2.3.1 Conventional Spatial-Correlated Channel Model
Consider a single-cell massive SU-MIMO system with an NT-antenna BS and an NR-antenna UE, where NT NR. The signal received by the UE can be expressed as
y = Hx + n (2.7)
where H = [hij] is the NR× NT SSFC channel matrix with complex Gaussian entries, hij’s, x is the transmitted signal, and n ∼ CN (0, INR) represents the white noise.
Let Φ, ΦT, and ΦR be the spatial correlation matrices of vec(H)
Φdef= E
vec(H)vec(H)H
, (2.8)
and those of the Tx and Rx antennas, respectively.
In general, a spatial-correlated Rayleigh fading MIMO channel can be modeled as
vec(H) = Φ12vec(Hw), (2.9)
where Hw is NR× NT with i.i.d., zero-mean, unit-variance complex Gaussian entries.
For a massive MU-MIMO channel where the receiving end consists of many single-antenna receivers at various locations, the above model must be modified in an user-by-user sense. The small-scale fading channel seen by user k can in general be written as
hk = Φ
1 2
kh˜k, (2.10)
where Φk is the transmit spatial correlation matrix with respect to the kth user and h˜k ∼ CN (0M, IM).
2.3.2 Kronecker Model
Kronecker model [9] is commonly used which assumes that the spatial correlations among Tx antennas and those among Rx antennas are separable so that
Φ = ΦT⊗ ΦR = Φ12(Φ12)H, (2.11) where the square root matrix Φ12 has a similar decomposition
Φ12 = Φ
This model is reasonable accurate only when the main scattering is locally rich at the transmitter and receiver sides, respectively [?]. There are many field measurements and experiments that report inconsistencies with this model.
For a massive MU-MIMO channel where the receiving end consists of K single-antenna MSs, the Kronecker model is equivalent to conventional spatial-correlated chan-nel model in 2.10.
2.3.3 Virtual Channel Representation
To solve the deficiencies of the Kronecker model, Sayeed [?] suggested a so-called virtual channel representation that takes the Tx-Rx cross-correlation into account. This model expands the spatial correlations by unitary matrices, ΦT, ΦR, relate the Tx and Rx spatial modes by a coupling matrix. Using the Fourier basis, one obtains
H = FR(Ω Hw)FHT, (2.14)
where FR and FT are DFT matrices of dimension NR and NT and Ω is the coupling matrix. However, this model is only applicable for single-polarized ULAs and the Fourier basis is far from the optimal choice.
2.3.4 Weichselberger Model
An obvious optimal choice of basis that incurs no approximation error can be readily obtained by performing eigen-decomposition on the spatial correlation matrices. Weich-selberger et al. therefore suggested the model [?]
H = UR
Ω H˜ w
UHT, (2.15)
where UT and UR are the eigenbases of ΦT, and ΦR, respectively, i.e.,
ΦR = URΛRUHR, ΦT = UTΛTUHT. (2.16)
with ΛR and ΛT being the diagonal matrices consist of the (nonnegative) eigenvalues of ΦR and ΦT. ˜Ω is the element-wise square root of the coupling matrix Ω = [ωij] in which each entry specifies the average energy coupled from a transmit eigenmode to a receiver eigenmode [?].
Let uR,i and uT,j be the ith and jth column vector of UR and UT, respectively. Then
ωij = En uHR,iHuT,j 2o
, i = 1, · · · , NR, j = 1, · · · , NT (2.17)
The Weichselberger model is perhaps more convenient to generate the SSFC matrix H and for evaluating the channel capacity of correlated MIMO channels as the coefficient matrix has independent entries. It is also useful to analyze MIMO system performance.
However, it is not suitable for channel estimation applications because the number of parameters, including the unknown eigenbases, is even larger than that of H.
2.3.5 Rank-reduced Channel Representation
The rank-reduced (RR) model introduced in [12] is reviewed in this section. Singular value decomposition (SVD) of H gives
H = UΛVH, (2.18)
Let Q1 and Q2 be two predefined unitary matrices. Then
Q1 = UP1, Q2 = VP2, (2.19)
where Λ is a NR× NT rectangular diagonal matrix with nonnegative entries and both P1 and P2 are also unitary. This unitary transform leads to
H = Q1P−11 Λ(P−12 )HQH2 = Q1CQH2 , (2.20)
with a random matrix C characterizing the cross-coupling effect. It can be verified that all the aforementioned models are special cases of (2.20) which is valid for all slow-varying narrowband MIMO channels without any attached pre-assumptions needed.
It is equivalent to the Kronecker model if C satisfies
vec(C) = (ΞT⊗ ΞR)vec(Hω), (2.21)
where ΞTand ΞR are obtained via Gram-Schmidt orthonormalization with Φ
1 2
T = Q2ΞT and Φ
1 2
R = Q1ΞR.
If Q1and Q2are chosen to be composed of columns of DFT matrices, this generalized model is compatible with the virtual channel representation [?]. Finally, the RR model is related to the Weichselberger model via
UT= Q2PHT, UR = Q1PHR (2.22)
with PT and PR being the eigenbasis matrices of E{CCH} and E{CTC∗} whose eigen-values are the same as those of E{HHH} and E{HTH∗}.
The use of pre-determined basis matrices avoids the need for the channel estimator to perform the extra work of determining the eigenbasis (matrices) and provides a con-venient way for RR representation. Furthermore, when the AoD spread is small, we can show that (2.20) can be expressed as
H = QRCQHTW, (2.23)
where W = Diag[w1, w2, . . . , wM] and wi = exp(−j2π(i−1)ξλ sinφ), ξ being the distance between neighboring elements at the BS linear array, and φ is the mean AoD.
As UL MU-MIMO introduced in Section 2.1.2 being considered, we can summarize the SSFC matrix (2.20) and (2.23) into the lemma below:
Lemma 2.3.1 (RR representations). The channel vector seen by kth user can be repre-sented by
h = Q(I)mc(I) (2.24)
or alternately by
h = W(φ)Q(II)m c(II) (2.25)
where Q(I)m, Q(II)m ∈ RM ×m are predetermined (unitary) basis matrices and c(I), c(II) ∈ Cm×1 are the channel vectors with respect to bases Q(I)m and Q(II)m for the user k-BS link and W(φ) is diagonal with unit magnitude entries. The two equalities hold only if m = M and become approximations if m < M . Furthermore, if the AS is small, [W(φ)]ii = exp
−j2π(i−1)ξλ sin φ
with ξ and λ being the antenna spacing and signal wavelength and φ is the mean AoA.
That is, Lemma 2.3.1 suggests that the mean AoA (which is close to the incident angle of the strongest path) of each user is extractable if its AS is small. In addition, due to the large aperture massive MIMO antenna array has offered, good AoA resolution and thus accurate mean value extraction are guaranteed [1].