Interference Cancellation and Detection for More than Two Users - Interference Cancellation Using Space-Time Processing and Precoding Design - page 43

⎛
⎝

⎞
⎠

⎛
⎝

⎞
⎠

⎛
⎝

⎞
⎠

t 1
t 2
t 3
t 4

s 1 z 1

s 2 z 2

We l e t , , , denote the detected signals of Users 2, 3, 4, respec-

s 3 z 3

s 4 z 4

⎛
⎝

⎞
⎠

⎛
⎝

⎞
⎠

⎛
⎝

⎞
⎠

t 1
t 2

s 1 z 1

s 2 z 2

tively. We subtract the term H 2 , H 3 , H 4 , from Equation ( 3.20 )

s 3 t 3 z 3

s 4 t 4 z 4

to remove the effect of Users 2, 3, 4 to have

⎛ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎞ ⎛ ⎞

t 1
t 2
t 3
t 4

s 1 z 1 c 1

c 2
c 3
c 4

E s E s H 1

⎝ ⎝ ⎠ − ⎝ ⎠ − ⎝ ⎠ ⎠ = ⎝ ⎠ +

s 2 z 2

y − H 2 H 3 H 4 n + σ

s 3 z 3

s 4 z 4

(3.52)

σ = σ 1 + σ 2 + σ 3 and

where

⎛ ⎛ ⎞ ⎛ ⎞ ⎞

s 1
s 2
s 3
s 4

s 1

E s H 2

⎝ ⎝ ⎠ − ⎝ ⎠ ⎠ ,

s 2

=

σ 1

s 3

s 4

⎛ ⎛ ⎞ ⎛ ⎞ ⎞

t 1
t 2
t 3
t 4

t 1
t 2
t 3
t 4

⎝ ⎝ ⎠ − ⎝ ⎠ ⎠ ,

σ 2 = H 3

⎛
⎝

⎛
⎝

⎞
⎠ −

⎛
⎝

⎞
⎠

⎞
⎠

z 1
z 2
z 3
z 4

z 1

z 2

σ 3 = H 4 (3.53)

z 3

z 4

denote the residual error. Then we can multiply both sides of Equation ( 3.52 )by H †
1
and use the same method in Sect. 3.2 to detect the signals of User 1. In what follows,
we first show that the method still provides full diversity to User 1. There are two
factors that result in an error for User 1. The first one is error in decoding symbols
of User 1 after removing the effect of other users and the second one is the error
in detecti ng the symbols of other users at the first time, i.e., error propagation. Let
Pr denote the pairwise error probability for User 1, we separate these two

events to have

( d 1 → d 1 )

Pr ( d 1 → d 1 ) = Pr { d 1 → d 1 | σ = 0 } Pr { σ = 0 }

+ Pr { d 1 → d 1 | σ = 0 } Pr { σ = 0 }

= Pr { d 1 → d 1 | σ = 0 } ( 1 − Pr { σ = 0 } )

+ Pr { d 1 → d 1 | σ = 0 } Pr { σ = 0 } (3.54)