2015 IEEE 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP)
An Array Processing Approach to Pilot Decontamination for Massive MIMO Karthik Upadhya and Sergiy A. Vorobyov Department of Signal Processing and Acoustics Aalto University Espoo, FI-00076 AALTO, Finland Emails:
[email protected] and
[email protected] Abstract—We address the problem of pilot decontamination for massive MIMO systems for finite dimensional channels, wherein the channel vectors between the users and the base station are composed of a finite number of discrete paths. The pilot decontamination problem is addressed using high-resolution parametric spectral estimation methods, such as root multiplesignal classification (root-MUSIC), for resolving paths based on their angle-of-arrivals. The paths corresponding to the desired user are identified based on their amplitudes and then the paths are used to form the channel estimate. Our simulations show an improved performance in terms of bit-error rate compared to the existing approaches in certain critical scenarios. Keywords—Pilot Decontamination, Massive MIMO, Array Processing, Spectral Analysis.
I.
II.
I NTRODUCTION
In massive multiple-input multiple-output (MIMO) systems [1], channel estimates are obtained by training using orthogonal pilot sequences, wherein each user in a cell is assigned a different pilot sequence. However, since the number of orthogonal sequences is limited, the orthogonal pilots are reused in different cells. As a result, the channel estimates of the users in a cell are corrupted by the channel vectors of users in adjacent cells, introducing interference and degrading the overall performance of the system. For an asymptotically orthogonal channel, the channel vectors corresponding to the desired and interfering users occupy orthogonal subspaces of the channel autocorrelation matrix [2] and subspace based methods can be used to separate the desired and interfering users. In [3], the eigenvectors of the autocorrelation matrix correspond to the channel vectors of each user, and a method for blind pilot decontamination is developed based on this observation. In [4], another blind pilot decontamination method is developed by projecting the received channel matrix onto an interference-free subspace spanned by the channel vectors of the desired user. In [5], a minimum mean-squared error (MMSE) based approach is proposed for pilot decontamination. In [6] and [7], the authors analyze the performance of the MMSE method for the finite dimensional channel and derive bounds for the sum-rate of the channel, assuming that the angle-of-arrivals (AoAs) of all the paths are known a priori. In this paper, we propose a method for pilot decontamination when the channel vector between a user and the base station (BS) is composed of a finite number of discrete paths.
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We estimate the AoAs of these discrete paths using highresolution parametric spectral estimation methods, such as root multiple-signal classification (root-MUSIC). A decontaminated channel can then be constructed by singling out only paths corresponding to the desired users. This classification can be performed on the basis of the amplitude of the received paths, wherein we assume that there is a power margin between the desired user and the interfering users. We propose to use a mean-squared error (MSE) optimal scheme for classification, which is shown to be equivalent to the maximum likelihood (ML) approach under certain conditions. Simulation results show that the proposed method has a superior performance in the finite dimensional channel scenario when compared with the existing algorithms in [3] and [4].
465
S YSTEM M ODEL
We consider the uplink scenario in a multi-user MIMO (MU-MIMO) system consisting of K users per cell with M K antennas at the BS. The k th user transmits a K length orthogonal pilot sequence Φk to the BS for channel estimation. We assume, without loss of generality, that the 0th cell is the desired one and that there are L − 1 cells whose users use the same set of pilot sequences as that of the 0th cell. As a result, the channel estimate of the desired user is corrupted by the channel vectors of users belonging to the L − 1 interfering cells. Assuming, without loss of generality, that all the users transmit at the same power, the received vector in the uplink at the 0th BS can be written as Y=
L−1 X
f H` ΦT + W
(1)
`=0
where (·)T stands for the matrix transpose, ΦK×K is an orthogonal matrix, i.e., ΦH Φ = IK with (·)H denoting the Hermitian transpose and IN stands for the N × N identity f M ×K is the additive white Gaussian noise at the matrix, W receiver whose columns are distributed as CN (0, σ 2 IM ), and [H` ]M ×K is the matrix of channel vectors from the users in the `th cell to the 0th BS with the (m, k)th element [H` ]m,k = hm,l,k being the channel coefficient from the k th user in the `th cell to the mth antenna of the 0th BS. In addition, H` , [h`,0 , . . . , h`,K−1 ]
(2)
where h`,k is the vector of channel values from the k th user in the `th cell at each antenna of the 0th BS. The estimate of
2015 IEEE 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP)
the channel at the 0th BS is given as b 0 = YΦ∗ = H0 + H
L−1 X
H` + W
(3)
`=1
f ∗ and (·)∗ denotes the complex conjugate. where W , WΦ A. Channel Model We assume that the BS is equipped with a uniform linear array (ULA) of M antennas with spacing d = λ/2 where λ is the wavelength of the transmitted signals. The channel vector from each user is assumed to be composed of a finite number of discrete paths, similar to the channel models in [5]–[7]. The channel vector h`,k can be then written as
A. Channel Estimation
P`,k −1
X 1 1 A`,kα `,k h`,k = p a(θ`,k,p )α`,k,p = p P`,k p=0 P`,k
(4)
where P`,k is the number of discrete paths between the k th user and the `th BS, θ`,k,p is the AoA of the pthppath from the k th user in the `th cell and α`,k,p = g`,k,p β`,k is the corresponding attenuation of the path. The term β`,k models the path-loss and shadowing effects and g`,k,p models fast-fading. The latter is assumed to be a zero-mean Complex h Gaussian irandom variable with unit-variance such that ∗ E g`,k,p gm,n,q = δ(p, q)δ(l, m)δ(k, n) with δ(i, j) being the Kronecker delta function, A`,k , [a(θ`,k,0 ), . . . , a(θ`,k,P −1 )], T α `,k , [α`,k,0 , . . . , α`,k,P −1 ] and the vector a(θ) denotes the steering vector corresponding to the direction θ and is given for T 2πd 2πd a ULA as a(θ) , 1 ej λ cos(θ) . . . ej λ cos(θ)(M −1) . From (2) and (4), the channel matrix H` can be written as " # 1 1 H` = p A`,0α `,0 , . . . , p A`,K−1α `,K−1 . P`,0 P`,K−1 (5) From (3) and (5), the estimate of the channel vector of the k th b k is given as user h h i bk , H b h
∗,k
A0,kα 0,k = p + P0,k
L−1 X `=1
A`,kα l,k p + wk . P`,k
(6)
It can be seen that the channel vector of the k th user in the 0th cell is contaminated by the channels of the k th user in each of the L − 1 interfering cells. It is assumed that the channel vectors of the desired users and the interferers 1 are mutually asymptotically orthogonal, i.e., M HH i Hj → 0 as M → ∞ when i 6= j. For the channel model (4), the asymptotic orthogonality condition implies that any two users in any of the L cells have no common AoAs at the 0th BS. The proof of this statement has been skipped due to space constraints, and will appear in the follow-up journal paper. III.
be estimated. This AoA estimation can be accomplished using parametric spectral estimation algorithms. However, these methods do not differentiate between desired and interfering users and estimate the AoAs of all users that use the same orthogonal pilot. It is therefore, also necessary to classify the estimated paths into those belonging to the desired user and those belonging to the interfering users. The channel estimate is reconstructed from only the paths corresponding to the desired user, while the paths corresponding to the interfering users are discarded. This classification can be performed based on the amplitude of the received paths, provided that there is an adequate power margin between the desired and interfering users.
PARAMETRIC C HANNEL E STIMATION
For the rest of the paper, we use the root-MUSIC algorithm to estimate the AoA and assume that the AoAs are unknown constants. In addition, it is assumed that the number of paths of the desired user P0,k and the total number of paths from the L−1 P L cells γk , P`,k are known at the receiver. In practice, `=0
P0,k can be estimated using a data-aided approach and γk can be obtained from the channel autocorrelation matrix using methods such as [8]. Using N autocorrelationolags of the n b channel vector hk , i.e., rhˆ k (0), . . . , rhˆ k (N − 1) , an N × N autocorrelation matrix can be obtained, which is given as Rk = A0,k D0,k AH 0,k +
L−1 X
2 A`,k D`,k AH `,k +σ IN
|
{z
}
Rk,s
where D`,k , diag |α`,k,0 |2 /P`,k , . . . , |α`,k,P −1 /P`,k |2 . The matrix Rk has rank M , whereas the matrix Rk,s corresponding to the subspace of the desired and interfering users has rank γk . The eigenvalue decomposition of Rk is given as . Λs Rk = Us .. Uw 0
H Us σ 2 IN −γk · · · UH w 0
466
(8)
where the columns of Us and Uw are the eigenvectors of Rk and Λs is a diagonal matrix of eigenvalues corresponding to Rs . Since Rk is a Hermitian matrix, the eigenvectors Us and Uw are mutually orthogonal. Therefore, if M ≥ N > γk , the columns of Us span the γk dimensional subspace corresponding to the channel vectors of the desired and interfering users and the columns of Uw span the N − γk dimensional orthogonal subspace. As a result UH w A`,k = 0 , ∀` = {0, . . . , L − 1} .
(9)
Using (9), we then construct the polynomial aH (z)Uw UH w a(z) = 0 .
Based on (6), the observed channel vector for each user can be seen as a collection of complex sinusoids in white Gaussian noise. The channel estimation problem then becomes an AoA estimation problem, wherein the AoAs and the amplitudes of the paths corresponding to only the desired users need to
(7)
`=1
(10)
For the ideal autocorrelation matrix Rk , the polynomial has γk roots on the unit circle since the signal subspace is of dimension γk . The n o angles of the roots are the estimated AoAs θˆ0 , . . . , θˆγk −1 .
2015 IEEE 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP)
The amplitudes of the paths corresponding to these directions can be obtained via least squares (LS), which can be formulated as
2
1 b
b
ˆ = arg min hk − p α (11) α Aα α
P0,k b , a(θˆ0 ) . . . a(θˆγ −1 ) . The solution to (11) is where A k given as α ˆ0 −1 p bk . b HA b bH h ˆ = ... = P0,k A α A (12) α ˆ γk −1 b k , i.e., In lags of h n practice, the N autocorrelated o rˆ (0), . . . , rhˆ k (N − 1) are replaced by their estimates n hk o rbhˆ k (0), . . . , rbhˆ k (N − 1) , which are obtained from the M × b k . Moreover, the noise in the estimated autocorre1 vector h lation matrix will cause the roots of (10) to lie inside the unit circle. Therefore, the AoA estimates are obtained by picking γk roots that are closest to the unit circle instead. B. Amplitude Based Classification The paths that belong to the desired users can be determinedn using aothresholding scheme, i.e., if τ > 0 is a threshold, then α ˆ p , θˆp ∈ U0 if |ˆ αp |> τ where U` is the set of all paths th from the ` user. Note that the index k has been dropped for notational convenience. Neglecting the estimation error of the AoAs, i.e., (θˆp = θp ), the amplitude of the pth path α ˆ p can be found as −1 p b b HA b b Hh P A A (13) α ˆ p = eH 0 p where ep is the pth column of the identity matrix. For large M , −1 b H ≈ aH (θp )/M , and the b HA b ≈ M IM , e H A b HA b A A p √ H noise term ( P0 /M )a (θp )w has variance (P0 /M )σ 2 and can be assumed to be zero. Therefore, for large M , α ˆ p ≈ αp . Then, the optimal threshold can be obtained by minimizing the MSE (
2 )
1
1 b √ A0 α 0 − h MSE(τ ) , Eα
M P0 X 1 1 X = Eα |αp |2 + Eα |αp |2 P0 P0 {αp ,θp }∈U0 |αp |>τ X 2 2 − Eα |αp | P0 |αp |>τ {αp ,θp }∈U0
=
The slope of the MSE is 0 at three points, of which two points (τ = 0, τ = ∞) do not correspond to a minima. Then, the optimal threshold satisfies the equation L−1 X P` β0 −τ 2 1 − 1 e opt β` β0 = 1 . (16) P0 β ` `=1
Note that the proposed thresholding scheme requires that the BS has knowledge of the number of paths from users in the interfering cells. While P0 can be estimated at the receiver using a data-aided method, it is in general difficult for the L−1 BS to acquire information corresponding to {P` }`=1 . As a workaround, a sub-optimal threshold can be obtained using the maximum-likelihood (ML) approach. The pth path with amplitude estimate and AoA (αp , θp ) is classified to be arriving from the desired user if U0
Pr ({αp , θp } ∈ U0 ) ≷ Pr ({αp , θp } ∈ / U0 ) where U ` is the complement of U` is the probability that the pth path angle θp correspond to the user in Pr ({αp , θp } ∈ / U0 ) can be simplified
− τβ
β0 − (β0 + τ )e
2
0
+
L−1 X `=1
2
e
− τβ
`
(β` + τ 2 )
P` . (14) P0
To find the threshold that minimizes the MSE (14), differentiate (14) with respect to τ . Then we have ! L−1 X P` β0 −τ 2 1 − 1 d 2τ 3 − τβ2 β` β0 MSE (τ ) = e 0 1− e . dτ β0 P0 β ` `=1 (15)
467
and Pr ({αp , θp } ∈ U` ) with amplitude αp and the `th cell. The term as
Pr ({αp , θp } ∈ / U0 ) = Pr ({αp , θp } ∈ U1 . . . {αp , θp } ∈ UL−1 ) L−1 X = Pr ({αp , θp } ∈ U` ) . (18) `=1
The events ({αp , θp } ∈ U` ) , p = (0, . . . , γ − 1) are mutually exclusive, i.e., a given path can only correspond to one user. If θp is uniformly distributed in the interval [0, π], then the probability that the pth path from a user in the `th cell has amplitude x and AoA θ can be written as Pr ({x, θ} ∈ U` ) = pα (x; `)pθ (θ; `) 1 1 − β1 |x|2 ` e = πβ` π 1 − 1 |α |2 Pr ({αp , θp } ∈ U` ) = 2 e β` p π β`
(19)
where pα (x; `) and pθ (θ; `) are the probability density functions of the random variables αp and θp , respectively, when the path {αp , θp } corresponds to a user in the `th cell. From (19), the decision rule in (17) becomes 1 − |αβp |2 0 e β0 ⇔
1
U0
L−1 X
U0
`=1 L−1 X
≷
U0
≷
U0 2
(17)
U0
`=1
1 − |αβp |2 ` e β` β0 −|αp |2 e β`
1 β`
− β1
0
.
(20)
Comparing with (16), the ML decision rule is MSE optimal when P` = P0 , ∀` = 1, . . . , L − 1. The resulting channel estimate is obtained as ˆ0 h i α . b 0 = a(θˆ0 ), . . . , a(θˆP −1 ) h .. α ˆ P −1
(21)
2015 IEEE 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP)
Method in [1] EVD-based Method Parametric Method
102 MSE
10−0.5
BER
103
Method in [1] EVD-based Method Parametric Method
10−1
101
100 10−1.5 −20
−10
0
10
20
30
−20
SNR
−10
0
10
20
30
SNR
Fig. 1: Comparison of the BER performance of the EVD, Marzetta, parametric, and peak-selection methods for the finite-dimensional channel
Fig. 2: Comparison of the MSE performance EVD, Marzetta, parametric, and peak-selection methods for the finitedimensional channel.
where {ˆ α0 , . . . , α ˆ P −1 } are the Ph amplitude values that i cross the threshold given in (20) and a(θˆ0 ), . . . , a(θˆP −1 ) are the corresponding steering vectors.
users. We also develop an ML method for classifying the received paths based on their amplitudes. Simulation results show that, for the finite dimensional channel, the proposed method has improved BER performance when the interference is strong, i.e., when the desired users are near the cell edge. The performance is similar to the state-of-the-art methods when the interference is weak. However, it can be significantly improved by using an AoA based classification instead. The proposed method does not require any coordination between the base stations and performs pilot decontamination using an instantaneous autocorrelation matrix estimated from the observations. It does not require a large coherence time or long term statistics, making it useful in high-mobility scenarios.
IV.
S IMULATION
The method in [1], eigenvalue decomposition (EVD)-based method [4], and the proposed parametric method are compared for a system with L = 4 cells, M = 300 antennas at the BS, and K = 4 users per cell. The users are assumed to be uniformly distributed at a distance of 800m from the BS in a hexagonal cell with a cell-radius of 1 km. The path-loss coefficient is set to be 3. The number of paths from each user to the BS is set to be P = 10 and the AoAs of each path is assumed to be uniformly distributed in the interval [0, π]. The modulation scheme used is quaternary phase-shift keying (QPSK) and the receiver at the BS uses a matched filter for detection. For our method in [4], the coherence time is set to be 100 symbols. For the parametric method, the receiver uses an autocorrelation matrix of size N = 50. The bit-error rate (BER) and the MSE performances are plotted in Figs. 1 and 2, respectively. While the performance of the proposed method is better than the method in [1], it is similar to that of [4] in general. However, the proposed method outperforms the EVD-based method in situations wherein the desired and interfering users have similar power at the BS. Moreover, while the error in the channel vector of the proposed method is larger than the EVD based method (c.f. Fig 2), it is less correlated with the channel vectors of the interfering users, leading to an improvement in the BER. V.
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[2]
[3]
[4]
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[6]
C ONCLUSION
We have proposed a parametric subspace method for pilot decontamination in massive MIMO systems for finitedimensional channels. For the case where the channel vectors are asymptotically orthogonal and when the total number of paths at the BS is smaller than the number of antennas, we use high-resolution parametric spectral estimation methods, such as root-MUSIC, to estimate the AoAs of the paths from all
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[8]
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