A Simple Receive Diversity Technique for Distributed Beamforming

TL;DR

This paper introduces a simple space-time diversity transformation technique for distributed beamforming with a two-antenna destination, enhancing diversity order without needing channel state information at the receiver.

Contribution

It proposes a novel front-end operation at the destination that extends distributed beamforming to a two-antenna receiver, doubling diversity order without additional channel information.

Findings

Doubles the diversity order compared to single-antenna receivers.

Retains array gain while eliminating the need for receiver channel state information.

Applicable in ultra-reliable, low-latency communication scenarios.

Abstract

A simple method is proposed for use in a scenario involving a single-antenna source node communicating with a destination node that is equipped with two antennas via multiple single-antenna relay nodes, where each relay is subject to an individual power constraint. Furthermore, ultra-reliable and low-latency communication are desired. The latter requirement translates to considering only schemes that make use of local channel state information. Whereas for a receiver equipped with a single antenna, distributed beamforming is a well known and adequate solution, no straightforward extension is known. In this paper, a scheme is proposed based on a space-time diversity transformation that is applied as a front-end operation at the destination node. This results in an effective unitary channel matrix replacing the scalar coefficient corresponding to each user. Each relay node then inverts its associated channel matrix, which is the generalization of undoing the channel phase in the classical case of distributed beamforming to a single-antenna receiver, and then repeats the message over the resulting "gain-only" channel. In comparison to a single-antenna destination node, the method doubles the diversity order without requiring any channel state information at the receiver while at the same time retaining the array gain offered by the relays.

Equations

$$r_j(t)=h_j^{s\to r}x(t)+n_j(t),$$

$${\mathbf{h}}_j\triangleq {\mathbf{h}}_j^{r\to d}\triangleq {\left[\begin{array}{cccc}{h}_{1j}& h_{2j}& \cdots & h_{N_{r}j}\end{array}\right]}^T,$$

$${\mathbf{s}}_{\mathrm{dest}}(t)=\sum _{j=1}^{M^{\prime }}{\mathbf{h}}_j{x}_j(t)+\mathbf{n}(t),$$

$$x_j(t)=g_{j}x(t),t=1,\dots ,T,$$

$$\mathbf{s}(t)=\sum _{j=1}^{M^{\prime }}{g}_j{\mathbf{h}}_{j}x(t)+\mathbf{n}(t).$$

$$\alpha \triangleq \sqrt{P_r/P_s},$$

$$I({\text{SNR}}_{\mathrm{scheme}})\triangleq \log (1+{\text{SNR}}_{\mathrm{scheme}}).$$

$$\mathrm{Pr}\left(I({\text{SNR}}_{\mathrm{scheme}})<R_{\mathrm{tar}}\right).$$

$$s_j(t)=h_{j}x(t)+n_j(t).$$

$$\mathbf{s}=[s_{1R}(1)s_{1I}(1)s_{2R}(1)s_{2I}(1)s_{1R}(2)s_{1I}(2)s_{2R}(2)s_{2I}(2){]}^T,$$

$$\mathbf{x}=[x_R(1)x_I(1)x_R(2)x_I(2){]}^T.$$

$$\displaystyle\mathbf{G}=\frac{1}{\sqrt{2}}\left[{\begin{array}[]{cccccccc}1&0&0&0&0&0&1&0\\ 0&1&0&0&0&0&0&-1\\ 0&0&1&0&-1&0&0&0\\ 0&0&0&1&0&1&0&0\end{array}}\right].$$

$$\mathbf{r}=\frac{\Vert \mathbf{h}\Vert }{\sqrt{2}}\mathbf{O}(h_1,h_2)\mathbf{x}+{\mathbf{n}}^{\prime },$$

$$\displaystyle\mathbf{O}(h_{1},h_{2})=\frac{1}{\|\mathbf{h}\|}\left[{\begin{array}[]{cccc}h_{1R}&-h_{1I}&h_{2R}&-h_{2I}\\ h_{1I}&h_{1R}&-h_{2I}&-h_{2R}\\ h_{2R}&-h_{2I}&-h_{1R}&h_{1I}\\ h_{2I}&h_{2R}&h_{1I}&h_{R1}\end{array}}\right]$$

$$\mathbf{y}$$

$${\mathbf{x}}_j$$

$$=\alpha \mathbf{O}(h_{1j},h_{2j}{)}^T\mathbf{x}$$

$$\mathbf{y}$$

$$=\sum _{j=1}^{M^{\prime }}\alpha \frac{\Vert {\mathbf{h}}_j\Vert }{\sqrt{2}}\mathbf{x}+{\mathbf{n}}^{\prime }.$$

$$\text{SNR}=\frac{{\left({\sum }_{j=1}^{M^{\prime }}\Vert {\mathbf{h}}_j\Vert \right)}^2}{2}{P}_r.$$

$${\mathbf{H}}_j={\mathbf{U}}_j\left[\begin{array}{cccc}{d}_{i1}& 0& \cdots & 0\\ 0& d_{i2}& \cdots & 0\end{array}\right]{\mathbf{V}}_j^H$$

$${\mathbf{H}}_j\mathbf{p}$$

$$={\mathbf{U}}_j\left[\begin{array}{cccc}{d}_{j1}& 0& \cdots & 0\\ 0& d_{j2}& \cdots & 0\end{array}\right]{\left[\begin{array}{ccc}{\tilde{p}}_1& \cdots & {\tilde{p}}_{N_t}\end{array}\right]}^T$$

$$={\mathbf{U}}_j\left[\begin{array}{c}{d}_{j1}{\tilde{p}}_1\\ d_{j2}{\tilde{p}}_2\end{array}\right]$$

$$\text{SNR}={\left(\sum _{j=1}^{M^{\prime }}|h_{1j}|\right)}^2{P}_r$$

$$\text{SNR}=\underset{j\in \{1,\dots ,M^{\prime }\}}{\max }\sum _{i=1}^2|h_{ij}{|}^2{P}_r.$$

$$\text{SNR}=\underset{i\in \{1,2\}}{\max }{\left(\sum _{j=1}^{M^{\prime }}|h_{ij}|\right)}^2{P}_r.$$

Full text

I Introduction

Cooperative diversity is a means to boost the reliability of communication over a wireless channel where adjacent devices collaborate and share their antennas to facilitate communication between a source and destination node. Different approaches and transmission protocols have been investigated over the years to address this goal.

The potential of using multiple single-antenna relay nodes as a means of forming a virtual antenna array has been recognized and studied in depth since the pioneering work of [1, 4]. Depending on the assumptions made on the availability of channel state information (CSI) at the relays, the virtual antenna array can serve either to provide diversity alone or to obtain also array (power) gain. The former goal does not require (forward channel, from relay to destination) CSI at the relays and may be achieved via distributed space-time coding as suggested in [4], or by means of opportunistic relay selection as suggested in [5]. In contrast, achieving the potential array gain offered by the relays requires at least local CSI to be available to the relays.

It is well known that in the case of a system where all nodes are equipped with a single antenna and each relay knows the channel gain between itself and the destination, distributed (phase-only) beamforming offers both diversity and array gain [1]. In fact, only a small power loss, with respect to the full (centralized) array gain, is incurred by the availability of only local CSI. Specifically, the loss is identical to that incurred in the dual (receiver side) scenario of performing equal-gain combining in place of maximal-ratio combining (MRC). Further, it has been shown in [20] that given a per-relay power constraint, such phase-only beamforming is optimal (in the sense of maximizing the receive SNR). In the present paper, we extend distributed beamforming to a scenario where the destination node is equipped with two receive antennas.

Recently, the need for transmission protocols that can provide ultra reliable communication while maintaining low latency has become apparent. While it is obvious that increasing the number of relays and/or antennas at each link potentially enables to attain higher diversity and array gain, utilizing these while meeting stringent latency constraints introduces substantial challenges, one of which is the need for acquiring channel state information rapidly.

A reasonable approach is to have the relays acquire local CSI (of both source-to-relay and relay-to-destination links) via channel reciprocity, employing time-division duplex (TDD). See further discussion in [5]. Such an approach fits well a scenario where there is a large number of “potential” relays, but only a rather small subset will be active in a given communication round. Thus, it would be highly inefficient for the source and destination nodes to try to acquire CSI and then feed the CSI back to the relays. We also note that similar considerations lead to TDD being advocated for use in massive (non-distributed) MIMO systems; see, e.g., [18].

Accordingly, the proposed technique does not require the transmitter nor the destination node to have access to any CSI. Rather, we assume that perfect, yet local only, CSI is available at each relay. We show that provided that the destination node applies the universal space-time diversity combining transformation recently introduced in [21], then a unitary MIMO channel is induced between each relay and the destination node. Each relay can thus invert the channel with no power loss. We demonstrate that this allows to attain outage probabilities that are comparable to those attained when the relays perform centralized beamforming on the maximum singular vector of the joint channel, up to a moderate power loss. The channel inversion operation at the relays can be seen as the analogue of undoing the phase, as performed in distributed beamforming to a single-antenna receiver.

II System model

We assume transmission between a source node and a destination node via an array of $M$ relays where the source is equipped with a single antenna and the receiver has $N_{r}$ antennas. Our focus will be on the case of $N_{r}=2$. As for the relays, they may be equipped with an arbitrary number of antennas but for sake of exposition, we will begin by assuming that each is equipped with a single antenna. The extension of the scheme to a scenario where the relays are equipped with multiple antennas is described in Section III-C.

We largely assume the system setup as described in [5] which we therefore only briefly recall, highlighting mostly the difference in assumptions, and referring the reader to the latter works for more details on the general problem formulation. The main differences in the assumptions is that in the present work, perfect synchronization of all nodes is assumed and we strive to achieve array gain in addition to diversity gain.

We consider a two-phase protocol. In the first phase, the source node transmits the coded message and all relays are in listening mode. As for the second phase, the proposed scheme can equally fit a decode-and-forward (DaF) or an amplify-and-forward (AaF) mode of operation. For simplicity, we will describe a DaF protocol where all nodes that have successfully decoded the message participate in performing distributed beamforming as described next. We denote the (random) number of relays that successfully decode the message by $M^{\prime}$.

We assume that the channel coefficients do not change over the entire transmission period of $2T$ symbols, where each phase occupies $T$ symbols. All nodes are assumed to operate in half-duplex mode and for simplicity we assume there is no direct link between the source and destination.

As for CSI, we assume that before transmission begins, both source node and destination node send a beacon (clear-to-send, ready-to-receive) signal, from which the relays obtain local CSI (which is assumed to be perfect) by invoking channel reciprocity. Thus, we assume that transmission during both phases takes place over the same frequency band, i.e., we assume TDD.

The source node encodes the data to form the transmitted signal $x(t)$, $t=1,2,\ldots,T$, where $T$ is the blocklength. The transmitted signal must satisfy the power constraint $\mathbb{E}\{|x(t)|^{2}\}\leq P_{s}$. The received signal at relay $j$ is

[TABLE]

where $n_{j}(t)$ is $\mathcal{CN}(0,1)$ and is i.i.d. over time and between relays. The channel coefficients are distributed in the same manner. Therefore, we may define the nominal SNR between the source and a relay node by \text{\mathsf{SNR}}^{s\rightarrow r}\triangleq P_{s}.

Now each link from a relay to the destination is a single-input multiple-output (SIMO) channel with coefficients

[TABLE]

for $j=1,\ldots,M^{\prime}$. Without loss of generality, we assume that the relays with indices $1,\ldots,M^{\prime}$ are the “successful” relays.

We denote the symbols sent from the relays by $x_{j}(t)$ where each relay is subject to the power constraint $\mathbb{E}\{|x_{j}|^{2}\}=P_{r,i}$. For simplicity, we further assume that $P_{r,i}=P_{r}$ for all $j$. Thus, signal received at the destination is given by

[TABLE]

for $t=1,\ldots,T$, and where $n_{j}(t)$ is i.i.d. $\mathcal{CN}(0,1)$ (over space and time). We note that with a slight abuse of notation we let $t$ run form $1$ to $T$ in both phases of transmission. We define the nominal SNR between a relay node and the destination by \text{\mathsf{SNR}}^{r\rightarrow d}\triangleq P_{r}.

We now describe the second phase of transmission. Each of the $M^{\prime}$ relays that have successfully decoded the message has access to the transmitted symbols $x(t)$, $t=1,\ldots,T$. We will only consider relaying operations that amount to applying a linear transformation to the received codeword. We do assume that buffering of symbols is possible and hence linear space-time modulation can be applied at the relay.

Nonetheless, for simplicity, we first describe the simplest setting (without buffering), in which case the operation done at each relay amounts to multiplying each codeword symbol by some complex number which we take to be independent of $t$. We denote this scalar by $g_{j}$, $j=1,\ldots,M^{\prime}$. Thus, the output of each relay is simply

[TABLE]

and hence the destination node receives

[TABLE]

Defining

[TABLE]

it follows that the gains $g_{j}$ should be chosen such that $|g_{j}|=\alpha$. When considering more general space-time processing at the relays, (4) is replaced with a corresponding matrix variant as described in the sequel.

We will compare the outage probability attained by different schemes and take as a figure of merit, the receive SNR attained at the destination node. This can be directly translated to an outage probability for either uncoded transmission or coded transmission, depending on the stringency of the latency constraints. In particular, in order to provide simple performance bounds, we will analyze the mutual information attained by a scheme, defined by

[TABLE]

Correspondingly, for coded transmission (with long blocklength), outage is defined as the event where the mutual information is below the target rate $R_{\rm tar}$, i.e.

[TABLE]

III New distributed beamforming protocol

The proposed method utilizes a recently introduced diversity combining transformation [21] that is performed as a front-end operation at the destination node. This transformation, briefly recalled next, may be viewed as the dual of Alamouti modulation [22].

III-A *Universal Diversity Combining Transformation

Consider a $2\times 1$ single-input multiple-output (SIMO) channel, with channel coefficients $h_{1}$ and $h_{2}$. The signal received at antenna $j=1,2$, at discrete time $t$, is

[TABLE]

We assume that the noise $n_{j}(t)$ is $\mathcal{CN}(0,1)$ and is i.i.d. over space and time. We further assume the transmitted symbols are subject to the power constraint $\mathbb{E}\{|x|^{2}\}=P$.

The scheme works on batches of two time instances and for our purposes, it will suffice to describe it for time instances $t=1,2$. Let us stack the four complex samples received over $T=2$ time instances, two over each antenna, into an $8\times 1$ real vector:

[TABLE]

where $x_{R}$ and $x_{I}$ denote the real and imaginary parts of a complex number $x$. We similarly define the stacked noise vector $\mathbf{n}$. Likewise, we define

[TABLE]

Next, we form a real vector with $4$ elements $\mathbf{r}$ by applying to the vector $\mathbf{s}$ the transformation $\mathbf{r}=\mathbf{G}\mathbf{s}$ where

[TABLE]

Note that unlike conventional diversity-combining schemes, here the combining matrix $\mathbf{G}$ is universal, i.e., it does not depend on the channel coefficients.

In [21] it is shown that the following relation holds,

[TABLE]

where

[TABLE]

is an orthonormal matrix for any realization of $h_{1},h_{2}$, and where $\mathbf{n^{\prime}}$ is i.i.d. and Gaussian with variance $1/2$.111The variance is $1/2$ as we chose above to normalize the complex noise to have unit power.

III-B Proposed Distributed Beamforming Scheme

We consider now the scenario of a $2\times K$ MIMO multiple-access channel (MIMO-MAC) where $M^{\prime}$ users, each equipped with a single antenna, transmit to a common receiver that is equipped with two antennas. The input/output relation of this MIMO-MAC is given in (3).

Now assume that the receiver applies as a front end the universal diversity combining transformation described in Section III-A, applied over two consecutive time instances. Then by (3) and (15), in a MAC scenario, the receiver output is

[TABLE]

where $\mathbf{h}_{k}=\begin{bmatrix}h_{1k}&h_{2k}\end{bmatrix}^{T}$ and $\mathbf{O}(h_{1j},h_{2j})$ is given by (20).

Since we assume that each relay has perfect local CSI, at the expense of adding an additional delay of one symbol, in the proposed scheme, each relay that decoded the message successfully, simply “undoes” its channel matrix and forwards the received (noiseless) symbols. Specifically, each relay transmits

[TABLE]

where ${\bf x}$ is defined in (9) and should be interpreted as two consecutive symbols transmitted from the source node, $\mathbf{O})(\cdot,\cdot)$ is defined in (20), and $\alpha$ is defined in (5).

Thus, the destination sees the effective channel

[TABLE]

The latter is a set of independent scalar channels with

[TABLE]

The attained SNR is quite pleasing as we obtain both the maximal diversity gain while also enjoying transmit-side array gain (but not receive-side MRC gain).

Equivalently, using the SVD of the channel between the $j$’th relay and the destination $\mathbf{h}_{j}=\mathbf{U}_{j}\begin{bmatrix}d_{j}\\ 0\end{bmatrix}v_{j}^{*}$, we may rewrite (24) as \text{\mathsf{SNR}}=\displaystyle{\frac{\left(\sum_{j=1}^{M^{\prime}}d_{j}\right)^{2}}{2}P_{r}}.

III-C Extension to Multi-Antenna Relays

When the relays have multiple antennas, we note that any beamforming vector (meeting the per-relay power constraint) can be applied at each relay in conjunction with the universal dimension reduction operation at the destination node. Assuming for simplicity that the relays are equipped with the same number of antenna, denote by $\mathbf{H}_{j}$ the $2\times N_{t}$ channel from the $j$’th relay to the destination, with SVD

[TABLE]

Applying beamforming vector $\bf{p}=\begin{bmatrix}p_{1}&\cdots&p_{N_{t}}\end{bmatrix}$T (where $\sum|p_{j}|^{2}=1$), we get

[TABLE]

where $|\tilde{p}_{1}|^{2}+|\tilde{p}_{2}|^{2}\leq 1$ (the result of a unitary transformation on the unit norm vector $\bf{p}$). The application of the beamforming vector transforms the $2\times N_{t}$ MIMO channel to a $2\times 1$ SIMO channel to which the universal dimension reduction transformation can be applied. Furthermore, it is readily seen that the singular value of this MISO channel is $\tilde{d}_{j}=\sqrt{(d_{j1}\tilde{p}_{1})^{2}+(d_{j2}\tilde{p}_{2})^{2}}$. It is easily shown that the optimal beamforming vector is the right singular vector of $\mathbf{H}_{j}$ corresponding to the largest singular value.

IV Numerical Performance Evaluation

We compare the performance of the proposed method with several different schemes considering an i.i.d. Rayleigh fading environment where each of the relays has a single transmit antenna. We first consider a scenario in which exactly $M^{\prime}=4$ relays participate in the second phase of transmission. Figure 2 depicts the outage probability of the mutual information. As simple benchmarks that are compatible with low latency constraints, we consider arbitrary antenna selection at the destination, which results in

[TABLE]

and opportunistic relaying, which results in222Note that for the latter, in order to benefit from the MRC gain of the two antennas at the receiver, additional training required to estimate the channel, once the best relay is chosen.

[TABLE]

Both methods suffer a significant penalty in terms of the transmit power required to meet a given outage probability as compared with centralized beamforming. As a further benchmark (which may not meet the low latency constraint) we take optimal (receiver) antenna selection which results in

[TABLE]

As can be seen, the proposed method suffers only a small loss with respect to optimal (receiver) antenna selection.

We next simulated the end-to-end performance when both phases of transmission are in operation. That is, we now include the first hop in the simulation. Figures 3 show the performance of the different schemes as a function of the maximal possible number of relays $M$, where we set $P_{s}=20$ dB and $P_{r}=0$ dB. In these figures, the number of active relays $M^{\prime}$ is a random variable that depends on the SNR of the links between the source node and the relays. It is worth noting that increasing the (effective) number of active relays boosts the performance of schemes that enjoy array gain in the second transmission phase. In contrast, opportunistic relaying, which does not exploit transmit-side CSI in the second phase, can only gain from enhanced diversity when the number of relays grows, resulting in very poor performance.