Interference Cancellation and Detection for More than Two Users - Interference Cancellation Using Space-Time Processing and Precoding Design - page 36

3.2 Decoding

Using our precoders, Equation ( 3.5 ) becomes

⎛
⎝

⎞
⎠ +

⎛
⎝

⎞
⎠ +

⎛
⎝

⎞
⎠ +

⎛
⎝

⎞
⎠ ) +

c 1
c 2
c 3
c 4

s 1
s 2
s 3
s 4

t 1
t 2
t 3
t 4

z 1
z 2
z 3
z 4

E s (

y = H 1 H 2 H 3 H 4 n (3.20)

where

⎛ ⎞

h i ( k 1 h i ( k 2 h i ( k 3 h i ( 1 , 1 ) 1 , 1 ) 1 , 1 ) 1 , 1 )

h i ( k 1 h i ( k 2 h i ( k 3 h i (

⎝ , ) , ) , ) , ) ⎠

2 1 2 1 2 1 2 1

h i ( k 1 h i ( k 2 h i ( k 3 h i ( 3 , 1 ) 3 , 1 ) 3 , 1 ) 3 , 1 )

h i ( k 1 h i ( k 2 h i ( k 3 h i (

4 , 1 ) 4 , 1 ) 4 , 1 ) 4 , 1 )

h i ( )) ∗ − ( h i ( )) ∗ h i ( )) ∗ − h i ( )) ∗

k 1 ( 1 , 1 1 , 1 k 3 ( 1 , 1 k 2 ( 1 , 1

h i ( )) ∗ − ( h i ( )) ∗ h i ( )) ∗ − h i ( )) ∗

k 1 ( 2 , 1 2 , 1 k 3 ( 2 , 1 k 2 ( 2 , 1

h i ( )) ∗ − ( h i ( )) ∗ h i ( )) ∗ − h i ( )) ∗ k 1 ( 3 , 1 3 , 1 k 3 ( 3 , 1 k 2 ( 3 , 1

h i ( )) ∗ − ( h i ( )) ∗ h i ( )) ∗ − h i ( )) ∗

k 1 ( 4 , 1 4 , 1 k 3 ( 4 , 1 k 2 ( 4 , 1

H i = (3.21)

k 2 h i ( k 3 h i ( h i ( k 1 h i ( 1 , 1 ) 1 , 1 ) 1 , 1 ) 1 , 1 )

k 2 h i ( k 3 h i ( h i ( k 1 h i ( 2 , 1 ) 2 , 1 ) 2 , 1 ) 2 , 1 )

k 2 h i ( k 3 h i ( h i ( k 1 h i ( 3 , 1 ) 3 , 1 ) 3 , 1 ) 3 , 1 )

k 2 h i ( k 3 h i ( h i ( k 1 h i ( 4 , 1 ) 4 , 1 ) 4 , 1 ) 4 , 1 )

h i ( )) ∗ − h i ( )) ∗ h i ( )) ∗ − ( h i ( )) ∗

k 3 ( 1 , 1 k 2 ( 1 , 1 k 1 ( 1 , 1 1 , 1

h i ( )) ∗ − h i ( )) ∗ h i ( )) ∗ − ( h i ( )) ∗ k 3 ( 2 , 1 k 2 ( 2 , 1 k 1 ( 2 , 1 2 , 1

h i ( )) ∗ − h i ( )) ∗ h i ( )) ∗ − ( h i ( )) ∗

k 3 ( 3 , 1 k 2 ( 3 , 1 k 1 ( 3 , 1 3 , 1

h i ( )) ∗ − h i ( )) ∗ h i ( )) ∗ − ( h i ( )) ∗

k 3 ( 4 , 1 k 2 ( 4 , 1 k 1 ( 4 , 1 4 , 1

Here y and n are the same with y and n in Equation ( 3.5 ). Note that using ou r
pr ec od ers, each column of array H 1 is orthogonal to each column of matrices H 2 ,
H 3 , H 4 .
In ord er to decode symbols fromUser 1, we multiply both sides of Equation ( 3.20 )
by array H 1 to achieve

⎛
⎝

⎞
⎠ +

c 1
c 2
c 3
c 4

E s H 1 H 1

H 1 y H 1 n = (3.22)

Note that the noise elements of H †

1 n are correlated with covariance matrix H †

1 H 1 .
We can w h ite n this noise vector by multiplying both sides of Equation ( 3.22 )bythe
matrix H †

1
2 as follows

1 H 1 ) −

(