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... well-known that when the likelihood function is a multivariate Gaussian pdf CN (m(θ),Σ) with the dependence on the parameters in the mean, then the Fisher information matrix J(θ) has (a, b) elements =-=[43]-=- Ja,b(θ) = 2Re {( ∂m(θ) ∂θa )H Σ−1 ∂m(θ) ∂θb } . In our case, we have Σ = N0I, m(θ) = Ξ(δξ)S∇(δµ)h, and the vector of parameters θ has elements θ1 = δξ, θ2 = δµ, θ2p+3 = Re{hp}, θ2p+4 = Im{hp}, p = 0,...

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...y channel resource by eliminating multiuser interference via spatial precoding, using a very large number of antennas at each base station site [1], [2]. The latter pertains to the use of small cells =-=[3]-=-, such that multiple short-range low-power links can co-exist on the same time-frequency channel resource thanks to sufficient spatial separation. Distributed Multiuser MIMO (MUMIMO) unifies these two...

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...ignals with negligible inter-carrier interference (ICI). This can be implemented by completely standard schemes used in current WLAN technology and forming a well-known and thoroughly studied subject =-=[9]-=-–[13]. Hence, such blocks need not be treated explicitly in this paper. In contrast, synchronization at the AP transmitter side plays a critical role in the distributed MU-MIMO architecture, since it ...

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...fers to serving multiple users on the same time-frequency channel resource by eliminating multiuser interference via spatial precoding, using a very large number of antennas at each base station site =-=[1]-=-, [2]. The latter pertains to the use of small cells [3], such that multiple short-range low-power links can co-exist on the same time-frequency channel resource thanks to sufficient spatial separatio...

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...libration) needed to enable such coherent DL MU4MIMO transmission. OFDM is assumed throughout this paper, since it is widely used in current WLAN (802.11a/g/n/ac) and cellular (LTE/802.16m) standards =-=[7]-=-. An UL-pilot/DL-precoded transmission cycle works as follows. At first, UL pilots are transmitted by the User Terminals (UTs). After receiving the UL pilots, each AP individually processes its observ...

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...data symbols to be sent to the Nu UTs, and G[m, ν] is the Nu×Na MU-MIMO precoding matrix for OFDM symbol m and subcarrier ν. In this work we consider linear Zero-Forcing Beamforming (ZFBF) [2], [28], =-=[33]-=- and conjugate beamforming [1], [2], [28], the details of which are given later on in this section. The two matrices Θ[m, ν] and Φ[m, ν] are diagonal of dimension Nu×Nu and Na×Na, respectively, with d...

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...bration have been treated to some extent in the existing literature. The synchronization of multiple transmitters for multiple-input singleoutput (MISO) cooperative beamforming is considered in [18], =-=[19]-=-, and software-defined radio (SDR) implementations of such ideas are presented in [20]–[22]. SDR implementations of MU-MIMO have been recently presented by a few research teams (including our own) in ...

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...speed from/to the CS, but not able to distribute a common clock (i.e., timing and phase information) to the APs, beyond the limitations of standard network synchronization protocols such as IEEE 1588 =-=[4]-=-. In order to achieve large spectral efficiencies through MU-MIMO downlink spatial precoding, channel state information at the transmitter (CSIT) is needed. Following the massive MIMO approach [1], [2...

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...roblem is given by Fcal(c, λ) = Jcal(c)− λ(‖c‖2 − 1). (27) where λ is the Lagrangian multiplier. Differentiating Fcal with respect to c∗i , treating ci and c ∗ i as if they were independent variables =-=[40]-=-, and then setting the partial derivatives to zero, we obtain ∂ ∂c∗i Fcal(c1, c2, . . . , cNa , λ) = ∑ j:(i,j)∈Fu ( ci|Yi→j |2 − cjY ∗i→jYj→i )− λci. (28) In matrix form, we obtain Ac = λc, where A is...

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...y calibration have been treated to some extent in the existing literature. The synchronization of multiple transmitters for multiple-input singleoutput (MISO) cooperative beamforming is considered in =-=[18]-=-, [19], and software-defined radio (SDR) implementations of such ideas are presented in [20]–[22]. SDR implementations of MU-MIMO have been recently presented by a few research teams (including our ow...

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...work implementations and it is not widely treated in the literature (see the existing-results overview provided below). In fact, in most theoretical works on distributed MU-MIMO (see for example [2], =-=[14]-=-–[17]) perfect transmitter synchronization is assumed without specifying how this is achieved. As shown in Section II-A, uncompensated synchronization errors at the AP transmitter side cannot be undon...

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...ls with negligible inter-carrier interference (ICI). This can be implemented by completely standard schemes used in current WLAN technology and forming a well-known and thoroughly studied subject [9]–=-=[13]-=-. Hence, such blocks need not be treated explicitly in this paper. In contrast, synchronization at the AP transmitter side plays a critical role in the distributed MU-MIMO architecture, since it needs...

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...e the presence or collaboration of the UTs, which may be legacy devices not necessarily implementing the protocol. A novel TDD reciprocity calibration method, referred to as “Argos”, was presented in =-=[28]-=- as part of an SDR implementation of a TDD reciprocity-based massive MIMO base station. This method enables calibration by exploiting two-way signaling that involves only base station antennas, exchan...

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... software-defined radio (SDR) implementations of such ideas are presented in [20]–[22]. SDR implementations of MU-MIMO have been recently presented by a few research teams (including our own) in [23]–=-=[25]-=-, where the feasibility of overthe-air synchronization with enough accuracy to guarantee near-ideal multiuser interference suppression is demonstrated. These works are based on a master-slave protocol...

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... We require that each anchor node is able to receive the pilot bursts from its neighboring anchor nodes without significant co-pilot interference. This is possible by finding an L(1, 1)-labeling (see =-=[35]-=- and references therein) of the subgraph (A, E(A)), and associating pilot bursts to the labeling colors. An L(1, 1)-labeling of a graph is a coloring scheme for which all neighbors of any node i ∈ A h...

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.... 6. 18Backing off the transmit power in order to meet the per-AP power constraint is suboptimal. The optimal MU-MIMO zeroforcing precoding with per-antenna power constraint is discussed in detail in =-=[42]-=-, and its optimization requires non-uniform power allocation per DL data stream and can be obtained using the theory of generalized inverses and convex relaxation. Here, we chose to use a more practic...

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...implementations and it is not widely treated in the literature (see the existing-results overview provided below). In fact, in most theoretical works on distributed MU-MIMO (see for example [2], [14]–=-=[17]-=-) perfect transmitter synchronization is assumed without specifying how this is achieved. As shown in Section II-A, uncompensated synchronization errors at the AP transmitter side cannot be undone at ...

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..., and software-defined radio (SDR) implementations of such ideas are presented in [20]–[22]. SDR implementations of MU-MIMO have been recently presented by a few research teams (including our own) in =-=[23]-=-–[25], where the feasibility of overthe-air synchronization with enough accuracy to guarantee near-ideal multiuser interference suppression is demonstrated. These works are based on a master-slave pro...

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...refore, reciprocity for such systems typically does not hold. 3the transmitter and receiver hardware are not, and introduce an unknown amplitude scaling and phase shift between the UL and DL channels =-=[6]-=-. Furthermore, in order to have all the remote APs behave as a single distributed multiple antenna system, the APs must maintain relative timing and phase synchronism throughout the duration of a DL p...

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...on of multiple transmitters for multiple-input singleoutput (MISO) cooperative beamforming is considered in [18], [19], and software-defined radio (SDR) implementations of such ideas are presented in =-=[20]-=-–[22]. SDR implementations of MU-MIMO have been recently presented by a few research teams (including our own) in [23]–[25], where the feasibility of overthe-air synchronization with enough accuracy t...

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...iently large” means above some design threshold determined by the system designer. 7the corresponding Cramer-Rao Bound (CRB) as a good approximation of the estimation MSE achievable in practice [29], =-=[30]-=-. The rest of this paper is organized as follows. Section II presents a simplified but sufficiently accurate model for OFDM in the presence of timing, sampling frequency and carrier frequency offsets,...

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... multiple transmitters for multiple-input singleoutput (MISO) cooperative beamforming is considered in [18], [19], and software-defined radio (SDR) implementations of such ideas are presented in [20]–=-=[22]-=-. SDR implementations of MU-MIMO have been recently presented by a few research teams (including our own) in [23]–[25], where the feasibility of overthe-air synchronization with enough accuracy to gua...

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...dices, if (i, j) ∈ E then also (j, i) ∈ E . Without loss of generality, we assume 15 that the graph (T , E) is connected.12 For the synchronization protocol, we define A ⊆ T to form a connected cover =-=[34]-=-, i.e., the subgraph formed by the APs i ∈ A and their associated edges E(A) = {(i, j) : i, j ∈ A} is connected and any other AP i′ /∈ A has at least one neighbor in A. The APs in A are referred to as...

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... with sufficiently large power and thus limiting the size and topology of the network. Extensions to networks of arbitrary size and topology based on distributed consensus protocols are considered in =-=[26]-=-, [27]. The slow rate of convergence, however, makes such schemes impractical for large networks. 3 TDD reciprocity was considered in [6], presenting a calibration technique based on exchanging pilot ...

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...AP are driven by a common clock and therefore need only to be calibrated but not mutually synchronized. Hierarchical calibration in multiple steps is naturally suited to this purpose, as evidenced in =-=[8]-=-, [38]. It is also interesting to notice that the proposed synchronization scheme calculates the timing and frequency correction factors relative to some particular anchor AP (denoted by AP 1 in Secti...

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...sufficiently large power and thus limiting the size and topology of the network. Extensions to networks of arbitrary size and topology based on distributed consensus protocols are considered in [26], =-=[27]-=-. The slow rate of convergence, however, makes such schemes impractical for large networks. 3 TDD reciprocity was considered in [6], presenting a calibration technique based on exchanging pilot signal...

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...nce time for typical wireless local area networks with nomadic users may span ∼ 0.1s to 1s or more. In fact, 800 msec is commonly used as the simulation assumption for 801.11ac IEEE standard for WLAN =-=[5]-=-. In WARP software-radio based experiments, we observed the coherence time to be about 1 second. In contrast, Frequency Division Duplexing schemes have the UL and the DL separated by tens of MHz, well...

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...r (ADC), anti-aliasing filter, etc.). Since the CFO between node i and node j is small with respect to the signal bandwidth, the complex baseband signal after demodulation can be written as (see also =-=[31]-=-) ỹi→j(t) = (∑ n N−1∑ k=−L x̃i[n, k]gi→j (t− (τi − τj)− (n(N + L) + k)Ts,i) ) × exp(j2pi(f0,i − f0,j)t), (3) where we define 5 gi→j(t) = (Πi ⊗ hi→j)(t), with Πi(t) = Π(t/Ts,i). The ADC at receiver j ...

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...by the frame structure of Fig. 3. The calibration slot is formed by pilot symbols arranged on the time-frequency plane induced by OFDM, as commonly 22 implemented for channel estimation purposes [7], =-=[36]-=-, [37]. Since the non-reciprocal elements of the channel (due to the modulation/demodulation hardware) are smooth over the signal bandwidth both in amplitude and in phase, the calibration pilots can b...

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...iting the correlation between the calibration coefficients at different subcarriers can be exploited to improve estimation accuracy. More details on efficient calibration pilot design are provided in =-=[38]-=-, [39] and go beyond the scope of this work. Here, for the sake of clarity, we focus on a single subcarrier and drop the subcarrier index ν from the notation. In order to motivate the need for TDD rec...

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...the correlation between the calibration coefficients at different subcarriers can be exploited to improve estimation accuracy. More details on efficient calibration pilot design are provided in [38], =-=[39]-=- and go beyond the scope of this work. Here, for the sake of clarity, we focus on a single subcarrier and drop the subcarrier index ν from the notation. In order to motivate the need for TDD reciproci...