Transceivers with orthonormal precoders - Signal Processing and Optimization for Transceiver Systems - page 488

Proof of Theorem 15.2. As explained in Sec. 12.4.2 we have σ h,k ≥ μ k

for

0 ≤ k ≤ M − 1 . Using this we see that

M− 1 M− 1

1
μ k ≥

1
σ h,k

E mse = σ q σ q (15 . 42)

k =0 k =0

The lower bound is achieved by choosing U f = V h so that U f H † HU f =
Σ h Σ h . Thus the optimal precoder F canbewrittenintheform(15.40)
indeed. The equalizer can be computed from the zero-forcing condition:
GHF = I . Taking G to be the minimum-norm left inverse of HF we get
G = ( HF ) † HF

− 1 ( HF ) † .

(15 . 43)

Since

I M = U h Σ h I M

0

= U h ( Σ h ) M
0

HF = U h Σ h V h V h (15 . 44)

0

we have

⎡ ⎤

Σ h ) M

0 ] U h U h

⎣ ⎦ =( Σ h ) 2 M ,

( HF ) † HF =[( Σ h ) M

0

where ( Σ h ) M is the M M diagonal matrix containing the M dominant

singular values of the channel. Substituting into Eq. (15.43), the equalizer
G takes the final form in Eq. (15.40). The MSE expression then achieves
the lower bound in Eq. (15.42), which proves Eq. (15.41).

×

15.3.2 Pure-MMSE transceivers

We now extend the results of Sec. 15.3.1 to the case where the zero-forcing
condition is not imposed. From Sec. 13.3 we know that given an arbtirary
precoder F , the MMSE equalizer has the closed form expression
G = σ s F † H † σ s HFF † H † + σ q I

− 1 ,

(15 . 45)

and the corresponding reconstruction error is

J )+ σ s Tr

σ q HFF † H † − 1 .

I J + σ s E mse = σ s ( M − (15 . 46)