I Introduction
Wireless communications systems employing multiple antennas have the advantage of increasing the overall throughput without increasing the required bandwidth. For this reason, multipleantenna systems are at the core of several wireless communications standards such as WiFi, Long Term Evolution (LTE) and the fifth generation (5G). However, such wireless systems suffer from multiuser interference (MUI). In order to mitigate MUI, transmit processing techniques have been employed in the downlink (DL), allowing accurate recovery of the data at the receivers. In general, a precoding technique maps the symbols containing the information to the transmit antennas so that the information arrives at the receiver with reduced levels of MUI. Due to its benefits, linear [1, 2, 3, 4, 5, 6] and nonlinear [7, 8, 9] precoding techniques have been extensively reported in the literature. The design of effective precoders demands very accurate channel state information at the transmitter (CSIT), which is an extremely difficult task to accomplish in actual wireless systems. Hence, the transmitter typically only has access to partial or imperfect CSIT. As a result, the precoder cannot handle MUI as expected, resulting in residual MUI at the receiver. This residual MUI can degrade heavily the performance of wireless systems since it scales with the transmit power employed at the base station (BS) [10].
Ia Prior and related work
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), powerdomain NonOrthogonal 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 multipleinput singleoutput (MISO) systems, where it was shown that RS provides gains in terms of DegreesofFreedom (DoF) with respect to conventional multiuser multipleinput multipleoutput (MUMIMO) 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 sumrate performance under perfect and imperfect CSIT assumptions. Another approach has been presented in [31] where the maxmin fairness problem has been studied. In [32]
a Kcell 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 sumrate performance. A successive nullspace precoder, that employs nullspace basis vectors to adjust the interuserinterference 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 RSbased 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 MUMIMO systems is an NPhard 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 closedform expressions for RSbased 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 realtime systems. For this reason, there is a strong demand for costeffective power allocation techniques for RS systems.
IB Contributions
In this paper, we present novel efficient robust and adaptive power allocation techniques [46] for RSbased MUMIMO systems. In particular, we develop a robust adaptive power allocation (APAR) strategy based on stochastic gradient learning [47, 48, 49, 50] and the minimization of the meansquare 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 APAR 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 sumrate of an RS system employing adaptive power allocation outperforms conventional MUMIMO systems under imperfect CSIT assumption. The contributions of this work can be summarized as:

Costeffective APAR and APA techniques for power allocation are proposed based on stochastic gradient recursions and knowledge of the variance of the channel errors for RSbased and standard MUMIMO systems.

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

A simulation study of the proposed APA and APAR, and the existing power allocation techniques for RSbased and standard MUMIMO systems.
IC Organization
The rest of this paper is organized as follows. Section II describes the mathematical model of an RSbased MUMIMO system. In Section III the proposed APAR technique is presented, the proposed APA approach is obtained as a particular case and sumrate expressions are derived. In Section IV, we present an analysis of convexity and stability of the proposed APA and APAR techniques along with a bound on the MSE of APA and APAR 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.
ID Notation
Column vectors are denoted by lowercase boldface letters. The vector stands for the th row of matrix . Matrices are denoted by uppercase boldface letters. Scalars are denoted by standard letters. The superscript represents the transpose of a matrix, whereas the notation stands for the conjugate transpose of a matrix. The operators , and
represent the Euclidean norm, and the expectation w.r.t the random variable
, respectively. The trace operator is given by . The Hadamard product is denoted by . The operator produces a diagonal matrix with the coefficients of located in the main diagonal.Ii System Model
Let us consider the RSbased MUMIMO system architecture depicted in Fig. 1, where the BS is equipped with antennas, serves users and the th UE is equipped with antennas. Let us denote by the total number of receive antennas. Then, . For simplicity, the message intended for the th user is split into a common message and a private message. Then, the messages are encoded and modulated. The transmitter sends one common stream and a total of private streams, with . The set contains the private streams, that are intended for the user , where . It follows that .
The vector
, which is assumed i.i.d. with zero mean and covariance matrix equal to the identity matrix, contains the information transmitted to all users, where
is the common symbol and contains the private symbols of all users. Specifically, the vector contains the private streams intended for the th user. The system is subject to a transmit power constraint given by , where is the transmitted vector and denotes the total available power. The transmitted vector can be expressed as follows:(1) 
where represents the power allocation matrix and is used to precode the vector of symbols . Specifically, and , where denotes the power allocated to the common stream and allocates power to the th private stream. Without loss of generality, we assume that the columns of the precoders are normalized to have unit norm.
The model established leads us to the received signal described by
(2) 
where
represents the uncorrelated noise vector, which follows a complex normal distribution, i.e.,
. We assume that the noise and the symbols are uncorrelated, which is usually the case in real systems. The matrix denotes the channel between the BS and the user terminals. Specifically, denotes the noise affecting the th user and the matrix represents the channel between the BS and theth user. The imperfections in the channel estimate are modelled by the random matrix
. Each coefficient offollows a Gaussian distribution with variance equal to
and . Then, the channel matrix can be expressed as , where the channel estimate is given by . From (2) we can obtain the received signal of user , which is given by(3) 
Note that the RS architecture contains the conventional MUMIMO as a special case where no message is split and therefore is set to zero. Then, the model boils down to the model of a conventional MUMIMO system, where the received signal at the th user is given by
(4) 
In what follows, we will focus on the development of power allocation techniques that can costeffectively compute and .
Iii Proposed Power Allocation Techniques
In this section, we detail the proposed power allocation techniques. In particular, we start with the derivation of the ARAR approach and then present the APA technique as a particular case of the APAR approach.
Iiia Robust Adaptive Power Allocation
Here, a robust adaptive power allocation algorithm, denoted as APAR, 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 , i.e., . It follows that the available power for the private streams is reduced to . We remark that the length of is greater than that of since the common symbol is superimposed to the private symbols. Therefore, we consider the vector , where is a transformation matrix employed to ensure that the dimensions of and match, and is given by
(5) 
All the elements in the first row of matrix 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
(6) 
where the received signal at the th antenna is described by
(7) 
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
(8) 
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 zeroforcing (ZF) and the matched filter (MF) precoders [2], where three private streams and one common stream are transmitted and the parameter 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 . Then, the MSE is given by
(9) 
where and for all . The proof to obtain the last equation can be found in appendix A. The partial derivatives of (9) with respect to and are expressed by
(10) 
(11) 
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 APAR technique are given by
(12) 
where the parameter 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 , where to ensure that the power constraint is satisfied. Algorithm 1 summarizes the proposed APAR algorithm.
IiiB Adaptive Power Allocation
In this section, a simplified version of the proposed APAR 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
(13) 
In this case, the MSE is equivalent to
(14) 
where we considered that and for all . The proof to obtain (14) can be found in appendix B. Taking the partial derivatives of (14) with respect to the coefficients of we arrive at
(15)  
(16) 
The power allocation coefficients are adapted using (15) and (16) in the following recursions:
(17) 
IiiC SumRate Performance
In this section, we derive closedform expressions to compute the sumrate performance of the proposed algorithms. Specifically, we employ the ergodic sumrate (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
(18) 
It follows that the instantaneous SINR while decoding the common symbol is given by
(19) 
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 th private stream, which is given by
(20) 
Considering Gaussian signaling, the instantaneous common rate can be found with the following equation:
(21) 
The private rate of the th stream is given by
(22) 
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
(23) 
respectively. With the ASR we can obtain the ergodic sumrate (ESR), which quantifies the performance of the system over a large number of channel realizations and is given by
(24) 
Iv Analysis
In this section, we carry out a convexity analysis and a statistical analysis of the proposed algorithms along with an assessment of their computational complexity in terms of floating point operations (FLOPs). Moreover, we derive a bound that establishes that the proposed APAR algorithm is superior or comparable to the proposed APA algorithm.
Iva Convexity analysis
In this section, we perform a convexity analysis of the optimization problem that gives rise to the proposed APAR and APA algorithms. In order to establish convexity, we need to compute the second derivative of with respect to and , and then check if it is greater than zero [47], i.e.,
(25) 
Let us first compute using the results in (10):
(26) 
Now let us compute using the results in (11):
(27) 
Since we have the sum of the strictly convex terms in (26) and (27) the objective function associated with APAR 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 and remains strictly convex.
IvB Bound on the MSE of APA and APAR
Let us now show that the proposed APAR 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 that minimize the error are found. However, under imperfect CSIT the transmitter is unaware of and the adaptive algorithm performs the power allocation by employing instead of . This results in a power allocation given by which originates an increase in the MSE obtained. It follows that
(28) 
On the other hand, the robust adaptive algorithm finds the optimal that minimizes and therefore takes into account that only partial knowledge of the channel is available. Since the coefficients and are different, we have
(29) 
Note that under perfect CSIT equation (9) reduces to (14). In such circumstances and therefore both algorithms are equivalent. In the following, we evaluate the performance obtained by the proposed algorithms. Specifically, we have that , where is the error produced from the assumption that the BS has perfect CSIT. Then, we have
(30) 
which is a positive quantity when and . The inequalities hold as long as and . As the error in the power allocation coefficients grows the lefthand side of the two last inequalities increases. This explains why the proposed APAR performs better than the proposed APA algorithm.
IvC Statistical Analysis
The performance of adaptive learning algorithms is usually measured in terms of its transient behavior and steadystate 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
(31) 
Let us define the error between the estimate of the power coefficients and the optimal parameters as follows:
(32) 
where represents the optimal allocation for the th coefficient.
By subtracting (32) from (31), we obtain
(33) 
where . Expanding the terms in (33), we get
Rearranging the terms of the last equation, we obtain
(34) 
Equation (34) can be rewritten as follows
(35) 
where
(36) 
Bu multiplying (35) by and taking the expected value, we obtain
(37) 
where we consider that .The previous equation constitutes a geometric series with geometric ratio equal to . Then, we have
(38) 
Note that from the last equation the step size must fulfill
(39) 
with .
For the common power allocation coefficient we have the following recursion: