Asymptotic Analysis in MIMO MRT/MRC Systems

In this paper, through the analysis of the probability density function of the squared largest 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.


I. Introduction
Multi-input multi-output (MIMO) systems can be exploited for spatial multiplexing or diversity gains. For a MIMO diversity 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 selection combining on both ends (SC/SC), joint maximum ratio transmission and maximum ratio combining (MRT/MRC), and various hybrid selection combining schemes in between.
For diversity usage, MRT/MRC systems provide the optimal performance reference [1]- [5], but its analysis is also more involved than others (see relevant distribution functions in Section II), which 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 signalto-noise ratio (SNR) of a MRT/MRC system is upper and lower bounded in [1], based on which the average symbol error rate (SER) and diversity 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 receive 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 equivalent channel gain max σ , the largest singular value of H . For Rayleigh fading channels, the distribution of 2 max σ , already derived in [6], is revisited in [2] and expressed in an alternative form -a linear combination of Gamma functions. Based on this expression, the exact system SER is derived for general modulation schemes in [2]. The distribution of 2 max σ for Ricean fading is obtained in [4]. Unfortunately, results in [2] and [4] don't easily lead one to an insightful understanding of the impact of the system parameters, including the number of transmit and receive antennas M and N , on performance. For example, in [2], the authors make two observations on MIMO MRT/MRC systems through simulation results: one is that when M N + keeps fixed, the antennas distribution 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 authors do not provide a rigorous justification for both observations. Some similar observations are also made in [4]. In a multiuser wireless network, there is another form of diversity called multiuser diversity, which reflects the fact of independent 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 general 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, asymptotic analysis also helps reveal some fundamental relationship of key system parameters, which may be concealed in the finite case by random fluctuations and other transient properties of channel matrices. This paper comprises two sub-topics: error performance in the single-user scenario and capacity 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 investigation of the approximate behavior of the distribution of 2 max σ 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 2 max σ at the origin, we obtain the asymptotic average SER (in terms of SNR) for MIMO MRT/MRC systems. As applications we verify the two observations made in [2].
2). By studying the behavior of the distribution function of 2 max σ at the tail, we obtain the asymptotic system capacity (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 representative 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 II, we give our model for MIMO MRT/MRC systems. Then we provide our asymptotic analysis for the average SER and system capacity in Section III and IV respectively, together with some numerical results for illustration purpose. Conclusion is given in Section V.

II. System model
We assume a narrowband MIMO MRT/MRC system with M transmit antennas and N receive antennas, modeled as: , t u = + = + y Hx n Hw n is applied on y to obtain a decision statistic for u , chosen as the principal left singular vector of H here. Other diversity schemes can be equivalently represented with t w and r w appropriately defined.
The cumulative distribution function (CDF) of 2 max γ σ = is given by [ a γ β is the incomplete Gamma function defined as +∞ . The probability density function (PDF) of x can be derived as In the remainder of this paper, we adopt the following notations for the limiting behaviors of two functions ( ) When convergence of a sequence of random variables is involved, shorthand notation "D" stands for in distribution and "P" for in probability.

III. Asymptotic Average SER -Single-User Scenario
In this section, we will derive a succinct expression for average SER at high SNR. The conditional SER for lattice-based modulations can be represented as ( ) ( ) where n M is the number of the nearest neighboring constellation points, ( ) Q ⋅ is the Gaussian tail Q-function, and κ is a positive fixed constant determined by the modulation and coding schemes [5]. At high transmit SNR t γ , the system average SER { ( )} s s P E P = H will be dominated by the low-probability outage event that γ becomes small [10]. Therefore, only the behavior of To this end, the following result is crucial.
Proof: By Maclaurin Series expansion 1 1 , we can obtain the approximation of ( ) c x Ψ at 0 x + = after some manipulation as The determinant of Λ can be obtained in a similar fashion as that of a Hilbert matrix. After some algebra we get and it follows from (2) that ■ With Lemma 1, we establish the following result for the asymptotic average SER for MIMO MRT/MRC systems following Proposition I in [10].
Proposition 1: For MIMO MRT/MRC systems, the asymptotic average SER is given by The validity of (8) is demonstrated in Fig. 1 for uncoded BPSK systems. Based on (8) s t s t + > + , we can obtain 1 2 2 1 s s t t < < < . As it is equivalent to show that The left hand side of (12) can be rewritten as 1 1 ( 1 ) (1) Similarly the right hand side of (12) can be represented as 2 2 ( 1 ) (1) It is not difficult to get . Therefore, after canceling out the same factors in (13) and (15), we can see that (13) is surely larger than (15). ■ From the asymptotic SER expression in (8), we have verified 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 diversity order for MIMO STBC/MRC and SC/SC systems (whose asymptotic average SERs assume the same forms as (8)).
Without loss of generality, we assume that the adopted space-time block coding scheme achieves the full rate and the transmit power is equally allocated among the transmit antennas.
In this case, the normalized effective link SNR for a generic user is given by Similarly the corresponding parameters for the coding gain and diversity order for MIMO STBC/MRC systems can be obtained as 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 the receiver is , whose PDF can be easily obtained as We can obtain the corresponding parameters for the coding gain and diversity order for MIMO SC/SC systems as ( / ) ( / ) , 1.

MN q MN
Comparing (9), (18) and (20) we can see that all these MIMO diversity schemes achieve the same diversity order. Nonetheless, their error performances could still be dramatically different owing to different coding gains, as exhibited in Fig. 2 ) for uncoded BPSK systems at high SNR, which agree well with simulation results (see Fig.3 at SER 5 10 − ). It is also observed that for the same diversity order, the performance of STBC/MRC worsens with the increase of the number of transmit antennas.

IV. Asymptotic System Capacity -Multiuser Scenario
In this section, we consider a homogeneous downlink multiuser MIMO communication scenario, which is envisioned to be of crucial importance for emerging wireless networks. We will explore how the average (ergodic) system capacity of a multiuser MIMO MRT/MRC system scales with the number of users K when opportunistic scheduling is employed, and how the number of antennas M and N come into play. Assume the normalized effective link SNR for user k is k γ , whose PDF and CDF are denoted by ( ) The closed-form expression for (22) is rather complicated, especially for MIMO MRT/MRC systems. We therefore resort to the theory of order statistics for asymptotic analysis [11] [12]. Some related pioneer study on spatial multiplexing systems can be found in [13]. To this end, the tail behavior of    : : 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 ( / )

MRT MRC K C
is given by where ( is solved through ( and is given by Proof: See Appendix. Remark: The following result is often invoked to indicate that 1 max k k K γ ≤ ≤ "grows like" K b in a coarse sense, and is widely used in the study of opportunistic communications involving extreme values and order statistics (e.g., [7][14]): where 1 ( ( )) .
This result can actually be strengthened from existing literature [11] [12]: Nonetheless, our result (28) is a yet stronger one, which is concerned with the convergence of the expected values of functions of In a similar fashion, we can obtain the asymptotic system capacity for multiuser MIMO STBC/MRC and SC/ SC systems, which are dictated by ( ) ( ) as K → ∞ . From the asymptotic system capacities of the joint spatial diversity and multiuser diversity systems, we can make some interesting observations. From (31), 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 number of receive antennas N, though in a second-order 2 sense, which is not clear from previous results in literature. It is also observed that the detrimental effect of multiple transmit antennas can be avoided with the closed-loop spatial diversity schemes, as seen in (29) and (33) 3 . Also from (29) and (33), 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 Fig. 4

V. Conclusions
In this paper, through the analysis of the distribution of the squared largest singular value of a complex Gaussian matrix at the origin and tail, we obtain two asymptotic results related to MIMO MRT/MRC systems. One is the asymptotic error performance in the single-user scenario at high transmit 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 performance reference for MIMO diversity systems and facilitate various tradeoff studies in terms of system parameters and designs.
On the other hand, ) log 1 ,