Single-input single-output transceiver optimization - Signal Processing and Optimization for Transceiver Systems - page 339

1. Unrealizability. H c ( jω ) is an ideal filter with passband gain equal to T and
stopband gain equal to zero. So the filter can only be approximated in
practice.

2. Multiband property. H c ( jω ) may have more than one passband, but the
total bandwidth, counting all passbands, is ω s =2 π/T .

3. Nyquist(T) property. H c ( jω ) satisfies the Nyquist( T ) property, that is,

h c ( nT )= δ ( n ) , (10 . 24)

or equivalently

∞

H c ( j ( ω + kω s )) = T. (10 . 25)

k = −∞

4. Alias-free property. Since no two terms in Eq. (10.25) overlap (owing to
the defining equation (10.22)), the filter H c ( jω ) has the alias-free property.
So, a signal y ( t ) which is bandlimited to the passband of H c ( jω )can
be reconstructed from its samples y ( nT ) by passing the sampled version
through the filter H c ( jω ) . The passband region of H c ( jω )isanexample
of alias-free( T ) regions described in Appendix G.

Example 10.1: Optimum compaction filters

Figure 10.2(a) shows an example of the effective channel H ef f ( jω ). In this
example h ef f ( t ) is complex, since

is not symmetric. The symbol
frequency ω s =2 π/T is indicated in the figure. The total bandwidth of
H ef f ( jω )is σ +3 ω s , and is considerably larger than ω s . To construct the
filter H c ( jω ) described in Theorem 10.2, imagine that the frequency axis is
divided into consecutive intervals of length ω s . Four such intervals overlap
with the passband of H ef f ( jω ) as indicated in the figure. To construct the
compaction filter we proceed as follows: given any ω 0 in [0 ,ω s ]wecompare
the magnitudes of H ef f ( jω ) at the frequencies ω 0 + kω s for integer values of
k. For fixed ω 0 , one such frequency falls inside each region in Fig. 10.2(a).
We simply choose one of the samples with maximum magnitude (i.e., pick
one integer k which maximizes the magnitude of H ef f ( j ( ω 0 + kω s ))), and
discard the rest. In this way, we isolate the dominant part of H ef f ( jω ),
as demonstrated in Fig. 10.2(b). The optimal compaction filter H c ( jω )by
definition is an ideal filter whose passband coincides with this chosen band,
as shown in Fig. 10.2(c). The magnitudes of optimal pre- and postfilters
(10.5) and (10.6) (up to scale) are shown in Fig. 10.2(d), assuming S qq ( jω )
is constant.
If the effective channel has real impulse response h ef f ( t )thenacom-
paction filter can be constructed with real impulse response h c ( t ) . This is
because when h ef f ( t )isreal,

| H ef f ( jω ) |

is symmetric, and H c ( jω ) will turn
out to be symmetric too. Figure 10.3(a) shows an example. Proceeding as
in the preceding example, we arrive at the optimal compaction filter shown
in Fig. 10.3(c).

| H ef f ( jω ) |