Transceivers with decision feedback equalizers - Signal Processing and Optimization for Transceiver Systems - page 610

In Appendix 19.C at the end of the chapter we show that when G is as in Eq.
(19.69), the preceding expression for R ee can be rewritten in the following form:
R ee = σ s ( B + I )

− 1 ( B + I ) † ,

I + σ s

σ q F † H † HF

(19 . 72)

which is more useful for optimization. The goal is now to optimize the matrices
F and B to minimize the trace of Eq. (19.72) subject to the power constraint
and the constraint that B be strictly upper triangular. Thus the objective to be
minimized is
E mse = σ s Tr ( B + I )

− 1 ( B + I ) † ,

I + σ s

σ q F † H † HF

(19 . 73)

subject to the power constraint
σ s Tr

FF † = p 0

(19 . 74)

and the upper triangular constraint

b km =0 , k ≥ m. (19 . 75)

19.4.3 Bound on the error

We now find a lower bound on the reconstruction error E mse in terms of quantities

that depend only on F and H . For the reader who is eager for a preview, this
bound is shown in Eq. (19.88), where σ f,k are the singular values of the precoder
F , and σ h,k are the first M dominant singular values of the channel H . Depending
on the available power, this bound can further be refined as in Eq. (19.98), where
K and λ are quantities that depend on power p 0 . Finally we show in Sec. 19.4.5
how this bound can be achieved by choice of F , G , and B .
Now for the details. The trace is the sum of the eigenvalues λ k of R ee ,
whereas the determinant is the product of λ k . Since R ee is positive definite, we
have λ k > 0, and the AM-GM inequality can be applied to obtain
Tr R ee
M

(det R ee ) 1 /M . ≥ (19 . 76)

That is,

M det ( B + I )

− 1 ( B + I ) † 1 /M

I + σ s

E mse
σ s ≥

σ q F † H † HF

= M det

− 1 /M

I + σ s

σ q F † H † HF

,

where we have used the fact that B + I is upper triangular with diagonal elements
equal to unity. So

Mσ s det

− 1 /M

I + σ s

σ q F † H † HF

E mse ≥ . (19 . 77)