Performance Analysis and Bit Allocation of Cell-Free Massive MIMO Network With Variable-Resolution ADCs

This paper concentrates on cell-free massive multiple-input and multiple-output (MIMO) network with variable-resolution analog-to-digital converters (ADCs). In such an architecture, all ADCs equipping at any access point (AP) can use arbitrary bit resolution to realize adaptive quantization and reduce power consumption. Under this circumstance, we first introduce a quantization-aware channel estimator based on linear minimum mean-square error (LMMSE) theory. On this basis, intra-AP and inter-AP bit allocation problems are investigated to maximize channel estimation quality subject to the total number of quantization bits. By leveraging the statistical characteristics of the estimated channels and estimation errors, we then derive the theoretical expressions of the achievable uplink spectral efficiency (SE) for maximal ratio combining (MRC) and minimum mean-square error (MMSE) combining, respectively. Furthermore, to maximize the sum SE under the constraint of total ADC quantization bits, we also investigate intra-AP and inter-AP bit allocation problems for both single-user and multi-user scenarios. Finally, simulation results confirm that our theoretical analyses are correct and accurate. In addition, we resort to numerical results to achieve some new insights and verify the advantages and conclusions pertinent to the proposed bit allocation techniques.

tems [1], by invoking a considerable amount of antennas. Thanks to its merits, e.g., more spatial degree-of-freedoms (DoFs) and higher array gains, massive MIMO has also been perceived as a cornerstone many other technologies build on. In the sense of a cell-free architecture, Ngo et al. [2] coined a terminology of cell-free massive MIMO in which a large number of geographically distributed access points (APs) with large-scale antenna arrays serve many randomly distributed user equipments (UEs) over a wide area. In such an architecture, there is no concept of cell and all APs cooperate with the assistance of central processing units (CPUs) and fronthauls [3], [4], [5], [6], [7]. On the positive side, cell-free massive MIMO has ability to enhance ubiquitous coverage, provide uniform service quality, boost channel capacity, reduce interference, and increase link reliability. On the negative side, a tremendous amount of power consumption and hardware cost, caused by numerous radio frequency (RF) chains followed by high-resolution analog-to-digital converters (ADCs), becomes one of the major drawbacks. To overcome this practical barrier, it was substantiated by [8] and [9] that reducing ADC's resolution is a prospective solution. Along this direction, low-resolution ADCs have been incorporated into cell-free massive MIMO networks [4], [5], [6], [7].
The capacity of fronthaul links is limited [4], 1 under which it is not so clear how to efficiently use low-resolution ADCs. Using one-bit ADCs for all antennas might satisfy the constraint of fronthaul capacity, but the performance would be poor because of such coarse quantization. Alternatively, using a few high-resolution ADCs could also meet the fronthaul capacity, but doing so loses MIMO gain because only a few antennas are exploited to obtain high resolution signal. It is not straightforward to find the appropriate use of low-resolution ADCs because the optimal resolution dynamically changes depending on the channel gain, fronthaul capacity, and transmit power. If each ADC's resolution is predetermined, it is difficult to use varying optimal resolution, resulting in SE degradation. To overcome this issue, the use of variable resolution ADC is a feasible solution. As pointed out by [10], performance can be improved relying on a variable-resolution ADC or resolution-adaptive ADC architecture, which provides extra degree-of-freedoms for system design and optimization [11]. By using bit allocation within and between APs, it is possible to improve the sum SE of cell-free massive MIMO under the constraint of fronthaul capacity. These observations are the main motivations of our work.
Regarding variable-resolution quantization, we have the following brief review. In cloud radio access networks, Park et al. [12] investigated how a remote radio head selects the appropriate ADCs in a mixed-ADC pool comprised of multiple ADC units with various resolutions. More recently, bit allocation has been investigated in [13], [14], [15], [16], and [17] for mmWave systems. In particular, to minimize the mean square quantization error of received analog signals, Choi et al. [13] proposed two bit allocation strategies under a total ADC power constraint, but the bit allocation algorithms are designed based on perfect CSI. Wang et al. in [14] optimized the pilot sequences, the number of quantization bits and the hybrid receiver combiner by leveraging fractional programming (FP) techniques. To maximize system performance and reduce total receiver power consumption, Chen [15] designed the ADC resolution profile. The authors in [16] maximized the system throughput of the scheduled users by optimizing the number of quantization bits along with transmit power and hybrid combiners. Ahmed et al. [17] presented an optimal energy-efficient bit allocation algorithm under a power constraint. Considering RIS-aided massive MIMO systems, Wang et al. in [18] optimized the ADC's resolution, the transmit power, the passive reflection coefficients of the RIS and the hybrid combiner via the Lagrangian dual transform and FP method. Although it has been shown that adopting variable-resolution ADCs can improve system performance, these studies have not investigated the use of variable-resolution ADCs over cell-free networks. To fill this gap, the authors in [19] investigated ADC bit allocation (BA) and demonstrated that cell-free massive MIMO systems benefit the most from optimizing the ADC bit allocation.
Inspired by the preceding observations, we focus on cell-free massive MIMO networks with variable-resolution ADCs and derive the theoretical achievable uplink SE for both MRC and MMSE combiners. On this basis, we further investigate the bit allocation problems for channel estimation and data transmission, respectively. The main contributions of this article are summarized as follows: • In the context of variable-resolution ADCs, we first introduce a quantization-aware channel estimator based on linear minimum mean-square error (LMMSE) theory.
To improve channel estimation quality subject to the constraint of total quantization bits, we then investigate the bit allocation problem within one AP by minimizing the total weighted estimation errors. After that, we further solve the bit allocation problem among all concerned APs. Regarding the two types of optimization problem, we finally achieve the corresponding bit allocation strategies by solving the relevant Karush-Kuhn-Tucker (KKT) conditions. • Under imperfect CSI, we derive the theoretical expression of achievable uplink SE in closed form for MRC. When it comes to MMSE combining, it is challenging to derive the exact closed-form achievable uplink SE due to the existence of matrix inversion. Nevertheless, we resort to the additive quantization noise model (AQNM), use-andthen-forget (UatF) technique, and random matrix theory to facilitate the derivations of an asymptotic alternative. On the basis of theoretical SEs, we not only investigate the bit allocation problem within one AP by maximizing the sum SE over all users, but also study the bit allocation problem among all APs for single-user and multi-user scenarios, respectively. For both types of optimization problem, we also obtain the corresponding bit allocation strategies by transforming the associated optimization problems into convex ones. • We use simulation results to corroborate that our theoretical analyses are correct and tight. Moreover, in the case of uncorrelated channels, we also provide the simulation results with respect to bit allocation for both uplink channel estimation and data transmission. Given the total number of quantization bits, it is shown that equal ADC bit allocation is preferable within one AP. In regard to the bit allocation amongst different APs, we show that the proposed bit allocation outperforms the fixed one and more bits should be assigned to the AP with larger aggregated effect of received signals. To the authors' best knowledge, this is the first study to undertake the intra-AP and inter-AP bit allocation optimization for both channel estimation and data transfer based upon theoretical analyses. In comparison with the existing literature [4], [5], [6], [7] that focuses on fixed-resolution ADCs, we consider variable-resolution ADCs at all APs and investigate the bit allocation problems which are not discussed in [4], [5], [6], and [7]. Unlike [13], where the bit allocation builds on instantaneous CSI, we derive the theoretical achievable uplink SEs with respect to estimated CSI and optimize ADC quantization bits based on statistical estimation quality and SEs. In comparison with [19], we optimize ADC bits using different criteria, i.e., weighted normalized mean square error (WNMSE) for channel estimation and theoretical sum achievable SEs for both MRC and MMSE. The bit allocation problems in [19] are based on pilot distortion and achievable SE obtained by Monte-Carlo simulation. In contrast to [13], [14], [15], [16], [17], and [18], wherein RIS-aided and hybrid mmWave systems are considered, we pay attention to cell-free massive MIMO; accordingly, the corresponding theoretical analysis and bit optimization are different.
Organization: The remainder of this paper proceeds as follows: In Section II, we introduce the cell-free massive MIMO model with variable-resolution quantization and the procedures of signal transmission. Section III investigates channel estimation and the associated bit allocation. For both MRC and MMSE combiners, Section IV provides the performance analysis in terms of SE. In Section V, we investigate the bit allocation problems for data transmission. Simulation results are presented in Section VI. The major conclusions are drawn in Section VII.
Notations: Throughout this paper, the superscripts (·) −1 , (·) H , and (·) T represent matrix inverse, Hermitian transpose, Illustration of cell-free massive MIMO network with variableresolution/resolution-adaptive ADCs. transpose, respectively. I N denotes a N × N identity matrix and ||A|| indicates the Euclidean norm. x ∼ CN (0, Σ) denotes a complex Gaussian stochastic vector with mean 0 and covariance Σ. E{·} signifies the expectation and tr{·} refers to the trace. diag{A 1 , · · · , A L } is a block diagonal matrix with A 1 , · · · , A L on the main diagonal. a denotes the largest integer no greater than a and a denotes the smallest integer no less than a.

II. SYSTEM MODEL
As illustrated in Fig. 1, we concentrate on a cell-free massive MIMO network, in which L APs with M antennas jointly serve K single-antenna users within a large area. All APs are connected to a CPU via fronthaul links with capacity constraints. We consider that only partial APs serve one concerned user because some APs might be far away from this user and the associated channel conditions would be poor. We further assume that only partial users are served by one considered AP due to limited pilot resource. Accordingly, we let M k ⊂ {1, 2, · · · , L} and |M k | =L k denote the subset and the number of APs serving user k, respectively. The subset of users served by AP l is represented by D l ⊂ {1, 2, · · · , K}. We assume in the sequel that variable-resolution ADCs are used at each AP for uplink transmission. One reason is that the capacity of fronthaul links is limited and variable-resolution ADCs make the implementation of low-resolution quantization possible, particularly for large-scale MIMO networks. In contrast, high-precision quantization results in a tremendous amount of high-resolution data exchanged between the CPU and each AP. The volume of these data might exceed the capacity of fronthaul links. The other reason is that adaptive bit allocation can be realized by employing variableresolution ADCs, which has the advantage of flexibility and can cut power consumption and hardware cost compared with the usage of fixed high-resolution ones. Additionally, because wireless propagation channels are assumed to be constant throughout one coherence time interval, the time division duplex (TDD) protocol can be adopted. In this case, one coherence block with τ c symbols is divided into uplink pilot transmission (τ p symbols), uplink date transmission (τ u symbols) and downlink data transmission (τ d symbols). In this study, we focus on the first two transmission stages.

A. Uplink Pilot Transmission
During the stage of uplink pilot transmission, we assume that user k transmits pilot sequence where p k denotes the transmit power chosen by user k. N p l ∈ C M×τp represents the additive white Gaussian noise (AWGN) with independent and identically distributed (i.i.d.) elements following the distribution CN (0, σ 2 ). Moreover, h lk in (1) denotes the spatially correlated Rayleigh fading channel from user k to AP l and is given by Note that g lk ∈ C M×1 ∼ CN (0, I M ) denotes the small-scale fading channel, while R lk ∈ C M×M represents the spatial correlation matrix associated with user k and AP l. On this basis, the large-scale fading coefficient from user k to AP l can be calculated as β lk = tr(R lk )

M
. In the special case of uncorrelated channels, we have that R lk = β lk I M .
To approximate the nonlinear effect of low-resolution quantization, we resort to the well-known AQNM [4], [5], [6], [7], which can be regarded as a special case of the Bussgang decomposition for some distortion functions [20]. The AQNM can approximately transform nonlinear distortion into a linear operation. Thus, the quantized version of Y p l is given bỹ where D l = diag{α p l1 , · · · , α p lM } denotes the distortion caused by the ADCs at AP l during the stage of uplink pilot transmission. If antenna m at AP l uses b p lm bits for quantization, the distortion coefficient α p lm is given by Table I [4], [5], [6], [7]. According to the AQNM, we assume that Q p l is the additive Gaussian quantization noise and is statistically uncorrelated with Y p l [20].

B. Uplink Data Transmission
During the stage of uplink data transfer, all users simultaneously transmit their uplink information signals. Suppose that s k represents the information symbol transmitted by user k and is randomly generated according to a complex Gaussian distribution under constraints E{s k } = 0 and E{|s k | 2 } = 1. Thus, at AP l, the received signal is given by where n d l ∼ CN (0, σ 2 I M ) is the uplink AWGN during uplink data transmission. Subsequently, y d l is quantized by ADCs and the corresponding quantized version is modeled as where A l = diag{α d l1 , · · · , α d lM } denotes the distortion originating from the ADCs at AP l during data transmission. The relation between α d lm and b d l1 is similar to Section II-A. The covariance of quantization noise q d l is expressed as Recalling from (4), one has that E{y d whereby R q d l can be further simplified as with q l = K i=1 p i β li + σ 2 . Note that the cross-correlation of quantization noise in R q d l can be safely neglected in massive MIMO when there are sufficiently many users or i.i.d. Rayleigh fading channels are considered [21].

III. UPLINK CHANNEL ESTIMATION AND BIT ALLOCATION
For the MRC and MMSE combiners to be investigated, it is crucial to acquire the knowledge of the channel responses from the users to the APs serving them. Therefore, this section provides the techniques for CSI acquisition and bit allocation under the involvement of variable-resolution ADCs. Towards this purpose, we consider the quantize-and-estimate (QE) approach mentioned in [4] and [24]. Specifically, all APs first quantize the received pilot sequences and then transmit the quantized versions to the CPU where channel estimation is performed.

A. MMSE-Based Channel Estimator
To retain the orthogonality between pilot sequences, we consider that the pilot sequence φ k ∈ C τp×1 assigned to user k satisfies φ H k φ k = τ p and φ H k φ k = 0 for k = k. In pragmatic systems, the number of users might exceed that of pilot resource reserved for uplink training within one coherence block so that some users would share the same pilot. We denote by t k ∈ {1, · · · , τ p } the index of pilot assigned to user k and let the set of users that utilize pilot t k be We shall provide the scheme of pilot assignment and the creation of set M k prior to channel estimation. In the case of K > τ p , we suppose that user k utilizes pilot k when k τ p . For user k > τ p in the remaining users, the index of the master AP associated with this user is given by l k = arg max l∈{1,··· ,L} (β lk ).
To reduce the impact of pilot contamination, we select the pilot that causes minimum total interference from and to other users which might use the same pilot as the considered user.
For user k > τ p , its master AP identifies the index of the pilot according to where t i denotes the pilot used by user i, while L k = {l |t l j = t, j = 1, · · · , k − 1} is a subset of APs. Note that t l j denotes the pilot used by user j, which is served by AP l . The pilot determined by (8) is assigned to user k and the procedure then continues with the next user. After that, each AP goes through each pilot and identifies which of the users have the largest channel gain among those using that pilot. To be specific, for l = 1, · · · , L and t = 1, · · · , τ p , we find user that AP l serves on pilot t according to k = arg max i∈{1,··· , K},ti=t When it comes to channel estimation, to remove the interference from other users, AP l first correlates the received signal (3) with the associated pilot signal φ k to obtainỹ p lk = 1 √ τpỸ p l φ k , which is given bỹ According to the AQNM, the covariance of q p lk at AP l is approximated as where Similarly, the cross-correlation of quantization noise is also neglected.
Subsequently, we ask for the LMMSE estimation theory in [25] to obtain CSI based onỹ p lk . As a result, h lk can be estimated according tô where Note that the estimator in (11) is suboptimal because the quantization error is approximated to be Gaussian. Again, on the basis of the LMMSE estimation theory in [25], the estimation error of h lk is given byh lk = h lk −ĥ lk . It follows from [3] thatĥ lk andh lk are uncorrelated with each other. and The estimation quality of h lk is characterized through the NMSE, which is defined as [3] If user i and user k use the same pilot and R lk is invertible, then we haveĥ This implies thatĥ li andĥ lk are correlated with each other in the presence of pilot contamination and the corresponding cross-correlation matrix is given by Remark 1: Based on [32, Sec. 3.1.1], all pilot matrices that satisfy φ H k φ k = τ p and φ H k φ k = 0 for k = k are equivalent in terms of estimation performance, but the choice has an impact on the practical implementation. In fact, only the mutual orthogonality and norm ||φ k || 2 determine the estimation accuracy.
Remark 2: In the special case of uncorrelated channels, we have that R lk = β lk I M , with which Ψ lk in (12) can be simplified as (10) and (13), one has Accordingly, (15) can be simplified as . The estimation quality of each user is different because the path loss measured at a concerned AP will differ from user to user. At this point, to give more preference to the users with better channel conditions, it is reasonable to consider weighted NMSE (WNMSE). Suppose that w lk is the weight for user k ∈ D l and w lk = 0 if k / ∈ D l . Thus, the sum weighted NMSE (SWNMSE) over all APs and users is given by It should be pointed out that g lk (α p lm ) in (18) is not always convex or concave with respect to α p lm because the sign of the term i∈P k p i τ p β li +σ 2 −q l is not deterministic. So, we shall simplify g lk (α p lm ). To this end, when i∈P k p i τ p β li + σ 2 q l , it follows that g lk (α p lm ) the SWNMSE in (19) has a upper bound via In the remaining parts of this section, we plan to investigate bit allocation problems by minimizing the upper bound given by (20) during the phase of uplink channel estimation.

B. Bit Allocation Within One AP
Without loss of generality, we here concentrate on the bit allocation between different antennas within AP l. According to (20), we minimize the upper bound of SWNMSE over the users in D l . For this purpose, we should maximize Therefore, ρ l is a monotonically decreasing and symmetric function with respect to b lm for m = 1, · · · , M. Accordingly, under the constraint which reveals that the quantization bits should be equally allocated to all antennas within one AP. This conclusion is in accordance with our intuition because the aggregated large-scale fading coefficients are identical for all antenna elements which are co-located at one concerned AP.

C. Bit Allocation Among APs
When it comes to the bit allocation between APs, we consider all antennas at AP l use b l bits for quantization, i.e., D l = α p l I M . Suppose that all APs aim to minimize the SWNMSE over all the served users, i.e., (19). We denote by B the total number of quantization bits. Thus, the optimization problem relating to bit allocation among APs can be formulated as To tackle the non-convex discrete constraint of quantization bit, we relax it into a continuous constraint, i.e., b p l 1, and use α p l = 1−a4 −b p l . According to (20), we intend to minimize the upper bound of SWNMSE and let for notational conciseness. Thus, the bit optimization problem in (21) can be reformulated as The objective function and constraints in (23) are convex, and we have the following theorem. Theorem 1: The ADC quantization bit number related to the lth AP is given by Proof: The proof is available in Appendix B. To obtain an integer solution of quantization resolution, (b p l ) should be rounded to its nearby integer via [13], [14] b where 0 ε 1 is selected such that constraint Remark 3: It should be pointed out that Theorem 1 is applicable to any method of weighting. If one seeks to improve the estimation quality for the users with strong channel conditions, these users will be given more advantage. In this sense, we could let Accordingly, we can conclude from (24) that more quantization bits should be assigned to AP l with larger u l . These conclusions will be verified by the simulation in Section VI.

IV. UPLINK SPECTRAL EFFICIENCY
Because there exists matrix inverse in the MMSE based combiner, the exact theoretical analysis is challenging, which is the main motivation of this study. In addition, by introducing variable-resolution ADCs, performance derivations are difficult to some extent. Specifically, in a technical aspect, compared with un-quantized cases, there are quantization errors which are related to quantization bits and should be considered during the derivations. In a mathematical aspect, leveraging the AQNM, we model the impacts of variable-resolution ADCs by diagonal matrix A l , which is also involved into quantization errors. Considering these difficulties, in this section, we derive the theoretical expressions of the achievable uplink SE in closed form for both MRC and MMSE combiners with the assistance of Appendix A.

A. Common Received SINR
Without loss of generality, we focus on analyzing the achievable uplink SE for user k which is served by the APs in M k . Let the total number of antennas serving user k bẽ M k =L k M . According to (5), at the CPU, the collective uplink data signal from the APs in M k can be written as are, respectively, stacked byỹ d l , h li , n d l and q d l for all l ∈ M k in a vector form. By left-multiplying Suppose that the CPU uses combining vector v k to recover the signal s k for user k. Accordingly, the estimateŝ k is calculated asŝ k = v H kȓ d k , which can be unfolded aŝ In the right-hand side (RHS) of (29), the first term is regarded as the desired signal, the second term represents the aggregated interference originating from the other users, the third term indicates the effect of the entire channel estimation errors, the fourth term is the influence of the noise, and the fifth term denotes the impact of quantization errors. Since the estimated CSI, the channel estimation errors, the AWGN, the quantization noise, and the data symbols transmitted by different users are mutually uncorrelated, the second term to the fifth term in the RHS of (29) can be treated as the effective noise. We assume the effective noise as an equivalent Gaussian random variable to consider the worst case. As such, an achievable SE of user k is which is a lower bound on the ergodic capacity [3] and the expectation is with respect to the channel estimates. Moreover, the instantaneous effective signal-to-interferenceand noise ratio (SINR) in (30) is calculated as where We also assume that the cross-correlation in R d k can be safely neglected [21]. Based on the Lemma 1 in [26] and by referring to the use-and-then-forget (UatF) methodology [2], [3], an alternative achievable SE is given by which is an approximation of the ergodic capacity. Moreover, the statistical effective SINR is where (32) and (33) and the expectations are with respect to the channel estimates or estimation errors.
Remark 4: Intuitively, SE R,UatF k should be smaller than SE R k due to A R k E{A R k }. This conjecture is hard to prove analytically but has been verified by numerical experiments [3].

B. Closed-Form SE for MRC
If MRC is applied, one has that v k =ĥ k k . Whilst this combiner cannot mitigate interference, it has the lowest computational complexity. Using the covariances of estimated channels and estimation errors in Section III-A, we have the following Theorem 2.
Theorem 2: For the MRC technique, the terms in (33) can be obtained in closed form. Concretely, A MRC k is given by Considering pilot contamination, B MRC k is expressed as The other terms in (33) are, respectively, given by Proof: The proof is available in Appendix C. Note that, for i = 1, · · · , K, (33) and (32) yields the closed-form SE for user k in the case of using MRC.

C. Closed-Form SE for and MMSE
In an effort to recover the information s k for user k, if the APs in M k attempt to suppress the interference components stemming from the other users, MMSE combining can be adopted. In this sense, one has that where C i = diag{C li } l∈M k . Applying the covariances of estimated channels and estimate errors in Section III-A, we have the following Theorem 3.
Proof: The proof is available in Appendix D. It should be noted that Γ k is obtained via Lemma 3 by . Moreover, Γ ki ,Γ ki , Γ k,n , and Γ k,q are obtained via Lemma 4 by letting respectively. Plugging (40)-(44) back into (33) and (32) yields an asymptotic achievable uplink SE for user k in the context of MMSE combining. Note that the asymptotic results depend on the statistical characterization of channels rather than the instantaneous CSI. In addition, if we consider the effect of quantization noise in the MMSE combiner, (39) can be rewritten as The corresponding closed-form SE can be obtained analogous to what we did in Theorem 3. Because the impacts of quantization errors are coupled in both (45) and the expression of SINR and there exists matrix inversion, it is considerably challenging to explicitly model the impacts of quantization errors for the MMSE-based combiner in (45). Therefore, when it comes to the bit allocation in the sequel, we will concentrate on the MMSE combiner in (39).

V. BIT ALLOCATION FOR DATA TRANSMISSION
Because the major purpose of this section is to study the bit allocation associated with uplink data transmission, we assume that the bit allocation discussed in Section III is applied to acquire the imperfect CSI, which is then used for data detection. In addition, to make mathematical analysis tractable, we also concentrate on spatially uncorrelated channels. In this sense, we separately investigate bit allocation problems within one AP and among different APs.

A. Bit Allocation Within One AP
Without loss of generality, we here concentrate on the bit allocation between different antennas within AP l. In the case of uncorrelated channels, one has It is evident that B lk is a diagonal matrix and all diagonal elements are identical. Before proceeding to optimize ADC bits, we first rearrange the expressions of the statistical effective SINR for both MRC and MMSE techniques. Regarding the MRC scheme for user for simplicity in notation and recalling from (7) and (46), we can further convert (33) into where γ l = M m=1 As for the MMSE combiner for user k ∈ D l , by referring to (44), we find that Γ k,q and Γ k are both block-diagonal matrices. Accordingly, we let Γ k,q = diag{Γ lk,q } l∈M k and Γ k = diag{Γ lk } l∈M k . By separating the contributions of AP l and the other APs in E MRC k and defining for notational convenience, we can reformulate (33) as where γ l = M m=1 1−α d lm α d lm andB lk = B lk + Ξ lk . 2 Subsequently, we move on to optimize ADC bits within AP l. It is observable from (48) and (51) that all the users served by AP l have the same γ l given any bit allocation 2 In the case of uncorrelated channels, Ξ lk = 1 scheme within AP l. Because we assume that the quantization bits are predetermined for channel estimation, A R k , B R k , C R k , and D R k are unrelated to the quantization for data transmission. Moreover, because we here focus on the bit allocation within AP l, E R,c k is also unrelated to the bit allocation schemes selected by AP l. In this context, if γ l arrives at its minima, all users served by AP l will obtain the maximal SE. By relaxing the discrete constraint of quantization bit into a continuous one and using α d lm = 1 − a4 −b d lm in γ l , the bit allocation problem within AP l is formulated as The objective function and constraints in (52) are convex, which gives the following conclusion. Theorem 4: To achieve the maximal SE for both MRC and MMSE combiners during uplink data transmission, the number of quantization bit related to the mth antenna at AP l is Proof: The proof is available in Appendix E. Remark 5: In the case of uncorrelated channels, Theorem 4 reveals that the quantization bits should be equally allocated to all antennas within one AP during the stage of data transmission. This conclusion is consistent with our intuition because the aggregated strength of received signals is almost identical for all antenna elements which are co-located at one AP.

B. Bit Allocation Among APs Under Single-User Scenario
When turning our attention to the bit allocation between all L APs, we use the conclusion in Theorem 4 and consider that every antenna at AP l uses b d l bits for quantization, i.e., A l = α d l I M . Suppose that there exists only one user served by all APs in coverage area and B denotes the total number of quantization bits for all APs. Because we assume that the quantization bits are predetermined for channel estimation, A R 1 , B R 1 , C R 1 , and D R 1 are independent of the quantization resolution during data transfer. In this regard, it is straightforward to conclude that this single user obtains the maximum SINR when E R 1 arrives at its minima. Thus, the optimization problem of bit allocation between all considered APs can be formulated as where We also relax the non-convex discrete constraint of quantization bit into a continuous constraint, i.e., b d l 1, and use 2 . Therefore, using (7) and (50), we can further transform (54) into where The objective function and constraints in (56) are convex and we have the following results.
Theorem 5: The KKT conditions of (56) are given by , the Lagrangian function is expressed as Then, solving ∂L ∂b d l = 0 and ∂L λ = 0 leads to (57) and (58), respectively.

Remark 6:
In view of (57), we have λ Thus, we can conclude that the optimal ADC bits are allocated based on κ R l1 , and larger κ R l1 results in more quantization bits allotted to AP l. Note that Theorem 5 holds for any R ∈ {MRC, MMSE}.

C. Bit Allocation Among APs Under Multi-User Scenario
Under a multi-user scenario, we also assume that every antenna at AP l uses b d l bits for quantization, i.e., A l = α d l I M . Prior to formulating the optimization problem, we first rewrite the SINR for MRC and MMSE, respectively. To be specific, with regard to MRC, if we let As for MMSE, we let Ψ MMSE Based on (50), we can convert (33) into Based upon the preceding discussions, if we define then (59) and (60) can be compacted into Subsequently, recalling from (32), one can obtain the sum achievable SE according tō Due to the fact that the logarithmic functions in (62) are monotonically increasing, it holds that Finally, we seek to maximize a lower bound on the sum achievable SE. Because A R k in (61) is not related to quantization, maximizing the RHS of (63) is equivalent to mini- . We also relax discrete quantization bits into continuous ones, i.e., b d l 1, and use By doing so, the bit allocation problem is reformulated as Theorem 6: The objective function in (64) is convex with respect to b d l for all l = 1, · · · , L. Proof: The proof is available in Appendix F. Because the objective function and constraints in (64) are all convex, we have the following KKT conditions. Theorem 7: The KKT conditions of (64) are expressed as Proof: The Lagrangian function is expressed as Then, computing ∂L ∂b d l = 0 and ∂L λ = 0 leads to (65) and (66), respectively.
Remark 7: The analytical solutions cannot be obtained in closed form by solving the KKT conditions in (65) and (66). Fortunately, we can achieve the feasible solutions via CVX tool because (64) is a convex optimization problem. Similarly to (25), the solutions obtained by CVX tool should be rounded to their nearby integers, respectively. Note that Theorem 7 holds for any R ∈ {MRC, MMSE}.

D. Implementation of Variable-Resolution Quantization
The decision of bit allocation at the CPU can be sent to each AP via fronthaul links without extra hardware costs. Firstly, the rate of bit allocation information is significantly lower compared with that of quantized data. Therefore, it is feasible to deliver the bit configuration to each AP with few signaling loads. Secondly, as described previous, passive optical networks can be used as fronthaul networks. Therefore, the delay associated with sending bit configuration can be neglected. Thirdly, because the bit allocation results depend on long-timescale information, e.g., the large-scale fading coefficients, the transmit power, and the fronthaul capacity, it is workable to transmit the bit configuration in a framebased manner. In other words, it is unnecessary to frequently update the bit configuration.
After receiving the bit configuration, each AP changes the resolution of ADCs equipping at every antenna. It is possible to vary their resolution based on the bit configuration because variable-resolution ADCs are used at all APs. To the authors' best knowledge, there is an infineon technologies company named Cypress which have produced the variable-resolution ADCs. Moreover, concerning the hardware design of variableresolution ADCs, the authors in [27], [28], [29], [30], and [31] developed many types of ADCs that can dynamically change their resolution with low power and cost. These ADCs are capable of operating at variable resolution and make it possible to modify their quantization bits according to the bit configuration.

VI. NUMERICAL RESULTS AND DISCUSSIONS
In the simulation to follow, all APs and users are uniformly distributed within a square of size 0.4km×0.4km. We consider τ c = 200 and assume that all active users transmit with equal power. The large-scale fading coefficient β lk in dB is computed as β lk [dB] = −30.5 − 36.7 log 10 (d lk ) + sh lk [3], where d lk (in m) denotes the distance between AP l and user k and sh lk ∼ N (0, 4 2 ) models the shadowing effect. Given one realization of the random locations of APs and users, Monte-Carlo simulation results are obtained over 500 realizations of small-scale fading channel and AWGN, whereas  Fig. 2 compares the theoretical results and the simulated ones for MRC and MMSE combiners. Note that "Ana." and "Sim." represent the analytical and simulation results, respectively. To validate the technical assumptions related to quantization noise in analysis, during the Monte-Carlo simulation, the covariance of q d l is obtained by averaging H over 1000 realizations of channel and AWGN. Additionally, all antennas use the same quantization bit to obtain the results in Fig. 2. For the MRC detection, it can be observed from the left part of Fig. 2 that the theoretical results basically coincide with the simulation ones despite the small gap. Meanwhile, it is evident from the right part of Fig. 2 that the asymptotic results of MMSE combining match with the corresponding simulation results in a high degree of accuracy. These observations approve the tightness of our theoretical analyses relating to Theorem 2 and Theorem 3. Furthermore, under a cell-free massive MIMO scenario, MRC exhibits worse performance because it cannot effectively mitigate the interference caused by other users. This combiner preserves as much energy content of the data stream of interest as possible but at the cost of possibly facing strong interference. On the contrary, the MMSE combiner can obtain a balance between achieving high signal power and suppressing interference. Moreover, we also provide the simulations of large-scale fading decoding (LSFD) [3] used in the distributed cell-free operation. With the same configuration of ADCs, it can be observed that the centralized operation discussed in this study basically has a performance gain compared with the distributed one. This benefit comes with the price of more fronthaul signaling loads and higher computational complexity. Fig. 3 compares the results of (30) and (33), which validates the tightness of (33) by considering various system parameters. The SE in (30) is obtained by Monte-Carlo simulation. It can be observed from Fig. 3 that (33) has relative lower SE values compared with (30), which confirms the conclusion in Remark 4. For the MRC combiner, the gap between (30) and (33) is large because the interference cannot be effectively mitigated. By contrast, the gap is almost negligible for the MMSE combiner. As the number of antennas at each AP increases, the gap shrinks due to higher degree of channel hardening with more antennas. Moreover, it is beneficial to deploy more APs with less antennas on the condition that the total number of antennas is fixed. For instance, configuration "L = 20, M = 5" leads to better SE than configuration "L = 5, M = 20". The main reason is that using more AP can provide more macro diversity and the distance between users and APs can be shortened to a certain extent. Fig. 4 manifests the performance of channel estimation in terms of WNMSE for different intra-AP bit allocation strategies. Assume that the ADCs at every AP are equally divided into three groups. As a result, each group has 10 numbers given that M = 30. In the legends of Fig. 4, [a 1 , a 2 , a 3 ] denotes that group 1 to group 3 use a 1 , a 2 , and a 3 bits for quantization, respectively. Providing that the total number of quantization bits is fixed, e.g., 90 for each AP in the simulation, it can be seen from Fig. 4 that the equal ADC bit allocation is optimal under spatially uncorrelated channels. This observation is in line with the conclusion in Section III-B. The main reason is that the aggregated large-scale fading coefficients are equal for all antenna elements which are co-located in the big array at one concerned AP. Moreover, we can also find from Fig. 4 that the smaller the variation between a 1 , a 2 , and a 3 , the better the WNMSE. In addition, as we increase the transmit power, the WNMSE eventually saturates because the impacts of pilot contamination and quantization errors. Fig. 5 evaluates the performance of channel estimation in terms of WNMSE for different inter-AP bit allocation (BA) strategies. We here use (26) to weight the NMSE and perform bit allocation. Notice in Fig. 5   in the case where the total quantization bits are significantly limited. The main reason is that assigning larger weight to the users with strong channel conditions leads to larger u l for the APs serving these "superior" users. As such, it is more advantageous to allocate more quantization bits to these APs. In addition, when the budget of quantization bits increases, the gap between the proposed bit allocation and the fixed counterpart reduces because all APs can use more bits. Fig. 6 compares the estimation quality for various pilot assignment schemes. Scheme 1 denotes that only the interference k−1 i=1,ti=t β l k i , originating from other users which utilize the same pilot as user k, in (8) is considered, Scheme 2 represents that only the leakage of interference l ∈L k β l k , originating from user k to other users which use the same pilot as user k, in (8) is considered. Scheme 3 denotes that both types of interference in (8) are considered. Scheme 4 is the greedy pilot assignment from [2]. Because of K = 10 and τ p = 5, pilot contamination is inevitable. It can be observed that the pilot assignment scheme mentioned in this study has the best WNMSE among such four schemes because we take into consideration two types of interference during pilot assignment stage. Moreover, for both proposed and fixed bit allocation techniques, the proposed pilot assignment is efficiently able to mitigate pilot contamination to a certain extent in comparison with the other three schemes.  In what follows, we turn our attention to the bit allocation for data transmission after achieving bit configuration during channel estimation stage. Note that we use the analytical results to conduct the investigation owing to the agreement of the theoretical results with the simulated ones. Fig. 7 compares the sum achievable SEs for different intra-AP bit allocation strategies. Assume that the ADCs at every AP are equally divided into three groups. As a result, each group has 10 antennas given M = 30. In the legends of Fig. 7, [a 1 , a 2 , a 3 ] denotes that group 1 to group 3 use a 1 , a 2 , and a 3 bits for quantization, respectively. Note that the total number of quantization bits is 90 for each AP in the simulation. It is noticeable from Fig. 7 that the equal ADC bit allocation is optimal under spatially uncorrelated channels. This finding validates the conclusion in Theorem 4. The main reason is that the aggregated strength of received signals is almost identical for all antenna elements which are co-located at one AP, as mentioned in Remark 5. From the viewpoint of statistic, we conclude in this investigation that the achievable SE is maximum when all antennas at each AP use the same quantization bits. In contrast, as mentioned in [10], from the perspective of instantaneous CSI, more bits can be allotted to the antennas with higher instantaneous channel gains to attain the best performance. However, switching quantization resolution based on instantaneous CSI is more challenging because it varies faster than statistical and long-timescale characteristics such as spatial channel correlation and large-scale fading coefficients. Moreover, we can also observe from Fig. 7 that the smaller the variation between a 1 , a 2 , and a 3 , the larger the sum achievable SE. Fig. 8 assesses the sum achievable SEs for different inter-AP bit allocation strategies. Notice in Fig. 8  For MRC, on the one hand, the proposed bit allocation can achieve remarkable improvement compared with the fixed one in the case where the total quantization bits are significantly limited. A possible explanation is that the impact of quantization error dominates performance loss compared with inter-user interference and can be alleviated by the proposed bit allocation. On the other hand, as the budget of quantization bits increases, the impact of quantization errors decreases and inter-user interference will dominate performance loss. Therefore, the performance gain of the proposed bit allocation eventually saturates in the case of adequate quantization bits. By contrast, the proposed bit allocation can provide more gains for MMSE combining because the interference caused by other users can be effectively suppressed and quantization errors have greater influence. For instance, when B = 60, the proposed bit allocation still has considerable benefit.
To illustrate how the number of users affects the results of bit allocation, Fig. 9 shows the empirical CDF curves of the optimal bit allocation given by Theorem 5 and Theorem 7 for single-user and multi-user scenarios. Here, we consider that B/M = L l=1 b d l = 360 and the solutions do not be rounded to their nearby integers. As the number of users increases under both MRC and MMSE combiners, we find that the optimized ADC bits approach to averaged bit to a certain extent. This behavior suggests the following possible interpretation. When there exists only one user, it has different distances to different APs. Therefore, the optimal ADC bits become more spread. When there exist more users, they are randomly distributed over a considered area; and the aggregated strength of received signals at any AP is determined by the powers and channel conditions of all users. In general, the variance of received signal between different APs reduces as the number of users grows. As a result, the variation of optimized bits reduces as well.

VII. CONCLUSION
Focusing on cell-free massive MIMO network with variableresolution ADCs, we introduced a quantization-aware channel estimator based on LMMSE theory and investigated the bit allocation problem to maximize the weighted estimation quality subject to the total number of quantization bits. By leveraging the estimated CSI, we derived the theoretical expressions of the achievable uplink SE for MRC and MMSE combiners. Simulation results demonstrated that our theoretical analyses are correct and tight. Moreover, to maximize the sum SE, we also investigated the bit allocation problem for both single-user and multi-user cases under the constraint of total ADC resolution bits. Given the total number of quantization bits, this study has found that equal ADC bit allocation is preferable within one AP. Concerning the bit allocation amongst different APs, we have shown that more bits should be assigned to the AP with larger aggregated effect of received signals.

Lemma 1 (Matrix Inversion Lemma):
Suppose that A is a Hermitian invertible matrix and A + αbb H is also invertible for any vector b and scalar α. It holds that b H A + αbb H −1 = b H A −1 1+αb H A −1 b . Lemma 2: Suppose that A has uniformly bounded spectral norm and x ∼ CN (0, R x ). It follows that E{x H Ax} = tr(R x A).
Lemma 3: Suppose that F ∈ C M×M and J ∈ C M×M are nonnegative definite Hermitian matrices and G ∈ C M×K has random column vectors g i ∼ CN (0, 1 M Δ i ). Moreover, F, J, and Δ i (∀i = 1, . . . , K) have bounded spectral norms. Then, for positive α, it follows from [ with δ k being the solution of fixed-point equation Lemma 4: On the basis of Lemma 3, for a nonnegative definite Hermitian matrix K ∈ C M×M with bounded spectral norm, it follows from [32]  and [x] k = 1 M tr(Δ k ΓKΓ), respectively. Lemma 5 (Rank-1 Perturbation Lemma): Suppose that A ∈ C M×M , q ∈ C M×1 , and that G ∈ C M×M is a nonnegative Hermitian matrix. With α and α two given positive real numbers, it follows from [33, Theorem 3.9] that tr A(G + αI M ) −1 − A(G + α gg H + αI M ) −1 ≤ ||A|| α . Lemma 6: Suppose that a ∼ CN (0, A) and that B ∈ C M×M is a diagonalizable matrix. It holds that E{|a H Ba| 2 } = |tr(BA)| 2 + tr(BAB H A).

APPENDIX B PROOF OF THEOREM 1
The optimization problem of (23) is convex, and the corresponding Lagrangian function is given by By solving (68) and (69), we finally achieve the results of (b p lm ) in Theorem 1.

APPENDIX C PROOF OF THEOREM 2
According to (13), we have A MRC k = ( l∈M k p k E{ĥ H lkĥ lk }) 2 = ( l∈M k p k tr(B lk )) 2 based on Lemma 2. In what follows, we try to derive the closed-form expressions for the other terms. 1) Compute B MRC k : If we define b ki = E{|(ĥ k k ) Hĥk i | 2 } = E{(ĥ k k ) Hĥk i (ĥ k i ) Hĥk k }, the following two cases should be separately discussed due to pilot contamination. i) When i / ∈ P k ,ĥ k k andĥ k i are mutually uncorrelated and assumed to be independent. In this case, it follows that with which we have (35) for the case when i / ∈ P k . ii) When i ∈ P k ,ĥ k k andĥ k i are mutually correlated and k with R i = diag{R li } l∈M k for i = 1, · · · , K. In this case, it follows that APPENDIX D PROOF OF THEOREM 3 For notation brevity in what follows, we letM k =L k M and define Λ = 1 obviously convex. As a result, the optimization problem of (52) is convex, and the corresponding Lagrangian function is expressed as By solving (80), we have b d l1 = · · · = b d lm = · · · = b d lM . Using M m=1 b d lm = B l , we finally achieve the results of (b d lm ) in Theorem 4.