MMSE transceivers without zero forcing - Signal Processing and Optimization for Transceiver Systems - page 450

1. Compute the singular values of the channel.

2. With the transceiver designed to minimize the mean square

error as summarized in Sec. 13.5, what is the mean square recon-
struction error assuming the power is p 0 =2?
3. Now assume

{ F , G }

is the MMSE solution with the ZF constraint as
in Chap. 12. Then what is the mean square reconstuction error?

{ F , G }

4. What is the ratio of the MSE values obtained in the above two meth-
ods? This is the gain obtained by giving up the ZF constraint in

the jointly optimal transceiver system.

G

13.4. Repeat Problem 13.3 for the channel
H = 33
32 . 7

.

Is the gain G higher or lower compared to Problem 13.3? Explain.

13.5. With all other quantities as in Problem 13.3, find an example of a channel
H such that the gain G obtained (by giving up the ZF constraint) is at

least 4 .

13.6. Consider Eq. (13.64), which defines the optimal diagonal elements σ f,k of
Σ f in the precoder. Let M = 3, and let the dominant channel singular
values be σ h, 0 =1 ,σ h, 1 =0 . 1 , and σ h, 2 =0 . 01 . The integer K represents
the number of nonzero multipliers σ f,k . Ifthepower-to-noiseratio p 0 /σ q is
large enough, then we will have K = 3 (all three channel modes are used).
If p 0 /σ q is too small, then we will have K = 1 (only the dominant channel
mode is used). Find three examples of p 0 /σ q such that K =3 , 2 , and 1 ,
respectively.

13.7. The main results of this chapter have been derived assuming that R ss =
σ s I . We now examine the possibility of making R ss a more general diag-
onal matrix Λ s . In this case it makes more sense to minimize the sum of
error-to-signal ratios (ESR)

M− 1

E|e k ( n ) | 2
E

ESR =

| s k ( n ) | 2

k =0

rather than the sum of errors. For, it is the error-to-signal ratios at the
detectors that determine the performance of the receiver. We now show
that the ESR minimization problem can readily be converted to the original
problem (Eqs. (13.8) and (13.9)) which assumed Λ s = σ s I . Let R ee denote
the error covariance, and define

s R ee Λ − 1 / s . So we have
to minimize Tr ( R new ) . Note that the diagonal elements of R ee and Λ s
represent E

Λ − 1 / 2 R new =

| 2 and E | 2 , respectively.

| e k ( n ) | s k ( n )