 Research
 Open Access
Uncoordinated pilot decontamination in massive MIMO systems
 Jesper H. Sørensen^{1}Email authorView ORCID ID profile and
 Elisabeth de Carvalho^{1}
https://doi.org/10.1186/s1363801709397
© The Author(s) 2017
Received: 5 April 2017
Accepted: 4 September 2017
Published: 19 September 2017
Abstract
This work concerns wireless cellular networks applying time division duplexing (TDD) massive multipleinput multipleoutput (MIMO) technology. Such systems suffer from pilot contamination during channel estimation, due to the shortage of orthogonal pilot sequences. This paper presents a solution based on pilot sequence hopping, which provides a randomization of the pilot contamination. It is shown that such randomized contamination can be significantly suppressed through appropriate filtering. The resulting channel estimation scheme requires no intercell coordination, which is a strong advantage for practical implementations. Comparisons with conventional estimation methods show that the MSE can be lowered as much as an order of magnitude at low mobility. Achievable uplink and downlink rates are increased by 42 and 46%, respectively, in a system with 128 antennas at the base station.
Keywords
 Massive MIMO
 Pilot contamination
 Kalman filter
1 Introduction
Mulipleinput multipleoutput (MIMO) technology [1] is finding its way into practical systems, like LTE and its successor LTEAdvanced. It is a key component for these systems’ ability to improve the spectral efficiency. The success of MIMO technology has motivated research in extending the idea of MIMO to cases with hundreds, or even thousands of antennas, at transmitting and/or receiving side. This is often termed massive MIMO. In mobile communication systems, like LTE, the more realistic scenario is to have a massive amount of antennas only at the base station (BS), due to the physical limitations at the user equipment (UE). It has been shown that such a system [2], in theory, can eliminate entirely the effect of smallscale fading and thermal noise, when the number of BS antennas goes to infinity. The only remaining impairment is intercell interference, caused by imperfect channel state information (CSI), which is a result of nonorthogonality of training pilots used to gather the CSI. This is often referred to as pilot contamination. It is considered as one of the major challenges in massive MIMO systems [3].
Mitigation of pilot contamination has been the focus of several works recently. These fall into two categories: one with coordination among cells and one without. The first category includes [4], where it is utilized that the desired and interfering signals can be distinguished in the channel covariance matrices, as long as the angleofarrival spreads of desired and interfering signals do not overlap. A pilot coordination scheme is proposed to help satisfying this condition. The work in [5] utilizes coordination among BSs to share downlink messages. Each BS then performs linear combinations of messages intended for users applying the same pilot sequence. This is shown to eliminate interference when the number of BS antennas goes to infinity.
The category without coordination also includes notable contributions. A multicell precoding technique is used in [6] with the objective of not only minimizing the mean squared error of the signals of interest within the cell but also minimizing the interference imposed to other cells. In [7], it is shown that channel estimates can be found as eigenvectors of the covariance matrix of the received signal when the number of BS antennas grows large and the system has “favorable propagation.” The work in [8–11] is based on examining the eigenvalue distribution of the received signal to identify an interference free subspace on which the signal is projected. It is shown that an interference free subspace can be identified when certain conditions are fulfilled concerning the number of BS antennas, user equipment antennas, channel coherence time, and the signaltointerference ratio. Recently, in [12], a combination of the solutions in [4] and [8–11] was proposed. The resulting solution unites the strengths of these solutions leading to a more robust pilot decontamination.
In this paper, we propose pilot decontamination, which does not require intercell coordination and is able to exploit past pilot signals. It is based on pilot sequence hopping performed within each cell. Pilot sequence hopping means that every user chooses a new pilot sequence in each time slot. Thus, in every time slot, the pilot signal of a user is contaminated by a different set of interfering users, which means channel estimation is affected by a different set of interfering channels. If channel estimation is carried out based solely on the pilot sequence of the current slot, then pilot sequence hopping does not bring any gain. The key in our solution is a channel estimation that incorporates multiple time slots so that it can benefit from randomization of the pilot contamination. Recent work utilizing temporal correlation for channel estimation is found in [13], although not in combination with pilot hopping and not with the purpose of mitigating pilot contamination. Random selection of pilot sequences is also explored in [14] and [15]. Both works consider the random access problem in cellular networks. In [14], pilot contamination is avoided through a distributed collision detection algorithm, which enables users with weak channels to detect that they are contaminators of a user with a strong channel and as a result postpone their transmission. The work in [15] considers codeword transmissions that are spread across multiple time slots, each with a different contaminator. This decorrelates the contamination within a single codeword, which improves performance.
When the channel is timevariant and correlated across time slots, it is possible to exploit the information about the channel across time slots by an appropriate filtering and benefit from contamination randomization. In this paper, channel estimation across multiple time slots is performed using a modified version of the Kalman filter, which is capable of tracking the channel and the channel correlation. The level of contamination suppression depends on the channel correlation between slots of the UE of interest as well as the contaminators. In LTE, channel correlation between time slots is large even at mediumhigh speeds, making the proposed solution very efficient.
This work is an extension of the work in [16], where the concept of pilot sequence hopping in combination with a Kalman channel tracker is introduced. In this paper, the work is extended with more sophisticated mobility models and a generalization of the estimation algorithm, which allows higher order Kalman process models. Furthermore, the Bayesian CramerRao lower bound is derived for the estimation problem at hand and applied as a benchmark in the numerical evaluations.
The remainder of this paper is organized as follows. Section 2 presents the applied channel and mobility models and the problem of pilot contamination. The proposed solution is described in Section 3 and analyzed in Section 4. Section 5 provides numerical results and a comparison to existing solutions. Finally, conclusions are drawn in Section 6.
2 System model
In this work, we denote scalars in lower case, vectors in bold lower case, and matrices in bold upper case. A superscript “T” denotes the transpose, and a superscript “H” denotes the conjugate transpose.
2.1 Channel model
where N _{ s } is the number of fixed scatterers associated with all BS/UE pairs, f _{ d } is the maximum Doppler shift, α _{ m } and ϕ _{ m } are the angle of arrival and initial phase, respectively, of the wave from the mth scatterer. Both α _{ m } and ϕ _{ m } are independent and uniformly distributed in the interval [−π,π), which results from random scatterer locations. Furthermore, \(f_{d}=\frac {v}{c}f_{c}\), where v is the speed of the UE, c is the speed of light and f _{ c } is the carrier frequency.
where \(\boldsymbol {z}^{k0}_{n}=\left [z^{k0}_{n}(1)z^{k0}_{n}(2) \hdots z^{k0}_{n}(\tau)\right ]^{T}\) and \(z^{k0}_{n}(j)\) are circularly symmetric Gaussian random variables with zero mean and unit variance for all j. Here, only signals leading to contamination are included in the sum term since any \(h^{ij}_{n} \boldsymbol {x}^{ij}_{n}\) ∀\(i,j\notin \mathcal {C}_{n}^{k\ell }\) are removed when correlating with the applied pilot sequence. Hence, all contributions from the sum term are undesirable and will contaminate the CSI. Without loss of generality, we focus on the channel estimation for a single user in a single cell. Hence, in the remainder of the paper, we omit the superscript k for ease of notation.
2.2 Mobility model

M1: In this mobility model, the UE moves at a constant speed, v _{1}, for T _{1} seconds.

M2: (Train) This model emulates the mobility experienced in a train. Initially, the speed is zero for T _{2,1} seconds. Then, the speed increases linearly, i.e., with constant acceleration, δ _{2,+}, until a specified maximum speed, v _{2}. This speed is maintained for T _{2,2} seconds, after which the speed is decreased linearly, with deceleration, δ _{2,−}, until mobility has seized. Finally, the speed is kept at zero for T _{2,3} seconds.

M3: (Car) The third mobility model emulates the behavior of a car for T _{3} seconds. The model operates with a vector of possible speeds, v=[v _{0} v _{1}…v _{max}], where the individual speeds are uniformly spaced between zero and v _{max}. The initial speed is v _{0}=0. In every time slot, the speed is increased with probability p _{+} and decreased with probability p _{−} and remains constant with probability 1−p _{+}−p _{−}. Acceleration and deceleration are constant at δ _{3,+} and δ _{3,−}, respectively. Speed changes always occur to the nearest speed in v, and both acceleration from v _{max} and deceleration from v _{0} result in no change.
3 Pilot decontamination
 1.
Pilot sequence hopping: This component refers to random shuffling of the pilots applied within a cell. This shuffle occurs between every time slot. The purpose of this component is to decorrelate the contaminating signals. When pilots are shuffled, the set of contaminating users will be replaced by a new set, whose channel coefficients are uncorrelated with those of the previous set.
 2.
Kalman filtering: The autocorrelation of the channel coefficient of the user of interest is high at low mobility. This means that information about the value of the current channel coefficient exists not only in the most recent pilot signal but also in past pilot signals. This can be extracted using a filter. Since the channel coefficients are timevarying, we are dealing with a tracking problem. For this purpose, a Kalman filter is attractive due to its excellent tracking capability and recursive structure, which provides good performance at low complexity. Since the contaminating signals have been decorrelated, the Kalman filter will suppress the impact of these signals, leading to pilot decontamination.
3.1 Pilot sequence hopping
where \(P\left (t_{c}^{k}=w\right)\) is the probability that the collision distance is w and p is the probability of a given UE being the next contaminator. We then have \(\mathbb {E}\left [t_{c}^{k}\right ]=K\), i.e., the expected collision distance increases with the number of users/pilots per cell, which follows intuition. Note that the collision distance is a userspecific measure, which holds for all potential contaminators in the system. Hence, the analysis still holds when considering systems with more than one neighboring cell.
The maximization of \(\mathbb {E}\left [t_{c}^{k}\right ]\) leads to a decorrelation of the contaminating signals. The benefit of this is reaped using appropriate filtering techniques. For this purpose, we have chosen a modified version of the Kalman filter, which is described next.
3.2 Modified Kalman filter
where \(\boldsymbol {v}_{n}^{m}\) is the measurement noise, which is zero mean circularly symmetric Gaussian with covariance matrix \(\sigma ^{2}_{o} \boldsymbol {I}_{\tau } + \sigma ^{2}_{c} \boldsymbol {X}_{n} \boldsymbol {X}_{n}^{H}\). Here, \(\sigma ^{2}_{o}\) and \(\sigma ^{2}_{c}\) are noise power and total contamination power (average over time), respectively, which are both assumed known.
where I _{ τ } is the τ×τ identity matrix and \(\boldsymbol {\hat {h}}_{n}\) is the estimate of h _{ n }.
For the tracking of the AR coefficients, an approach similar to the one in [19] is taken. In [19], the inclusion of a firstorder AR coefficient tracker is presented for a Kalman predictor, i.e., a filter with the purpose of predicting the channel, h _{ n }, based on all observations until y _{ n−1}. In this work, we extend this approach to higher order AR models taking all observations until y _{ n } into account.
where μ is a parameter adjusting the convergence speed and the brackets denote truncations. The truncation to ν is for avoiding dramatic adjustments in situations with a high slope. The need for this will be explained in Section 5. In addition to the truncation, we enforce z _{ j }<0.999, where z _{ j } are the roots of the polynomial \(z^{d+1}  \sum _{j=1}^{d+1} a_{n}^{j} z^{d+1j}\). This ensures a stationary AR process.
In the following subsection, we derive the lower bound on the MSE of an estimate of the channel coefficients. It serves as a benchmark in the numerical evaluations in Section 5.
4 Analysis
In this case, the error in the estimate is solely composed of the average of the contaminating signals, which are independent and have variance \(\sigma _{c}^{2}\). Hence, \(\mathbb {E}\left [\left (h\bar {\hat {h}}_{n}\right)^{2}\right ]=\frac {\sigma _{c}^{2}}{n+\sigma _{c}^{2}}\), if prior knowledge on h is a standard Gaussian. If pilot sequence hopping had not been performed, the MSE had remained \(\frac {\sigma _{c}^{2}}{1+\sigma _{c}^{2}}\) since h n′ would be constant. Note that the MSE goes towards zero for n→∞, when pilot sequence hopping is performed. This is a result of the fact that a pilot signal in the infinite past carries as much information about the current channel as the most recent pilot signal, in the ideal example of a constant channel. Note also that for finite τ (and K) and thereby finite \(\mathbb {E}\left [t_{c}^{k}\right ]\), the MSE is lower bounded by \(\frac {\sigma _{c}^{2}}{K+\sigma _{c}^{2}}\) since only a maximum of K independent estimates can be achieved. In a more practical example with a timevarying channel, the amount of information carried in a pilot signal decays over time. We can account for this in a more elaborate Bayesian analysis, which is described next.
4.1 Bayesian analysis
Equations (26) and (27) provide the optimal coefficients of a Bayesian filter and the corresponding covariance. The lower right corner element of V is then the variance of a causal filter estimating the most recent channel coefficient, h _{ n }.
With Eqs. (25) to (27) as a starting point, we can analyze filters based on different assumptions on the underlying model of the channel.
In the following subsection, we derive the lower bound on the MSE for arbitrary AR model order. Along with the bound for firstorder AR models in (36), it serves as a benchmark in the numerical evaluations in Section 5.
4.2 CramerRao lower bound
where c _{1} and c _{2} are constants with independence from Y and h, Σ ^{−1} is the inverse of the n×n covariance matrix of h, and C ^{−1} is the inverse of the τ×τ observation error covariance matrix. The CRLB at time n is the corresponding submatrix of J ^{−1}; it gives a lower bound on channel estimation at time n accounting for the past observations.
where ⊗ denotes the Kronecker product. Furthermore, expression (43) allows a continuity with the case of no mobility for which the channel estimate of time n is the result of an average (see Eq. (23)).
5 Numerical results
Simulation parameters
Parameter  Value  Description 

\(\sigma _{o}^{2}\)  0.2  Noise variance 
L  7  Number of cells 
K  96  Users per cell 
τ  96  Pilot length 
μ  10^{−5}  Convergence speed 
ν  100  Derivative cap 
f _{ c }  1.8 GHz  Carrier frequency 
N _{ s }  20  Number of scatterers 
t _{ s }  0.5 ms  Time between pilots 
\(\boldsymbol {\hat {h}}_{0}\)  0  Initial estimate 
q _{0}  0  Initial differentiated estimate 
P _{1}  0  Initial error covariance 
S _{1}  0  Initial differentiated error covariance 
6 Conclusions
We have presented a solution to pilot contamination in channel estimation, which is a major challenge in TDD massive MIMO systems. It is based on a combination of a pilot sequence hopping scheme and a modified Kalman filter. The pilot sequence hopping scheme involves random shuffling of the assigned pilot sequences within a cell, which ensures decorrelation in the time dimension of the contaminating signals. This is essential since it enables subsequent filtering to suppress the contamination. For this filtering, the Kalman filter has been chosen, due to its ability to track a timevarying state. However, a conventional Kalman filter is not able to adapt to changes in the underlying model, which is necessary when users have unknown and varying levels of mobility. For this problem, we have presented a modified Kalman filter, which can adapt the underlying model based on a minimization of the mean squared error.
Numerical evaluations show that the proposed solution can suppress a significant portion of the contamination at low and moderate levels of mobility. Even at high mobility, i.e., car speeds of 100 to 130 km/h, the proposed solution can provide a noticeable gain over conventional estimation methods.
Declarations
Acknowledgements
The research presented in this paper was supported by the Danish Council for Independent Research (Det Frie Forskningsråd) DFF  1335−00273.
Authors’ contributions
Both authors have contributed to the design and analyis of the proposed methods as well as the writing of this manuscript. Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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