Simple Bounds for the Symmetric Capacity of the Rayleigh Fading Multiple Access Channel

TL;DR

This paper derives tight bounds on the probability that a simple, feedback-based communication protocol attains the sum capacity in a Rayleigh fading MAC, providing insights into the diversity-multiplexing tradeoff and proposing an effective coding scheme.

Contribution

It introduces tight bounds on the distribution of the symmetric rate in Rayleigh fading MACs and proposes a practical integer-forcing and space-time precoding scheme.

Findings

Bounds characterize the probability of achieving sum capacity.

Protocol's performance closely relates to the diversity-multiplexing tradeoff.

Proposed coding scheme is effective for the considered scenario.

Abstract

Communication over the i.i.d. Rayleigh slow-fading MAC is considered, where all terminals are equipped with a single antenna. Further, a communication protocol is considered where all users transmit at (just below) the symmetric capacity (per user) of the channel, a rate which is fed back (dictated) to the users by the base station. Tight bounds are established on the distribution of the rate attained by the protocol. In particular, these bounds characterize the probability that the dominant face of the MAC capacity region contains a symmetric rate point, i.e., that the considered protocol strictly attains the sum capacity of the channel. The analysis provides a non-asymptotic counterpart to the diversity-multiplexing tradeoff of the multiple access channel. Finally, a practical scheme based on integer-forcing and space-time precoding is shown to be an effective coding architecture for this communication scenario.

Equations

$$\mathbf{y}=\sum _{i=1}^N{\mathbf{H}}_i{\mathbf{x}}_i+\mathbf{n}$$

$$\sum _{i\in \mathcal{S}}{R}_i$$

$$\triangleq \log \det \left({\mathbf{I}}_{N_r}+\sum _{i\in \mathcal{S}}{\mathbf{H}}_i{\mathbf{H}}_i^H\right),$$

$$C$$

$$=\log \det \left({\mathbf{I}}_{N_r}+\sum _{i=1}^N{\mathbf{H}}_i{\mathbf{H}}_i^H\right).$$

$$C_{\Sigma -\mathrm{sym}}$$

$$\mathrm{Pr}(C_{\Sigma -\mathrm{sym}}<R|C).$$

$$d_{\mathrm{sym}}^{\ast }(r)=\{\begin{array}{l}{d}_{N_t,N_r}^{\ast }(r),r\le \min (N_t,\frac{N_r}{N+1})\\ d_{N\cdot N_t,N_r}^{\ast }(N\cdot r),r\ge \min (N_t,\frac{N_r}{N+1})\end{array}$$

$$y=\sum _{i=1}^N{h}_i{x}_i+n$$

$$C_{\Sigma -\mathrm{sym}}=\underset{\mathcal{S}\subseteq \{1,2,\dots ,N\}}{\min }\frac{N}{|\mathcal{S}|}\log \left(1+\sum _{i\in \mathcal{S}}|h_i{|}^2\right).$$

$$\mathrm{Pr}(C_{\Sigma -\mathrm{sym}}<R|C)=2\cdot \frac{2^{R/2}-1}{2^C-1}.$$

$$C_{\Sigma -\mathrm{sym}}=\min \left\{2C(\{1\}),2C(\{2\}),C\right\}.$$

$$2C(\{1\})$$

$$=2\log \left(1+|{\mathbf{U}}_{1,1}{|}^2(2^C-1)\right).$$

$$\mathrm{Pr}(2C(\{1\})<R|C)$$

$$=\mathrm{Pr}\left(|{\mathbf{U}}_{1,1}{|}^2\in \left[0,\frac{2^{R/2}-1}{2^C-1}\right)\right)$$

$$\mathrm{Pr}(2C(\{1\})<R|C)=\frac{2^{R/2}-1}{2^C-1}.$$

$$|{\mathbf{U}}_{1,1}{|}^2+|{\mathbf{U}}_{1,2}{|}^2=1.$$

$$\mathrm{Pr}(2C(\{2\})<R|C)$$

$$=\mathrm{Pr}\left(1-|{\mathbf{U}}_{1,1}{|}^2<\frac{2^{R/2}-1}{2^C-1}\right)$$

$$=\mathrm{Pr}\left(|{\mathbf{U}}_{1,1}{|}^2\in \left(1-\frac{2^{R/2}-1}{2^C-1},1\right]\right)$$

$$\mathrm{Pr}(C_{\Sigma -\mathrm{sym}}<R|C)=2\cdot \frac{2^{R/2}-1}{2^C-1}.$$

$$\mathrm{Pr}\left(C_{\Sigma -\mathrm{sym}}=C|C\right)$$

$$=1-2\cdot \frac{2^{C/2}-1}{2^C-1}.$$

$$\displaystyle P_{\rm out}(k,R|C)\triangleq{\rm Pr}\left(\frac{N}{k}C(\mathcal{S})<R\Big{|}C\right)$$

$$P_{\mathrm{out}}(k,R|C)$$

$$\mathcal{B}(x;a,b)={\int }_0^x{u}^{a-1}(1-u{)}^{b-1}du$$

$$C(\{1,2,\dots ,k\})$$

$$=\log \left(1+\left(2^C-1\right)\sum _{i=1}^k|U_{1,i}{|}^2\right).$$

$$\displaystyle{\rm Pr}\left(\frac{N}{k}C(\mathcal{S})<R\Big{|}C\right)$$

$$=\mathrm{Pr}\left(X<\frac{2^{R\frac{k}{N}}-1}{2^C-1}\right).$$

$$\frac{N}{k}C(\{1,2,\dots ,k\})=\frac{N}{k}\log \left(1+(2^C-1)\mathbf{U}(k{)}_1\mathbf{U}(k{)}_1^H\right)$$

$$\displaystyle{\rm Pr}\left(\frac{N}{k}C(\mathcal{S})<R\Big{|}C\right)$$

$$=\frac{\mathcal{B}\left(\frac{2^{Rk/N}-1}{2^C-1};k,N-k\right)}{\mathcal{B}(1;k,N-k)},$$

$$\underset{k}{\max }{P}_{\mathrm{out}}(k,R|C)$$

Full text

I Introduction

In this paper we consider communication over the slow (block) fading i.i.d. Rayleigh multiple access channel (MAC). For a given realization of the channel gains, the channel reduces to the classical Gaussian MAC, the capacity region of which is well known, see e.g., [1].

A basic criterion for analyzing the performance of different access methods is the gap from the sum-capacity (the maximal total rate that can be achieved by all the users). We note however, that in many cases, the rate distribution between different users is also of interest and in many applications, fairness is sought and a scheme which provides (maximal) equal rate to all users is desired.

The maximal rate that can be achieved in a system where all users have equal rate is denoted as the symmetric capacity. In case the symmetric and the sum capacity coincide (alternatively the case where the dominant face of the MAC capacity region contains a symmetric-rate point), fairness can be achieved without sacrificing performance. As this is a very desirable working point, it is of interest to investigate what is the probability of this being the case.

Some intuition to that question can be inferred from the diversity multiplexing tradeoff (DMT) of the i.i.d. Rayleigh fading MAC which provides an asymptotic analysis of the symmetric capacity [2]. As we show next, at high values of signal-to-noise ratio (SNR), the symmetric capacity approaches the sum capacity with high probability.

In this paper we characterize the behaviour of the symmetric capacity for finite SNR. From this characterization, the probability of getting fairness for “free” for all SNRs can be easily deduced.

Another motivation for studying the symmetric capacity comes from another design criterion which is the amount of coordination needed by the protocol. High level of coordination results in high throughput loss when finite block length coding is taken into account or increased latency. As the number of users that are simultaneously transmitting increases the amount of coordination increases and thus its impact increases. This is a major issue for new applications being developed for next generation wireless networks (see, e.g., [3, 4]) where supporting high number of users is required along with guaranteeing low latency.

In theory, transmission at rates approaching the symmetric capacity requires minimal coordination; namely a single parameter, the common code rate all users should use. Nonetheless, when is comes to practical schemes that are able to approach this operating point, hitherto practical applicable transmission schemes have relied on a much higher degree of coordination.

Specifically, both time sharing of the points achievable via successive interference cancellation as well as rate splitting are asymmetric between the users and thus require coordination. Furthermore, orthogonal multiple access techniques (e.g., time or frequency division multiple access) also require coordination to achieve its maximal achievable symmetric-rate point which further falls short of the symmetric capacity (unless the latter coincides with the sum capacity).

The contribution of the present work is two-fold:

Establishing bounds on the gap between the symmetric capacity and sum capacity for the Rayleigh-fading MAC. 2. 2.

Proposing a practical scheme that is able to approach the symmetric capacity with the minimal possible degree of coordination. i.e., specification of the common per-user transmission rate.

These two points have immediate practical implications. Specifically, we are able to characterize the performance of a protocol where all users transmit at a rate just below the symmetric capacity (per user) of the channel. The underlying assumption is that the latter rate is dictated to the users by the base station, utilizing a minimal amount of feedback (which does not scale with the number of users).

Our first result is an exact characterization of the performance of the suggested communication protocol, when assuming an optimal (maximum-likelihood) receiver, for the two-user case where all nodes are equipped with a single antenna. We then extend the analysis to the scenario of an $N$-user Rayleigh-fading MAC where all nodes are equipped with a single antenna. For this scenario, we provide inner and outer bounds on performance. We then further extend the analysis to a general symmetric i.i.d. Rayleigh multiple-input multiple-output (MIMO) MAC.

The derived tight characterization of the distribution of the symmetric capacity can serve as a basis for deriving other figures of merit (such as the ergodic capacity or the outage probability for any target rate). It is worthwhile noting that although the problem studied is by now quite classical, and the derivation relies only on elementary techniques, the obtained results–given in the from of simple closed-form expressions–appear to have eluded previous studies.

Since the complexity of maximum-likelihood (ML) receiver is prohibitive, we also consider the performance attained by a practical integer-forcing (IF) receiver, demonstrating that it performs quite well in the considered scenario. Interestingly, we observe that in order to approach the symmetric capacity with an IF receiver, another lesson from the MIMO-MAC DMT analysis should be followed. Specifically, it is necessary to apply “space-time” precoding at the transmitters (see, e.g., [5]).

The rest of this paper is organized as follows. Section II provides the problem formulation. Section III recounts the DMT of the i.i.d. Rayleigh-fading MIMO-MAC. As mentioned above, this asymptotic analysis provides intuition and tools that are subsequently refined to a full characterization of the considered communication protocol. In Section IV, the performance of the protocol is analyzed for the case where all terminals are equipped with a single antenna. In Section V, bounds are derived for the general case of $N$ users, where each user has $N_{t}$ antennas and the receiver is equipped with $N_{r}$ antennas. In Section VI, it is demonstrated that an IF receiver combined with (structured or random) space-time precoding yields performance that is close to the established theoretical limits of the proposed communication protocol. Finally, Section VII concludes the paper.

II Problem Formulation and Preliminaries

To simplify derivations, we will assume throughout that all users are equipped with the same number of transmit antennas. The results can easily be extended to a more general scenario.

Accordingly, we consider a MIMO-MAC with $N$ users, where each transmitter has $N_{t}$ antennas and the receiver is equipped with $N_{r}$ antennas. The channel model can be expressed as

[TABLE]

where $\mathbf{H}_{i}$ is the channel matrix between user $i$ and the receiver. We assume an i.i.d. Rayleigh-fading model so that \mathbf{H}_{i}\sim\mathcal{CN}(0,\text{\mathsf{SNR}}\cdot\mathbf{I}_{N_{r}}) and $\mathbf{n}\sim\mathcal{CN}(0,\mathbf{I}_{N_{r}})$, where there is no statistical dependence over space nor time.111The time index remains implicit since it plays no role in the analysis. Of course, coding over large blocklength is needed to approach the information-theoretic limits. We assume that the transmitted data $\mathbf{x}_{i}\in\mathbb{C}^{N_{t}\times 1}$ is isotropic (“white”) for each user and that all users are subject to the same power constraint $P$ where the SNR is absorbed in the channel gains.

Define a subset of users by $\mathcal{S}\subseteq\{1,2,\ldots,N\}$. Then, the capacity region of the channel is given by (see, e.g., [1]) all rate vectors $(R_{1},\ldots,R_{N})$ satisfying

[TABLE]

for all subsets $\mathcal{S}$ in the power set of $\{1,2,\ldots,N\}$. The sum capacity is given by

[TABLE]

If we impose the constraint that all users transmit at the same rate, then the maximal achievable rate is given by substituting $R_{i}=C_{\Sigma-\rm sym}/N$ in (2), from which it follows that the symmetric capacity $C_{\Sigma-\rm sym}$ is dictated by the bottleneck:

[TABLE]

We study the conditional “cumulative distribution function”:222We use quotation marks since we impose strict inequality in $C_{\Sigma-\rm sym}<R$.

[TABLE]

The latter quantity provides a full statistical characterization of the performance of the transmission protocol considered. Another interpretation of (6) is as a conditional outage probability in an open-loop scenario; that is, in a scenario where all users (when they are active) transmit at a common target rate $R$. For a given number of active users $N$, the outage probability is then given by $\mathbb{E}[{\rm Pr}(C_{\Sigma-\rm sym}<N\cdot R|C)]$ where the expectation is over $C$ and is computed w.r.t. an i.i.d. Rayleigh distribution.

III Lessons from the DMT

Some insight into the performance of the considered protocol may be obtained by considering the DMT of the symmetric Rayleigh-fading MIMO-MAC channel, which was studied in [2]. As a special case, the scenario where all users transmit at the same rate was considered in detail, for which a simple expression for the DMT was derived.

Specifically, the DMT of the Rayleigh MIMO-MAC with $N$ users, where each transmitter has $N_{t}$ antennas and the receiver has $N_{r}$ antennas, and where all users transmit at the same rate, is given by

[TABLE]

where $d^{*}_{N_{t},N_{r}}(r)$ is the DMT of the i.i.d. single-user Rayleigh-fading MIMO channel with $N_{t}$ transmit antennas and $N_{r}$ receive antennas (provided that the block length $l\geq N_{t}+N_{r}+1$); see, e.g., [6]). The function $d^{*}_{N_{t},N_{r}}(r)$ is a piecewise linear curve such that $d^{*}_{N_{t},N_{r}}(r)=(N_{t}-r)(N_{r}-r)$ for every integer $r\leq\min(N_{t},N_{r})$.

Although the DMT analysis is asymptotic in nature, instructive lessons may nonetheless be drawn from it. First, it is clear that in the limit of high SNR, the ratio of the symmetric capacity and sum capacity approaches one in probability (since the DMT is strictly positive for any multiplexing gain smaller than the maximal attainable degrees of freedom).

More importantly, the analysis of the typical error events in the Rayleigh-fading MAC (with equal-rate transmission) reveals that with high probability, outage occurs either as if all users were considered as a single one (“antenna pooling”) or as a result of a single-user constraint constituting the bottleneck [2]. These two regimes are reflected in the two cases appearing in (7).

Further, it can be easily shown that for a scalar MAC ($N_{r}=N_{t}=1$) with two or more users, the antenna polling bottleneck amounts to the probability that the sum-capacity is below the target rate. As for a (symmetric) transmission protocol where the target rate is set to just below the sum capacity, the latter type of outage event cannot occur, it follows that the diversity gain at the maximal multiplexing gain (the maximal attainable degrees of freedom) is strictly positive. This in turn implies that the ratio between the symmetric capacity and the sum capacity will approach $1$ quite fast as the SNR grows.

In fact, in the case of two users, the DMT of which is depicted in Figure 1, we show that perfect fairness may be gained “for free” with high probability. This is, the probability that the symmetric capacity is equal to the sum capacity approaches $1$ rather fast as a function of the SNR; hence, validating the intuition gained from the DMT.

IV I.I.D. Rayleigh-fading MAC with Single-Antenna Terminals

When all terminals are equipped with a single antenna, the Rayleigh-fading MAC is described by

[TABLE]

and the symmetric capacity is given by

[TABLE]

IV-A *Two-user i.i.d. single antenna Rayleigh-fading MAC *

We begin by analyzing the simplest case of a two-user scalar MAC, for which we obtain an exact of characterization of (6).

Theorem 1**.**

For a two-user i.i.d. Rayleigh-fading MAC with sum capacity $C$, for any rate $R\leq C$,

[TABLE]

Proof.

Given $C$, $\mathbf{h}\triangleq(h_{1},h_{2})$ is uniformly distributed over a two-dimensional complex sphere of radius $\sqrt{2^{C}-1}$. Hence, $\mathbf{h}/\|\mathbf{h}\|$ can be viewed as the first row of a random (Haar) unitary matrix $\mathbf{U}$.

By (9), we obtain

[TABLE]

We start by analyzing $2C(\{1\})$, which is given by

[TABLE]

It follows that

[TABLE]

Since (see, e.g., [7]) for a $2\times 2$ matrix drawn uniformly with respect to the Haar measure, we have $|\mathbf{U}_{1,1}|^{2}\sim\rm{Unif}([0,1])$, it follows that

[TABLE]

Now, since $\mathbf{U}_{1,1}$ and $\mathbf{U}_{1,2}$ are the elements of a row in a unitary matrix, we have

[TABLE]

Hence,

[TABLE]

Since for any rate $R\leq C$, the intervals appearing in (13) and (16) are disjoint and of the same length, it follows that

[TABLE]

∎

We note that the probability in (17) is strictly smaller than $1$ at $R=C$. Thus, the probability that the symmetric capacity coincides with the sum capacity is strictly positive.

Figure 2 depicts the capacity region for three different channel realizations for which the sum capacity equals $2$. The probability that the symmetric capacity coincides with the sum capacity amounts to the probability that the symmetric rate line passes through the dominant face of the capacity region and is given by

[TABLE]

As an example, for $C=2$, this probability is $1/3$.

Figure 3 depicts the probability density function of the symmetric capacity of a two-user i.i.d. Rayleigh-fading MAC given that the sum capacity is $C=2$. The probability in (18) manifests itself as a delta function at the sum capacity.

IV-B Extension to the $N$-user i.i.d. scalar Rayleigh-fading MAC

Theorem 1 may be extended to the case of $N>2$ users. However, rather than obtaining an exact characterization of the distribution of the symmetric capacity, we will now be content with deriving lower and upper bounds for it.

Let us define

[TABLE]

We begin with the following lemma which is the key technical step from which Theorem 2 follows.

Lemma 1**.**

For an $N$-user i.i.d. Rayleigh-fading MAC with sum capacity $C$, and for any subset of users $\mathcal{S}\subseteq\{1,2,\ldots,N\}$ with cardinality $k$, we have

[TABLE]

where $0\leq R\leq C$ and

[TABLE]

is the incomplete beta function.

Proof.

Similar to the case of two users, $\mathbf{h}\triangleq(h_{1},\ldots,h_{N})$ is uniformly distributed over an $N$-dimensional complex sphere of radius $\sqrt{2^{C}-1}$ and hence $\mathbf{h}/\|\mathbf{h}\|$ may be viewed as the first row of a unitary matrix $\mathbf{U}$ drawn at random according to the Haar measure.

By symmetry, for any set $\mathcal{S}$ with cardinally $k$, the distribution of $C(\mathcal{S})$ is equal to that of

[TABLE]

Denoting the partial sum of $k$ entries as $\displaystyle{X=\sum_{i=1}^{k}|\mathbf{U}_{1,i}|^{2}}$, we therefore have

[TABLE]

We note that the vector $(|\mathbf{U}_{1,1}|^{2},\ldots,|\mathbf{U}_{1,N}|^{2})$ follows the Dirichlet distribution and a partial sum of its entries has a Jacobi (also referred to as MANOVA) distribution. To see this, we note that (20) can be written as

[TABLE]

where $\mathbf{U}(k)_{1}$ is a vector which contains the first $k$ elements of the first row of $\mathbf{U}$. Noting that since $\mathbf{U}(k)_{1}$ is a submatrix of a unitary matrix, its singular values follow (see, e.g., [8]) the Jacobi distribution, and more specifically, $X$ has Jacobi distribution with rank 1. We thus obtain

[TABLE]

where $\mathcal{B}(x;a,b)$ is the incomplete beta function defined.

∎

Theorem 2**.**

For an $N$-user scalar i.i.d. Rayleigh-fading MAC, we have

[TABLE]

where $P_{\rm out}(k,R|C)$ is defined in (19) and given in Lemma 1.

Proof.

To establish the left hand side of the theorem, first note that $C_{\Sigma-\rm sym}\leq C(\mathcal{S})$ for any subset $\mathcal{S}$ and hence

[TABLE]

It follows that

[TABLE]

The right hand side follows by the union bound. ∎

Figures 4 and  5 illustrate the theorem for the case of four users (where the markers indicate the height of the delta functions). As can be seen from Figure 4, already at not very high values of capacity, the single-user constraints already constitute the bottleneck. We further observe from Figure 5 that the union bound is quite tight.

V Upper Bound on the Outage Probability for the symmetric $N$-user Rayleigh-fading MIMO MAC

We now consider the symmetric MIMO-MAC scenario where each of the $N$ users is equipped with $N_{t}$ antennas and the receiver is equipped with $N_{r}$ antennas. In this case, the channel as described by (1) can be rewritten as

[TABLE]

where

[TABLE]

and

[TABLE]

Therefore, the symmetric capacity (5) can be expressed as

[TABLE]

where $\mathcal{H}_{\mathcal{S}}$ is defined as the submatrix of $\mathcal{H}_{\mathcal{S}}$ generated from taking only the channel matrices $\mathbf{H}_{i}$ corresponding to user indices $i$ such that $i\in\mathcal{S}$.

In order to leverage the bounds derived for the scalar MAC scenario, we may use the simple bound (see, e.g. [9], Equation (5))

[TABLE]

where $\|\mathbf{A}\|_{F}$ denotes the Frobenius norm of a matrix $\mathbf{A}$. Denote the “Frobenius-norm mutual information” by

[TABLE]

and

[TABLE]

It follows that for any channel realization $C\geq\tilde{C}$ and similarly, for any subset of users $\mathcal{S}$, we have $C(\mathcal{S})\geq\tilde{C}(\mathcal{S})$.

Considering now the performance of a protocol where all users transmit at a rate that is just below $\tilde{C}/N$, the counterparts of Lemma 1 and Theorem 2 are the following.

Lemma 2**.**

For a symmetric $N$-user $N_{r}\times N_{t}$ Rayleigh-fading MIMO-MAC with Frobenius-norm mutual information $\tilde{C}$, for any subset of users $\mathcal{S}\subseteq\{1,2,\ldots,N\}$ with cardinality $k$, we have

[TABLE]

where $\mathcal{B}(x;a,b)$ is the incomplete beta function, $\tilde{C}(\mathcal{S})$ is defined in (29) and $\tilde{C}$ is defined in (30).

Proof.

Denoting by $\mathbf{h}_{\rm vec}$ the vectorization of $\mathcal{H}$, we have

[TABLE]

As noted in the previous section, conditioned on $\tilde{C}$, $\mathbf{h}_{\rm vec}\triangleq(h_{1},\ldots,h_{N_{r}N_{t}N})$ is uniformly distributed over an $N_{r}N_{t}N$-dimensional complex sphere of radius $\sqrt{2^{\tilde{C}}-1}$ and hence $\mathbf{h}_{\rm vec}/\|\mathbf{h}_{\rm vec}\|$ may be viewed as the first row of a unitary matrix $\mathbf{U}$ drawn at random according to the Haar measure.

By symmetry, for any set $\mathcal{S}$ with cardinally $k$, the distribution of $\tilde{C}(\mathcal{S})$ is equal to that of

[TABLE]

Denoting the partial sum of $k$ entries as $\displaystyle{X=\sum_{i=1}^{N_{r}N_{t}k}|U_{1,i}|^{2}}$, we therefore have

[TABLE]

the rest of the proof follows the footsteps of the proof of Lemma 1. ∎

Theorem 3**.**

For a symmetric $N$-user $N_{r}\times N_{t}$ Rayleigh-fading MIMO-MAC, we have

[TABLE]

Proof.

By (28) and (29), for every $\mathcal{S}$ and channel realization, it holds that

[TABLE]

Therefore,

[TABLE]

and similar to Theorem 1, applying the union bound, we get

[TABLE]

∎

Figure 6 depicts a comparison between the empirical outage probability and the upper bound provided by Theorem 3 for the case of two users, each equipped with $2$ antennas and a receiver equipped with $3$ antennas, where the target rate is set to $3$ bits. The outage probability was evaluated empirically by Monte-Carlo simulation. To calculate the bound, the Frobenius norm of each channel matrix drawn was calculated.

It can seen that at high $\mathsf{SNR}$, the slope of the bound is similar to that of the empirical results. Recalling the MIMO-MAC DMT, we note that since the target rate is fixed (is not a function of the $\mathsf{SNR}$), the slope at high $\mathsf{SNR}$ is in fact the maximal diversity offered in this configuration, i.e., the diversity corresponding to zero multiplexing gain. Recalling the DMT of the symmetric capacity (7), the latter is $N\cdot N_{r}\cdot N_{t}$ which matches the slope given by Theorem 2. On the other hand, relying on the Frobenius norm results in a loose bound at low values of SNR (high outage probabilities).

We may obtain a tighter bound for low SNR values, for the special case of $N=2$ users, where each is equipped with a single antenna and the receiver is equipped with $N_{r}\geq 2$ antennas. Specifically, in [10] a different upper bound for the outage probability was derived in the context of a randomly precoded compound single-user $N_{r}\times 2$ MIMO channel. It is easy to verify that the derived bound carries over to the setting considered in the present paper, when rewritten as follows:333The main step is to recall that the SVD decomposition of an i.i.d. circularly-symmetric complex Gaussian matrix yields left and right singular vector matrices that are uniformly (Haar) distributed, as is the precoding matrix considered in the analysis of [10].

Theorem 4** (Theorem 2 in [10]).**

For a two-user i.i.d. Rayleigh-fading MAC where each user is equipped with a single antenna and the receiver is equipped with $N_{r}$ antennas,

[TABLE]

The main advantage Theorem 4 Figure with respect to Theorem 3 is that the conditioning is on $C$, the true sum capacity of the channel, rather than its Frobenius-norm counterpart.

Figure 7 depicts the empirical outage probability, the upper bound of Theorem 3 and the bound of Theorem 4 for the case of a symmetric two-user $6\times 1$ Rayleigh-fading MAC, where the target rate is set to $3$ bits. It can be seen that at low SNR, Theorem 4 provides a tighter bound than Theorem 3 but it becomes loose rapidly as it does not capture the maximal diversity offered by the system.

VI Practical Realization of the Communication Protocol via Precoded Integer Forcing

In this section we empirically demonstrate the effectiveness of the integer-forcing (IF) receiver when used in conjunction with unitary space-time precoding as a practical transmission scheme in the context of the considered communication protocol. Due to space limitations, we refer the reader to [11] for a description of the integer forcing framework and its implementation.

When it comes to fading channels, it has been shown in [11] that the IF receiver achieves the DMT over i.i.d. Rayleigh-fading channels where the number of receive antennas is greater or equal to the number of transmit antennas.

We observe that this does not hold in the general case; in particular, IF does not achieve the DMT for the case of a MAC where all terminals are equipped with a single antenna. Specifically, Figure 8 depicts (in logarithmic scale) the empirical outage probability of the IF receiver and the exact outage probability for optimal communication (Gaussian codebooks and ML decoding), as given by Theorem 1, for the two-user i.i.d. Rayleigh-fading MAC. The symmetric rate achieved by a given scheme is denoted by $R_{\rm scheme}$.

It is evident that the slopes are different. This raises the question of whether IF is inherently ill-suited for the MAC channel. A negative answer to this question may be inferred by recalling some further lessons from the DMT analysis of the MAC.

While the optimal DMT for the i.i.d. Rayleigh-fading MAC was derived in [2] using Gaussian codebooks of sufficient length, it was subsequently shown that the DMT of the MAC can be achieved using structured codebooks by combining uncoded QAM constellations with space-time unitary precoding (and ML decoding). Specifically, such a MAC-DMT achieving construction is given in [12]. This raises the possibility that the sub-optimality of the IF receiver observed in Figure 8 may at least in part be remedied by applying unitary space-time precoding at each of the transmitters. We note that each transmitter applies precoding only to its own data streams so the distributed nature of the problem is not violated.

Following this approach, we have implemented the IF receiver with unitary space-time precoding applied at each transmitter. We have employed random (Haar) precoding (with independent matrices drawn for the different users) over two ($T=2$) time instances as well as deterministic precoding using the matrices proposed in [13].444When using an ML receiver, this space-time code is known to achieve the DMT for multiplexing rates $r\leq\frac{1}{5}$. As detailed in [12], whether this code achieves the optimal MAC-DMT also when $r>\frac{1}{5}$ remains an open question.

These matrices can be expressed as

[TABLE]

where

[TABLE]

We also replot Figure 8 in terms of PDF (rather than CDF) as Figure 9, but without random Haar space-time precoding (so as to avoid “clutter”). As can be seen, the precoding matrices in (40) improve the outage probability for most target rates.

We further note that in addition to standard IF, we also implemented a variant that incorporates successive interference cancellation, referred to as IF-SIC [14]. As can be seen, IF-SIC results in a significant improvement for all precoding schemes used.

In Figure 10 we study the average symmetric rate achieved by different schemes w.r.t. a two-user i.i.d. Rayleigh-fading channel when we condition on the sum capacity of the channel. We plot the fraction of the sum capacity attained by the various schemes. We first observe that IF-SIC combined with space-time precoded linear codes achieves a large fraction of the symmetric capacity. Further, as can be seen, the fraction of the sum capacity achieved by all the different schemes considered approaches one as the sum capacity grows.

Finally, in Figure 11, we plot the fraction of the sum capacity that is achieved, allowing for a fixed outage probability, by the proposed protocol where we consider both the ideal performance achieved as captured by the symmetric capacity and the rate achieved using IF in conjunction with SIC. As can be observed, the performance of IF-SIC for small outage probabilities is very close to the theoretical limits of the considered transmission protocol. We note, however, that as the number of users increases (and also, as the sum-capacity increases), the problem of finding a “good” integer matrix (as required in IF equalization) becomes computationally difficult and may result in compromised rates when using practical sub-optimal algorithms such as the LLL algorithm to find candidate integer matrices.

VII Conclusions

We analyzed the performance of a simple communication protocol for transmission over the scalar Rayleigh-fading MAC, where all users transmit at just below the symmetric capacity (normalized per user) of the channel. Tight bounds were established on the distribution of the achievable rate of the protocol. The derived bounds may be viewed as a significant tightening of the diversity multiplexing tradeoff analysis of the channel. It was further demonstrated that integer-forcing equalization in conjunction with “space-time” precoding (over the time axis only) offers a practical means to approach the theoretical limits of the proposed protocol.