Transceivers with orthonormal precoders - Signal Processing and Optimization for Transceiver Systems - page 493

Problems

15.1. Consider the circulant channel

⎡ ⎤

10 . 1
0 . 11

1 −

⎣ ⎦ .

H = − 1

− 10 . 11

Assume that the 3 3 precoder is orthonormal and that zero forcing is in

place. Assume σ s =1 ,σ q =0 . 01 , and p 0 =10 .

×

1. Find a pair of optimal precoder and equalizer matrices F and G to
minimize the mean square reconstruction error.
2. Compute the mean square errors (per symbol s k ( n )) for the optimal
system. What would this mean square error be for the optimal system
where the precoder is not restricted to be orthonormal? What is the
ratio G of these two errors? This ratio measures the loss due to

orthonormality restriction.

15.2. Repeat Problem 15.1 for the circulant channel

⎡
⎣

⎤
⎦ .

411
141
114

H =

What is the ratio of the two errors? How does it compare with the

answer in Problem 15.1? Explain why it is so different.

G

15.3. Repeat Problem 15.1 for the case where the zero-forcing constraint is re-
moved.

15.4. Repeat Problem 15.2 for the case where the zero-forcing constraint is re-
moved.