Interference Cancellation and Detection for MAC with Two Users - Interference Cancellation Using Space-Time Processing and Precoding Design - page 11

g 11 (

= η 2 −

a 12
a 22

g 12 b 12 ) ∗

h 21 − h 22

(2.24)

g 21 g 22

b 22 ) ∗

( h 11 h 12

with the normalization conditions of the precoders represented by

1
2

a 11 |

2

a 21 |

2

b 11 |

2

b 21 |

2

| +| =| +| = (2.25)

1
2

a 12 |

2

a 22 |

2

b 12 |

2

b 22 |

2

| +| =| +| = (2.26)

wherewehaveusedEqs.( 4.15 ) and ( 2.18 ). Note that Eqs. ( 2.25 ) and ( 2.26 ) are non-
linear equations, if numerical algorithms are used to solve these equations directly,
the encoding complexity will be increased exponentially with respect to the number
of users and antennas. So we need to find a low-complexity method to determine the
precoder parameters. First, we consider Eqs. ( 2.23 ) and ( 2.25 ).
From ( 2.23 ), we have

(

= η 1 g 11

− 1 − a 11

a 21

b 11 ) ∗ g 12 h 21 − h 22

(2.27)

b 21 ) ∗

g 21 g 22

h 11 h 12

(

Let

g 11 − 1 −

g 12

h 21 − h 22

Q = (2.28)

g 21 g 22

h 11 h 12

By ( 2.25 ) and ( 2.27 ), we have

η 1 Q a 11

2
F =

1
2

b 11 |

2

b 21 |

2

| +| = (2.29)

a 21

Now, let us consider the Singular Value Decomposition of matrix Q , i.e.,

V † V † Q = U Σ = U diag (λ 1 ,λ 2 ) (2.30)

where U and V are unitary matrices and Σ is a diagonal matrix with nonnegative

diagonal elements { λ 1 ,λ 2 } in decreasing order. Replacing ( 2.30 )in ( 3.8 ) results in

η 1 U

V † a 11
a 21

2
F =

1
2

Σ (2.31)

Multiplying by a unitary matrix does not change the norm of a vector, so we have
η 1 Σ

V † a 11
a 21

2
F =

1
2

(2.32)