B.2.3.2 MIMO Correlation Matrices using Cross Polarized Antennas (X-pol)
38.521-43GPPNRPart 4: PerformanceRadio transmission and receptionRelease 17TSUser Equipment (UE) conformance specification
The MIMO channel correlation matrices defined in B.2.3.2 apply for the antenna configuration using cross polarized (XP/X-pol) antennas at both gNB and UE. The cross-polarized antenna elements with +/-45 degrees polarization slant angles are deployed at gNB and cross-polarized antenna elements with +90/0 degrees polarization slant angles are deployed at UE.
For the 2D cross-polarized antenna array at eNodeB, the N antennas are indexed by 
, and total number of antennas is 
, where
– 
is the number of antenna elements in first dimension with same polarization,
– 
is the number of antenna elements in second dimension with same polarization, and
– 
is the number of polarization groups.
For the 2D cross-polarized antennas at gNB, the N antennas are labelled such that antennas shall be in increasing order of the second dimension firstly, then the first dimension, and finally the polarization group. For a specific antenna element at p-th polarization, n1-th row, and n2-th column within the 2D antenna array, the following index number is used for antenna labelling:
where N is the number of transmit antennas, p is the polarization group index, n1 is the row index, and n2 is the column index of the antenna element.
For the linear (single dimension, 1D) cross-polarized antenna, the N antennas are labelled following the above equations with N2=1.
B.2.3.2.1 Definition of MIMO Correlation Matrices using cross polarized antennas
For the channel spatial correlation matrix, the following is used:
where
– 
is the spatial correlation matrix at the UE with same polarization,
– 
is the spatial correlation matrix at the gNB with same polarization,
– 
is a polarization correlation matrix, and
– 
denotes transpose.
The matrix 
is defined as:
A permutation matrix P elements are defined as:
.
where Nt and Nr is the number of transmitter and receiver respectively. This is used to map the spatial correlation coefficients in accordance with the antenna element labelling system described in B.2.3.2.
For the 2D cross-polarized antenna array at gNB, the spatial correlation matrix at the gNB is further expressed as following for 2D cross-polarized antenna array at gNB:
where
– 
is the correlation matrix of antenna elements in first dimension with same polarization, and
– 
is the correlation matrix of antenna elements in second dimension with same polarization.
For the 2D cross polarized antenna array at gNB side, the spatial correlation matrices in one direction of antenna array are as follows:
– For 1 antenna element with the same polarization in one direction,
.
– For 2 antenna elements with the same polarization in one direction,
.
– For 3 antenna elements with the same polarization in one direction,
.
– For 4 antenna elements with the same polarization in one direction,
.
where the index i = 1,2 stands for first dimension and second dimension respectively.
For the 1D cross-polarized antenna array at gNB, the matrix of
is determined by follow the equations for 2D cross-polarized antenna array and letting
, i.e.
The spatial correlation matrices at UE side are as follows:
– For 1 antenna element with the same polarization,
.
– For 2 antenna elements with the same polarization,
.
B.2.3.2.2 MIMO Correlation Matrices using cross polarized antennas
The values for parameters α1, α2. β and γ for the cross polarized antenna models are given in Table B.2.3.2.2-1.
Table B.2.3.2.2-1: The α and β parameters for cross-polarized MIMO correlation matrices
|
Correlation Model |
α1 |
α2 |
β |
|
|
Medium Correlation |
0.3 |
0.3 |
0.6 |
0.2 |
|
High Correlation |
0.9 |
0.9 |
0.9 |
0.3 |
|
Note 1: Value of α1 applies when more than one pair of cross-polarized antenna elements in first dimension at gNB side. Note 2: Value of α2 applies when more than one pair of cross-polarized antenna elements in second dimension at gNB side. Note 3: Value of β applies when more than one pair of cross-polarized antenna elements at UE side. |
||||
For the 1D cross polarized antenna array at gNB side, the correlation matrices for high spatial correlation and medium correlation are defined in Table B.2.3.2.2-2 and Table B.2.3.2.2-3 as below.
The values in Table B.2.3.2.2-2 have been adjusted to ensure the correlation matrix is positive semi-definite after round-off to 4 digit precision. This is done using the equation:
or 
Where the value "a" is a scaling factor such that the smallest value is used to obtain a positive semi-definite result. For the 8(4,1,2)x2 high spatial correlation case, a=0.00010.
Table B.2.3.2.2-2: MIMO correlation matrices for high spatial correlation
|
4(2,1,2)x2 case |
|
|
2(1,1,2)x4 case |
|
|
4(2,1,2)x4 case |
|
|
8(4,1,2)x2 case |
|
Table B.2.3.2.2-3: MIMO correlation matrices for medium spatial correlation
|
2(1,1,2)x2 case |
 |
B.2.3.2.3 Beam steering approach
For the 2D cross-polarized antenna array at gNB, given the channel spatial correlation matrix in B.2.3.2.1 and B.2.3.2.2, the corresponding random channel matrix H can be calculated. The signal model for the k-th slot is denoted as:
And the steering matrix is further expressed as following:
Where:
– H is the Nr×Nt channel matrix per subcarrier.
– 
is the steering matrix,
– 
is the steering matrix in first dimension with same polarization,
– 
is the steering matrix in second dimension with same polarization,
– 
is the number of antenna elements in first dimension with same polarization,
– 
is the number of antenna elements in second dimension with same polarization,
For antenna array with only one direction, number of antenna element in second direction 
equals 1.
For 1 antenna element with the same polarization in one direction,
.
For 2 antenna elements with the same polarization in one direction,
.
For 3 antenna elements with the same polarization in one direction,
.
For 4 antenna elements with the same polarization in one direction,
.
where the index 
stands for first dimension and second dimension respectively.
controls the phase variation in first dimension and second dimension respectively, and the phase for k-th subframe is denoted by
, where 
is the random start value with the uniform distribution, i.e. 
, 
is the step of phase variation, which is defined in Table B.2.3.2.3-1, and k is the linear increment of 2-μ for every slot throughout the simulation, the index 
stands for first dimension and second dimension respectively.
– 
is the precoding matrix for Nt transmission antennas,
– y is the received signal, x is the transmitted signal, and n is AWGN.
– 
corresponds to subcarrier spacing configuration, 
For the 1D cross-polarized antenna array at gNB, the corresponding random channel matrix H can be calculated by letting N2=1, i.e.
Table B.2.3.2.3-1: The step of phase variation
|
Variation Step |
Value (rad/ms) |
|
|
1.2566×10-3 |
B.2.3.2.3A Beam steering approach with dual cluster beams
For the 2D cross-polarized antenna array at gNB, given the channel spatial correlation matrix in B.2.3.2.1 and B.2.3.2.2, the corresponding random channel matrix H can be calculated. The signal model for the k-th slot is denoted as
And the steering matrix is further expressed as following:
where
– 
,
are independent channels for the first beam and second beam with the Nr xNt channel matrix per subcarrier.
– 
, 
are the steering matrix for first beam and second beam
– 
is the steering matrix in first dimension with same polarization,
– 
is the steering matrix in second dimension with same polarization,
– 
is the number of antenna elements in first dimension with same polarization,
– 
is the number of antenna elements in second dimension with same polarization,
– For antenna array with only one direction, number of antenna element in second direction 
equals 1，
– 
is the relative power ratio of the second beam to the first beam, the value of 
is specific to a test case,
For 1 antenna element of the same polarization in one direction, 
.
For 2 antenna elements of the same polarization in one direction, 
.
For 3 antenna elements of the same polarization in one direction,
.
For 4 antenna elements of the same polarization in one direction, 
.
where the index 
stands for first dimension and second dimension respectively.
– 
controls the phase variation in first dimension and second dimension respectively, and the phase for k-th subframe is denoted by
, where 
is the random start value with the uniform distribution, i.e., 
, 
is the step of phase variation, which is defined in Table B.2.3.2.3A-1, and k is the linear increment of 2-μ for every slot throughout the simulation, the index 
stands for first dimension and second dimension respectively.
– 
is the precoding matrix for Nt transmission antennas,
– y is the received signal, x is the transmitted signal, and n is AWGN.
– 
corresponds to subcarrier spacing configuration, 
For the 1D cross-polarized antenna array at gNB, the corresponding random channel matrix H can be calculated by letting N2=1, i.e.,
Table B.2.3.2.3A-1: The step of phase variation
|
Variation Step |
Value (rad/subframe) |
|
|
1.2566×10-3 |
|
|
2.5132×10-3 |
