MMSE transceivers without zero forcing - Signal Processing and Optimization for Transceiver Systems - page 449

Problems

Note. Unless mentioned otherwise, σ s =1 .

13.1. Consider the case of a real scalar channel with real-valued precoder and
equalizer. In this case, F = f, G = g, and H = h (scalar multipliers), and
the mean square error (13.8) simplifies to

E mse = σ s ( ghf 1) 2 + σ q g 2 .

−

By setting ∂ E mse /∂g =0 , derive an expression for the optimal equalizer g

for fixed f, h. Also find an expression for the mimimzed mean square error.

13.2. In Sec. 13.3 we showed that the MMSE equalizer matrix G has the form
(13.12) and results in the mean square error (13.13). For the case where
R ss = σ s I and R qq = σ q I , we showed that this simplifies to
E pure = σ q Tr

− 1 .

F † H † HF + σ q

σ s I

If the equalizer G is chosen to be the zero-forcing equalizer (instead of the
MMSE equalizer) then (from Chap. 12) the mean square error is
E zf = σ q Tr

− 1 .

F † H † HF

Let σ k denote the singular values of the product HF .

1. Express E zf in terms of σ k .

2. Consider an example where σ q /σ s =0 . 01, and HF is 2

E pure and

2. If the
singular values of HF are given by σ 0 =1 ,σ 1 =0 . 1, what is the ratio
E zf /

×

E pure ? (This number represents the advantage of going from a
zero-forcing system to a pure-MMSE system.)
3. Repeat for the case where σ 1 =0 . 001.

In this problem we have compared the ZF-MMSE and pure-MMSE equal-
izers for fixed precoder F . If we optimize the precoder, then F is different
for the two cases, and the ratio E pure is different from what we found

above. This is addressed in the next few problems.

E zf /

13.3. Consider the MIMO channel

H = 33
32

with noise variance σ q =0 . 1 .