Rate-Splitting Multiple Access for Downlink Multiuser MIMO: Precoder Optimization and PHY-Layer Design

In light of the information theoretic results of [21], in a symmetric setup with $M$ transmit antennas and $Q$ receive antennas at each user, $\min(M,Q)$ common streams should be transmitted in RS to achieve the information theoretical optimal DoF with imperfect CSIT. Motivated by this result and the performance gain of RS over SDMA, NOMA and DPC in terms of rate and sum-DoF with imperfect CSIT in MISO BC, we fill the aforementioned research gaps and make the following contributions:

• •

  We introduce a general framework of RS in symmetric MIMO BC with the same number of receive antennas at all users. The setting is general in the sense that RS can have arbitrary number of common streams between $1$ and $\min(M,Q)$ inclusive. This is the first work that allows flexibility in the number of common and private streams to be transmitted in RS.

• •

  At high SNR, we derive the achievable sum-DoF for the proposed RS framework in MIMO BC with imperfect CSIT and show the influence of multiple common streams on the sum-DoF of RS. Even with a single common stream, the sum-DoF of RS is shown to be greater than the sum-DoF of MU–MIMO and MIMO NOMA. This sum-DoF of RS increases as the number of common streams increases. We show that by transmitting multiple common streams, the sum-DoF gain of RS over MU–MIMO and MIMO NOMA increases. The assertions are further testified through the Ergodic Sum Rate (ESR) performance using Monte-Carlo simulations. This is the first paper to compare the DoF of MU-MIMO, MIMO NOMA and RS in a MIMO setting as opposed to current comparisons as in [13] which are limited to MISO settings.

• •

  We propose to utilize vectorization and Weighted Minimum Mean Square Error (WMMSE)-based Alternative Optimization (AO) algorithm to optimize the precoders for RS in MIMO BC with the aim of maximizing the WSR subject to the transmit power constraint. The proposed optimization framework addresses the challenge of intractable optimization introduced due to matrix variables. To the best of our knowledge, this is the first work that studies the precoder optimization and the benefits of transmitting multiple commons streams in RS-assisted MIMO BC with perfect and imperfect CSIT.

• •

  Under the assumption of Gaussian signalling and infinite block lengths, we demonstrate that the Ergodic Rate (ER) region of RS with optimized precoders always outperforms the ER regions of MU–MIMO and MIMO NOMA in MIMO BC with both perfect and imperfect CSIT. When CSIT is perfect, we also demonstrate that the ER region of RS comes closer to the capacity region achieved by DPC than MU–MIMO and MIMO NOMA. This is the first work to demonstrate such benefits of RS in MIMO settings.

• •

  To demonstrate the performance of RS in practical systems, we design the Physical (PHY)-layer architecture of RS with finite constellation modulation schemes, finite length polar codes and Adaptive Modulation and Coding (AMC). We show via the Link Level Simulations (LLS) that RS achieves significant throughput gain over MU–MIMO and MIMO NOMA in MIMO BC. This is the first work to design the PHY-layer architecture and to provide the LLS of RS in MIMO settings.

The rest of the paper is organized as follows. In Section II, the system model and CSIT assumptions are delineated. Problem is formulated in Section III. Section IV contains the proposed methodology to solve the optimization problem. Section V describes the PHY-layer architecture for RS. Simulation results are illustrated in Section VI and Section VII concludes the paper. Appendix A contains the derivation of the achievable sum-DoF for RS, MU-MIMO and MIMO NOMA schemes.

Matrices are denoted by boldface uppercase letters, column vectors are denoted by boldface lowercase letters and scalars are denoted by standard letters. Trace and determinant of matrix $\mathbf{A}$ are denoted by $tr(\mathbf{A})$ and $\det(\mathbf{A})$, respectively. $diag(\mathbf{A})$ denotes the diagonal entries of the matrix. $\mathbf{A}^{T}$ and $\mathbf{A}^{H}$ denote the Transpose and Hermitian operators on the matrix $\mathbf{A}$, respectively. Euclidean norm of a vector $\mathbf{a}$ is denoted as $\lVert\mathbf{a}\rVert$. $\otimes$ denotes the kronecker product and $vec(\mathbf{A})$ denotes vectorization of matrix $\mathbf{A}$. $\mathbb{E}_{X}\{Y\}$ is expectation of $Y$ w.r.t random variable $X$. $\mathbb{C}^{M\times N}$ and $\mathbb{R}^{M\times N}$ denote the set of all $M\times N$ dimensional matrices with complex-valued and real-valued entries, respectively. The Circularly Symmetric Complex Gaussian (CSCG) distribution with mean $\mu$ and variance $\sigma^{2}$ is denoted as $\mathcal{CN}(\mu,\sigma^{2})$.

The overall channel state can be denoted as $\mathbf{H}=[\mathbf{H}_{1},\mathbf{H}_{2},\ldots,\mathbf{H}_{K}]$ $\in\mathbb{C}^{M\times(QK)}$, where the fading channel varies according to an ergodic stationary process during the time of transmission. The probability density function of the stationary process is ${f}_{\mathrm{\mathbf{H}}}(\mathbf{H})$. Practical limitations in CSI acquisition such as quantized feedback [45], feedback and processing delay [46], [47], hardware impairments [48] and channel estimation [49] result in partial knowledge of the CSI at the BS given by $\widehat{\mathbf{H}}=[\widehat{\mathbf{H}}_{1},\widehat{\mathbf{H}}_{2},\ldots,\widehat{\mathbf{H}}_{K}]$ and is modeled as $\mathbf{H}=\widehat{\mathbf{H}}+\widetilde{\mathbf{H}}$. We assume that the joint distribution of the channel state and its estimate $\{\mathbf{H},\widehat{\mathbf{H}}\}$ is ergodic and stationary [20]. The conditional density ${f}_{\mathrm{\mathbf{H}}|\mathrm{\widehat{\mathbf{H}}}}(\mathbf{H}|\widehat{\mathbf{H}})$ is assumed to be known at the BS while $\mathbf{H}$ is unknown over the entire transmission. Error in the estimation is defined by the channel estimation error matrix $\widetilde{\mathbf{H}}=[\widetilde{\mathbf{H}}_{1},\widetilde{\mathbf{H}}_{2},\ldots,\widetilde{\mathbf{H}}_{K}]$ in which each element of $\widetilde{\mathbf{H}}_{k}$ is an independent and identically distributed (i.i.d) complex Gaussian distribution variable with zero mean. Whereas, $\mathbb{E}\{\widetilde{\mathbf{H}}_{k}\widetilde{\mathbf{H}}_{k}^{H}\}=\mathbf{R}_{e,k}$ is the covariance matrix of the error matrix, independent of $\widehat{\mathbf{H}}_{k}$. Furthermore, the average CSIT error power is defined as $\sigma_{e,k}^{2}\triangleq\mathbb{E}_{\widetilde{\mathbf{H}}_{k}}\big{\{}\lVert\widetilde{\mathbf{H}}_{k}\rVert^{2}\big{\}}=\frac{1}{M}tr(\mathbf{R}_{e,k})$. $\sigma_{e,k}^{2}$ is allowed to scale as $\mathcal{O}(P_{t}^{-\alpha})$ felicitating the scaling of the CSIT quality with SNR, where $\alpha\in[0,\infty)$ is the quality scaling factor representing the quality of CSI at the BS in the high SNR regime [20, 45, 46, 47]. Consequently, we write $\sigma_{e,k}^{2}=\mathcal{O}(P_{t}^{-\alpha})$ such that the error variance is assumed to scale exponentially with SNR. For $\alpha=\infty$, the average CSIT error power is equal to zero as $\sigma_{e,k}^{2}=0,\>\forall\,k\,\in\,\mathcal{K}$, resulting in a perfect CSIT scenario. On the other extreme, for $\alpha=0$, the CSIT quality remains invariant w.r.t SNR. Thus, a finite non-zero $\alpha$ leads to CSIT quality improvement as SNR increases, e.g., increasing the number of feedback bits with SNR. Here we truncate $\alpha\in[0,1]$. From a DoF perspective $\alpha=1$ corresponds to perfect CSIT [20].

Here we delineate the RS framework proposed for MIMO BC.

To better illustrate RS, we consider a two-user case. At the BS, $Q_{k},\>k\in\{1,2\}$ messages of both users denoted by $\mathbf{w}_{1}=\{W_{1}^{1},\ldots,W_{Q_{1}}^{1}\}$ and $\mathbf{w}_{2}=\{W_{1}^{2},\ldots,W_{Q_{2}}^{2}\}$ are respectively split into sub-messages as $\mathbf{w}_{1}=\big{\{}\{W_{1}^{c,1},W_{1}^{p,1}\},\ldots,\{W_{Q_{1}}^{c,1},W_{Q_{1}}^{p,1}\}\big{\}}$ and $\mathbf{w}_{2}=\big{\{}\{W_{1}^{c,2},W_{1}^{p,2}\},\ldots,\{W_{Q_{2}}^{c,2},W_{Q_{2}}^{p,2}\}\big{\}}$. Sub-messages $\{\{W_{1}^{c,k},\ldots,W_{Q_{k}}^{c,k}\}\mid\forall k\in\,\{1,2\}\}$ are combined and jointly encoded into a common stream vector $\mathbf{s}_{c}$ of size $Q_{c}$. Whereas, the private sub-messages $\{W_{1}^{p,1},\ldots,W_{Q_{1}}^{p,1}\}$ and $\{W_{1}^{p,2},\ldots,W_{Q_{2}}^{p,2}\}$ are respectively encoded into streams $\mathbf{s}_{1}$ of size $Q_{1}$ and $\mathbf{s}_{2}$ of size $Q_{2}$. For instance, in our simulations, we consider $M=4$, $Q=2$, $Q_{1}=Q_{2}=2$ and $Q_{c}=2$. The transmit signal is formed by precoding and superposing the encoded data stream vectors $\mathbf{s}_{c}$, $\mathbf{s}_{1}$, $\mathbf{s}_{2}$, which is expressed as $\mathbf{x}=\mathbf{P}_{c}\mathbf{s}_{c}+\mathbf{P}_{1}\mathbf{s}_{1}+\mathbf{P}_{2}\mathbf{s}_{2}$. At the user side, both users decode their intended streams using SIC. At user-$1$, first $\mathbf{s}_{c}$ is decoded by treating $\mathbf{s}_{1}$ and $\mathbf{s}_{2}$ as noise. Assuming $\mathbf{s}_{c}$ is successfully decoded, user-$1$ then removes it from the received signal and decodes $\mathbf{s}_{1}$ by treating $\mathbf{s}_{2}$ as noise. Similarly, user-$2$ decodes its intended common and private streams sequentially. The two-user RS case reduces to the conventional MU–MIMO strategy by simply turning off the common streams, i.e., allocating no power to the common stream vector. Considering the other extreme of fully decoding the interference, we look at the MIMO NOMA strategy. By encoding the entire $\mathbf{w}_{2}$ into $\mathbf{s}_{c}$, allocating no power to $\mathbf{s}_{2}$ and encoding $\mathbf{w}_{1}$ into $\mathbf{s}_{1}$, the two-user RS reduces to MIMO NOMA with decoding order $1\rightarrow 2$, where the message of user-$2$ is decoded by both users and the message of user-$1$ is decoded by user-$1$ only. Table I illustrates the mapping of messages to streams.

DoF is the total number of interference free streams that can be transmitted simultaneously in a single channel use [15]. The sum-DoF for RS is defined as

where $\widehat{R}_{RS}(P_{t})$ is the Average SR (ASR) and is equal to $\widehat{R}_{RS}(P_{t},\boldsymbol{\mu})$ for equal user weights, i.e., $\mu_{k}=1,\,\forall k\in\mathcal{K}$. We aim at establishing the sum-DoF achieved by the RS scheme for symmetric MIMO BC transmission with imperfect CSIT under the assumption that the channel $\widehat{\mathbf{H}}_{k}$ is full rank and CSIT error matrix $\widetilde{\mathbf{H}}_{k}$ is isotropically distributed. It should be noted that these assumptions are not necessary for optimization. With $Q$ receive antennas at each user, $Q_{p}$ being the total number of private streams transmitted and $Q_{c}$ as the number of common streams, the sum-DoF achieved by the RS precoding scheme is

For comparison, we consider the conventional MU–MIMO and MIMO NOMA schemes with sum-DoF expressed as

The procedure to obtain the sum-DoF achieved by all schemes is relegated to Appendix A. Following the principle of SC-SIC, the ASR achieved by MIMO NOMA (SC–SIC) is limited by the decodability of all users messages decoded in the last place. This restricts the sum-DoF to a maximum value of $\min(M,Q)$. For MU–MIMO, the sum-DoF can attain a maximum value of $Q_{p}$ when $\alpha=1$. As $\alpha$ falls below $1$, detrimental effects of interference lead to a decrease in its sum-DoF. Once $\alpha$ goes further down, CSIT quality deteriorates to a point where it is not conducive enough to support MU transmission and transmitting to a single user yields a better sum-DoF. However, in the RS scheme, the presence of common messages allows the transmitter to adjust the power allocated to the private stream vectors in a way that the interference is always at the level of noise. Thus, the DoF of the private stream vectors is maintained at $Q_{p}\alpha$ by scaling down the power allocated to the private stream vectors to $\mathcal{O}(P_{t}^{\alpha})$. The remaining power which scales as $\mathcal{O}(P_{t})$ is allocated to the common stream vector. The DoF gain achieved by the common streams is $Q_{c}(1-\alpha)$. For $\alpha\,\in(0,1)$ and $Q_{c}=\min(M,Q)$, the sum-DoF of RS is strictly greater than the sum-DoF of both MU–MIMO and MIMO NOMA.

Though optimization does not improve the achievable sum-DoF, it does play a significant role in improving the rate performance. As $\widehat{R}_{\textrm{MU-MIMO}}(P_{t})$ can be obtained by switching off the common streams, the inequality $\widehat{R}_{\textrm{RS}}(P_{t})\geq\widehat{R}_{\textrm{MU-MIMO}}(P_{t})$ is guaranteed for the entire range of SNR. The results in Section VI validate the theoretical assertions.

We first consider a set of $N$ i.i.d channel samples indexed $\mathcal{N}=\{1,2,\ldots,N\}$ drawn from a distribution with density $f_{\mathrm{\mathbf{H}}|\mathrm{\widehat{\mathbf{H}}}}\big{\{}\mathbf{H}|\mathbf{\widehat{H}}\big{\}}$. Therefore, for a given channel estimate $\widehat{\mathbf{H}}$ we have $N$ channel samples denoted as $\mathbb{H}^{(N)}\triangleq\big{\{}\mathbf{H}^{(n)}=\widehat{\mathbf{H}}+\widetilde{\mathbf{H}}^{(n)}\mid\widehat{\mathbf{H}},n\in\mathcal{N}\big{\}}$. Using the channel realizations and Sample Average Functions (SAFs) defined as: $\widehat{R}_{z,k}^{(N)}\triangleq\frac{1}{N}\sum_{n=1}^{N}R_{z,k}^{(n)},\,z\in\{c,p\}$, we approximate the average rates. Here, $R_{z,k}^{(n)}\triangleq R_{z,k}(\mathbf{H}^{(n)}),\,z\in\{c,p\}$ are the common and private rates associated with the $n^{th}$ realization. The SAA of problem (11) is

where $\widehat{R}_{k,tot}^{(N)}=\widehat{R}_{p,k}^{(N)}+\widehat{C}_{k}$ and $\widehat{R}_{c}^{(N)}\triangleq\min_{j}\{\widehat{R}_{c,j}^{(N)}\}_{j=1}^{K}$. The rates obtained here are bounded [20] and therefore by applying LLN, it can be inferred that

The set $\mathbb{P}$ defined as $\{tr(\mathbf{P}\mathbf{P}^{H})\leq P_{t}\mid\mathbf{P}\in\mathbb{P}\}$ is the feasible set of precoders for which the rate functions are bounded, continuous and differentiable in $\mathbf{P}$, thereby making convergence in (17) uniform in $\mathbf{P}$. The ARs are also continuous and differentiable [20] and therefore using (17) we obtain

Based on (17), (18), we obtain that as $N\rightarrow\infty$, the optimum solutions of the SAA in problem (16) converges to the solution of the stochastic problem in (11)[50, 20].

In this subsection, we aim at solving the sample average approximated problem (16) by using the methods adopted in [4] to transform problem (16) into an equivalent WAMMSE form.

First we define Augmented Weighted Mean Square Error (AWMSE) for common and private stream vectors as

where $\mathbf{U}_{z,k},\,z\in\{c,p\}$ are instantaneous weights introduced for common and private MSE matrices of user-$k$. Next we aim to establish the Rate-WMMSE relationship by optimizing the AWMSEs w.r.t equalizers (filters) and weights. By solving $\frac{\partial\xi_{z,k}(\mathbf{H},\widehat{\mathbf{H}})}{\partial\mathbf{G}_{z,k}}=0$, the optimum equalizers are obtained as $\mathbf{G}_{z,k}^{*}=\mathbf{G}_{z,k}^{\mathrm{MMSE}}$, $z\in\{c,p\}$. Substituting the optimum equalizers in (19), we get

By solving $\frac{\partial\xi_{z,k}\big{(}\mathbf{G}_{z,k}^{\mathrm{MMSE}}\big{)}}{\partial\mathbf{U}_{z,k}}=0$, the optimum MMSE weights are obtained as $\mathbf{U}_{z,k}^{*}=\mathbf{U}_{z,k}^{\mathrm{MMSE}}\triangleq(\mathbf{E}_{z,k}^{\mathrm{MMSE}})^{-1}$, $z\in\{c,p\}$. Subsituting the obtained optimum weights for the weights in equation (20), the instantaneous Rate-WMMSE relationship is established as

Based on the principle of SAA, the AR-WAMMSE relationship is derived by taking the expectation over the conditional distribution of channel $\mathbf{H}$ for a given channel estimate $\widehat{\mathbf{H}}$ and is written as

Next, we use the SAFs to obtain the deterministic equivalent relations of (22). Taking the $N$ i.i.d channel samples, the average AWMSEs are $\widehat{\xi}_{z,k}^{(N)}\triangleq\frac{1}{N}\sum_{n=1}^{N}{\xi}_{z,k}^{(n)}$, $z\in\{c,p\}$, where ${\xi}_{z,k}^{(n)}$, $\mathbf{G}_{z,k}^{(n)}$, $\mathbf{U}_{z,k}^{(n)}$, $z\in\{c,p\}$ are all associated with the $n^{th}$ realization in $\mathbb{H}^{(N)}$. For ease of notation, we use $\mathbf{G}$ to represent the set of equalizers for common and private stream vectors, i.e., $\mathbf{G}\triangleq\{\mathbf{G}_{z,k}\mid\forall\,k\in\mathcal{K}\}$, where $\mathbf{G}_{z,k}\triangleq\{\mathbf{G}_{z,k}^{(n)}\mid\forall\,n\in\mathcal{N},\,z\in\{c,p\}\}$. Following the same method, we obtain the set of weights for common and private streams, denoted as $\mathbf{U}$.

Following the LLN as in (17) and the approach used to obtain (21), the AR-WAMMSE in (22) is written as

The sets of optimum MMSE equalizers and weights associated with equation (23) are defined as $\mathbf{G}^{\mathrm{MMSE}}\triangleq\big{\{}\big{(}\mathbf{G}_{z,k}^{\mathrm{MMSE}}\big{)}^{(n)}\mid z\in\{c,p\},\forall\,n\in\mathcal{N},\,\forall k\in\mathcal{K}\big{\}}$, $\mathbf{U}^{\mathrm{MMSE}}\triangleq\big{\{}\big{(}\mathbf{U}_{z,k}^{\mathrm{MMSE}}\big{)}^{(n)}\mid z\in\{c,p\},\forall\,n\in\mathcal{N},\,\forall k\in\mathcal{K}\big{\}}$ respectively. Motivated by the AR-WAMMSE relationship in (23), the deterministic WAMMSE optimization problem is formulated as

where $\widehat{\xi}_{c}^{(N)}=\max\{\widehat{\xi}_{c,k}^{(N)}\}_{k=1}^{K}$, $\widehat{\xi}_{k,tot}^{(N)}=\widehat{\xi}_{p,k}^{(N)}+\widehat{X}_{k}$ and $\widehat{\mathbf{x}}=\{\widehat{X}_{1},\widehat{X}_{2},\ldots,\widehat{X}_{K}\}=-\widehat{\mathbf{c}}$. (24) is optimized w.r.t $(\mathbf{U},\mathbf{G})$ by minimizing individual AWMSEs shown in (23) as the AWMSEs are decoupled in their corresponding weights and equalizers. This can be validated by showing that for a given precoder $\mathbf{P}$, the KKT optimality conditions of (24) are satisfied by the MMSE solution $\{\mathbf{U}^{\mathrm{MMSE}},\mathbf{G}^{\mathrm{MMSE}}\}$. Consequently, it can be shown that for the MMSE solution, (24) boils down to (16) and the Rate-WMMSE relationship is not just limited to the global optimum solution but can be extended to the entire set of stationary points. For any point $\{\mathbf{U}^{*},\mathbf{G}^{*},\mathbf{P}^{*},\widehat{\mathbf{x}}^{*}\}$ satisfying the KKT optimality conditions of (24), also satisfies the KKT optimality conditions of (16), with $\widehat{\mathbf{c}}=-\widehat{\mathbf{x}}$ [20]. Therefore, as $N\rightarrow\infty$, solving (24) yields a solution for (16), which in turn, converges to a solution of the WASR problem in (11).

Problem (24) is non-convex for the joint optimization of variables $\mathbf{G},\mathbf{U}$, $\widehat{\mathbf{x}}$ and $\mathbf{P}$ but it is convex for each block of variables if the other two are fixed. Therefore, we utilize the AO algorithm with 2 steps, 1) updating the equalizers $\mathbf{G}$ and weights $\mathbf{U}$ by using (fixing) the precoders $\mathbf{P}$ from the previous iteration and 2) updating the precoders $\mathbf{P}$ and the message split $\mathbf{x}$ by solving the optimization problem for a given $\mathbf{G}$ and $\mathbf{U}$. Unlike the MISO case in [10], optimization in (24) encounters difficulties of optimizing matrix variables. Furthermore, the presence of $\mathbf{R}_{z,k}^{-1},\,z\in\{c,p\}$ matrices in the AWMSE expressions makes the optimization intractable. Bearing that in mind, we first use vectorization and deduce the objective function into a tractable form. Let us consider the augmented AWMSE expression (19) for the common stream. In step 2 of the AO, the weights are fixed. However, the calculation of the term $tr(\mathbf{U}_{c,k}\mathbf{E}_{c,k})$ introduces difficulties due to the aforementioned reasons and thus we try to simplify the expression and consequently the entire objective function into a solvable form. Expanding $tr(\mathbf{U}_{c,k}\mathbf{E}_{c,k})$ it follows,

Simplifying⁴⁴4Applying matrix manipulation $tr(\mathbf{AB})=tr(\mathbf{BA})$ and $tr(\mathbf{ABC})=vec(\mathbf{A}^{H})^{H}(\mathbf{I}\otimes\mathbf{B})vec(\mathbf{C})$ we transform (25) to (26). the expanded expression in (25), we get

Similarly, the resultant expression for $tr(\mathbf{U}_{p,k}\mathbf{E}_{p,k})$ is

where $\mathbf{p}_{c}=vec(\mathbf{P}_{c})$, $\mathbf{p}_{k}=vec(\mathbf{P}_{k})$, $\mathbf{A}_{c,k}^{{}^{\prime}}=\mathbf{I}_{Q_{c}}\otimes\mathbf{H}_{k}\mathbf{G}_{c,k}^{H}\mathbf{U}_{c,k}\mathbf{G}_{c,k}\mathbf{H}_{k}^{H}$, $\mathbf{A}_{z,k}=\mathbf{I}_{Q_{k}}\otimes\mathbf{H}_{k}\mathbf{G}_{z,k}^{H}\mathbf{U}_{z,k}\mathbf{G}_{z,k}\mathbf{H}_{k}^{H}$, $\mathbf{a}_{z,k}=vec(\mathbf{U}_{z,k}\mathbf{H}_{k}\mathbf{G}_{z,k}^{H})$ and ${\Phi}_{z,k}=\sigma_{n}^{2}tr(\mathbf{U}_{z,k}\mathbf{G}_{z,k}\mathbf{G}_{z,k}^{H})+tr(\mathbf{U}_{z,k})-\log\det(\mathbf{U}_{z,k})$, $z\in\{c,p\}$. Next, we calculate the SAFs of the AWMSEs following (26) and (27).