Downlink deprecoding onto transmission layers
For transmission schemes 'CDD'
, 'SpatialMux'
,
and 'MultiUser'
, and degenerately 'Port0'
,
Precoding involves multiplying a P-by-v precoding matrix, F, by a v-by-NSYM matrix, representing NSYM symbols on each of v transmission layers. This multiplication yields a P-by-NSYM matrix, representing NSYM precoded symbols on each of P antenna ports. Depending on the transmission scheme, the precoding matrix can be composed of multiple matrices multiplied together. But the size of the product, F, is always P-by-v.
For the 'TxDiversity'
transmission scheme,
A P 2-by-2v precoding
matrix, F, is multiplied by a 2v-by-NSYM matrix,
formed by splitting the real and imaginary components of a v-by-NSYM matrix
of symbols on layers. This multiplication yields a P 2-by-NSYM matrix
of precoded symbols, which is then reshaped into a P-by-PNSYM matrix
for transmission. Since v is P for
the 'TxDiversity'
transmission scheme, F is
of size P 2-by-2P,
rather than P 2-by-2v.
When v is P in 'CDD'
, 'SpatialMux'
,
and 'MultiUser'
transmission schemes, and when P and v are
2 in the 'TxDiversity'
transmission scheme,
The precoding matrix, F, is square. Its size is 2P-by-2P for the transmit diversity scheme and P-by-P otherwise. In this case, the deprecoder takes the matrix inversion of the precoding matrix to yield the deprecoding matrix F –1. The matrix inversion is computed using LU decomposition with partial pivoting (row exchange):
Perform LU decomposition PxF = LU.
Solve LY = I using forward substitution.
Solve UX = Y using back substitution.
F –1 = XPx.
The degenerate case of the 'Port0'
transmission
scheme falls into this category, with P = v = 1.
For the 'CDD'
, 'SpatialMux'
,
and 'MultiUser'
transmission schemes,
The deprecoding is then performed by multiplying F –1 by the transpose of
the input symbols
(symbols
is
size NSYM-by-P,
so the transpose is a P-by-NSYM matrix).
This multiplication recovers the v-by-NSYM (equals P-by-NSYM)
matrix of transmission layers.
For the 'TxDiversity'
transmission scheme,
The deprecoding is performed, multiplying F –1 by the transpose of
the input symbols
(symbols
is
size PNSYM-by-P,
so the transpose is a P-by-PNSYM matrix),
having first been reshaped into a 2P-by-NSYM matrix.
This multiplication yields a 2v-by-NSYM,
matrix which is then split into two v-by-NSYM matrices.
To recover the v-by-NSYM matrix
of transmission layers multiply the second matrix by j and
add the two matrices together (thus recombining real and imaginary
parts).
For the other cases, specifically 'CDD'
, 'SpatialMux'
,
and 'MultiUser'
transmission schemes with v ≠ P and the 'TxDiversity'
transmission
scheme with P = 4,
The precoding matrix F is not square.
Instead, the matrix is rectangular with size P-by-v,
except in the case of 'TxDiversity'
transmission
scheme with P = 4,
where it is of size P 2-by-(2P = 16)-by-8. The number of rows is always
greater than the number of columns in the matrix F is
size m-by-n with m > n.
In this case, the deprecoder takes the matrix pseudo-inversion of the precoding matrix to yield the deprecoding matrix F +. The matrix pseudo-inversion is computed as follows.
Perform LU decomposition PxF = LU.
Remove the last m − n rows of U to give .
Remove the last m − n columns of L to give .
(the matrix inversions are carried out as in the previous steps).
F + = XPx
The application of the deprecoding matrix F + is the same process as described for deprecoding the square matrix case with F + in place of F –1.
This method of pseudo-inversion is based onLinear Algebra and Its Application [3], Chapter 3.4, Equation (56).
[1] 3GPP TS 36.211. “Evolved Universal Terrestrial Radio Access (E-UTRA); Physical Channels and Modulation.” 3rd Generation Partnership Project; Technical Specification Group Radio Access Network. URL: https://www.3gpp.org.
[2] 3GPP TS 36.213. “Evolved Universal Terrestrial Radio Access (E-UTRA); Physical layer procedures.” 3rd Generation Partnership Project; Technical Specification Group Radio Access Network. URL: https://www.3gpp.org.
[3] Strang, Gilbert. Linear Algebra and Its Application. Academic Press, 1980. 2nd Edition.