Physical-Layer Multicasting by Stochastic Transmit Beamforming and Alamouti Space-Time Coding

Physical-Layer Multicasting by Stochastic Transmit Beamforming and Alamouti Space-Time Coding

Sissi Xiaoxiao Wu, Wing-Kin Ma, and Anthony Man-Cho So Copyright (c) 2012 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to pubs-permissions@ieee.org.This work was supported in part by the Hong Kong Research Grant Council (RGC) General Research Fund (GRF) Project CUHK 416012, and in part by The Chinese University of Hong Kong Direct Grant No. 2050506. Part of this work was presented at ICASSP 2011 and ICASSP 2012.Sissi Xiaoxiao Wu is with the Department of Electronic Engineering, The Chinese University of Hong Kong, Shatin, Hong Kong S.A.R., China. E-mail: xxwu@ee.cuhk.edu.hk.Wing-Kin Ma is the corresponding author. Address: Department of Electronic Engineering, The Chinese University of Hong Kong, Shatin, Hong Kong S.A.R., China. E-mail: wkma@ieee.org.Anthony Man-Cho So is with the Department of Systems Engineering and Engineering Management, and, by courtesy, Department of Computer Science and Engineering and the CUHK-BGI Innovation Institute of Trans-omics, The Chinese University of Hong Kong, Shatin, Hong Kong S.A.R., China. E-mail: manchoso@se.cuhk.edu.hk.
Abstract

Consider transceiver designs in a multiuser multi-input single-output (MISO) downlink channel, where the users are to receive the same data stream simultaneously. This problem, known as physical-layer multicasting, has drawn much interest. Presently, a popularized approach is transmit beamforming, in which the beamforming optimization is handled by a rank-one approximation method called semidefinite relaxation (SDR). SDR-based beamforming has been shown to be promising for a small or moderate number of users. This paper describes two new transceiver strategies for physical-layer multicasting. The first strategy, called stochastic beamforming (SBF), randomizes the beamformer in a per-symbol time-varying manner, so that the rank-one approximation in SDR can be bypassed. We propose several efficiently realizable SBF schemes, and prove that their multicast achievable rate gaps with respect to the MISO multicast capacity must be no worse than  bits/s/Hz, irrespective of any other factors such as the number of users. The use of channel coding and the assumption of sufficiently long code lengths play a crucial role in achieving the above result. The second strategy combines transmit beamforming and the Alamouti space-time code. The result is a rank-two generalization of SDR-based beamforming. We show by analysis that this SDR-based beamformed Alamouti scheme has a better worst-case effective signal-to-noise ratio (SNR) scaling, and hence a better multicast rate scaling, than SDR-based beamforming. We further the work by combining SBF and the beamformed Alamouti scheme, wherein an improved constant rate gap of  bits/s/Hz is proven. Simulation results show that under a channel-coded, many-user setting, the proposed multicast transceiver schemes yield significant SNR gains over SDR-based beamforming at the same bit error rate level.

Index terms physical-layer multicasting, multicast capacity, transmit beamforming, semidefinite relaxation, semidefinite programming
EDICS: MSP-CODR (MIMO precoder/decoder design), MSP-STCD (MIMO space-time coding and capacity), MSP-CAPC (MIMO capacity and performance)

Final Version, May 2013

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I Introduction

In recent years, the explosive growth in the demand for various wireless data services has motivated a vast amount of research on resource-efficient techniques for massive content delivery. One scenario that has received significant attention is physical-layer multicasting, in which a base station broadcasts common information to a prespecified group of users. For instance, in the long term evolution (LTE) standard, a particular work item called multimedia broadcast multicast service (MBMS) [1] is being actively considered as a preferred mass media streaming option.

The scenario of interest in this paper is that of physical-layer multicasting in multiuser multiple-input single-output (MISO) downlink, assuming channel state information at the transmitter side (CSIT). A central problem in this context is to develop efficient and physically realizable transceiver techniques. Currently, a popularized approach is multicast beamforming, in which the physical-layer transmit strategy is single-stream beamforming, and the beamformer is designed so that users can simultaneously receive good quality of service (QoS). The idea of using beamforming as a transmit strategy for physical-layer multicasting can be traced back to a 1998 paper by Narula et al. [2], although more appropriate beamforming optimization formulations, namely, the QoS-constrained problem and the max-min-fair problem, appeared later [3, 4]. As it turns out, both of these formulations are NP-hard in general [5, 6]. To circumvent such intractability, a state of the art approach is semidefinite relaxation (SDR) [5]. The main observation behind this approach is that the beamforming problem can be reformulated as a rank-one constrained semidefinite program (SDP). Thus, by dropping the non-convex rank constraint, one obtains a convex and tractable SDR problem, whose solution can then be used to generate a rank-one approximate solution to the original beamforming problem. The viability of SDR-based multicast beamforming has been proven by both empirical evidence [5] and theoretical analysis [6, 7]. In fact, SDR-based multicast beamforming has sparked much interest in the area, where we have seen the same fundamental idea of SDR being applied to many different beamforming scenarios; see, e.g., the references in [8]. In addition, we should note that the significance of multicast beamforming as demonstrated through SDR has motivated the development of many other competing optimization methods [9, 10, 11, 12, 13].

While our main interest is in multicast transceiver designs under CSIT, we should also briefly mention the no-CSIT case. A common transmit strategy without CSIT is to transmit isotropically, which is called the open-loop strategy in the literature and may physically be implemented by space-time coding [3]. Open-loop system capacity analyses have been considered in [7, 14]. The work [14] also considers antenna subset selection for striking a balance between the full CSIT and no CSIT settings. Moreover, in [15], a diagonally precoded extension of the space-time coding approach was proposed for the full CSIT case.

I-a Motivations and Contributions

The now popularized SDR-based multicast beamforming scheme has been shown to be capable of providing accurate approximations in a variety of practical regimes, most notably, in the cases where there is a small or moderate number of users. However, the analyses in [6] also reveal an inherent limitation, namely, the SDR approximation accuracy may degrade as the number of users increases. The focus of this work is to pursue alternative physical-layer multicasting strategies that can deliver good performance even in the presence of large number of users. Our endeavor is motivated from an information theoretic perspective—the SDR solution under the max-min-fair formulation is equivalent to the optimal transmit covariance of the multicast capacity [5, 7]. Hence, instead of extracting a rank-one approximate solution from the SDR solution, which is the case in multicast beamforming, we consider altering the transmit structure to embrace the non-rank-one nature of the multicast optimal transmit covariance (or the SDR solution). Specifically, we propose two new physical-layer strategies for multicasting in this paper.

I-A1 Stochastic Beamforming

The first strategy, called stochastic beamforming (SBF), is to employ a per-symbol time variant, randomly generated, beamformer. The underlying intuition of SBF is to use time-varying spatial randomizations to mimic the multicast capacity-optimal transmit covariance, thus producing “rank- beamforming” in a virtual manner for any . A distinguishing characteristic of our SBF framework is that channel coding (which is usually present in practical systems) is utilized to approach some kind of ergodic achievable rate metric. We will develop three efficiently and practically implementable schemes under the SBF framework. Numerical simulations show that they can have significant bit-error-rate (BER) performance gains over SDR-based beamforming. On the theoretical side, we prove that the achievable rate gaps of the proposed SBF schemes with respect to (w.r.t.) the multicast capacity must be no worse than  bits/s/Hz, irrespective of any factors such as the number of users. From a practical viewpoint, this implies that even when there is a large number of users, SBF can still perform reasonably well.

I-A2 Alamouti-Assisted Rank-Two Beamforming

Our second strategy is to develop rank- generalizations of beamforming, both fixed and stochastic, through the use of the Alamouti space-time code. To motivate this strategy, we should first note that in the point-to-point multiple-input multiple-output (MIMO) literature, there has been interest in combining beamforming and space-time coding (STC) to provide rank- beamforming; see, e.g., [16, 17, 18]. However, developing a combined beamforming and STC scheme is a scenario-dependent challenge, as evidenced in the above referenced work. The reason is that many available space-time codes are designed for performance metrics in point-to-point CSIT-uninformed scenarios, such as diversity order or diversity multiplexing tradeoff, and those merits do not always carry forward to another MIMO scenario that has a different performance metric. In Section 4.2 of the companion technical report [19], we provide simulation results that demonstrate a direct combination of beamforming and STC based on intuition would lead to poor performance in the multicast scenario.

There is however an exception where beamformed STC designs can be tractable, namely, when the class of orthogonal space-time block codes (OSTBCs) is used. OSTBCs are well known to be simple to implement, and, more importantly, their performance can be easily characterized by an explicit signal-to-noise ratio (SNR) expression. Some representative point-to-point beamformed STC designs are, in fact, based on OSTBCs [16, 17, 18]. On the other hand, one must note that full-rate OSTBCs do not exist for dimensions higher than two [20].

In view of the above discussion, we will consider beamformed STC based on the two-dimensional full-rate OSTBC, that is, the well-known Alamouti space-time code. We first develop an SDR-based fixed beamformed Alamouti scheme, which is a rank-two generalization of the previous (rank-one) SDR-based beamforming framework. Our analysis shows that in terms of the effective worst-user SNR, the worst-case approximation accuracy of the beamformed Alamouti scheme degrades only at a rate of , where is the number of users. This is an improvement over the previous beamforming scheme, where the provable worst-case approximation accuracy degrades at the higher rate of  [6]. Next, we combine the SBF strategy and the beamformed Alamouti scheme; that is, we produce virtually rank- beamforming from physically rank-two beamforming. By analysis, we show that the SBF Alamouti schemes have a worst-case multicast achievable rate gap of  bits/s/Hz, which is better than the previous bits/s/Hz bound for the SBF schemes. The SBF Alamouti schemes also yield the best coded BER performance by simulations when compared to beamforming and other proposed schemes.

I-B Related Works

We should mention some existing works that might seem related to SBF, and contrast the differences. At first sight, using randomness in beamforming may remind one of the opportunistic beamforming (OBF) technique [21]. However, OBF deals with user scheduling in a multiuser TDMA setting, which is a very different scenario from multicasting. By closely examining OBF and SBF, one would find that the ways randomness is used also have much difference: OBF is a per-frame randomized approach without CSIT, while SBF is per-symbol random with CSIT. For a similar reason, SBF is different from the randomized space-time coding approach for cooperative communication [22]—the latter is per-frame random without CSIT, with an aim to harvest cooperative diversity. Moreover, it is interesting to note that the philosophical possibility of randomizing the beamfomer was vaguely alluded to in a study of the unicast scenario [23], although no further investigation was provided. In fact, the authors there never needed to—they showed that in unicasting, SDR always has a rank-one solution, i.e., transmit beamforming is sufficient in unicasting. However, this result does not apply to multicasting [5, 6]. In this study, the idea of utilizing channel coding, and the subsequent ergodic rate characterization for multicasting, are new.

We should also describe related work on our fixed beamformed Alamouti scheme. As mentioned earlier, the beamformed Alamouti structure, or, more generally, the beamformed OSTBC structure, has previously appeared in the point-to-point MIMO literature, e.g., [16, 17, 18]. Also, in the multicast scenario, there is an early work [15] where the authors considered a diagonally precoded OSTBC scheme with per-antenna power allocation (rather than beamforming). The issue that is different in the present scenario is the beamformer designs, where the restriction of rank-two beamforming for full-rate transmission results in a multicast design optimization problem that is NP-hard. The significance of our development lies not only in proposing a rank-two SDR framework for the beamformer design, but also in generalizing the theoretical analysis of SDR-based beamforming in a non-trivial manner. In particular, we are able to establish for the first time a worst-case performance bound for the NP-hard rank-two beamforming problem. We should bring readers’ attention to the work [24, 25], wherein the authors independently introduced the same Alamouti-assisted rank-two SDR idea at about the same time when a preliminary version of this work [26] was presented. What distinguishes our work is that we also provide performance analysis of the resulting scheme.

I-C Organization and Notations

The organization of this paper is as follows. In Section II we provide the problem formulation and review the SDR-based multicast beamforming scheme. The SBF framework is developed and described in Section III. Section VI provides the simulation results, and the paper is concluded in Section VII.

Our notation is standard: is the set of all complex -dimensional vectors; is the set of all complex Hermitian matrices; means that is elementwise non-negative; means that is positive semidefinite; is the vector Euclidean norm; , , , and stand for the trace, rank, the largest eigenvalue, and the smallest non-zero eigenvalue of , resp.; and are the all-zero and all-one vectors, resp.; is a unit vector with the nonzero element at the th entry; denotes the -by- identity matrix; is statistical expectation; (resp. ) is used to denote the circularly symmetric complex Gaussian distribution (resp. the real Gaussian distribution) with mean vector and covariance matrix ; and means that the random variables and have the same distribution.

Ii Problem Formulation and Background Review

This section describes the physical-layer multicasting problem formulation and gives a review of multicast beamforming.

We consider a standard multicast scenario [7] where a base station transmits a common message to users under slow channel fading. To be specific, the base station is equipped with transmit antennas, while the users a single antenna. The channel of each user is assumed to be frequency flat and slow faded in the sense that its coherence time is larger than the data frame or packet transmission period. Under this setting, the signal model for one data frame transmission can be described by

(1)

where is the received signal of user at time (or th channel use), is the data frame length, which is assumed to be large, is the channel from the base station to user , denotes the multi-antenna transmit signal, and is zero mean unit variance complex Gaussian noise. We denote the transmit covariance by .

The subject of interest is to provide good multicast rate performance for each frame transmission, assuming knowledge of , or channel state information at the transmitter (CSIT). From an information theoretic perspective, it is known that the multicast capacity under model (1) and in the presence of CSIT is given by

(2)

where is the maximum allowable transmit power and is natural logarithm (and thus is in units of nats/s/Hz) [7]. Note that we do not assume any physical-layer transmit structure on at this point. By the change of variable , we can rewrite (2) as

where

(3)

In particular, an optimal solution to (2) can be constructed from the optimal solution to (MC) via . Problem (MC) is an SDP, which is convex and polynomial-time solvable [27]. Alternatively, one may employ low-complexity heuristics specially designed for (MC); see, e.g., [11].

An important question is how physical-layer schemes should be designed to practically approach the information rate promised by the multicast capacity . From such a realizable transceiver design viewpoint, there seems to have no report on a practical multicast capacity-achieving scheme that has been successfully implemented and demonstrated in physical layer. Currently, a widely adopted scheme is transmit beamforming, which is efficiently realizable but generally suboptimal. In transmit beamforming, the transmit signal is constrained to take the form

where is a transmit beamforming vector, is again the maximum allowable transmit power, and is a stream of data symbols with unit power (i.e., ). In beamforming, the received signal in (1) reduces to a single-input single-output (SISO) model , and we can characterize the performance by the signal-to-noise ratios (SNRs) of the received symbols, namely, , where . Consequently, the multicast beamforming problem can be formulated as

(4)

where

represents the multicast achievable rate of a given beamformer  [5, 7]. Note that this rate can be practically approached by applying an ideal channel code to 111The common, tacit, understanding is that Turbo codes or low density parity check codes should provide near-ideal scalar channel coding performance in practice.. Now, it is known that Problem (4) is equivalent to the max-min-fair (MMF) problem

which is NP-hard in general [5, 6]222Note that the MMF problem was originally formulated from a QoS perspective [5], where the aim is to maximize the worst user’s QoS under a power constraint . The QoS commonly refers to the SNR defined here, although other measures of QoS, such as the long-term average SNR [28], can also be considered.. To circumvent this intractability, an arguably de facto solution is to apply semidefinite relaxation (SDR) to approximate (MMF). In the SDR approach, one first substitute into (MMF) and use the equivalence

to obtain the following equivalent formulation of (MMF):

(5)

The rationale behind such a reformulation is that one can then drop the nonconvex rank constraint in (5) to obtain a convex relaxation problem, viz.

which is an SDP. Some rank-one approximation procedure is then used to convert the solution of (SDR) to a rank-one, feasible, solution to (MMF); see [5, 8, 27] for details. It is interesting to note that (SDR) and (MC) are exactly the same. Hence, the SDR approach essentially uses the multicast capacity-optimal transmit covariance to find a good rank-one beamforming solution.

Empirically, it has been shown that SDR-based multicast beamforming offers good performance, especially for a small to moderate number of users. In fact, theoretical results quantifying the extent to which SDR can perform are available, and they are briefly summarized as follows. Let

(6)

denote the worst-user effective SNR associated with , which appears in the objective functions of (5) and (SDR). By noting that the optimal solution to (MC) is also optimal for (SDR), we have the following:

Fact 1
  • ([29]) When , there is a polynomial-time procedure that can generate from an optimal solution to (MMF). Also, is a solution to (MC).

  • ([6, 5]) When , by using a Gaussian randomization procedure (which runs in randomized polynomial time), one can generate from a feasible solution to (MMF) that satisfies

    with probability at least . In particular, after independent runs of the randomization procedure, one can boost this probability to at least .

Fact 1(a) states that the generally NP-hard (MMF) is equivalent to the convex, polynomial-time solvable (SDR) when the number of users is no greater than .333As an aside, note that for , a closed-form solution to (MMF) can be derived [11]. Thus, in view of the equivalence of (SDR) and (MC), we conclude that transmit beamforming is guaranteed to be a multicast capacity-optimal physical-layer strategy for . As for Fact 1(b), it reflects how the performance of SDR-based beamforming scales with the number of users in a worst case sense. Specifically, consider the achievable rate gap of SDR-based beamforming relative to the multicast capacity, i.e., . From the derivations above, one can readily deduce the following bound for :

(7)

Note that for large , the right-hand side of (7) is approximately equal to , which implies that SDR-based beamforming may suffer from a rate loss that increases logarithmically with the number of users. Hence, the beamforming strategy is only effective when there are not too many users.

Iii Multicast Stochastic Beamforming

In view of the above mentioned drawbacks of beamforming, in this section we propose an alternative physical-layer multicasting strategy based on stochastic beamforming.

Iii-a System Model

Consider the following transmit structure:

(8)

where is a time-varying beamformer weight vector, and the other notations are the same as those in the beamforming strategy discussed above. At each time , is randomly generated according to a common distribution . To distinguish this random-in-time beamforming endeavor from the conventional beamforming scheme, we will henceforth call the former stochastic beamforming (SBF), and the latter fixed beamforming. The SBF strategy is motivated by the observation that the transmit covariance of (8) is given by . In particular, if we choose so that the beamformer covariance and the multicast capacity-optimal transmit covariance are equal, i.e.,

then the SBF should have a better multicast performance than the fixed beamformer, especially when has high rank.

Let us now consider the receiver side. Substituting (8) into (1), the received SBF signals can be written as

(9)

As seen in (9), each user has an instantaneous SNR given by , which fluctuates in time. Hence, we apply channel coding (presumably ideal) across the symbols within the data frame to “average out” the fluctuations caused by SBF. Interestingly, this receiver approach is the same as how one uses channel coding in fast fading channels to exploit time diversity [30]. We assume coherent reception, which means that all the users are assumed to know deterministically (as well as ). This can be made possible by having the transmitter sending the random seed for generating and the multicast optimal transmit covariance , either as part of the preamble of the transmitted frame or via a feedback channel. We should also note that SBF receivers involve simple coherent symbol reception (without inter-symbol interference) and channel decoding, and hence are as efficient as those of fixed beamforming with channel coding.

The SBF system description is complete. Now, several natural questions arise: What distribution should we use to generate the random beamformer weights? How can we characterize the performance of an SBF scheme? These aspects are considered in the subsequent subsections.

Iii-B SBF Achievable Rate

We employ an achievable rate view to study the SBF strategy. For notational simplicity, we use the random variable to denote the randomly generated beamformer weight vector . Under the SBF system model in (9), where channel coding is applied across with sufficiently large, the achievable rate of each user, say, user , can be expressed as

(10)

where denotes the (given) distribution for generating . We should mention that the capacity expression in (10) is deduced in the same spirit as the ergodic capacities for fast fading channels without CSIT, as described or used frequently in the literature; see, e.g., [30, 31]. However, we should emphasize that in this study, it is not the channels that are random, but the beamformer . Moreover, studies in fast fading channels have suggested that the rate (10) may practically be approached by near-ideal scalar channel codes; see, e.g., [30, p. 2627]. Based on (10), the multicast achievable rate of SBF can be formulated as

(11)

Note that must satisfy , so that the power constraint holds.

Before we proceed, let us discuss the key underlying assumption behind the SBF achievable rate metric above—that should be large. In practice, the frame length is constrained by the coherence time of the channels. As such, the rate metric above is more suitable for slow fading scenarios. In our simulations, we found that the idea works well when is the same as that of the coded symbol length for a fixed beamforming channel (or a standard scalar Gaussian channel), which is typically on the order of hundreds in wireless standards.

To facilitate the SBF design and rate analysis, we first derive an alternative expression for . Set

(12)

(see (3) for the definition of ). Clearly, if satisfies the capacity-optimal transmit covariance property , then . Then, we can rewrite (11) as

(13)

The above SBF rate characterization reveals that the SBF performance depends on the “fading” distribution of . The following properties can be derived for (13):

Fact 2

Suppose that are identically distributed. Let for any .

  • The SBF multicast achievable rate (13) can be simplified to .

  • Suppose, in addition, that . Then, the function , where , is nondecreasing in .

Fact 2(a) is simply a consequence of the monotonicity of the log function. For a proof of Fact 2(b), see Appendix A-A.

Iii-C The Gaussian SBF Scheme

Let us now turn our attention to the choice of the beamformer distribution . The most desirable choice of would be that of maximizing the multicast achievable rate under the power constraint. However, this may be too difficult to solve analytically. Hence, we seek simple, easy-to-generate, beamformer randomizations that can yield provably good multicast rate performance.

A simple way to generate is to use the circularly symmetric complex Gaussian distribution:

(14)

We will call the resulting SBF scheme the Gaussian SBF scheme . Gaussian SBF aims at using a simple beamformer generation to satisfy the optimal transmit covariance property . It can be analytically shown that even such a simple beamformer randomization possesses desirable multicast achievable rate properties. From (12), we see that for Gaussian SBF, every follows an exponential distribution with mean . Therefore, the premises of Fact 2 are satisfied, and by Fact 2(a) we can express the Gaussian SBF achievable rate as

(15)

As it turns out, the expression in (15) is identical to that for the ergodic capacity of a scalar Rayleigh channel, which is known to admit the explicit expression

(16)

where , , is the exponential integral of the first order [32]. Now, we are interested in extracting insight from the explicit rate expression (16)—how far away is (16) from the multicast capacity ? Towards that end, consider the achievable rate gap

We then have the following result:

Theorem 1

The achievable rate gap of the Gaussian SBF scheme satisfies

Moreover, the bound is tight when .

Proof:  By Fact 2(b), is nondecreasing in . Moreover, it can be shown that ; see Section 1.1 of the companion technical report [19]. Hence, we conclude that for all .

The implication of Theorem 1 is meaningful—the Gaussian SBF rate is at most  bits/s/Hz () away from the multicast capacity ; otherwise it has the same scaling as the multicast capacity, irrespective of the number of users. This is unlike the SDR-based fixed beamforming scheme reviewed in Section II, where the rate gap may increase with the number of users; cf. (7).

Iii-D The Elliptic SBF Scheme

As shown in the previous subsection, even with just a simple Gaussian SBF scheme, we can achieve a rate that is within less than 1 bit/s/Hz of the multicast capacity. From a practical viewpoint, however, the Gaussian SBF scheme has a drawback—its instantaneous beamformer power, which is given by , can have a large spread. Indeed, since is a chi-square random variable, the instantaneous power can in principle take any non-negative values. Hence, while Gaussian SBF is interesting from a fundamental viewpoint, where a theoretically provable rate gap of less than one bit w.r.t. the multicast capacity can be established, it may not be desirable for practical implementation. To remedy this, we consider an alternative SBF scheme, in which the beamformer weight is generated by

(17)

where and is a square root decomposition of , i.e., . Note that (17) is simply a Gaussian SBF normalized by the factor ; cf. (14). Intuitively, such a normalization serves to limit the instantaneous beamformer power. More precisely, since , by the Courant-Fischer min-max theorem, we have with probability 1. As it turns out, the random vector also satisfies the capacity-optimal transmit covariance property:

Fact 3

[33] The random vector in (17) follows an elliptic symmetric distribution with covariance matrix .

Motivated by Fact 3, we shall call the resulting SBF scheme the elliptic SBF scheme. Now, just as in the case of the Gaussian SBF scheme, we are interested in determining the achievable rate of the elliptic SBF scheme. Towards that end, consider the non-negative random variables

(18)

see (12). Naturally, we would like to use Fact 2 to characterize the elliptic SBF rate. However, this entails understanding the distribution of . Fortunately, as we shall see shortly, the distribution of admits a simple closed form expression. We begin with the following lemma, which generalizes [34, Lemma 1] and whose proof can be found in Appendix A-B:

Lemma 1

Let be a fixed vector and be independent random vectors. Then, the CDF of the non-negative random variable

is given by

where

From (18), we see that if we take and in Lemma 1, then . In particular, upon differentiating the corresponding CDF w.r.t.  and observing that , we obtain the following:

Proposition 1

Consider the elliptic SBF scheme. The PDF of , where , is given by

(19)

where .

Proposition 1 has two important implications. First, it shows that the random variables are identically distributed, and hence by (19) and Fact 2(a) the elliptic SBF rate can be readily computed via

Secondly, we have by Fact 3. Hence, by Fact 2(b), the achievable rate gap of the elliptic SBF scheme, which is given by

is nondecreasing in .

To further understand the behavior of , let us first derive an explicit formula for .

Proposition 2

For any ,

The proof of Proposition 2 can be found in Section 1.2.1 of the companion technical report [19]. Armed with this formula, we can establish the following result:

Theorem 2

The achievable rate gap of the elliptic SBF scheme satisfies

Moreover, the bound is tight when .

Proof:  We have already shown that is nondecreasing in . Moreover, it can be shown that ; see Section 1.2.2 of the companion technical report [19]. Hence, we conclude that for all .

Since the function is nondecreasing and tends to as (see, e.g., [35, Formula 0.131]), an important corollary of Theorem 2 is that the worst-case rate gap of the elliptic SBF scheme is no worse than that of the Gaussian SBF scheme. For comparison, we compute the worst-case rate gap of the elliptic SBF scheme for various values of and summarize the results in Table I.

     
rate gap in nats
rate gap in bits
TABLE I: The worst-case rate gap of the elliptic SBF scheme

Iii-E The Bingham SBF Scheme

In the previous subsection, we have illustrated that a proper normalization of the Gaussian beamformer randomization not only helps to limit the instantaneous beamformer power spread effects, but also improves the multicast achievable rate. Now, let us consider another beamformer randomization

(20)

The motivation behind (20) is straightforward—we want , or in other words, zero instantaneous beamformer power spread. Curiously, the kind of randomization in (20) has been studied in the statistics literature—it is known that follows the Bingham distribution [36]. For that reason, we will call the resulting SBF scheme the Bingham SBF scheme.

Unlike the previous two SBF schemes, Bingham SBF may not satisfy the capacity-optimal transmit covariance property . Moreover, the achievable rate analysis of Bingham SBF is different from that of Gaussian and elliptic SBF—a key component of the latter is to derive the distribution of in (12), and this appears to be hard for Bingham SBF. We therefore resort to a different analysis approach. Consider the following proposition, whose proof can be found in Appendix A-C:

Proposition 3

For the Bingham SBF scheme, the rate of user can be expressed as

(21)

Here, is given by

(22)

where is a random vector with independent and identical (i.i.d.) unit-mean exponentially distributed components, contains the positive eigenvalues of , and contains the eigenvalues of .

As it turns out, one can derive an explicit expression for .

Proposition 4

Let be as in (22). Organize as

where are such that , and for all . Then, we have

where , ,

The proof of Proposition 4 can be found in Section 2 of the companion technical report [19]. The idea behind the proof of Proposition 3 is somewhat similar to that in [37, Theorem 1], where the authors there dealt with a different scenario (unicast). While Proposition 4 gives an explicit expression for (22), which in turn provides a way of computing the Bingham SBF achievable rate efficiently (in contrast with Monte Carlo simulations), it is too complicated for the purpose of extracting insights. This difficulty motivates us to turn to the stochastic majorization technique for Bingham SBF rate gap characterization:

Fact 4

Consider , where is a random vector with arbitrary i.i.d. components.

  • ([38, Theorem 2.15, Example 2.2]) For any with ,

  • ([39]) Suppose that every follows a unit-mean exponential distribution. Then, we have

Applying Fact 4 to (21), we obtain

where the first inequality follows from Fact 4(a) and the observation that (this is implied by the structure of (MC)), and the second inequality is due to Fact 4(b). The derivations above show that user-’s Bingham rate is lower bounded by , which lead us to a neat conclusion:

Theorem 3

The achievable rate gap of the Bingham SBF scheme satisfies

Surprisingly, the worst-case rate gap of the Bingham SBF scheme as proven above is exactly the same as that of the elliptic SBF scheme (cf. Theorem 2). It follows that the worst-case rate gap of the Bingham SBF scheme is also no worse than that of the Gaussian SBF scheme.

Iii-F Summary of the SBF Schemes

We now summarize the characteristics of our proposed SBF schemes in Table II. It can be seen that all three schemes exhibit a multicast achievable rate gap that is no worse than  bits/s/Hz, irrespective of any factors such as the number of users. In fact, the elliptic and Bingham SBF schemes can perform better than  bits/s/Hz, depending on the transmit covariance rank ; see Table I. In terms of the instantaneous beamformer power spread effects, the Gaussian SBF scheme is, by nature, the worst. The elliptic SBF scheme is better than the Gaussian SBF scheme, limiting the instantaneous beamformer power to within . The Bingham SBF scheme has zero instantaneous beamformer power spread. On the other hand, the Gaussian and elliptic SBF schemes achieve the multicast capacity-optimal transmit covariance , while the Bingham SBF scheme may not.

scheme generation has MC-opt. covariance ? instantaneous beamformer power spread worst-case rate gap upper bound
Gaussian yes large bits/s/Hz
elliptic where ; is a square root factor of ; yes better than Gaussian; with probability optimal when
Bingham where . no zero; same as elliptic
TABLE II: Summary of the SBF schemes

Iv Multicast Beamformed Alamouti Space-Time Coding

In this section, we describe our second physical-layer multicasting strategy—transmit beamformed Alamouti space-time coding. Compared to SBF, which uses time randomizations to enable rank- transmit covariance structures, the beamformed Alamouti strategy adopts a rank-two transmit covariance structure in a fixed or deterministic way. This will motivate a rank-two generalization of SDR.

Iv-a System Model

We describe the system model for (fixed) beamformed Alamouti space-time coding. Like the beamforming case, we aim at transmitting a stream of unit-power data symbols, denoted by . The data symbol stream is parsed into blocks via . In block , we transmit by a transmit beamformed Alamouti space-time code:

(23)

Here, is a transmit beamforming matrix and is the Alamouti space-time block code, i.e.,

From the basic model in (1), we have

(24)

where . Using a key property introduced by the special structure of the Alamouti code (see, e.g., [40]), Eq. (24) can be turned into an equivalent SISO model, where each symbol can be independently detected and user ’s SNR of the received symbols can be characterized by Hence, for the beamformed Alamouti strategy, we can formulate the following achievable rate problem:

(25)

where

Note that is assumed to be ideally channel-coded (just like in the beamforming case), and the constraint is equivalent to the total power constraint . In the next subsection, we will study how SDR can be employed to deal with the above achievable rate optimization problem.

Iv-B A Generalization of SDR for the Fixed Beamformed Alamouti Strategy

Our strategy for tackling (25) expands on the ideas used to reformulate the beamforming multicast achievable rate problem (4) into a rank-constrained SDP; see Section II. To begin, observe that

Hence, Problem (25) can be equivalently formulated as

(26)

At this point, it is worth noting that the achievable rate problem for the beamforming scheme (5) is a restriction of that for the beamformed Alamouti scheme (26). This suggests that our proposed design should have a performance no worse than that of the beamforming scheme. In fact, as we shall see shortly, the worst-case performance gain can be quantified.

Now, upon removing the nonconvex rank constraint in (26), we obtain exactly the same convex relaxation as that of the fixed beamforming problem discussed in Section II, namely, Problem (SDR). Let denote an optimal solution to (SDR). Since may not satisfy , we need to develop a procedure that can generate from a feasible solution to (26). Moreover, since the generated solution need not be optimal for (26) in general, we are interested in quantifying the approximation quality of such a solution. To tackle these problems, we employ the SDR rank reduction theory (see, e.g., [29, 41]). Let us begin with the following proposition:

Proposition 5

Suppose that . Then, there is a polynomial-time procedure that can generate from an optimal solution to the fixed beamformed Alamouti problem (25).

Proposition 5 can be established using [5, Claim 2] and [29, Theorem 5.1] (see also [27] for an exposition of the latter). It implies that the fixed beamformed Alamouti problem (25) can be optimally solved by SDR for instances with users or less. By contrast, beamforming can guarantee the same result only for users or less; see Fact 1(a). Moreover, by the equivalence of (SDR) and (MC), we arrive at the important conclusion that fixed beamformed Alamouti space-time coding is a multicast capacity-optimal transmit strategy when there are no more than users.

For the case where , it may not be possible to generate an optimal solution to (25) from in polynomial time, as Problem (25) is NP-hard. However, we can still generate a feasible solution to (25) using the following Gaussian randomization procedure:

1:  Input: an optimal solution to (SDR), number of randomizations
2:  for  to  do
3:     generate two independent random vectors and define ;
4:     let
5:  end for
6:  let (see (6) for the definition of )
7:  Output:
Algorithm 1 Gaussian Randomization Procedure for (25)

Algorithm 1 is a generalization of the Gaussian randomization procedure used for the SDR-based beamforming scheme [5]. Regarding its worst-case approximation performance, we have the following result, whose proof can be found in Appendix A-D:

Theorem 4

With probability at least , the solution returned by Algorithm 1 satisfies

Theorem 4 has two important implications. First, with our fixed beamformed Alamouti scheme, the provable gap between the worst-user SNR and the best achievable worst-user SNR scales only on the order of . This is substantially better than the fixed beamforming case, where the provable gap scales on the order of (cf. Fact 1(b)). Secondly, for , the achievable rate gap of the SDR-based fixed beamformed Alamouti scheme relative to the multicast capacity is bounded above by

which for large is approximately equal to . This is strictly better than that of the SDR-based fixed beamforming scheme for all (cf. (7) in Fact 1(b)).

Before we proceed, several remarks are in order.

Remark 1: The techniques we developed for proving Theorem 4 can be used to obtain approximation bounds for a fairly general class of rank constrained SDPs. As such, they generalize the techniques in [6], which only apply to a certain class of rank-one constrained SDPs.

Remark 2: The approximation bound stated in Theorem 4 is only a worst-case bound. In practice, the solution returned by Algorithm 1 can have a much better performance. This will be confirmed by our simulation results; see Section VI.

Remark 3: In view of the development of the fixed beamformed Alamouti scheme, it is natural to ask whether the techniques can be extended to deliver a “rank-” beamforming scheme rather than just a “rank-” scheme as in the Alamouti case. Indeed, it is possible to extend the SDR techniques above to general -dimensional orthogonal space-time bock codes (OSTBCs). However, full rate OSTBCs do not exist for  [20], and the rate deduction (for ) can significantly outweigh the gain obtained from “rank-” beamforming. For example, consider a fixed beamformed OSTBC for dimension . Since the maximal-rate OSTBC for is  [20], the achievable rate should be formulated as

where , with . Our SDR analysis can be extended to show that the solution