Interference Cancellation and Detection for More than Two Users - Interference Cancellation Using Space-Time Processing and Precoding Design - page 37

⎛
⎝

⎞
⎠ +

c 1
c 2
c 3
c 4

E s (

H †

1

2 H †

H †

1
2

1 H 1 ) −

(

1 y

=

1 H 1 )

n (3.23)

H † H †

1
2

1 H 1 ) −

where n = ( (

1 n

) has uncorrelated elements ∼ CN ( 0 , 1 ) . Equation ( 3.23 )

can be further rewritten as

⎛
⎝

⎞
⎠

2 ⎛
⎝

⎞
⎠ +

x 1 x 2 x 3 x 4
x 2 x 5 x 6 x 7
x 3 x 6 x 8 x 9
x 4 x 7 x 9 x 10

c 1
c 2
c 3
c 4

E s

H 1 H 1 ) −

1

2 H 1 y

( = n (3.24)

where

k 1 b k 2 c k 3 d

x 1 =

+ + + ,

x 2 =

− + −

a k 1 a k 1 b k 2 k 3 c k 2 k 3 d

x 3 = k 2 a + k 1 k 3 b + k 2 c + k 1 k 3 d , x 4 = k 3 a − k 1 k 2 b + k 1 k 2 c − k 3 d

k 1 a k 3 c k 2 d

x 5 = + b + + , x 6 = k 1 k 2 a − k 3 b + k 3 c − k 1 k 2 d

k 2 a k 3 b k 1 d x 7 = k 1 k 3 a + k 2 b + k 1 k 3 c + k 2 d , x 8 = + + c +

k 3 a k 2 b k 1 c

x 9 =

− + − ,

x 10 =

+ + +

k 2 k 3 a k 2 k 3 b k 1 c k 1 d d (3.25)

4 4

h 1 (

2

h 1 (

2

a =

1 |

i , 1 ) | , b =

1 |

i , 1 ) | ,

i = i =

4 4

h 1 (

2

h 1 (

2

c =

1 |

i , 1 ) | , d =

1 |

i , 1 ) | (3.26)

i = i =

Now let

⎛
⎝

⎞
⎠

1
2

x 1 x 2 x 3 x 4
x 2 x 5 x 6 x 7
x 3 x 6 x 8 x 9
x 4 x 7 x 9 x 10

H

= (3.27)

From Equation ( 3.24 ), we can see that User 1 transmits 4 different codewords along
4 different equivalent channel vectors in the 4 time slots. So the rate is 1. If k 1 , k 2 , k 3
are all real, from ( 3.27 ), it is easy to see that the equivalent channel matrix H is real.
So if QAM is used, Equation ( 3.24 ) is equivalent to the following two equations

⎛ ⎞

c 1 R
c 2 R
c 3 R
c 4 R

E s H

⎝ ⎠ +

H − 1 Real

H † {

1 y

}= Real { n } (3.28)