Single-input single-output transceiver optimization - Signal Processing and Optimization for Transceiver Systems - page 347

So, in the pure-MMSE case the mean square error is bounded above by
the signal variance σ s .

4. MIMO channels. We will see that the basic mathematical formulation and
solution for the optimal transceiver given in this section makes repeated
appearances in more complicated scenarios. For example, when the SISO
digital communication system becomes a MIMO system, we will see similar
results (Chaps. 12 and 13). Instead of integrals involving H ( e jω ) , the

expressions will have summations involving the singular values of a MIMO
memoryless channel.

| |

5. Error probabilities. Even though we have minimized mean square errors in
this section, the more important quantity in practice is the symbol error
probability. It can be shown that the pure-MMSE system has smaller
symbol error probability than the ZF-MMSE system (see Appendix 16.D
of Chap. 16).

10.4 General forms of optimal filters

In Sec. 10.2 we considered the optimization of the transceiver of Fig. 10.6 under
the zero-forcing constraint. With the zero-forcing constraint removed the prob-
lem becomes more general, though more di cult. A number of such generalized
results were reported in the early literature. In the early 1970s Ericson observed
that in many of these problems the solutions have some common properties, and
showed that the optimal combination of the prefilter F ( jω ) and postfilter G ( jω )
in Fig. 10.6 can be restricted in certain ways without loss of generality [Eric-
son, 1971, 1973]. More specifically, for any objective function which depends
only on the noise spectrum at the detector input and the amount of transmit-
ted power, there is a certain standard form for the filters G ( jω )and F ( jω ) . In
this section we shall elaborate on this. One consequence of this observation is
that the optimal design of F ( jω )and G ( jω ) can be converted into an optimal
design problem involving an all-digital system like Fig. 10.5, as we shall explain.
Historically, Ericson's work unifies many earlier papers in transceiver literature
which obtained rather similar solutions for different optimization problems.
For simplicity we restrict our discussions to the case where the objective to
be minimized is the mean square error, subject to the average power constraint
(10.3). But these observations also hold for “any reasonable objective function”
as Ericson puts it. The contents of this section can be used primarily as a
reference. For this section, the reader should review Appendix G at the end of
the topic, especially notations such as

↓ T , jargon such as “alias-free( T ) bands,”

and the noble identities.