = max (M, N), and Ψ c (x)isan s × s Hankel matrix function with the (i, j)th entry given by {Ψ c (x)} i, j = γ(t − s + i + j − 1,x), for i, j = 1, 2, , s. Here γ(a, β) is the incomplete Gamma function defined as γ(a, β) = β 0 e −t t a−1 dt,andΓ(a) is the Gamma function de- fined as Γ(a) = γ(a,+∞). The probability density function (PDF) of x can be derived as f MRT/MRC γ (x) = F MRT/MRC γ (x)tr Ψ −1 c (x)Φ c (x) , x ∈ (0, +∞), (3) where Φ c (x)isans ×s matrix whose (i, j)th entry is g iven by {Φ c (x)} i, j = x t−s+i+ j−2 e −x . In the remainder of this paper, we adopt the following notations for the limiting behaviors of two functions f (x) and g(x) with lim x→∞ or x→0 g(x)/f(x) = c : g(x) = O( f (x)) for 0 < |c| < ∞ and specifically g(x) ∼ f (x)forc = 1; g(x) = o( f (x)) for c = 0. When convergence of a sequence of random variables is involved, shorthand notation “D” stands for in distribution and “P” for in probability. 3. ASYMPTOTIC AVERAGE SER: SINGLE-USER SCENARIO In this section, we will derive a succinct expression for aver- age SER at high SNR. The conditional SER for lattice-based modulations can be represented as P s (H) = M n Q( √ κγ t γ), where M n is the number of the nearest neighboring con- stellation points, Q( ·) is the Gaussian tail Q-function, and κ is a positive fixed constant determined by the modula- tion and coding schemes [5]. At high transmit SNR γ t , the system average SER P s = E{P s (H)} will be domi- nated by the low-probability outage event that γ becomes small [10]. Therefore, only the behavior of f MRT/MRC γ (x)at x → 0 + matters. To this end, the following result is cru- cial. Q. Zhou and H. Dai 3 Lemma 1. f MRT/MRC γ (x) ∼ MN s−1 k=0 k! s−1 k =0 (t + k)! x MN−1 , as x −→ 0 + . (4) Proof. By Maclaurin series expansion Ψ c (x) i, j = γ(t − s + i + j − 1, x) = 1 t − s + i + j − 1 x t−s+i+ j−1 + o x t−s+i+ j−1 , (5) we can obtain the approximation of |Ψ c (x)| at x = 0 + after some manipulation as Ψ c (x) =| Λ|x MN + o x MN ,(6) with {Λ} i, j = 1/(t − s + i + j − 1), for i, j = 1, 2, , s.The determinant of Λ can be obtained in a similar fashion as that of a Hilbert matrix. After some algebra we get |Λ|= s−1 k =0 (k!) 2 (t − s + k)! 2 2s−1 k =0 (t − s + k)! ,(7) and it follows from (2) that F MRT/MRC γ (x) = s−1 k =0 k! s−1 k=0 (t + k)! x MN + o x MN . (8) With Lemma 1, we establish the following result for the asymptotic average SER for MIMO MRT/MRC systems fol- lowing [10, Proposition I]. Proposition 1. For MIMO MRT/MRC systems, the asymp- toticaverageSERisgivenby P s = 2 q (MRT/MRC) M n α (MRT/MRC) Γ q (MRT/MRC) +3/2 √ π q (MRT/MRC) +1 × κγ t −(q (MRT/MRC) +1) + o γ −(q (MRT/MRC) +1) t , (9) where α (MRT/MRC) = MN s−1 k =0 k! s−1 k =0 (t + k)! , q (MRT/MRC) = MN − 1. (10) The validity of (9) is demonstrated in Figure 1 for un- coded BPSK systems. Based on (9), one readily concludes that the optimal diversity order for MIMO diversity systems is M ×N. Therefore, if we keep M +N fixed (a measure of sys- tem cost), even distribution of the number of transmit and receive antennas (more precisely a smallest |M − N|)maxi- mizes M × N, thus minimizing the system SER at high SNR. On the other hand, when comparing two MIMO MRT/MRC systems with the same diversity order M × N, the one with smaller α (MRT/MRC) yields larger coding gain and thus smaller SER (in this case, q (MRT/MRC) is a constant). We can conclude that in this scenario, the sum of transmit and receive anten- nas should be made as large as possible, with the optimum achieved at s = 1andt = M × N.Thisconclusionisbased on the following result regarding α (MRT/MRC) as a function of M and N (or equivalently of s and t). 10 7 10 6 10 5 10 4 10 3 10 2 SER 5 6 7 8 9 101112131415 SNR (dB) (1, 3) MRT/MRC asym. result (1, 3) MRT/MRC simulation (2, 2) MRT/MRC asym. result (2, 2) MRT/MRC simulation (2, 3) MRT/MRC asym. result (2, 3) MRT/MRC simulation Figure 1: Comparison between asymptotic and simulation results for BPSK under different antennas configurations (the notation (M, N) refers to MIMO systems with M transmit and N receive an- tennas). Lemma 2. Given four positive integers s 1 , t 1 , s 2 , t 2 , assume s 1 × t 1 = s 2 × t 2 , s 1 <t 1 , s 2 <t 2 ,ands 1 + t 1 >s 2 + t 2 , then α (MRT/MRC) (s 1 , t 1 ) <α (MRT/MRC) (s 2 , t 2 ). Proof. From s 1 + t 1 >s 2 + t 2 ,wecanobtains 1 <s 2 <t 2 <t 1 . As α (MRT/MRC) s 1 , t 1 = s 1 −1 k =0 k! s 1 −1 k =0 t 1 + k ! = 1 1 × 2 ×···×t 1 • 1 2 × 3 ×···× t 1 +1 •··· 1 s 1 ×···× s 1 + t 1 − 1 , (11) α (MRT/MRC) s 2 , t 2 = s 2 −1 k =0 k! s 2 −1 k =0 t 2 + k ! = 1 1 × 2 ×···×t 2 • 1 2 × 3 ×···× t 2 +1 •··· 1 s 2 ×···× s 2 + t 2 − 1 , (12) 4 EURASIP Journal on Wireless Communications and Networking it is equivalent to show that 1 ×···×t 1 ×···× s 1 ×···× s 1 + t 1 − 1 > 1 ×···×t 2 ×···× s 2 ×···× s 2 + t 2 − 1 . (13) The left-hand side of (13)canberewrittenas 1 f (1) × 2 f (2) ×···× s 1 + t 1 − 1 f (s 1 +t 1 −1) , (14) with f (i) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ i,1≤ i ≤ s 1 , s 1 , s 1 +1≤ i ≤ t 1 , s 1 + t 1 − i, t 1 +1≤ i ≤ s 1 + t 1 − 1. (15) Similarly the right-hand side of (13)canberepresentedas 1 g(1) × 2 g(2) ×···× s 2 + t 2 − 1 g(s 2 +t 2 −1) , (16) with g(i) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ i,1≤ i ≤ s 2 , s 2 , s 2 +1≤ i ≤ t 2 , s 2 + t 2 − i, t 2 +1≤ i ≤ s 2 + t 2 − 1. (17) It is not difficult to get s 1 +t 1 −1 i =1 f (i) = s 1 × t 1 = s 2 × t 2 = s 2 +t 2 −1 i =1 g(i). Therefore, after canceling out the same factors in (14)and(16), we can see that (14) is surely larger than (16). From the asymptotic SER expression in (9), we have ver- ified the two observations made in [2] rigorously at high SNR. Below we will follow a similar approach to compute the corresponding parameters for the coding gain and diver- sity order for MIMO STBC/MRC and SC/SC systems (whose asymptoticaverageSERsassumethesameformsas(9)). Without loss of generality, we assume that the adopted space-time block coding scheme achieves the full r a te and the tr ansmit power is equally allocated among the transmit antennas. In this case, the normalized effective link SNR for a generic user is given by γ = (1/M) N i =1 M j =1 |h i, j | 2 , whose PDF admits f STBC/MRC γ (x) = M MN (MN − 1)! x MN−1 e −Mx , x ≥ 0. (18) Similarly the corresponding par a meters for the coding gain and diversity order for MIMO STBC/MRC systems can be obtained as α (STBC/MRC) = M MN (MN − 1)! , q (STBC/MRC) = MN − 1. (19) For the SC/SC scheme, both the user and the base station choose one optimal antenna such that the resultant channel gain is maximized. Thus the normalized effective link SNR at 10 6 10 4 10 2 10 0 10 2 10 4 10 6 Value of α 12345678910 Number of transmit antennas M SC/SC, M N = 10, SC/SC, M N = 8 SC/SC, M N = 6 STBC/MRC, M N = 6, STBC/MRC, M N = 8 STBC/MRC, M N = 10 MRT/MRC, M N = 6, MRT/MRC, M N = 8 MRT/MRC, M N = 10 Figure 2: Coding gain parameter α with the number of transmit antennas for the same diversity order M × N. the receiver is γ = max 1≤i≤N,1≤j≤M (|h i, j | 2 ), whose PDF can be easily obtained as f SC/SC γ (x) = MNe −x 1 − e −x MN−1 , x ≥ 0. (20) We can obtain the corresponding parameters for the coding gain and diversity order for MIMO SC/SC systems as α (SC/SC) = MN, q (SC/SC) = MN − 1. (21) Comparing (10), (19), and (21) we can see that all these MIMO diversity schemes achieve the same diver- sity order. Nonetheless, their error performances could still be dramatically different owing to different coding gains, as exhibited in Figure 2.Forexample,whenM = 6 and N = 1, our asymptotic results predict an SNR gap of 4.7 dB between MRT/MRC (α (MRT/MRC) = 1/120) and SC/SC (α (SC/SC) = 6), and 7.8 dB between MRT/MRC and STBC/MRC (α (STBC/MRC) = 388.8)foruncodedBPSKsys- tems at high SNR, which agree well with simulation results (see Figure 3 at SER 10 −5 ). It is also observed that for the same diversity order, the performance of STBC/MRC wors- ens with the increase of the number of transmit antennas. 4. ASYMPTOTIC SYSTEM CAPACITY: MULTIUSER SCENARIO In this section, we consider a homogeneous downlink mul- tiuser MIMO communication scenario, which is envisioned Q. Zhou and H. Dai 5 10 7 10 6 10 5 10 4 10 3 10 2 10 1 SER 5 6 7 8 9 101112131415 SNR (dB) MRT/MRC SER SC/SC SER STBC/MRC SER Figure 3: Symbol error rate of the three MIMO diversity schemes for BPSK (M = 6, N = 1). to be of crucial importance for emerging wireless networks. We will explore how the average (ergodic) system capac- ity of a multiuser MIMO MRT/MRC system scales with the number of users K when opportunistic scheduling is em- ployed, and how the number of antennas M and N come into play. Assume the normalized effective link SNR for user k is γ k , w hose PDF and CDF are denoted by f γ (x)and F γ (x), respectively (same for all users). In the opportunistic scheduling scheme, the base station chooses the user k ∗ = arg max k (γ k ) K k =1 . Thus the resultant normalized system SNR seen by the base station is γ k ∗ with PDF f γ k ∗ (x) = Kf γ (x)F K−1 γ (x). (22) Assuming that average transmit SNR is γ t , average system ca- pacity obtained by opportunistic scheduling can be expressed as E log 1+γ t max 1≤k≤K γ k = +∞ 0 log 1+γ t x f γ k ∗ (x) dx. (23) The closed-form expression for (23) is rather compli- cated, especially for MIMO MRT/MRC systems. We there- fore resort to the theory of order statistics for asymptotic analysis [11, 12]. Some related pioneer study on spatial mul- tiplexing systems can be found in [13]. To this end, the tail behavior of f MRT/MRC γ (x) is required, which we state below. Lemma 3. f MRT/MRC γ (x)∼ 1 (M − 1)!(N − 1)! e −x x M+N−2 , as x −→ +∞. (24) Proof. When x → +∞, F MRT/MRC γ (x) → 1, and lim x→∞ Ψ c (x) i, j = lim x→∞ γ(t − s + i + j − 1, x) = (t − s + i + j − 2)!. (25) Assume λ = t −s, then Ψ c (+∞)isgivenby Ψ c (+∞) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ λ!(λ +1)! ··· (λ + s − 1)! (λ +1)! . . . . . . . . . . . . (λ + s − 1)! ··· ··· (λ +2s −2)! ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ s×s . (26) Since Φ c (x) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ x λ e −x x λ+1 e −x ··· x λ+s−1 e −x x λ+1 e −x . . . . . . . . . . . . . . . . . . . . . x λ+s+1 e −x . . . . . . x λ+2s−2 e −x ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 x ··· x s−1 x . . . . . . . . . . . . . . . . . . . . . x s−1 . . . . . . x 2s−2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ x λ e −x , (27) the tail behavior of f MRT/MRC γ (x) will be determined by that of Φ c (x), given by (where the coefficients {a i } come from linear combinations of elements in Ψ −1 c (+∞)) f MRT/MRC γ (x) ∼ tr Ψ −1 c (+∞)Φ c (x) = e −x a 1 x λ+2s−2 + a 2 x λ+2s−3 + ··· + a 2s−2 x λ+1 + a 2s−1 x λ = e −x x λ+2s−2 a 1 + O 1 x , (28) with a 1 = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ λ!(λ +1)! ··· (λ + s − 2)! (λ +1)! ··· ··· (λ + s − 1)! . . . . . . . . . . . . (λ + s − 2)! ··· ··· (λ +2s −4)! ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ Ψ c (+∞) = s−1 k=1 (t − k − 1)!(s − k −1)! s k=1 (t − k)!(s − k)! = 1 (t − 1)!(s − 1)! = 1 (M − 1)!(N − 1)! . (29) 6 EURASIP Journal on Wireless Communications and Networking With Lemma 3, we derive the asymptotic system capacity for multiuser MIMO MRT/MRC systems as follows. Proposition 2. When multiuser diversity is exploited in a K- user MIMO MRT/MRC system, the asymptotic average system capacity C (MRT/MRC) K is given by C (MRT/MRC) K = E log 1+γ t max 1≤k≤K γ (MRT/MRC) k −→ log 1+γ t b (MRT/MRC) K , as K −→ ∞, (30) where b (MRT/MRC) K is solved through F (MRT/MRC) γ (b K ) = 1 −1/K and is given by b (MRT/MRC) K = log K (M −1)!(N − 1)! +(M + N − 2)log log K (M − 1)!(N − 1)! + O(log log log K). (31) Proof. See the appendix. Remark 1. The following result is often invoked to indicate that max 1≤k≤K γ k “grows like” b K in a coarse sense, and is widely used in the study of opportunistic communications involving extreme v alues and order statistics (e.g., [7, 14]): max 1≤k≤K γ k − b K a K D −→ Λ(x) = exp − e −x , (32) where a K = (Kf γ (b K )) −1 . This result can actually be strengthened from existing literature [11, 12]iflim x→∞ ((1 − F γ (x))/f γ (x)) = c = 0, max 1≤k≤K γ k − b K P → 0, otherwise if lim x→∞ ((1 − F γ (x))/f γ (x)) = c>0max 1≤k≤K γ k /b K P → 1. Nonetheless, our result (30)isyetastrongerone,whichis concerned with the convergence of the expected values of functions of max 1≤k≤K γ k . In a similar fashion, we can obtain the asymptotic sys- tem capacity for multiuser MIMO STBC/MRC and SC/SC systems, which are dictated by C (STBC/MRC) K −→ log 1+γ t 1 M log cK + N − 1 M log log(cK) + O(log log log K) , (33) where c = M MN−1 (MN − 1)! , (34) and 1 C (SC/SC) K −→ log 1+γ t log(MNK) , (35) 1 This is a rare accurate expression. Note that in this case, the growth in transmit and receive antennas can be equivalently seen as an increase in the number of users (due to the i.i.d. assumptions). 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 Average system capacity (nats/s/Hz) 4 6 8 101214161820 Number of users: log 2(K) MRT/MRC simulation MRT/MRC approximation SC/SC simulation SC/SC approximation STBC/MRC simulation STBC/MRC approximation Figure 4: Average system capacity of opportunistic scheduling (γ t = 0dB,M = N = 2). as K →∞. From the asymptotic system capacities of the joint spatial diversity and multiuser diversity systems, we can make some interesting observations. From (33), a tradeoff between transmit diversity and multiuser diversity for an open-loop spatial diversity system is seen, which has also been observed by other researchers (e.g., [14, 15]). But in our paper, a more rigorous proof is provided and how the asymptotic system capacity is related to key system parameters is revealed. For example, our result does show the positive role of the num- ber of receive antennas N, though in a second-order 2 sense, whichisnotclearfrompreviousresultsinliterature.Itisalso observed that the detrimental effect of multiple transmit an- tennas can be avoided with the closed-loop spatial diversity schemes, as seen in (31)and(35) 3 . Also from (31)and(35), we can infer that for the general hybrid selection combining schemes, the scaling laws should only have differences in the second-order approximations. Numerical results in Figure 4 verify that log(1+γ t b K ) is a good approximation for the aver- age capacity of the STBC/MRC, SC/SC, and MRT/MRC sys- tems using the opportunistic scheduler. 5. CONCLUSIONS In this paper, through the analysis of the distribution of the squared largest singular value of a complex Gaussian matrix 2 We define the first-order approximation when truncated at log K,andthe second-order approximation when truncated at log log K. 3 The coefficient of Kis not important when K becomes large. In this sense, multiple antennas even help for the MRT/MRC scheme. Q. Zhou and H. Dai 7 at the origin and tail, we obtain two asymptotic results re- lated to MIMO MRT/MRC systems. One is the asymptotic error performance in the single-user scenario at high trans- mit SNR, and the other is the asymptotic system capacity in the multiuser scenario when multiuser diversity is exploited. Our results are rigorous and succinct, which provide a per- formance reference for MIMO diversity systems and facilitate various tradeoff studies in terms of system parameters and designs. APPENDIX A. PROOF OF PROPOSITION 2 Proof. For the purpose of brevity, we will use F(x)and f (x) to denote F MRT/MRC γ (x)and f MRT/MRC γ (x), respectively, in the following proof, and b K for b (MRT/MRC) K . Assume C K = log(1+ γ t (max 1≤k≤K γ k )). Define the growth function g(x) = (1 − F(x))/f(x), with Lemma 3 we hav e lim x→+∞ g(x) = lim x→+∞ −f (x) f (x) = 1. (A.1) Clearly F(x)in(2) is less than 1 for all finite x and is twice differentiable for all x.By(20)of[16], we can obtain the fol- lowing expansion at b K : log − log F K b K + xg b K =− x + x 2 2! g b K + x 3 3! g b K g (2) b K − 2g 2 b K ···+ ··· + e −x + ··· 2K + 5e −2x+··· 24K 2 + ···− 1 8K 3 e −3x + ···+ ···, (A.2) where b K is given by F(b K ) = 1 −1/K. Solving for b K we can get (for some constant c 1 ) b K = logc 1 K +(M + N − 2) log log c 1 K + O(log log log K) = O(log K). (A.3) A close examination of g (x) using Lemma 3 reveals g (x) = O 1 x ,lim K→∞ Kg b K = +∞. (A.4) Therefore, the terms in the second line of (A.2) starting with the term e −x /2K can be ignored [16]. Further exploit- ing (A.1), (A.3), and (A.4) in the first line of (A.2)with x =±log log K yields 4 Pr − log log K≤ max 1≤k≤K γ k − b K ≤log log K ≥ 1−O 1 log K . (A.5) 4 It can be shown that (A.4) still holds for a more general condition g (x) = O(1/x δ )withδ>0. Appling Chebyshov’s inequality, we have E C K ≥ P C K ≥ log 1+γ t b K − log log K × log 1+γ t b K − log log K ≥ 1 − O 1 log K × log 1+γ t b K − log log K = log 1+γ t b K − log log K − O log log K log K = log 1+γ t b K − o(1). (A.6) On the other hand, E C K = ∞ 0 P C K >x dx = log(1+γ t b K ) 0 P C K >x dx+ +∞ log(1+γ t b K ) P C K >x dx ≤ log 1+γ t b K + +∞ log(1+γ t b K ) P C K >x dx, (A.7) with P C K >x = 1 − P C K ≤ x = 1 − F K e x − 1 γ t . (A.8) We know lim x→∞ ((1 − F(x))/f(x)) = 1 > 0, therefore when x is large enough, we can find a positive constant c 2 and x 0 , such that 1 − F(x) <c 2 f (x), for any x>x 0 .Thusfor sufficiently large x 1 − F K e x − 1 γ t = 1 − F e x − 1 γ t × 1+F e x − 1 γ t +···+ F (K−1) e x − 1 γ t ≤ Kc 2 f e x − 1 γ t . (A.9) Therefore when K is large enough, we have +∞ log(1+γ t b K ) P C K >x dx ≤ +∞ log(1+γ t b K ) Kc 2 f e x − 1 γ t dx = +∞ b K Kc 2 f (x) γ t 1+xγ t dx ≤ c 2 γ t 1+γ t b K +∞ b K Kf(x) dx = O 1 log K × +∞ b K Kf(x) dx = O 1 log K × K × 1 − F b K = O 1 log K , (A.10) 8 EURASIP Journal on Wireless Communications and Networking where the last equality uses the fact (1 − F(b K )) = 1/K.So for sufficiently large K E C K ≤ log 1+γ t b K + O 1 log K . (A.11) Based on (A.6)and(A.11) we can conclude that lim K→∞ E log 1+γ t max 1≤k≤K γ k − log 1+γ t b K −→ 0. (A.12) REFERENCES [1] T. K. Y. Lo, “Maximum ratio transmission,” IEEE Transactions on Communications, vol. 47, no. 10, pp. 1458–1461, 1999. [2] P. A. Dighe, R. K. Mallik, and S. S. Jamuar, “Analysis of transmit-receive diversity in Rayleigh fading,” IEEE Transac- tions on Communications, vol. 51, no. 4, pp. 694–703, 2003. [3] P. A. Dighe, R. K. Mallik, and S. S. Jamuar, “Analysis of K- transmit dual-receive diversity with cochannel interferers over a Rayleigh fading channel,” Wireless Personal Communications, vol. 25, no. 2, pp. 87–100, 2003. [4] M. Kang and M S. Alouini, “Largest eigenvalue of complex Wishart matrices and performance analysis of MIMO MRC systems,” IEEE Journal on Selected Areas in Communications, vol. 21, no. 3, pp. 418–426, 2003. [5] M. K. Simon and M S. Alouini, Digital Communications over Fading Channels, John Wiley & Sons, New York, NY, USA, 2000. [6] C. G. Khatri, “Distribution of the largest or the smallest char- acteristic root under null hypothesis concerning complex mul- tivariate normal populations,” The Annals of Mathematical Statistics, vol. 35, no. 4, pp. 1807–1810, 1964. [7] P. Viswanath, D. N. C. Tse, and R. Laroia, “Oppor tunistic beamforming using dumb antennas,” IEEE Transactions on In- formation Theory, vol. 48, no. 6, pp. 1277–1294, 2002. [8] C. Mun, M W. Lee, J G. Yook, and H K. Park, “Exact ca- pacity analysis of multiuser diversity combined with transmit diversity,” Electronics Letters, vol. 40, no. 22, pp. 1423–1424, 2004. [9] C J. Chen and L C. Wang, “A unified capacity analysis for wireless systems with joint antenna and multiuser diversity in Nakagami fading channels,” in Proceedings of IEEE Interna- tional Conference on Communications, vol. 6, pp. 3523–3527, Paris, France, June 2004. [10] Z. Wang and G. B. Giannakis, “A simple and general parame- terization quantifying performance in fading channels,” IEEE Transactions on Communications, vol. 51, no. 8, pp. 1389– 1398, 2003. [11] H. A. David and H. N. Nagaraja, Order Statistics,JohnWiley & Sons, New York, NY, USA, 3rd edition, 2003. [12] J. Galambos, The Asymptotic Theory of Extreme Order Statis- tics, John Wiley & Sons, New York, NY, USA, 2nd edition, 1978. [13] M. Sharif and B. Hassibi, “On the capacity of MIMO broadcast channels with partial side information,” IEEE Transactions on Information Theory, vol. 51, no. 2, pp. 506–522, 2005. [14] J. Jiang, R. M. Buehrer, and W. H. Tranter, “Antenna diversity in multiuser data networks,” IEEE Transactions on Communi- cations, vol. 52, no. 3, pp. 490–497, 2004. [15] R. Gozali, R. M. Buehrer, and B. D. Woerner, “The impact of multiuser diversity on space-time block coding,” IEEE Com- munications Letters, vol. 7, no. 5, pp. 213–215, 2003. [16] N. T. Uzgoren, “ The asymptotic development of the distribu- tion of the extreme values of a sample,” in Studies in Mathe- matics and Mechanics Presented to Richard von Mises, pp. 346– 353, Academic Press, New York, NY, USA, 1954. Quan Zhou received the B.E. degree from Northern Jiaotong University, Beijing, China, in 1998, the M.S. degree from Ts- inghua University, Beijing, China, in 2001, and the Ph.D. degree from North Carolina State University, Raleigh, NC, in 2006, all in electrical engineering. Currently he is doing internship at Mitsubishi Electric Research Laboratories (MERL), Cambridge, Mass. His research interests are in the general areas of wireless MIMO communications with emphasis on link adaptation and multiuser scheduling. His current research focuses on the design, analysis, and implementation of 802.11n system. Huaiyu Dai received the B.E. and M.S. degrees in electrical engineering from Ts- inghua University, Beijing, China, in 1996 and 1998, respectively, and the Ph.D. de- gree in electrical engineering from Prince- ton University, Princeton, NJ, in 2002. He worked at Bell Labs, Lucent Technologies, Holmdel, NJ, during the summer of 2000, and at AT&T Labs-Research, Middletown, NJ, during the summer of 2001. Currently he is an Assistant Professor of Electrical and Computer Engineer- ing at NC State University. His research interests are in the general areas of communication systems and networks, advanced signal processing for digital communications, and communication the- ory and information theory. He h as worked in the areas of digital communication system design, speech coding and enhancement, DSL and power line transmission. His current research focuses on wireless sensor networks, cross-layer design (with a physical layer emphasis), space-time communications and signal processing, the turbo principle and its applications, multiuser detection, and the information-theoretic aspects of multiuser communications and networks. . electrical engineering from Ts- inghua University, Beijing, China, in 1996 and 1998, respectively, and the Ph.D. de- gree in electrical engineering from Prince- ton University, Princeton, NJ, in 2002 clearly reveal the interaction between these two forms of diversity. Aiming at obtaining succinct and insightful performance evaluation for MIMO MRT/MRC systems (more general MIMO diversity systems) ,. becomes large. In this sense, multiple antennas even help for the MRT/MRC scheme. Q. Zhou and H. Dai 7 at the origin and tail, we obtain two asymptotic results re- lated to MIMO MRT/MRC systems. One