Optimization of cyclic prefix transceivers - Signal Processing and Optimization for Transceiver Systems - page 554

precoder output and channel output can be assumed to be real. The signal and
noise variances (real Gaussian noise) are assumed to be

σ s = 1 σ q =0 . 01 . and

The system designed for this channel has M =16and L = 9, so the bandwidth
expansion ratio is

γ = M + L
M

= 25

16 ≈

1 . 56 .

Figure 17.17 shows the performance of the four optimal systems described above.
The top plot shows the error probability curves as a function of the transmitted
power p 0 . Actually we plot it against p 0 /M σ q , i.e., the transmitted-power-
to-noise ratio per symbol). A number of points should be noticed about this
example:

1. The channel frequency response has some deep nulls, as seen from the
middle plot.

2. The pure-MMSE system with precoder not constrained to be orthonormal
is clearly the best (smallest error probability for fixed power p 0 ).

3. The system with ZF constraint and orthonormal precoder constraint is the
worst. The other two systems have intermediate performaces as expected.

4. The ZF systems approach the pure-MMSE systems as p 0 /σ q increases. For
large powers ,the ZF constraint is not a serious loss of generality.

5. The orthonormality constraint makes a difference only at large values of
p 0 /σ q .For small powers , orthonormality is not a serious loss of generality.

6. The mean square error (MSE) plots are shown in Fig. 17.17 (middle). The
MSE shown is the average error per symbol, that is,

1
M E [

2 ] , s ( n ) − s ( n )

where M is the size of the vector s ( n ) . For the ZF-MMSE case note that
the MSE is inversely proportional to p 0 /M σ q , as seen from Eq. (17.60).
The log-log plot in Fig. 17.17 (middle) is therefore a straight line. For
the pure-MMSE case, the MSE is smaller, and it is a more complicated
function of p 0 /M σ q as seen from Eq. (17.61). Note that the MSE of
the pure-MMSE system never exceeds σ s , but the MSE of the ZF-MMSE
system is not bounded like that. It takes on very large values for low
power p 0 , as seen from the plots. The error probability of course does
not become arbitrarily large (even though the MMSE can be arbitrarily
large) because the error probability, by definition, is bounded by unity!
Thus, even though the ZF-MMSE system has much larger MSE than the
pure-MMSE system for low p 0 , both of these systems have unacceptably
low error probability for low p 0 .