Study of Robust Adaptive Power Allocation Techniques for Rate Splitting based MU-MIMO systems

In this context, Rate Splitting (RS) has emerged as a novel approach that is capable of dealing with CSIT imperfection [11] in an effective way. RS was initially proposed in [12] to deal with interference channels [13], where independent transmitters send information to independent receivers [14]. Since then, several studies have found that RS outperforms conventional schemes such as conventional precoding in spatial division multiple access (SDMA), power-domain Non-Orthogonal Multiple Access (NOMA) [15] and even dirty paper coding (DPC) [16]. Interestingly, it turns out that RS constitutes a generalized framework which has as special cases other transmission schemes such SDMA, NOMA and multicasting [17, 18, 19, 20]. The main advantage of RS is its capability to partially decode interference and partially treat interference as noise. To this end, RS splits the message of one or several users into a common message and a private message. The common message must be decoded by all the users that employ successive interference cancellation [21, 22, 23, 24, 25]. On the other hand, the private messages are decoded only by their corresponding users.

RS schemes have been shown to enhance the performance of wireless communication systems. In [26] RS was extended to the broadcast channel (BC) of multiple-input single-output (MISO) systems, where it was shown that RS provides gains in terms of Degrees-of-Freedom (DoF) with respect to conventional multiuser multiple-input multiple-output (MU-MIMO) schemes under imperfect CSIT. Later in [27], the DoF of a MIMO BC and IC was characterized. RS has eventually been shown in [28] to achieve the optimal DoF region when considering imperfect CSIT, outperforming the DoF obtained by SDMA systems, which decays in the presence of imperfect CSIT.

Due to its benefits, several wireless communications deployments with RS have been studied. RS has been employed in MISO systems along with linear precoders [29, 30] in order to maximize the sum-rate performance under perfect and imperfect CSIT assumptions. Another approach has been presented in [31] where the max-min fairness problem has been studied. In [32] a K-cell MISO IC has been considered and the scheme known as topological RS presented. The topological RS scheme transmits multiple layers of common messages, so that the common messages are not decoded by all users but by groups of users. RS has been employed along with random vector quantization in [33] to mitigate the effects of the imperfect CSIT caused by finite feedback. In [34, 35], RS with common stream combining techniques has been developed in order to exploit multiple antennas at the receiver and to improve the overall sum-rate performance. A successive null-space precoder, that employs null-space basis vectors to adjust the inter-user-interference at the receivers, is proposed in [36]. The optimization of the precoders along with the transmission of multiple common streams was considered in [37]. In [38], RS with joint decoding has been explored. The authors of [38] devised precoding algorithms for an arbitrary number of users along with a stream selection strategy to reduce the number of precoded signals.

Along with the design of the precoders, power allocation is also a fundamental part of RS-based systems. The benefits of RS are obtained only if an appropriate power allocation for the common stream is performed. However, the power allocation problem in MU-MIMO systems is an NP-hard problem [39, 40], and the optimal solution can be found at the expense of an exponential growth in computational complexity. Therefore, suboptimal approaches that jointly optimize the precoder and the power allocation have been developed. Most works so far employ exhaustive search or complex optimization frameworks. These frameworks rely on the augmented WMMSE [16, 41, 42, 43, 37], which is an extension of the WMMSE proposed in [44]. This approach requires also an alternating optimization, which further increases the computational complexity. A simplified approach can be found in [45], where closed-form expressions for RS-based massive MIMO systems are derived. However, this suboptimal solution is more appropiate for massive MIMO deployments. The high complexity of most optimal approaches makes them impractical to implement in large or real-time systems. For this reason, there is a strong demand for cost-effective power allocation techniques for RS systems.

In this paper, we present novel efficient robust and adaptive power allocation techniques [46] for RS-based MU-MIMO systems. In particular, we develop a robust adaptive power allocation (APA-R) strategy based on stochastic gradient learning [47, 48, 49, 50] and the minimization of the mean-square error (MSE) between the transmitted common and private symbols of the RS system and the received signal. We incorporate knowledge of the variance of the channel errors to cope with imperfect CSIT and adjust power levels in the presence of uncertainty. When the knowledge of the variance of the channel errors is not exploited the proposed APA-R becomes the proposed adaptive power allocation algorithm (APA). An analysis of the convexity and stability of the proposed power allocation algorithms along with a study of their computational complexity and theoretical bounds relating the power allocation strategies are developed. Numerical results show that the sum-rate of an RS system employing adaptive power allocation outperforms conventional MU-MIMO systems under imperfect CSIT assumption. The contributions of this work can be summarized as:

• •

  Cost-effective APA-R and APA techniques for power allocation are proposed based on stochastic gradient recursions and knowledge of the variance of the channel errors for RS-based and standard MU-MIMO systems.

• •

  An analysis of convexity and stability of the proposed power allocation techniques along with a bound on the MSE of APA and APA-R and a study of their computational complexity.

• •

  A simulation study of the proposed APA and APA-R, and the existing power allocation techniques for RS-based and standard MU-MIMO systems.

The rest of this paper is organized as follows. Section II describes the mathematical model of an RS-based MU-MIMO system. In Section III the proposed APA-R technique is presented, the proposed APA approach is obtained as a particular case and sum-rate expressions are derived. In Section IV, we present an analysis of convexity and stability of the proposed APA and APA-R techniques along with a bound on the MSE of APA and APA-R and a study of their computational complexity. Simulation results are illustrated and discussed in Section V. Finally, Section VI presents the conclusions of this work.

Column vectors are denoted by lowercase boldface letters. The vector $\mathbf{a}_{i,*}$ stands for the $i$th row of matrix $\mathbf{A}$. Matrices are denoted by uppercase boldface letters. Scalars are denoted by standard letters. The superscript $\left(\cdot\right)^{\text{T}}$ represents the transpose of a matrix, whereas the notation $\left(\cdot\right)^{H}$ stands for the conjugate transpose of a matrix. The operators $\lVert\cdot\rVert$, and $\mathbb{E}_{x}\left[\cdot\right]$ represent the Euclidean norm, and the expectation w.r.t the random variable $x$, respectively. The trace operator is given by $\text{tr}\left(\cdot\right)$. The Hadamard product is denoted by $\odot$. The operator $\text{diag}\left(\mathbf{a}\right)$ produces a diagonal matrix with the coefficients of $\mathbf{a}$ located in the main diagonal.

Here, a robust adaptive power allocation algorithm, denoted as APA-R, is developed to perform power allocation in the presence of imperfect CSIT. The key idea is to incorporate knowledge about the variance of the channel uncertainty [51, 52] into an adaptive recursion to allocate the power among the streams. The minimization of the MSE between the received signal and the transmitted symbols is adopted as the criterion to derive the algorithm.

Let us consider the model established in (2) and denote the fraction of power allocated to the common stream by the parameter $\delta$, i.e., $a_{c}^{2}=\delta E_{tr}$. It follows that the available power for the private streams is reduced to $\left(1-\delta\right)E_{tr}$. We remark that the length of $\mathbf{s}^{\left(\text{RS}\right)}$ is greater than that of $\mathbf{y}^{\left(\text{RS}\right)}$ since the common symbol is superimposed to the private symbols. Therefore, we consider the vector $\mathbf{y}^{\prime}=\mathbf{T}\mathbf{y}^{\left(\text{RS}\right)}$, where $\mathbf{T}\in\mathbb{R}^{\left(M+1\right)\times M}$ is a transformation matrix employed to ensure that the dimensions of $\mathbf{s}^{\left(\text{RS}\right)}$ and $\mathbf{y}^{\left(\text{RS}\right)}$ match, and is given by

All the elements in the first row of matrix $\mathbf{T}$ are equal to one in order to take into account the common symbol obtained at all receivers. As a result we obtain the combined receive signal of all users. It follows that

where the received signal at the $i$th antenna is described by

Let us now consider the proposed robust power allocation problem for imperfect CSIT scenarios. By including the error of the channel estimate, the robust power allocation problem can be formulated as the constrained optimization given by

which can be solved by first relaxing the constraint, using an adaptive learning recursion and then enforcing the constraint.

In this work, we choose the MSE as the objective function due to its convex property and mathematical tractability, which help to find an appropriate solution through algorithms. The objective function is convex as illustrated by Fig. 2 and analytically shown in Section IV. In Fig. 2, we plot the objective function using two precoders, namely the zero-forcing (ZF) and the matched filter (MF) precoders [2], where three private streams and one common stream are transmitted and the parameter $\delta$ varies with uniform power allocation across private streams.

To solve (8) we need to expand the terms and evaluate the expected value. Let us consider that the square error is equal to $\varepsilon=\lVert\mathbf{s}^{\left(\text{RS}\right)}-\mathbf{y}^{\prime}\rVert^{2}$. Then, the MSE is given by

where $\hat{\phi}^{\left(i,c\right)}=\hat{\mathbf{h}}_{l,*}\mathbf{p}_{c}$ and $\hat{\phi}^{\left(i,l\right)}=\hat{\mathbf{h}}_{i,*}\mathbf{p}_{l}$ for all $i,l\in\left[1,M\right]$. The proof to obtain the last equation can be found in appendix A. The partial derivatives of (9) with respect to ${a}_{c}$ and ${a}_{i}$ are expressed by

The partial derivatives given by (10) and (11) represent the gradient of the MSE with respect to the power allocation coefficients. With the obtained gradient we can employ a gradient descent algorithm, which is an iterative procedure that finds a local minimum of a differentiable function. The key idea is to take small steps in the opposite direction of the gradient, since this is the direction of the steepest descent. Remark that the objective function used is convex and has no local minimum. Then, the recursions of the proposed APA-R technique are given by

where the parameter $\mu$ represents the learning rate of the adaptive algorithm. At each iteration, the power constraint is analyzed. Then, the coefficients are scaled with a power scaling factor by $\mathbf{a}^{\left(\rm{RS}\right)}\left[n\right]=\beta\mathbf{a}^{\left(\rm{RS}\right)}\left[n\right]$, where $\beta=\sqrt{\frac{1}{\textrm{tr}\left(\textrm{diag}\left(\mathbf{a}^{\left(\rm{RS}\right)}\left[n\right]\odot\mathbf{a}^{\left(\rm{RS}\right)}\left[n\right]\right)\right)}}$ to ensure that the power constraint is satisfied. Algorithm 1 summarizes the proposed APA-R algorithm.

In this section, a simplified version of the proposed APA-R algorithm is derived. The main objective is to reduce the complexity of each recursion of the adaptive algorithm and avoid the load of computing the statistical parameters of the imperfect CSIT, while reaping the benefits of RS systems. The power allocation problem can be reformulated as the constrained optimization problem given by

In this case, the MSE is equivalent to

where we considered that $\phi^{\left(i,c\right)}=\mathbf{h}_{l,*}\mathbf{p}_{c}$ and $\phi^{\left(i,l\right)}=\mathbf{h}_{i,*}\mathbf{p}_{l}$ for all $i,l\in\left[1,M\right]$. The proof to obtain (14) can be found in appendix B. Taking the partial derivatives of (14) with respect to the coefficients of $\mathbf{a}^{\left(\text{RS}\right)}$ we arrive at

The power allocation coefficients are adapted using (15) and (16) in the following recursions:

In this section, we derive closed-form expressions to compute the sum-rate performance of the proposed algorithms. Specifically, we employ the ergodic sum-rate (ESR) as the main performance metric. Before the computation of the ESR we need to find the average power of the received signal, which is given by

It follows that the instantaneous SINR while decoding the common symbol is given by

Once the common symbol is decoded, we apply SIC to remove it from the received signal. Afterwards, we calculate the instantaneous SINR when decoding the $k$th private stream, which is given by

Considering Gaussian signaling, the instantaneous common rate can be found with the following equation:

The private rate of the $k$th stream is given by

Since imperfect CSIT is being considered, the instantaneous rates are not achievable. To that end, we employ the average sum rate (ASR) to average out the effect of the error in the channel estimate. The average common rate and the average private rate are given by

respectively. With the ASR we can obtain the ergodic sum-rate (ESR), which quantifies the performance of the system over a large number of channel realizations and is given by

In this section, we perform a convexity analysis of the optimization problem that gives rise to the proposed APA-R and APA algorithms. In order to establish convexity, we need to compute the second derivative of $\mathbb{E}\left[\varepsilon|\hat{\mathbf{H}}\right]$ with respect to $a_{c}$ and $a_{i}$, and then check if it is greater than zero [47], i.e.,

Let us first compute $\frac{\partial^{2}\mathbb{E}\left[\varepsilon|\hat{\mathbf{H}}\right]}{\partial a_{c}\partial a_{c}}$ using the results in (10):

Now let us compute $\frac{\partial^{2}\mathbb{E}\left[\varepsilon|\hat{\mathbf{H}}\right]}{\partial a_{i}\partial a_{i}}$ using the results in (11):

Since we have the sum of the strictly convex terms in (26) and (27) the objective function associated with APA-R is strictly convex [47]. The power constraint is also strictly convex and only scales the powers to be adjusted. In the case of the APA algorithm, the objective function does not employ knowledge of the error variance $\sigma_{e,i}^{2}$ and remains strictly convex.

Let us now show that the proposed APA-R power allocation produces a lower MSE than that of the proposed APA power allocation. The MSE obtained in (14) assumes that the transmitter has perfect knowledge of the channel. Under such assumption the optimal coefficients $\mathbf{a}_{o}$ that minimize the error are found. However, under imperfect CSIT the transmitter is unaware of $\tilde{\mathbf{H}}$ and the adaptive algorithm performs the power allocation by employing $\hat{\mathbf{H}}$ instead of $\mathbf{H}$. This results in a power allocation given by $\hat{\mathbf{a}}^{\left(\text{APA}\right)}$ which originates an increase in the MSE obtained. It follows that

On the other hand, the robust adaptive algorithm finds the optimal $\mathbf{a}_{o}^{\left(\text{APA-R}\right)}$ that minimizes $\mathbb{E}\left[\varepsilon\left(\mathbf{H},\mathbf{a}\right)\right|\hat{\mathbf{H}}]$ and therefore takes into account that only partial knowledge of the channel is available. Since the coefficients $\hat{\mathbf{a}}^{\left(\text{APA}\right)}$ and $\mathbf{a}_{o}^{\left(\text{APA-R}\right)}$ are different, we have

Note that under perfect CSIT equation (9) reduces to (14). In such circumstances $\mathbf{a}_{o}^{\left(\text{APA}\right)}=\mathbf{a}_{o}^{\left(\text{APA-R}\right)}$ and therefore both algorithms are equivalent. In the following, we evaluate the performance obtained by the proposed algorithms. Specifically, we have that $\hat{\mathbf{a}}^{\left(\text{APA}\right)}=\mathbf{a}_{o}^{\left(\text{APA-R}\right)}+\mathbf{a}_{e}$, where $\mathbf{a}_{e}=\left[a_{c,e},a_{1,e},\cdots,a_{M,e}\right]^{\text{T}}$ is the error produced from the assumption that the BS has perfect CSIT. Then, we have

which is a positive quantity when $-2a_{c,e}\sum_{i=1}^{M}\Re{\left\{\hat{\phi}^{\left(i,c\right)}\right\}}<2a_{c,e}^{2}\left(\sum_{i=1}^{M}\lvert\hat{\phi}^{\left(i,c\right)}\rvert^{2}+M\sigma_{e,i}^{2}\lVert\mathbf{p}_{c}\rVert^{2}\right)$ and $-2\sum_{j=1}^{M}a_{j,e}\Re{\left\{\hat{\phi}^{\left(j,j\right)}\right\}}<2\sum_{j=1}^{M}a_{j,e}^{2}\left(\sum_{l=1}^{M}\lvert\hat{\phi}^{\left(l,j\right)}\rvert^{2}+M\sigma_{e,i}^{2}\lVert\mathbf{p}_{j}\rVert^{2}\right)$. The inequalities hold as long as $a_{c,e}\left[\sum_{i=1}^{M}\left(\Re\left\{\hat{\phi}^{\left(i,c\right)}\right\}\right)^{2}+\sum_{i=1}^{M}\left(\Im\left\{\hat{\phi}^{\left(i,c\right)}\right\}\right)^{2}+M\sigma_{e,i}^{2}\lVert\mathbf{p}_{c}\rVert^{2}\right]>\sum_{i=1}^{M}\Re{\left\{\phi^{\left(i,c\right)}\right\}}$ and $\sum_{j=1}^{M}a_{j,e}\left[\sum_{l=1}^{M}\left(\Re\left\{\hat{\phi}^{\left(l,j\right)}\right\}\right)^{2}+\sum_{l=1}^{M}\left(\Im\left\{\hat{\phi}^{\left(l,j\right)}\right\}\right)^{2}+M\sigma_{e,i}^{2}\lVert\mathbf{p}_{j}\rVert^{2}\right]>\sum_{j=1}^{M}\Re{\left\{\phi^{\left(j,j\right)}\right\}}$. As the error in the power allocation coefficients grows the left-hand side of the two last inequalities increases. This explains why the proposed APA-R performs better than the proposed APA algorithm.

The performance of adaptive learning algorithms is usually measured in terms of its transient behavior and steady-state behaviour. These measurements provide information about the stability, the convergence rate, and the MSE achieved by the algorithm[53, 54]. Let us consider the adaptive power allocation with the update equations given by (17). Expanding the terms of (17) for the private streams, we get

Let us define the error between the estimate of the power coefficients and the optimal parameters as follows:

where $a_{j}^{\left(o\right)}$ represents the optimal allocation for the $j$th coefficient.

By subtracting (32) from (31), we obtain

where $f_{q,r}^{\left(j\right)}=\Re\left\{\left(\mathbf{h}_{q,*}\mathbf{p}_{j}\right)^{*}\left(\mathbf{h}_{r,*}\mathbf{p}_{j}\right)\right\}$. Expanding the terms in (33), we get

Rearranging the terms of the last equation, we obtain

Equation (34) can be rewritten as follows