Improving Sum-Rate of Cell-Free Massive MIMO with Expanded Compute-and-Forward

Let us suppose that all APs have full channel state information. To obtain the estimation of the integer combination with UEs’ original data ${{{\bf{\hat{u}}}}_{m}}$, the $m$th AP multiplies the received signal by an equalization factor $b_{m}$ by the received signal to obtain the effective channel as

After choosing $b_{m}$ to be the minimum mean-square error (MMSE) coefficient adopted at the $m$th AP, the minimum effective noise variance for parallel computation is given by

We denote ${\bf{A}}$ as the matrix of the coefficient vectors, ${\bf{A}}=\left[{{{\bf{a}}_{1,}}{{\bf{a}}_{2}},\ldots,{{\bf{a}}_{M}}}\right]$. Specifically, if the $m$th column of ${\bf{A}}$ is a null vector, the $m$th AP does not serve any UE; if the $l$th row of ${\bf{A}}$ is a zero vector, the $l$th UE is not served by any AP. When we remove such columns and rows from ${\bf{A}}$, we obtain ${\bf{A}}\in{\mathbb{Z}}{\left[i\right]^{L^{\prime}\times M^{\prime}}}$, where $L^{\prime}$ and $M^{\prime}$ refers to the number of effective UEs and APs. Due to the array gain, the sum-rate increases along with the value of $M^{\prime}$ increase. However, there is a trade-off between the values of $L^{\prime}$ and sum-rate performance, since the growth of effective UEs does not always lead to the increase in sum-rate [6]. According to the discussion of coefficient vector selection in Section III-A, the values of $M^{\prime}$ and $L^{\prime}$ are determined by the location of APs and UEs, therefore $M^{\prime}$ and $L^{\prime}$ can take the optimal value when the location of APs and UEs is optimal. Define the rank of ${\bf{A}}$ by ${{M^{\prime}}_{{\rm{rank}}}}\buildrel\Delta\over{=}{\rm{Rank}}\left({\bf{A}}\right)$. According to [18], all effective UEs’ data can be recovered if ${{M^{\prime}}_{{\rm{rank}}}}=L^{\prime}$. Therefore, we only need $L^{\prime}$ integer linear combinations among the whole $M^{\prime}$ combinations. In other words, only $L^{\prime}$ APs need to transmit signals to the CPU through fronthaul links. The computation rate region for the parallel computation is given by

where $a_{ml}=0$ means the $m$th AP doesn’t serve the $l$th UE.

If ${a_{ml}}\neq 0$, the computed achievable rate for the $l$th UE at the $m$th AP is given as

However, when recovering the data of the $l$th UE, the codeword spacing for that data should tolerate the maximum effective noise variance across APs, whose linear combinations involve that data. Therefore, the actual achievable rate of the $l$th UE is

Hence, the achievable sum-rate of $L^{\prime}$ UEs is

Recall that all UEs transmit with equal power in CF scheme. For fairness, we compare the performance of CF and ECF with the constraint of equal total transmit power. We aim at optimizing the power allocation to maximize the achievable sum-rate under the constraints on the total power consumption $P_{t}$. The optimization problem is formulated as follows:

UEs can share a total power budget which is the upper bound performance of each UE’s power constraint as their total maximum allowable transmit power [29]. Besides, (19) is handled at the CPU since the global information ${{\bf{a}}_{1}},\cdots,{{\bf{a}}_{M}}$ and ${{\bf{g}}_{1}},\cdots,{{\bf{g}}_{M}}$ are required.

As mentioned above, each AP decodes $\sum\nolimits_{l=1}^{L}{{a_{ml}}{{\mathbf{x}}_{l}}}$ as one regular codeword due to the lattice algebraic structure. All UEs served by the $m$th AP need to tolerate the same effective noise. If the linear integer combinations $\sum\nolimits_{l=1}^{L}{{a_{ml}}{{\mathbf{x}}_{l}}}$ can tolerate the effective noise variance ${{{\sigma_{{\rm{para}}}^{2}\left({{{\bf{P}}},{{\bf{g}}_{m}},{{\bf{a}}_{m}}}\right)}}}$, then all UEs served by the $m$th AP, which means ${a_{ml}}\neq 0$ can be successfully recovered from the linear combination with integer coefficient vector ${{\bf{a}}_{m}}$. In cell-free massive MIMO, we always emphasize a good quality-of-service for all users. However, directly improving the achievable sum rate cannot achieve a good balance of quality-of-service for all users [12]. Therefore, the goal of minimizing the maximum effective noise variance that can generally improve the achievable rate for most UEs is more suitable for our model. In other words, we could minimize the maximum effective noise variance as

According to (14), (III-B2) is equivalent to

According to [12, Lemma B. 4], the matrix inversion can be equivalently represented by

Therefore, the effective noise variance for the $m$th AP is

Aa a result, (III-B2) can be rewritten as

However, (III-B2) is NP-hard. To tackle this challenge, we first introduce three auxiliary variables. On this basis, we can build the following optimization problem by introducing three auxiliary variables $r$, $s$, and $t$:

In particular, variables $r$ and $s$ have limited searching space, respectively. For a given value of $r$, the variable $s$ should be smaller than $r$. Therefore, we employ a two-dimension of brute force search on these two scalars. For fixed $r$ and $s$, the optimization problem in (25) can be rewritten as a feasibility problem

where ${\bf{p}}={\left[{{P_{1}},\ldots,{P_{L^{\prime}}}}\right]^{T}}$, ${\bf{v}}={\left[{{{\left|{{g_{m1}}}\right|}^{2}},\ldots,{{\left|{{g_{mL^{\prime}}}}\right|}^{2}}}\right]^{T}}$. ${\bf{J}}$ is a $L^{\prime}\times L^{\prime}$ matrix whose ${\left({{l_{1}},{l_{2}}}\right)}$th element is given as ${{\bf{J}}_{\left({{l_{1}},{l_{2}}}\right)}}=2{\left|{{a_{m{l_{1}}}}}\right|\left|{{g_{m{l_{1}}}}}\right|\left|{{a_{m{l_{2}}}}}\right|\left|{{g_{m{l_{2}}}}}\right|}$. Clearly, ${\bf{J}}$ is a positive semi-definite matrix. Consequently, (26) can be solved efficiently by performing a brute force search on two scalars. In each step, a feasibility problem needs to be solved. Since transforming a nonconvex problem into its equivalent convex form is quite difficult if not possible, an off-the-shelf optimization solver, e.g. fmincon in Matlab, is adopted to obtain a suboptimal solution. Besides, simulations show that solving a nonconvex problem also brings obvious performance improvement, the computation cost is tolerable. Besides, actually (III-B2) is equal to (25) only if the two terms in the objective of (III-B2) are independent. However, at the optimal point, we only care about minimize the final maximum term. More specifically, Algorithm 1 can solve (25). The parameters in Step 1, e.g., ${r_{\min}}$ and ${r_{\max}}$, can be determined by solving another two feasibility problems:

respectively. When the search is completed, all APs achieve the same minimal effective noise variance $t$. The corresponding values of $r$ and $s$ can be denoted as $r_{\text{opt}}$ and $s_{\text{opt}}$. Finally, utilizing (18) and (23), the achievable sum-rate can be obtained. In Algorithm 1, there are at most

feasibility problems that need to be solved, where ${s_{sl}}$ and ${r_{sl}}$ refer to the step size for searching $r$ and $s$, respectively.

As mentioned above, recovering $L^{\prime}$ UEs’ original data only requires $L^{\prime}$ integer linear combinations. Therefore, we propose a low-complexity AP selection algorithm for two purposes. First, only $L^{\prime}$ effective noise variance participant in the brute force search leads to a reduction in computational complexity. Second, the noise tolerance on UEs’ data can be relaxed, which contributes to the improvement of the achievable sum-rate. Recall that (10) restricts the maximum value in the coefficient vector ${{{\bf{a}}_{m}}}$ and the search space is generally small. Hence, for different APs, it is the difference on the second term in (23) that leads to a significant deviation on the effective noise variance.

Note that the average channel gain is -70 dB while the noise power is -130 dBW. Therefore, the denominator of that term is close to 1 and is several orders of magnitude smaller than the numerator. Consequently, the main factor that affects the effective noise variance across different APs is the sum of all elements in ${\bf{J}}$. In other words, decoding the estimations ${{{\bf{\hat{u}}}}_{m}}$ of APs with a high sum value of ${\bf{J}}$ will obtain a higher achievable sum-rate. Therefore, we prefer selecting APs with a high sum value of ${\bf{J}}$. Furthermore, according to (1), $g_{ml}$ is a function of $\beta_{ml}$. Then, we propose an algorithm for AP selection based on large-scale fading coefficient ${\beta_{ml}}$, which generally stays constant for several coherence intervals.

We construct matrix ${\bf{J}}$ for each AP by replacing the channel coefficient $g_{ml}$ with ${\beta_{ml}}$. For each ${\bf{J}}$, we first sum all the elements and sort the sum values in ascending order. Then, we apply the greedy AP selection for message recovery stated in [17, Algorithm 1]. Compared with the AP selection in [17], we sort the APs by firstly replacing the channel coefficient $g_{ml}$ with $\beta_{ml}$, and then calculating the sum value of all the elements in the matrix $J$, which is independent of the power allocation. According to [17], we check the columns of ${\bf{A}}$ one by one, where ${\bf{A}}={\left[{{\bf{a}}_{1}\ldots,{\bf{a}}_{M}}\right]}$, until rank requirement satisfied, therefore the computational complexity of the proposed AP selection method is no more than ${\cal O}\left({M^{\prime}+M^{\prime}{{\log}_{2}}\left({M^{\prime}}\right)+M^{\prime}{{\left({M^{\prime}-1}\right)}^{3}}}\right)$.

In successive computation, AP $m$ sends the received signal ${\bf{y}}_{m}$ and the integer linear combinations of codewords ${\bf{a}}_{m}^{T}{\bf{X}}$ to the CPU, instead of decoding the received signal ${{\bf{y}}_{m}}$ into the combinations of UEs’ original data ${{{\bf{\hat{u}}}}_{m}}$ ⁴⁴4When applying with successive computation, the interference cancellation procedure takes place at the decoder equipped at the CPU. Note that the signaling exchanges occurred per coherent interval and the fronthaul load are $2M^{\prime}n$ and $4M^{\prime}n$ with parallel computation and successive computation, respectively. Besides, the data of UEs are conveyed from the CPU to the APs through fronthaul links and then distributed to the UEs with precoding.. Define ${{\bf{A}}_{m-1}}\buildrel\Delta\over{=}{\left[{{{\bf{a}}_{1}},\ldots,{{\bf{a}}_{m-1}}}\right]^{T}}$, the CPU applies equalization factor $b_{m}$ and vector ${\bf{c}}$ to the $m$th combination as

where ${\bf{c}}={\left[{{c_{1}},\ldots,{c_{m-1}}}\right]^{T}}$. (III-C1) shows that the decoded linear combinations $\left\{{{{\bf{y}}}_{1}},\ldots,{{{\bf{y}}}_{m-1}}\right\}$ can be used as side information for reducing the effective noise experienced in the latter decoding stages for other UEs.

After choosing $b_{m}$ and c to be the MMSE projection scalar and vector, the minimum effective noise variance with transmit power matrix P is given by

where

and

Then, the computation rate region for successive computation is given as

Among $M^{\prime}$ APs, we first select the effective UEs and APs that participate in the uplink transmission. Then, we try to find the candidate integer combinations with small effective noise utilizing the AP selection algorithm. The detailed procedure has been introduced in Section III-B3. According to (III-C1), the effective noise variance is related to the decoding order of integer linear combinations. Therefore, we propose an efficient method for determining the side information matrix ${{\bf{A}}_{m-1}}$ with power control.

The main idea of successive computation is to utilize the side information and the decoded integer linear combinations to reduce the effective noise. Note that the integer linear combination decoded firstly does not have any side information to exploit. Hence, the effective noise expression for the first decoding step is similar to that of parallel computation, which is given by

where $\delta\left(m\right)$ denotes the decoding order. As the remaining combinations can reduce their effective noise with the help of side information, we can select the integer linear combination which has the minimum effective noise to decode firstly. To determine ${\bf{a}}_{\delta\left(1\right)}^{T}{\bf{X}}$, we solve the following optimization problem to obtain its local sub-optimal power allocation. As the problem (III-C2) is non-convex and translate it into non-convex is quite difficult, an off-the-shelf optimization solver, e.g. fmincon in Matlab, is adopted to obtain a suboptimal solution. Note that although the obtained solution is suboptimal, the performance of the proposed framework is still superior compared with CF and MRC, which will be verified in the simulation section. Such that, for each ${\bf{a}}_{m}^{T}{\bf{X}}$, $m=1,\ldots,L^{\prime}$, we have

Then, we calculate the effective noise $\sigma_{{\rm{succ}},\delta\left(1\right)}^{2}$ for each ${\bf{a}}_{m}^{T}{\bf{X}}$ with its own power allocation matrix and select the combination which has the minimal effective noise as ${\bf{a}}_{\delta\left(1\right)}^{T}{\bf{X}}$.

After determining ${\bf{a}}_{\delta\left(1\right)}^{T}{\bf{X}}$, we begin to determine the remaining decoding order. For the $m$th step, we determine ${{\bf{a}}^{T}_{\delta\left(m\right)}}{\bf{X}}$ depending on the effective noise and the rank of the side information matrix. More specifically, we first calculate the effective noise variance for each of the remaining integer linear combinations according to (30) and sort them in ascending order. Then, in line with the order, in each turn add the corresponding coefficient vector to the side information matrix ${{\bf{A}}_{m-1}}$, which is known from step 1 through $m-1$, to form ${{\bf{A}}_{m}}=\left[{{{\bf{A}}_{m-1}};{\bf{a}}_{m^{\prime}}^{T}}\right]$, $m^{\prime}=1,\ldots,M^{\prime}-m+1$. Finally, we select the integer linear combination which meets the constraint

to update the side information matrix. The procedure for finding ${{\bf{a}}_{\delta\left(m\right)}^{T}}{\bf{X}}$ terminates when all integer linear combinations find themselves decoding orders. The detailed procedure for searching the decoding order of combinations in successive computation is summarized in Algorithm 2. To determine ${\bf{a}}_{\delta\left(1\right)}^{T}{\bf{X}}$, we need to solve the optimization problem in (III-C2) for $L^{\prime}$ times. Then, for searching the decoding order for the left $L^{\prime}-1$ combinations in terms of the number of complex multiplications is

In successive computation, the effective noise of UEs whose data is decoded in the latter decoding stages can be reduced. At the $m$th decoding step, it is possible to use the side information matrix ${{\bf{A}}_{m-1}}$ to reduce some known individual codewords and remove them from the integer linear information ${\bf{a}}_{\delta\left(m\right)}^{T}{\bf{X}}$ without changing the effective noise variance. In particular, if the $l$th UE’s data has been recovered at the $m$th step, the data only need to tolerate the maximum effective noise among $\left\{{{\bf{a}}_{\delta\left(1\right)}^{T}{\bf{X}},\ldots,{\bf{a}}_{\delta\left(m\right)}^{T}{\bf{X}}}\right\}$. Therefore, the decoding order of UEs also has an effect on the achievable sum-rate. Searching the decoding order has been studied in some works [31], [32] with SIC. However, our problem is generally intractable and the use of convex optimization for obtaining the optimal solution is not possible. Therefore, we propose several methods to determine the decoding order of UEs and select the best one as a suboptimal solution.