The lazy precoder with a zero-forcing equalizer - Signal Processing and Optimization for Transceiver Systems - page 312

Appendix to Chapter 8

8.A Monotonicity of noise gain

With A M denoting the ( M + L ) × M banded Toeplitz matrix (8.2), denote

R M = A † M A M .

(8 . 25)

Using Eq. (8.8) with T = A M ,wehave
A M 2 =Tr

A M ( A M ) † =Tr( A † M A M ) − 1

(8 . 26)

so that

M Tr

R − M .

1
M A M 2 =

1

noise gain = (8 . 27)

Proving that the noise gain is monotonically increasing is therefore equivalent
to proving that

M +1 Tr M Tr

R − M .

1 1

R − M +1 ≥ (8 . 28)

It should be recalled here that R M is positive definite and Toeplitz (see Eq.
(8.23)). We begin by proving the following lemma. The results in this appendix
are based on the work of Ohno [2006].

♠ Lemma 8.2. Let R M +1 be ( M +1) × ( M + 1) positive definite. Partition

it as

R M +1 = R M

,

v

(8 . 29)

v †

c

where R M is M × M .Then

[ R − M +1 [ R − M ] kk ] kk ≥ (8 . 30)

for 0 ≤ k ≤ M − 1. Similarly, suppose S M is the lower M × M principal

submatrix of R M +1 ,thatis,

R M +1 = d u †
uS M

.

(8 . 31)

Then we have

[ R − M +1 [ S − M ] kk ] k +1 ,k +1 ≥ (8 . 32)

for 0 ≤ k ≤ M − 1 . ♦

Note that since R M +1 is positive definite, R M and S M are positive definite, and
c, d > 0 . The Lemma does not assume that the matrices are Toeplitz.

Proof. Using the formula for inverses of matrices in partitioned form (Sec.
B.4.3, Appendix B) we get

⎡
⎣

⎤

= I − R − M v
0

R − M

.

0

I 0

R − M +1

1
c − v † R − M v

⎦

− v † R − M

0

1 1