Page 86 - Kaleidoscope Academic Conference Proceedings 2021

2021 ITU Kaleidoscope Academic Conference

operator, the inverse operator and the conjugate transpose temporal correlations, since the path delays vary much slowly
operator, respectively. CA(.) denotes the trace operator. A than the path gains, while the path gains vary continuously.
diagonal matrix with 0 1 , ..., 0 # on the main diagonal is Thus, the channel sparse pattern is nearly unchanged, and
denoted 3806(0 1 , ..., 0 # ). Bold uppercase and lowercase the path gains are also correlated. So, the CIRs of different
letters represent matrices and vectors, respectively. The transmit–receive antenna pairs share very similar scatterers.
Frobenius and spectral norms of a matrix x are denoted by Therefore, the CIRs share the same sparsity pattern. This
kxk and kxk 2 respectively. {.} denotes the expectation of channel property is referred as sparse common support (where
random variables within the brackets. A Gaussian stochastic the taps of nonzero gain of different CIR is identical) in the
variable > is then denoted by > ∼ #(A, @), where A is the sequel. The sparse communality property can be employed
mean and @ is the variance. Furthermore, a random vector to enhance the estimation accuracy of the proposed method
x having the prober complex Gaussian distribution of mean [14–16].
- and covariance is indicated by x ∼ #(x; -, ), where
#(x; -, ) = 1 4 −(x−-) −1 (x−-) , for simplicity we refer 3. SBL-BASED CHANNEL ESTIMATOR
34C ( c )
to #(x; -, ) as x ∼ #(-, ).
In Bayesian estimation methods the wireless channel is treated
2. MASSIVE MIMO SYSTEM MODEL as a random variable with a priori statistics known at the
mobile station where the estimation process is performed.
Consider an FDD single-cell massive MU-MIMO-OFDM While, on the other hand the non-Bayesian estimators that
system where the base station comprises of " antennas and known as parametric estimators require no prior channel
serves # single antenna users (" > #). At the beginning statistic that needs to be known at the expense of the estimation
of the transmission, a common downlink pilots channel are performance. In Bayesian estimation, the estimation of the
broadcast by the base station [12,13]. Let the average power unknown parameters of interest is the expectation of the
√
at each mobile user be ? A . The pilots are allocated randomly posterior probability. As such, to obtain the estimated channel,
in subcarriers while they share the same location in each we need to infer the posterior probability of the unknown
OFDM symbol in different transmit antenna, but each pilot parameters. While the posterior probability is proportional
x 8 sequence is unique. The Channel Impulse Response (CIR) to the prior probability distribution and the likelihood of the
vector between the 8-th transmitting antenna and the mobile channel based on Bayesian channel estimation philosophy.
)
users can be denoted as h 8 = [h 8 (1), h 8 (2)..., h 8 (!)] , where Then, the estimation of the channel in Bayesian channel
! is the number of the paths. Therefore, the received signal estimation is the expectation of the posterior probability
at a certain mobile station can be expressed as distribution [17,18]. More elaborations will be presented in
the forthcoming subsections where we present the proposed
"
√ Õ
y = ? A X 8 (F ! ) [ h 8 + n, (1) algorithm.
8=1
×1
where X 8 = 3806(x 8 ) with x 8 ∈ C denotes the 3.1 Bayesian Inference Model
transmitted pilot, F ! is the sub-matrix of size ×! consisting
Following the block sparse Bayesian learning model [18], the
of the first ! columns of the normalized Discrete Fourier
full posterior distribution of h over unknown parameters of
Transform (DFT), [ is the index set of the pilot that assigned
interest for the problem at hand can be computed as
randomly from the OFDM subcarrier, (F ! ) [ is the sub-matrix
consisting of the rows with indexes from [, % is the number of
%(h|y, ", W, H 8 ) = %(h|", H 8 )%(y|h, W), (4)
the pilots and n is the Additive White Gaussian Noise (AWGN)
2
with zero mean and f variance.
where W represents the inverse of the noise variance and " are
Let A denote the % × "! matrix as
non-negative hyperparameters controlling the sparsity of the
channel h. According to probability theory, the term %(y|h)
A = [X 1 (F ! ) [ , X 2 (F ! ) [ , ..., X " (F ! ) [ ], (2)
can be written as
Therefore, (1) can be written as
√ 1 ||y − Ah|| 2 2
y [ = ? A Ah + n [ , (3) %(y|h) = ( √ )4G?(− ). (5)
−1 2W −1
2cW
where h = [h , ..., h ] ) ∈ C " !×1 represents the CIR
)
)
1 " The statistical properties of the sparse multipath structure
aggregation from the " antennae that need to be estimated.
of the channel is following Gaussian distribution based on the
In a typical wireless communications scenario, the CIR by
central line theorem [19]. Thus, the Gaussian prior for each
its very nature is sparse due to several significant channel taps.
channel coefficient %(h|", H 8 ) is given by
Therefore, the CS-based estimators can be employed for the
proposed communication system model. This sparse property %(h|", H 8 ) ∼ N (0, "H 8 ), 8 = 1, ..., " (6)
can be exploited to reduce the necessary channel parameters
to be estimated. Accordingly, we can reduce the training where H 8 is a matrix that captures the temporal correlation
overhead through employing fewer pilots than the number of of the CIR that needs to be estimated. Since, the CIRs share
the unknown channel coefficients [14–16]. On the other the same common support, we can assume that the CIRs
hand, it has been shown that the wireless channels display have the same correlation structure as elaborated in Section
– 24 –