Interference Cancellation and Detection for MAC with Two Users - Interference Cancellation Using Space-Time Processing and Precoding Design - page 20

H i

) = H i ) = ··· = H i

( 1 ( 2 ( N ) (2.85)

G i

) = G i ) = ··· = G i ( 1 ( 2 ( N ) (2.86)

respectively. In order to make the symbols of Users 1 and 2 transmitted in two
orthogonal subspaces, i.e., the first N columns of H are orthogonal to the second N
columns of H ,welet

G i

= η i − H i

∗ (2.87)

( 1 , 1 ) ( 2 , 1 )

G i H i

( , ) ( , )

2 1 1 1

Step 3 : Designing low-complexity algorithms to calculate the parameters of the
precoders:
From ( 2.87 ), we have

⎛ ⎞ ⎛ ⎞

∗

b i 11
b i 21
.
b i N 1

a i 11
a i 21
.
a i N 1

g 11 g 12 ··· ∗

= η i −

⎝ ⎠ ⎝ ⎠

g 1 N h 21 − h 22 ··· − h 2 N

(2.88)

g 21 g 22 ··· g 2 N h 11 h 12 ··· h 1 N

with normalization equations

1
N

a i 11 |

2

a i 21 |

2

a i N 1 |

2

| +| +···+| =

1
N

b i 11 |

2

b i 21 |

2

b i N 1 |

2

| +| +···+| = (2.89)

Note that the channel matrices in ( 2.88 ) are not square matrices. Therefore, we cannot
use the reverse matrix directly as we did for the users with 2 transmit antennas in
Sect. 2.1 . Instead, in order to simplify the precoder design, at the first 2 time slots,
we let all the elements in complex vector

= a i 11 a i 21 ···

a i N 1 T a i

, = ,

i 1 2 (2.90)

be zero except for the first 2 elements and also let all the elements in

= b i 11 b i 21 ···

b i N 1 T

b i , i = 1 , 2 (2.91)

be zero except for the first 2 elements. By the above choices for a i and b i ,Eq.( 2.88 )
results in

g 11 g 12
g 21 g 22

∗ b i 11
b i 21

∗

= η i −

a i 11
a i 21

h 21 − h 22

(2.92)

h 11 h 12