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-*N*_{SYM}matrix, representing*N*_{SYM}symbols on each of*v*transmission layers. This multiplication yields a*P*-by-*N*_{SYM}matrix, representing*N*_{SYM}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*-by-2^{ 2}*v*precoding matrix,*F*, is multiplied by a 2*v*-by-*N*_{SYM}matrix, formed by splitting the real and imaginary components of a*v*-by-*N*_{SYM}matrix of symbols on layers. This multiplication yields a*P*-by-^{ 2}*N*_{SYM}matrix of precoded symbols, which is then reshaped into a*P*-by-*P**N*_{SYM}matrix for transmission. Since*v*is*P*for the`'TxDiversity'`

transmission scheme,*F*is of size*P*-by-2^{ 2}*P*, rather than*P*-by-2^{ 2}*v*.

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 2*P*-by-2*P*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

*P*=_{x}F*LU*.Solve

*LY*=*I*using forward substitution.Solve

*UX*=*Y*using back substitution.*F*=^{ –1}*XP*._{x}

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*by the transpose of the input^{ –1}`symbols`

(`symbols`

is size*N*_{SYM}-by-*P*, so the transpose is a*P*-by-*N*_{SYM}matrix). This multiplication recovers the*v*-by-*N*_{SYM}(equals*P*-by-*N*_{SYM}) matrix of transmission layers.

For the `'TxDiversity'`

transmission scheme,

The deprecoding is performed, multiplying

*F*by the transpose of the input^{ –1}`symbols`

(`symbols`

is size*P**N*_{SYM}-by-*P*, so the transpose is a*P*-by-*P**N*_{SYM}matrix), having first been reshaped into a 2*P*-by-*N*_{SYM}matrix. This multiplication yields a 2*v*-by-*N*_{SYM}, matrix which is then split into two*v*-by-*N*_{SYM}matrices. To recover the*v*-by-*N*_{SYM}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*-by-(2^{ 2}*P*= 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

*P*=_{x}F*LU*.Remove the last

*m*−*n*rows of*U*to give $$\overline{U}$$.Remove the last

*m*−*n*columns of*L*to give $$\overline{L}$$.$$X={\overline{U}}^{H}{\left(\overline{U}{\overline{U}}^{H}\right)}^{-1}{\left({\overline{L}}^{H}\overline{L}\right)}^{-1}{\overline{L}}^{H}$$ (the matrix inversions are carried out as in the previous steps).

*F*=^{ +}*XP*_{x}

The application of the deprecoding matrix *F ^{ +}* is the same process as described
for deprecoding the square matrix case with

This method of pseudo-inversion is based on*Linear
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.