DERIVATIONS OF (8)-(10) AND (12)-(14)
The following notations are used throughout this document. All boldface letters indicate vectors
(lower case) or matrices (upper case). (⋅) , (⋅) , and ‖⋅‖ stand for Hermitian transpose, matrix
inversion, and Frobenius norm, respectively. The notation (⋅,⋅) denotes complex Gaussian
distribution with mean and covariance. E{⋅} denotes statistical expectation. ℰ () denotes the
eigenvector of matrix corresponding to the largest eigenvalue λ () , i.e. ℰ () =
λ ()ℰ ().
1. Derivations of (8)-(10)
According to the definition of , it can be expressed as:
= ! . = ! + #
where #: (0, & ' () represents circularly symmetric complex additive white Gaussian noise
(AWGN) vector with power & ' ; > 0 denotes the average energy per symbol; ∈ ℂ denotes
the information symbol with unit energy, i.e. E{| |' } = 1.
Similarly, we have
Therefore, we have
/
0 = − . = + #.
/
(!
According to the definition of 0' , it can be expressed as:
0' = 2'22'2
Therefore, the objective function (7) is rewritten as:
− ! ) .
/
' ! 3
min =E>‖0 ‖' ? + E>‖0' ‖' ?@
!/789:;7<
=
min A
!/789:;7<
E BC
/
>!
+E BC2'22'2
'
− ! ? C D
F.
'
/
' ! C D
After applying first-order derivative to the above function, we have
!/ = (H + H ' ) H ! , (cf. (8))
H ≜
H ' ≜
' 2'22'2 2'22'2
,
' .
Ideally, the optimal un-quantized precoding vector ! = ℰ (
1). Then, we have = B
JKL
B! D . Hence, we define
JKL
JKL
! D
JKL
/N (
).
)
(cf. [6], lemma
As a result, H = !
JKL
JKL
H ≜ ! !
(cf. (9))
Furthermore, according to (6), the optimal solution for ! can be obtained by solving the
following optimization function:
O
max C2'22'2
JKL
O
?
Hence,>!
'
O
' ! C
O
O
? !
s. t.>!
=1
=
' 2'22'2 /‖
' 2'22'2 ‖.
Then, we have
O
O
>!
? (cf. (10)).
H ' ≜ !
2. Derivations of (12)-(14)
We can rewrite (11) as:
min
!/VWV:VWV
A
E BC2'22'2
/
'' >!2'22'2
+E BC
'
− !2'22'2?' C D
'
/
' !2'22'2 ' C D
F.
Notice that the rewritten version of (11) has the same pattern as the rewritten version of (7).
Therefore, (12)-(14) can be easily derived following the same derivation procedure shown in
Section 1.