Bit allocation and power minimization - Signal Processing and Optimization for Transceiver Systems - page 458

where Λ h is the diagonal matrix of J singular values of H in the order

σ h, 0 ≥ σ h, 1 ≥ ... ≥ σ h,M− 1 ≥ .... (14 . 19)

We then have the following result:

Lemma 14.1. Bound on φ. For a given channel H the quantity φ in Eq.
(14.17) is bounded as

♠

1
M− 1
k =0

φ ≥ (14 . 20)

σ h,k

The bound is achieved by choosing the M × J equalizer G to be

G =[ U h ] M×J , (14 . 21)

where the right-hand side denotes the submatrix defined by the first M rows of
U h . ♦

=0in
the denominator of Eq. (14.20). To proceed with the proof of the lemma, it is
convenient to represent G in terms of its SVD matrices:

Note that the assumption that H has rank ≥ M ensures that σ h,k

0 ] V g ,

G = U g [ Σ g (14 . 22)

where U g is M × M unitary, V g is J × J unitary, and Σ g is M × M diagonal

with positive diagonal elements (singular values of G ).

Proof of Lemma 14.1. Using Hadamard's inequality for positive definite
matrices (Appendix B) we have

M− 1 M− 1

det( GG † )
det( GHH † G † )

[( GHH † G † ) − 1 ] kk [ GG † ] kk ≥ φ = (14 . 23)

k =0 k =0

with equality if and only if ( GG † )and[( GHH † G † )] are diagonal. From
Eq. (14.22) we have

0 ] V g V g Σ g 0 ] Σ g

0

GG † = U g [ Σ g U g = U g [ Σ g U g

0

and

0 ] V g HH † V g Σ g GHH † G † = U g [ Σ g U g .

0

So

det( GG † )=det( Σ g Σ g ) (14 . 24 a )

and

det( GHH † G † )=det( Σ g Σ g )det( V g HH † V g ) M

(14 . 24 b )