MMSE transceivers with zero-forcing equalizers - Signal Processing and Optimization for Transceiver Systems - page 416

shows that the minimized error can be large for channels with very small
singular values σ h,k . Indeed there is no upper bound on how large the error
can get because the smallest nonzero singular value σ h,M− 1 can be arbi-
trarily small for very bad channels. A similar observation was also made
in Chap. 11 where we considered the special case of diagonal channels. In
Chap. 13, where we study the MMSE transceiver without the zero-forcing
constraint, we will see that this problem goes away.

5. Minimized error and noise power . Equation (12.66) also shows that the
minimized error is proportional to the channel noise variance σ q ,asone
would expect.

6. Minimized error and channel input power . From Eq. (12.66) we see that
E mmse decreases with increasing channel input power p 0 . This is intuitively
appealing of course, and it comes about as follows: if we incre ase the
channel power p 0 for fixed σ s , then the multipliers σ f,k = c/ √ σ h,k in
the transmit ter are larger (because c is scaled up). So the multipliers
σ g,k =1 /c √ σ h,k at the receiver will get correspondingly smaller, which
reduces the reconstruction error.

7. Signal-to-noise ratio . From Eq. (12.66) we see that

M− 1 2

= σ q
p 0

E mmse
σ s

1
σ h,k

.

k =0

This ratio is clearly independent of σ s .Infact, σ s can be set to unity in
all discussions without loss of generality.

8. Diagonal channel . Suppose the channel H is a diagonal matrix with real
non-negative diagonal elements. It then follows that the unitary matrices
in its SVD are U h = V h = I , so the optimal precoder and equalizer
(essentially Σ f and Σ g ) are also diagonal matrices. (In Chap. 11 this was
assumed without proof.) If P>M then the first M dominant diagonal
elements of H are used by the transceiver as seen in Fig. 12.5. If the
diagonal channel H has complex diagonal entries then the matrices U h
and V h are diagonal unitary matrices (i.e., their diagonal entries are of
the form e jθ k ), and can be combined with diagonal matrices Σ f and Σ g .
The result is again an optimal transceiver with diagonal precoder and
equalizer.

Example 12.2: ZF-MMSE transceiver

Consider the simple 2 × 2 channel

√ 3

1 + ( √ 3 / 16)

⎡
⎣

⎤
⎦ .

− (1 / 16)

1
2 √ 2

H =

√ 3+(1 / 16)

( √ 3 / 16)

1 −