The state diagram of the proposed approach is shown in Figure 2. In Figure 2, the proposed approach consists of three blocks. In this section, we will thoroughly explain the mechanism in each block.
3.1 Coarse timing and frequency synchronization
It can be easily observed that the proposed training sequence is composed of two identical parts. Therefore, a coarse timing offset (CTO) at the b th receiver τ
c, b
is obtained based on the cross-correlation function outputs as follows:
(8)
where denotes the complex conjugate of r
b
(n), |r
b
(n)| represents the absolute value of r
b
(n), Ω
d
is the observation interval, Ω
d
= {0, 1, …, D − 1}, D is the length of observation interval, and M
c,b
(d) is the coarse timing metric at the b th receiver. From Equations 2 and 8, M
c,b
(d) will give a maximum value when d is at the delayed timing of the line-of-sight (LOS) path in multipath fading channels. Then, the CFO is
(9)
where ℜ{x} and ℑ{x} describe the real and the imaginary parts of x, respectively. After coarse frequency synchronization, the CFO-compensated received signal at the b th receiver is
(10)
where denotes the residual CFO and .
3.2 Generalized maximum-likelihood-based channel estimation
Assume there is no timing offset and CFO in Equation 2, the received training sequence at the b th receiver can be expressed as follows:
where
(12)
(13)
(14)
(15)
N
′ = 2N
c
+ N
g
, , is a N
′ × N
′ identity matrix, {w(n), ∀n} is independent identically distributed, and h
ib
(k) is the k th tap CIR from the i th transmit antenna to the b th receive antenna. In Equation 14, h
b
contains different CIRs from all transmit antennas. In general, the received training sequence is
(16)
where Equation 12 is a special case of Equation 16 when τ = 0 and ε
b
= 0. Then, the likelihood function is given by
(17)
and the ML estimate of h
b
can be obtained by
(18)
where A
H is the Hermitian of A, B
− is the generalized inverse of B, S
† is the Moore-Penrose pseudo-inverse of S,
(19)
and diag(·) represents a diagonal matrix. For fixed pseudo-sequences c
i
, ∀i ∈ {1, …, N
T
}, we only need to compute S
† once, and different CIR estimates can be obtained simply by multiplying the received training sequence at each receiver with S
†.
3.3 Joint timing synchronization and channel estimation
In this subsection, we utilize the coarse timing offset and CIR estimate based on the generalized ML criterion to develop a joint timing synchronization and channel estimation algorithm. After coarse timing synchronization, a sliding observation vector v
b
at the b th receiver is applied to obtain an advanced timing, relative timing indices, and the corresponding CIR estimates, where
(20)
, the length of observation interval is 2L + 1, and L is a positive integer without any constraint. In Equation 20, v
b
consists of 2L + 1 timing indices. Based on these timing indices in v
b
, the corresponding CIR estimates with K
′ taps are obtained by Equation 18, where
(21)
(22)
(23)
(24)
, is the CIR estimate corresponding to the time index v
b
(l
1), and is the estimated (k + 1)th tap CIR from the i th transmit antenna to the b th receive antenna corresponding to the time index v
b
(l
1). In order to avoid any loss of the channel information, K
′ should be at least equal to or greater than K.
If N
g
≥ K, can be obtained in an efficient way by averaging two channel estimates, and , where
(25)
(26)
(27)
and
(28)
After we obtain 2L + 1 CIR estimates, the advanced timing at the b th receiver τ
a d,b
is given by
(29)
where is the first tap in Equation 22. Then, relative timing indices in the modified sliding observation vector and the corresponding CIR estimates are fed forward to perform fine time adjustment, where
(30)
and .
3.4 Fine time adjustment
In this subsection, we exploit the MMSE criterion to perform fine time adjustment in an iterative manner based on the information of relative timing indices and the corresponding CIR estimates. First, two thresholds on the sum of the first N
T
tap powers N
T
· γ
1 and the sum of the first 2N
T
tap powers 2N
T
· γ
1 are chosen in order to eliminate the AWGN effect and to reduce the computational complexity, where γ
1 should be less than the tap power of the first path in the channel model. In other words, if or , τ
a d,b
is the estimated timing offset; otherwise, the algorithm keeps running until or for some f, where .
Let denote the convolution of the training sequence and the corresponding CIR estimate , where
(31)
(32)
and is the CIR estimate from the i th transmit antenna to the b th receive antenna. Then, the mean squared error (MSE) of the timing index t
b
(l
2) is given by
(33)
Thus, the estimated timing offset can be obtained by solving
(34)
From Equations 31 and 33 and reasonable CIR estimates, we have
(35)
where δ ≠ 0, δ is an integer, and . Consider a set Ω
U
= {0, 1, ⋯, u − 1} composed of consecutive timing indices that satisfy the following conditions
(36)
Then, the estimated timing offset and the CIR estimate at the b th receiver based on the thresholds and Equation 33 can be expressed as follows:
(37)
The detailed procedure of fine time adjustment is described in Algorithm 1.
Algorithm 1 Fine time adjustment