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Downlink deprecoding onto transmission layers

`out = lteDLDeprecode(in,nu,txscheme,codebook)`

`out = lteDLDeprecode(enb,chs,in)`

For transmission schemes `'CDD'`

, `'SpatialMux'`

,
and `'MultiUser'`

, and degenerately `'Port0'`

,

Precoding involves multiplying a

-by-*P*precoding matrix,*v*, by a*F*-by-*v**N*_{SYM}matrix, representing*N*_{SYM}symbols on each oftransmission layers. This multiplication yields a*v*-by-*P**N*_{SYM}matrix, representing*N*_{SYM}precoded symbols on each ofantenna ports. Depending on the transmission scheme, the precoding matrix can be composed of multiple matrices multiplied together. But the size of the product,*P*, is always*F*-by-*P*.*v*

For the `'TxDiversity'`

transmission scheme,

A

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

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

When * v* is

`'CDD'`

, `'SpatialMux'`

,
and `'MultiUser'`

transmission schemes, and when `'TxDiversity'`

transmission scheme, The precoding matrix,

, is square. Its size is 2*F*-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*P**F*^{ –1}. The matrix inversion is computed using LU decomposition with partial pivoting (row exchange):Perform LU decomposition

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

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

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

The degenerate case of the `'Port0'`

transmission
scheme falls into this category, with * P* =

For the `'CDD'`

, `'SpatialMux'`

,
and `'MultiUser'`

transmission schemes,

The deprecoding is then performed by multiplying

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

(`symbols`

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

For the `'TxDiversity'`

transmission scheme,

The deprecoding is performed, multiplying

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

(`symbols`

is size*P**N*_{SYM}-by-, so the transpose is a*P*-by-*P**P**N*_{SYM}matrix), having first been reshaped into a 2-by-*P**N*_{SYM}matrix. This multiplication yields a 2-by-*v**N*_{SYM}, matrix which is then split into two-by-*v**N*_{SYM}matrices. To recover the-by-*v**N*_{SYM}matrix of transmission layers multiply the second matrix byand add the two matrices together (thus recombining real and imaginary parts).*j*

For the other cases, specifically `'CDD'`

, `'SpatialMux'`

,
and `'MultiUser'`

transmission schemes with * v* ≠

`'TxDiversity'`

transmission
scheme with The precoding matrix

is not square. Instead, the matrix is rectangular with size*F*-by-*P*, except in the case of*v*`'TxDiversity'`

transmission scheme with= 4, where it is of size*P*-by-(2*P*^{ 2}= 16)-by-8. The number of rows is always greater than the number of columns in the matrix*P*is size*F*-by-*m*with*n*>*m*.*n*In this case, the deprecoder takes the matrix pseudo-inversion of the precoding matrix to yield the deprecoding matrix

. The matrix pseudo-inversion is computed as follows.*F*^{ +}Perform LU decomposition

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

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

−*m*columns of*n*to give $$\overline{L}$$.*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. "Physical Channels
and Modulation." *3rd Generation Partnership Project;
Technical Specification Group Radio Access Network; Evolved Universal
Terrestrial Radio Access (E-UTRA)*. URL: http://www.3gpp.org.

[2] 3GPP TS 36.213. "Physical layer
procedures." *3rd Generation Partnership Project;
Technical Specification Group Radio Access Network; Evolved Universal
Terrestrial Radio Access (E-UTRA)*. URL: http://www.3gpp.org.

[3] Strang, Gilbert. *Linear Algebra
and Its Application*. Academic Press, 1980. 2nd Edition.

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