Digital Signal Processing Reference
In-Depth Information
and appears to be parabolic. Even for a reasonably small value of
M
the noise
gain is quite large. Note that the scale for the vertical axis is very different when
compared with Fig. 8.4. If the zeros are moved a little bit away from the unit
circle then the growth is not as drastic. This is shown by the middle and bottom
plots in Fig. 8.5 for
C
5
(
z
)=
c
(0)(1 + 0
.
95
z
−
1
)
2
and
C
6
(
z
)=
c
(0)(1 + 0
.
9
z
−
1
)
2
,
respectively. In these cases the growth slows down and tends to saturate. Finally
Fig. 8.6 shows the behavior for the channel
C
7
(
z
)=
c
(0)(1 +
z
−
1
)
3
,
which has a triple zero on the unit circle. The growth of noise gain is unaccept-
able indeed. Again notice that the vertical scale is significantly different from
the earlier plots. A couple of remarks are now in order:
1. Even though it is unlikely that a practical channel will have multiple zeros
on the unit circle, it is possible that the channel gain is quite small in some
frequency regions, thereby approximating the behavior of unit circle zeros.
So the observations made here have some practical relevance.
2. While the monotone increasing behavior of
A
#
2
/M
as
M
increases has
been proved (Appendix 8.A), the reason for the specific nature of the
growth outlined above (for unit circle zeros) has not been theoretically
studied.
8.7 Concluding remarks
The lazy precoder with a zero-forcing equalizer is admittedly a restricted special
case of the more general transceiver with precoder
R
and equalizer
T
. The noise
gain can be reduced considerably by optimization. For example we can follow
up
T
with a Wiener filter (Sec. 4.10). In fact we can reduce the reconstruction
error even further by jointly optimizing
R
and
T
for a given channel and noise
statistics (Chaps. 12 and 13). Another sophisticated method is called bit al-
location, whereby fewer bits are transmitted on parts of the channel with poor
frequency response (Sec. 14); this has the effect of selectively eliminating some
parts of the channel which have very low gain.
Even though optimization helps a lot, the zero-padded optimal transceiver
still exhibits a monotonically increasing reconstruction error as
M
increases!
This will be elaborated in Chap. 18. As we shall see in Chap. 17, the situation
is quite different for cyclic-prefix sytems (single as well as multicarrier).
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