Transceivers with decision feedback equalizers - Signal Processing and Optimization for Transceiver Systems - page 601

M M diagonal matrix Σ f has positive diagonal elements. Recall also that the

channel SVD is

×

V h
P×P

H = U h Σ h (19 . 31)

J×J J×P

Since this has to have rank M (in view of Eq. (19.13)), we can assume the

first M dominant singular values are positive:

≥

σ h, 0 ≥ σ h, 1 ≥ ... ≥ σ h,M− 1 > 0 .

Thus, the determinant in Eq. (19.24) can be written as

=d t

0 ] U f V h Σ h Σ h V h U f Σ f

det

F † H † HF V f V f [ Σ f

0

=d t [ Σ f

,

0 ] U f V h Σ h Σ h V h U f Σ f

0

where we have used the fact that det [ V f BV f ]=det B because V f is a square
unitary matrix. From the preceding we finally obtain the expression

=det

,

det Σ f U f V h Σ h Σ h V h U f F † H † HF (19 . 32)

M

where [ A ] M indicates the M × M leading principal submatrix of

A = U f V h Σ h Σ h V h U f .

(19 . 33)

We will establish the desired lower bound on the right-hand side of Eq. (19.24)
by establishing an upper bound on Eq. (19.32). Since U f V h is unitary, the
eigenvalues of the P P matrix A are the diagonal elements of Σ h Σ h , namely

×

σ h, 0 ≥ σ h, 1 ≥ σ h,M− 1 ≥ σ h,P− 1 . ... ≥ ... ≥ (19 . 34)

Let

η 0 ≥ η 1 ≥ ... ≥ η M− 1 > 0 (19 . 35)

be the M eigenvalues of [ A ] M (evidently positive since this matrix has rank
≥ M ). We now use the same technique that we used in Sec. 12.4.2, namely the

eigenvalue interlace property described in Fig. 12.4. Thus

σ h,k ≥ η k ,

0 ≤ k ≤ M − 1 . (19 . 36)

Since the determinant is the product of eigenvalues, we therefore have

det

=

M− 1 M− 1

F † H † HF σ f,k η k ≤ σ f,k σ h,k , (19 . 37)

k =0 k =0