Interference Cancellation and Detection for MAC with Two Users - Interference Cancellation Using Space-Time Processing and Precoding Design - page 14

2.3 Decoding

In this section, we focus on the decoding. We start with Eq. ( 2.20 ). Note that ( 2.20 )
can also be written as
⎛

⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞

y 1

h 11 h 11

g 11 g 11 n 1 c 1

c 2
s 1
s 2

E s

⎝ ⎠ = ⎝

( h 12 ) ∗ − ( h 12 ) ∗ (

⎠ ⎝ ⎠ + ⎝ ⎠

y 1 ) ∗
y 2

g 12 ) ∗ − ( g 12 ) ∗ n 1 ) ∗

n 2

( (

h 21 h 21

g 21 g 21

( h 22 ) ∗ − ( h 22 ) ∗ (

y 2 ) ∗ g 22 ) ∗ − ( g 22 ) ∗ n 2 ) ∗

( (

(2.50)

and we define

⎛
⎝

⎞
⎠ ,

⎛
⎝

⎞
⎠

h 11 h 11

g 11 g 11 n 1

H 1 G 1
H 2 G 2

( h 12 ) ∗ − ( h 12 ) ∗ ( g 12 ) ∗ − ( g 12 ) ∗ n 1 ) ∗

n 2

(

H = = n = (2.51)

h 21 h 21

g 21 g 21

( h 22 ) ∗ − ( h 22 ) ∗ ( g 22 ) ∗ − ( g 22 ) ∗ n 2 ) ∗ (

where

h 11 h 11
( h 12 ) ∗ − ( h 12 ) ∗

g 11 g 11

H 1 = , G 1 =

g 12 ) ∗ − ( g 12 ) ∗ (

h 21 h 21
( h 22 ) ∗ − ( h 22 ) ∗

g 21 g 21

H 2 = , G 2 = (2.52)

g 22 ) ∗ − ( g 22 ) ∗ (

Note that H has a quasi-orthogonal structure, i.e., the first two columns are o rthogonal
to the second two columns. If we multiply both sides of Eq. ( 2.50 ) with H † , we will
have

⎛
⎝

⎞
⎠ =

⎛
⎝

⎞
⎠ +

y 1

c 1
c 2
s 1
s 2

H 1 H 1 +

E s

H 2 H 2 y 1 ) ∗

y 2

( 0

H † H † n (2.53)

G † G † 0

1 G 1 + 2 G 2

y 2 ) ∗

(

Now we define

⎛ ⎞

y 1

⎝ ⎠

y 1 ) ∗
y 2

y 1 (

H †

y = = (2.54)

y 2

y 2 ) ∗ (

. Note that the noise elements of H † n are
correlated with covariance matrix H † H . We ca n whiten this noise vector by multi-
plying both sides of ( 2.54 ) by the matrix

y ( 1 , 1 ) y ( 3 , 1 )

where y 1 = , y 2 =

( , ) ( , )

y 2 1 y 4 1

H † H

1
2 as follows

) −

(