The capacity of cellular networks can be improved by the unprecedented array gain and spatial multiplexing offered by Massive MIMO. Since its inception, the coherent interference caused by pilot contamination has been believed to create a finite capacity limit, as the number of antennas goes to infinity. In this paper, we prove that this is incorrect and an artifact from using simplistic channel models and suboptimal precoding/combining schemes. We show that with multicell MMSE precoding/combining and a tiny amount of spatial channel correlation or large-scale fading variations over the array, the capacity increases without bound as the number of antennas increases, even under pilot contamination. More precisely, the result holds when the channel covariance matrices of the contaminating users are asymptotically linearly independent, which is generally the case. If also the diagonals of the covariance matrices are linearly independent, it is sufficient to know these diagonals (and not the full covariance matrices) to achieve an unlimited asymptotic capacity. The Shannon capacity of a channel manifests the spectral efficiency (SE) that it supports. Massive MIMO (multiple-input multiple-output) improves the sum SE of cellular networks by spatial multiplexing of a large number of user equipments (UEs) per cell [1] . It is therefore considered a key timedivision duplex (TDD) technology for the next generation of cellular networks [2]-[4]. The main difference between Massive MIMO and classical multiuser MIMO is the large number of antennas, M , at each base station (BS) whose signals are processed by individual radio-frequency chains. By exploiting channel estimates for coherent receive combining, the uplink signal power of a desired UE is reinforced by a factor M , while the power of the noise and independent interference does not increase. The same principle holds for the transmit precoding in the downlink. Since the channel estimates are obtained by uplink pilot signaling and the pilot resources are limited by the channel coherence time, the same pilots must be reused in multiple cells. This leads to pilot contamination which has two main consequences: the channel estimation quality is reduced due to pilot interference E. Björnson is with the utilizes semi-blind estimation to separate the subspace of desired UE channels from the subspace of interfering channels [15]-[19]. This method can fully remove pilot contamination if M and the size of the channel coherence block go jointly to infinity [18] . Unfortunately, the channel coherence is fixed and finite in practice (this is why we cannot give unique pilots to every cell), thus we cannot approach this limit in practice. The third category uses multiple pilot phases with different pilot sequences to successively eliminate pilot contamination [20], [21] , without the need for statistical information. However, the total pilot length is larger or equal to the total number of UEs, which would allow allocating mutually orthogonal pilots to all UEs and thus trivially avoiding the pilot contamination problem. This is not a scalable solution for networks with many cells. The fourth category is pilot contamination precoding that rejects interference by coherent joint transmission/reception over the entire network [22], [23]. This method appears to achieve an unbounded SE, but this has not been formally proved and requires that the data for all UEs is available at every BS, which might not be feasible in practice. In summary, it appears that pilot contamination is a fundamental issue that manifests a finite SE limit, except in unlikely special cases. We show in this paper that this is basically a misunderstanding, spurred by the popularity of analyzing suboptimal combining/precoding schemes, such as MR and S-MMSE, and focusing on unrealistic i.i.d. Rayleigh fading channels (as in the prior work [8], [9] on M-MMSE). We prove that the SE increases without bound in the presence of pilot contamination when using M-MMSE combining/precoding, if the pilot-sharing UEs have asymptotically linearly independent covariance matrices. Note that R 1 and R 2 in (1) are linearly independent if [a b c] T and [d e f ] T are non-parallel vectors, which happens almost surely for randomly generated covariance matrices. Hence, our results rely on a condition that is most likely satisfied in practice-it is the general case, while prior work on the asymptotics of Massive MIMO considered practically unlikely special cases. In contrast to prior work, no multicell cooperation is utilized herein and there is no need for orthogonal support of covariance matrices. In the conference paper [24], we proved the main result in a two-user uplink scenario. 1 In this paper, we prove the result for both uplink and downlink in a general setting. Section II proves and explains the intuition of the results in a two-user setup, while Section III generalizes the results to a multicell setup. The results are demonstrated numerically in Section IV and the main conclusions are summarized in Section V. 1 After submitting our conference paper [24] , the related work [25] appeared. That paper considers the mean squared error in the uplink data detection of a single cell with multiple UEs per pilot sequence. The authors show that the error goes asymptotically to zero for linearly independent covariance matrices. However, the paper [25] contains no mathematical analysis of the achievable SE.
However, recent developments REF show that even in the presence of pilot contamination, the capacity of massive MIMO systems is unbounded as the number of antennas grow, for the case of spatially correlated channels.
Massive MIMO has Unlimited Capacity
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Mathematics Computer Science