Optimization of zero-padded systems - Signal Processing and Optimization for Transceiver Systems - page 580

not. Repeating the expressions summarized in Chap. 15 we have

⎧
⎨

M− 1

Mσ q σ s
p 0

1
σ h,k

(orthonormal precoder)

k =0

E mmse = (18 . 5)

⎩ M− 1 2

σ q σ s
p 0

1
σ h,k

(unrestricted precoder),

k =0

for the zero-forcing case, and

⎨

M− 1
k =0

1

Mσ s σ q
p 0

(orthonormal precoder)

Mσ q
p 0

+ σ h,k

σ s σ q 1 /σ h,k 2

K− 1

E mmse =

⎩

k =0

K ) σ s

+( M − (unrestricted precoder),

K− 1

p 0 + σ q 1 /σ h,k

k =0

(18 . 6)

for the non-zero-forcing case. In the last expression K is the number of nonzero
multipliers q kk in the precoder (see Sec. 13.5). In all expressions σ h,k are the
M nonzero singular values of the Toeplitz matrix A in Eq. (18.2), ordered such
that σ h, 0 ≥ σ h,M− 1 .

In the examples we assume that the input is PAM, and that the channel
has real coe cients. Even though the equivalent baseband channel is usually
complex, one can use a time-domain equalizer first to get a shorter real equivalent
channel in the PAM case as mentioned in Sec. 2.A.2 (Appendix of Chap. 2).

σ h, 1 ≥ ... ≥

Example 18.1:

Consider the channel

C ( z )=0 . 0331 + 0 . 0894 z − 1 +0 . 1407 z − 2 +0 . 1522 z − 3 +0 . 1050 z − 4 +0 . 0827 z − 5

+0 . 0882 z − 6 +0 . 1389 z − 7 +0 . 1099 z − 8 +0 . 0598 z − 9 .

The order is L =9 . This channel was also used in the cyclic-prefix example in
Sec. 17.7. Its frequency response is reproduced in Fig. 18.3. As in the cyclic
prefix example we assume 2-bit PAM (i.e., 4-PAM) for the samples s ( n ). Since
the channel and the input are real, all the signals and matrices in Fig. 18.2 are
real. The signal and noise variances are assumed to be σ s = 1 and σ q =0 . 01 .
For each of the four types of optimal zero-padded transceivers, Fig. 18.4 shows
the symbol error probability. Three block sizes are shown in the figure, namely
M =16 , 4 , and 1 . A number of observations can be made from these plots: