I Introduction
The use of mobile phones and other portable devices is continuously increasing the demand for data in wireless networks [2, 3]. The cellular technology has evolved over time to cater for the increasing demand but although Massive MIMO (multipleinput multipleoutput) is now being used, beamforming can only mitigate the large pathloss variations in cellular deployments to a limited extent [4]. Cellfree Massive MIMO is a promising new technology to deal with the mediocre celledge performance by distributing the antennas over the coverage area and removing the cell edges by joint operation [5, 6, 7]
. Each distributed antenna location is called an access point (AP) and the APs transmit coherently in the downlink and process their received signals coherently in the uplink, leading to a higher signaltonoise ratio (SNR) without using more transmit power. The coherent joint transmission in Cellfree Massive MIMO is inherited from the classical coordinated multipoint (CoMP) beamforming design with a few colocated antenna arrays
[8, 9, 10, 11, 12]and gradually extended to scenarios with many distributed APs with few antennas. These previous network designs were mainly considering slowly fading channels where the smallscale fading realizations can be estimated perfectly and spectral efficiency (SE) utility metrics can be formulated as functions of one set of smallscale fading realizations. In contrast, a key novelty in the Cellfree Massive MIMO area is the analysis of practical fast fading channels, for which the ergodic SE is the preferred performance metric and the SE depends on imperfect channel state information (CSI) and pilot contamination. In scenarios with Rayleigh fading and some choices of linear processing, the ergodic SE can even be obtained in closed form, which makes it easier to formulate and solve practical spatial resource allocation problems.
There will be billion wirelessly connected devices by [13], which raises concerns about the power consumption and the corresponding energyrelated pollution. Cellular networks have been developed to maximize the SE and coverage, leading to the norm of transmitting at the maximum allowed power in the downlink [14]. This results in high power consumption at the base stations/APs, even when the traffic is low. Upcoming technologies should be redesigned to achieve a direct connection between power consumption and traffic load [15], so that the power is low when the users request low SEs. In the context of Cellfree Massive MIMO, energy efficiency optimization has been considered in [16, 17]. These papers considered how the fronthaul power consumption can be reduced by only serving each user by a subset of the APs, but all APs are assumed to be turned on continuously. As reported in [18, 19, 20] (and references therein), the energy efficiency of heterogeneous or cloud radio access networks can be substantially improved by also turning APs on and off. However, the operating point where the energy efficiency is maximized might not provide the service quality that the users need. Hence, [21] instead considered that each user has an SE requirement that the system must satisfy with minimum power consumption, considering both the transmit power and hardwareconsumed power of active APs. Hence, the goal of the resource allocation is for the system to deliver the required SEs with as low total power consumption as possible. These previous works considered cellular networks with slowly fading channels, where the channel takes one random realization throughout the entire transmission. To the best of our knowledge, there is no previous work on AP activation in Cellfree Massive MIMO networks, especially not with realistic fast fading channels where the ergodic SE must be considered. Moreover, the effects of practical aspects such as pilot contamination and different linear signal processing schemes have not been studied yet.
Ia Main Contributions
Motivated from the coexistence of multiple users using different services with stringent requirements, this paper considers that each user has a predetermined downlink SE requirement that the network must satisfy to not cause service interruption. The users and APs are arbitrarily distributed, thus it is likely that these SE requirements can be fulfilled without using all the APs. When minimizing the total power consumption in the downlink, we consider both the transmit power and the hardwareconsumed power. Bearing in mind that each user will mainly be served by its neighboring APs, we consider the possibility to turn off APs that are not needed to serve the current set of users. This is an important feature since CellFree Massive MIMO networks may have many APs [5, 6], where the large number is needed to provide consistent coverage but might not be needed at every time instant. We formulate the new optimization problem using rigorous closedform ergodic SE expressions for uncorrelated Rayleigh fading channels, linear precoding (either maximum ratio transmission (MRT) or fullpilot zeroforcing (FZF)), imperfect CSI, and pilot contamination. This allows us to optimize largescale networks with many APs and users. The main contributions are:

We formulate a total downlink power minimization problem, where the active APs and transmit power allocation are the optimization variables. This problem is nonconvex, but we still can obtain a globally optimal solution to both the transmit power allocation and the active APs topology by solving a mixedinteger second order cone (SOC) program.

Since algorithms that solve mixedinteger SOC programs are too complex for realtime applications, two heuristic lowcomplexity algorithms are developed by exploiting the structure of the optimization problem. The first algorithm utilizes both the optimized transmit power and sparsity, while the second algorithm only utilizes optimized transmit powers to determine which APs to turn off.

Numerical results demonstrate that there are scenarios where only a subset of the APs are needed to satisfy the SE requirements for all users and large power reductions can be achieved by turning off the remaining APs. Moreover, the lowcomplexity algorithms give total power consumptions close to the global minimum.
The rest of this paper is organized as follows: Section II gives the network model together with the downlink SE analysis. A power consumption model is introduced in Section III. Then, we formulate and solve the total power minimization problem to obtain the global optimum. Section IV propose two suboptimal algorithms with low complexity. Finally, Section V presents extensive numerical results and the main conclusions are given in Section VI.
Notations:
We use boldface lowercase and uppercase letters to denote vectors and matrices, respectively. The transpose is denoted by the superscript
and the Hermitian transpose is denoted by . The expectation operator is anddenotes a circularly symmetric complex Gaussian distribution. The Euclidean norm,
norm, and norm of a vector is denoted as , , and , respectively. Finally, the cardinality of the set is denoted by and represents the big notation.Ii System Model
We consider a Cellfree Massive MIMO network with APs and users that are arbitrarily distributed over the coverage area. A central processing unit (CPU) is connected to all APs via unlimited fronthaul links. Each AP is equipped with antennas, while there is a single antenna in each user device. We assume every user has a required SE value [b/s/Hz] that must be satisfied. At a given time instance, the users will be heterogeneously distributed and their SE requirements are likely in the interior of the capacity region of the network. Intuitively, each user will receive most of its downlink signal power from the closest APs while more distant APs typically have a negligible impact. Hence, it might suffice to only utilize a subset of the APs to satisfy the SE requirements. The remaining APs can be put into sleep mode to save power. The main goal of this paper is to find this subset and the corresponding transmit powers that satisfy the SE requirements while minimizing the total power consumption, taking the power dissipation in active APs into account.
Since the channels vary over time and frequency, the classic block fading channel model with a time division duplex (TDD) protocol is considered in this paper [22]. A coherence interval encompasses symbols and of them are dedicated to estimate the channels from uplink pilot signals. We focus on the downlink performance analysis, thus the remaining symbols are used for downlink data transmission. The channel response between AP and user is denoted by and is assumed to follow an independent and identically distributed Rayleigh fading model:
(1) 
where denotes the largescale fading coefficient involving both path loss and shadowing. Each channel takes an independent realization in each coherence interval. We assume the APs know the channel statistics, but the realizations need to be estimated from the uplink pilots.
Iia Uplink Pilot Transmission
In the uplink training phase, is a matrix gathering a set of orthonormal pilot signals that are assigned to the users. Specifically, user transmits the pilot signal with being the pilot index. We consider a fixed and arbitrary pilot assignment but note that many algorithms have been proposed in prior work [4, 23]. We let denote the subset of users assigned to the same pilot signal as user , thus it holds that
(2) 
The signal received at AP is a superposition of the transmitted pilot signals from all the users:
(3) 
where is the transmit pilot power of user and is additive noise where each element is independently distributed as . AP computes an estimate of from the sufficient statistics , which is obtained as
(4) 
Lemma 1.
The minimum mean square error (MMSE) estimate of the channel between user and AP is
(5) 
The channel estimate is distributed as
, in which the variance
is(6) 
Proof:
The proof is adopted from the standard MMSE estimation [24] to our notation. ∎
IiB Downlink Performance Analysis
In the downlink data transmission phase, each active AP constructs the precoding vectors based on their locally estimated channels that were computed using Lemma 1. Let us denote the precoding vector used by AP to steer the data signal to user as . Let denote the data symbol that is jointly transmitted to user by all the active APs and assume . The transmitted signal at AP to all users is
(7) 
where is the transmit data power that AP allocates to user . The received signal at user from all the active APs is
(8) 
where is independent additive noise with the zero mean and the variance . By using the capacity bounding technique described in [25, Section 2.3], [22, Section 4.3], a lower bound on the ergodic channel capacity of user is
(9) 
where , , and terms denote the desired signal, the beamforming uncertainty gain, and the interuser interference, respectively, which are expressed as
(10)  
(11)  
(12) 
We stress that the lower bound on the downlink channel capacity in (9) can be applied for any precoding scheme and any active AP set. To obtain closedform expressions that can be efficiently used for optimization, we now assume the active APs either use MRT or FZF precoding, which are defined for as
(13) 
where and is the
th column of identity matrix
.Lemma 2.
The downlink ergodic SE of user is
(14) 
where the effective SINR is
(15) 
The parameters and depend on the selection of precoding scheme. MRT gives and . For , FZF gives and .
Proof:
In (15), the numerator is proportional to , which is the array gain from the multiple antennas installed at each AP. The fact that the contributions from different APs are summed up inside the square is typical for coherent joint transmission. The first part in the denominator represents coherent interference from other users in the set , which is caused by pilot contamination. The remaining parts are the noncoherent interference and noise. If FZF precoding is used, each AP “sacrifices” antennas (i.e.,
spatial degrees of freedom) to cancel interference between users that have different pilots. We stress that the condition on the number of antennas
is essential for the validity of closedform SE expression if FZF precoding is utilized.The ergodic SE in (14) will hereafter be used to establish the SE constraint for each user in the network. Unlike the previous work [26, 27] that considered all APs in active mode , the new closedform SE expressions in (14) are multivariate functions of both the transmit powers and the set of active APs. One can observe that at least a single AP should be activated, say , when the network serves users with the nonzero SE requirements. We will use these expressions to formulate and solve a new total power consumption minimization problem for Cellfree Massive MIMO networks in the next sections.
Iii Total Power Minimization Problem
To maximize the energy efficiency of the network, we can minimize the power consumption while satisfying the SE requirements of the users. This section formulates a new total power consumption minimization problem subject to transmit power constraints at the APs and the required SEs of the users. The optimization variables are the active AP set and the transmit powers. The global optimum can be found by an exhaustive search, but it is extremely costly, in particular for large networks, since the problem contains both the continuous transmit power variables and discrete variables representing the active APs. We reduce the computational complexity by transforming this nonconvex problem into a mixedinteger SOC program, which is solved by the branchandbound approach.
Iiia Problem Formulation
The power consumption of the network consists of both the transmit power and power dissipation in the transceiver hardware of the active APs. Similar to [28, 16], we model the total power consumption from the all active APs in the network as
(16) 
where the first term in (16) is the total transmit power consumed by every active AP. The transmit power at AP is computed as , where the scaling factor determines the inefficiency of the power amplifiers. In the second term in (16), , models the power consumption of the transceiver chain connected to active APs and the trafficindependent power of the fronthaul connections and baseband processing. In the last part of (16), (measured in Watt per bit/s) is the trafficvarying power consumption (of the fronthaul and baseband processing) that is proportional to the SE and system bandwidth Hz.
The total power consumption minimization problem that we want to solve is
(17)  
where is the maximum downlink power of AP . The SE requirement of user is denoted as [b/s/Hz] and thus the SE in (14) must be larger or equal to this number. Note that all the transmit power variables affect all the SEs due to mutual interference.
Remark 1.
Similar optimization problems have been considered in [19, 29], but under less practical conditions. The previous optimization problems must be solved in every coherence interval since the instantaneous SE for given small scalefading realizations are considered and the operational meaning of such an SE questionable since the codeword that fits into a coherence interval is short, while the SE concept builds on (infinitely) long codewords. Such a problem formulation limits the ability to put APs into sleep mode since each AP needs to be turned on in the pilot phase and the optimization problems must be solved very rapidly. Additionally, it has previously been assumed that perfect CSI is available everywhere, thus channel estimation and pilot reuse were not considered. In contrast, we formulate problem (17) based on the ergodic SE, which is relevant in practical networks where there is channel coding over multiple coherence intervals. This reduces the power consumption since the smallscale fading variations are exploited rather than combatted. Since the optimization depends on largescale fading coefficients, instead of the smallscale fading coefficients, the solutions can be used over a relatively long period of time and inactive APs can be in sleep mode in the pilot transmission phase. Finally, we stress that our SE expressions take both channel estimation and pilot contamination effects into account.
In many scenarios, the network only needs to activate a subset of the APs to deliver the required SE to the users, meaning that . In order to study how many elements in are needed, we set and rewrite problem (17) with SINR constraints as
(18)  
where the total hardware power consumption at AP , , is simplified from (16) based on the fact that all the SINR constraints will be achieved with equality at an optimal solution [30]:
(19) 
We have reduced the computational complexity of problem (18) compared to (17) since the hardware power consumption (19) is now a constant, which transforms the objective function of problem (18) from a nonlinear to a linear function. To further simplify the problem, we introduce the notations
(20)  
(21)  
(22)  
(23)  
(24) 
where are the members of the active AP set (i.e., the indices of the active APs). In (22), is defined as for MRT precoding and for FZF precoding. The matrix in (23) has the th column and the th row is denoted as . In (24), are the indices of the users belonging to the set , and is the cardinality of the set . The operator denotes the Hadamard product. We can now obtain an equivalent epigraph representation of problem (18) as
(25a)  
(25b)  
(25c)  
(25d) 
The auxiliary variable moves the objective function of problem (18) to the first constraint in (25b). We observe that for a given , problem (18) reduces to an SOC program, as previously shown in [5, 6]. Hence, although (25) is nonconvex, it can be solved by making an exhaustive search over all possible selections of and solving each subproblem using convex optimization. Since at least one AP needs to be active if there is users with nonzero SE requirements, there are different selections of the APs that need to be considered in an exhaustive search. This naive approach to solving (18) will be very computationally costly even in a relatively small network.
IiiB Globally Optimal Solution to the Total Power Minimization Problem
Instead of making an exhaustive search, a global optimum to (25) can be achieved in a structured way by utilizing, for example, using the branchandbound approach [31]. That would result in a more efficient implementation but the computational complexity will still grow exponentially with the number of APs. However, it enables offline benchmarking in problems with up to tens of APs and users, as will be demonstrated numerically in Section V. Let the binary optimization variable mathematically characterize the on/off activity of AP . Instead of explicitly forcing the AP’s transmit powers to zero when , we can do it implicitly by replacing its maximum transmit power by . This gives the original value when the AP is active and is zero when the AP is turned off. This feature is exploited to formulate a mixedinteger SOC program as in Lemma 3.
Lemma 3.
Consider the mixedinteger SOC program
(26a)  
(26b)  
(26c)  
(26d)  
(26e) 
where is the th row of matrix and . Moreover, the vectors and are defined as
(27)  
(28)  
(29)  
(30) 
Problems (25) and (26) are equivalent in the sense that they have the same optimal transmit powers. If we denote by
an optimal solution to the binary variables
, which is obtained by solving problem (26), the optimal set of active APs in problem (25) is(31) 
Proof:
The binary variable behaves as an indicator function which uniquely determines the activity of AP . When , the related constraint (26d) is . Since , we obtain . Alternatively, AP will be turned off and it does not have any contribution to the total power consumption of the network as well as all terms that would have contained in the SINR expression are missing in (26c). By contrast, when , the related constraint (26d) becomes , which is a total transmit power constraint when AP is in active mode as shown in (25). For that reason, finding results is the same as optimizing the active APs set in problem (25) by utilizing (31). ∎
The new binary variables provide the explicit link between the hardware and transmit power consumption, which is an important factor to obtain the global optimum to problem (26). A key reason that we can preserve the SOC structure, despite adding the new binary variables, is that the binary variables are not involved in the SINR constraints (26c). Instead there is an implicit connection via the zero maximum transmit power for inactive APs. This is different from the previous approaches, e.g., [18], which also defined the on/off activity using but then included it in the SINR expressions, leading to nonconvex SINR constraints.
Problem (26) is a mixedinteger SOC program on standard form, thus a globally optimal solution can be obtained, for example, using CVX [32] in conjunction with the MOSEK solver [33]. This software applies the branchandbound approach [31] to deal with the binary variables. It is implemented in an iterative manner where the main cost of each iteration consisting three steps: finding a box, which gives a lower bound on the total power consumption, and splitting that box into the two new boxes; computing upper and lower bounds for the new generated boxes; and pruning boxes which cannot contain the optimum solution. The second step dominates the computational complexity of each iteration, while the third step decides the required number of iterations to reach the optimal solution. The following lemma provides an estimate of the computational complexity when solving problem (26).
Lemma 4.
By utilizing the standard interiorpoint method to solve a series of SOC programs, the computational complexity of the branchandbound approach to obtain a global optimum to problem (25) is in the order of
(32) 
where is the accuracy of solving SOC programs along the iterations.^{1}^{1}1For a given , the set of optimized variables is called solution to an optimization problem if the objective function at this point is at most away from the global optimum. denotes the number of iterations needed for the branchandbound approach to reach an optimal solution. Moreover and denote the cost of computing the lower and upper bounds (see Appendix A for the definitions of these bounds), which are given by:
(33)  
(34) 
Here, and denote the number of APs already in active and sleep modes, respectively, which are obtained from the previous iteration. The initial values are . Moreover, , , and the other parameters depend on the binary indices as
Parameter  
Proof:
The proof computes the computational complexity for solving SOC programs to achieve the upper and lower bounds that the branchandbound approach spends along iterations. A detailed derivation is provided in Appendix A. ∎
Lemma 4 shows that the computational complexity is the total cost of computing upper and lower bounds until reaching the global optimum. Even though the computational complexity per iteration varies as the change in both the optimization variables and the procedure needed in each iteration, (33) and (34) can exhibit such features by using the big notation. In the worst case, the branchandbound approach has the same computational complexity as an exhaustive search over all boxes with possible subsets of active APs. With a proper bounding rule, the average computational complexity can be significantly reduced by pruning many boxes. Nevertheless, an exponential growth with is expected. In Section V, we show that the branchandbound approach can find a globally optimal solution to a moderatesize network with APs.
Iv Two Suboptimal Algorithms With Lower Complexity
Motivated by the high computational complexity of solving the total power minimization problem using Lemma 3, we will now propose two algorithms that find good suboptimal solutions to problem (25) with a tolerable computational complexity and enabling implementation in large Cellfree Massive MIMO networks.
Iva Utilizing Sparsity to Turn Off APs
If the network does not need to turn on all the APs to provide the requested services from all the users, we know that many of the power variables will be zero. Hence, we can try to find the optimum AP subset by expressing (18) as a sparse reconstruction problem where we try to push many of the transmit power variables to become zero. To this end, we first reformulate problem (18) as a mixed norm optimization problem.
Lemma 5.
The original problem (18) has the same optimal transmit powers as the following problem
(35)  
where and each function is defined based on the transmit powers of AP as
(36) 
Moreover, if we denote by the optimal set of all transmit powers to (35), then the set
(37) 
is the optimal set of active APs to problem (18).
Proof:
When AP is in sleep mode, it assigns zero transmit power to all users (i.e., ). This AP has no contribution to the objective function of problem (35) due to and thus we can make the definition of as in (36). The optimal set of active APs is defined based on the groupsparsity structure as in (37). ∎
Lemma 5 shows that we do not need to define separate variables for optimizing the active APs set , but we can implicitly determine if AP is active or not by checking if or . The reformulated problem (35) reduces the number of optimization variables compared with (26), and in particular, all the optimization variables are now continuous. Nevertheless, problem (35) is still nonconvex due to norm in the second part of the objective function. However, we can relax norm to an norm for some . This is a standard relaxation technique that retains sparsity and we stress that it also gives better sparsity than an norm relaxation (cf. Figs. and in [34] for illustrations).^{2}^{2}2Strictly speaking, a value does not lead to norm since the subadditive property is not satisfied [35], but the “norm” terminology has anyway been used for many years and we adopt this convention. Therefore, we adopt the norm optimization to obtain a relaxation of problem (35) as^{3}^{3}3From the range of the considered norms, the condition as in (38) leads to .
(38)  
The objective function of problem (38) treats every vector as an entity in when seeking a sparse solution. We will utilize this groupsparse property of the transmit power coefficients to solve problem (38) in a novel way, which differs from previous works that considered elementbased [36] or beamformingvectorbased sparsity [21]. Even though problem (38) remains nonconvex after the norm relaxation, we can find a stationary point by adapting the iteratively reweighted least squares approach from [37], that was originally developed for problems with componentwise sparsity. Specifically, after removing the exponent and the hardware power consumption in the objective function, problem (38) can be recast as
(39)  
By noting that the groupsparse property implies the support of vector is the empty set, we can provide an iterative algorithm obtaining a stationary solution to problem (39).
Theorem 1.
Since the feasible set is convex, we can construct an iterative algorithm that starts with the given initial weight values and in iteration solves the SOC program
(40)  
to yield the solution , for which
(41) 
is the optimal transmit powers for AP at iteration . After that, the weight values are updated for the next iteration as
(42) 
where is a sufficiently small positive damping constant with . When , the proposed iterative process exhibits the properties below:

If an arbitrary AP has zero transmit power at the optimum of iteration , this AP will have zero transmit power in all the following iterations.
Proof:
The proof is based on the convergence property of the iteratively weighted least squares approach that has been adapted to our framework. The detailed proof is available in Appendix B. ∎
Theorem 1 guarantees a monotonically decreasing objective function and the main computational cost is to solve (40) in each iteration. The iterative process reaches a stationary point to problem (39). The second property supports turning off APs along iterations. The damping constant is introduced to cope with a numerical issue that can appear when updating the weight values (42), i.e., when . Even though the convergence properties in Theorem 1 are proved by the descending of along iterations, a sufficiently small constant value also works well in the simulations as reported in [38]. The stopping criterion can be selected by comparing two consecutive iterations. For a given accuracy , we can verify if , where is the difference of the objective function to problem (40):
(43) 
The stationary point achieved by Theorem 1 may not be a globally optimal solution to problem (35) due to the norm relaxation and the inherent nonconvexity. Consequently, we will not use the solution from Theorem 1 as the final solution but instead as an indication of which APs to further turn off. More precisely, we compute the transmit power that the APs utilize at the solution from Theorem 1 and reorder the APs in increasing power order.^{4}^{4}4We have implemented other possible orderings, for example, based on the total transmit power per AP or the relative maximum received power allocated to the users. For brevity, we are only considering the one that gave the best results.
Let us denote by the optimized transmit powers obtained by Theorem 1, for which a new parameter standing for the contribution of AP is defined. Specifically, is the total received power of the users that is transmitted by AP as:
(44) 
In order to classify the contribution of each AP in
to provide the required SINRs, we define a heuristic ascending order as^{5}^{5}5Note that APs that were inactive at the solution obtained from Theorem 1 are still considered. Since the numerical precision is limited, these APs will be assigned extremely small but nonzero power, which leads to a unique ordering in (45).(45) 
where is a permutation of . We will now decide how many APs to utilize and keep only those with the largest values using the ordering in (45). We further compute an auxiliary variable
(46) 
We begin by defining a range , with the condition . Specifically, the initial values are and , then we compute the middle point at iteration as
(47) 
where denotes the floor function. We now reorder the AP indices according to (45) and consider setting the first APs into sleep mode. Then, the active APs set is given by which has the cardinality . We now solve the following SOC program:
(48)  
where is an upper bound defined by the sublevel set in the epigraph representation of problem (18) when the APs are in active mode. From the solution to problem (48), the new upper or lower bounds on the number of inactive APs are updated as
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