Batch Processing versus Adaptive Processing
Depending on how the data are processed and on how the inference is made, most signal processing methods fall into one of two categories: batch processing and adaptive (i.e., sequential) processing. In batch signal processing, the entire data block Y is received and stored before it is processed, and the inference about X is made based on the entire data block Y. In adaptive processing, on the other hand, inference is made sequentially (i.e., online) as the data are being received. For example, at time t, after a new sample yt is received, an update on the inference about some or all elements of X is made. In this chapter we focus on optimal signal processing under the Bayesian framework for both batch and adaptive processing. We next illustrate batch and adaptive Bayesian signal processing, respectively, using the equalization example above.
Example 1: Batch Equalization
Consider the equalization problem mentioned above. Let 
be the
received signal and 
be the
transmitted symbols. Denote 
. An optimal batch processing procedure for this problem is as
follows. Assume that the unknown quantities g, s2, and X are independent of each other and have prior
densities p(g), p(s2), and p(X), respectively.
Since {n[i]} is a
sequence of i.i.d. Gaussian samples, the joint posterior density of these
unknown quantities (g,s2,X) based on
the received signal Y takes the form
The a posteriori probabilities of the transmitted symbols can then be calculated from the joint posterior distribution (8.6) according to
Clearly, the direct computation in (8.8) involves 2M-1 multidimensional integrals, which is certainly infeasible for most practical implementations in which M might be on the order of hundreds.
Example 2: Adaptive
Equalization Again consider the equalization problem above. Define 
and 
for any i. We now look at the problem
of online estimation of the symbol b[i] based on the received signals up to time 
for some fixed delay 
. This problem is one of making
Bayesian inference with respect to the posterior density
An online symbol estimate can then be obtained from the marginal posterior distribution
Again we see that direct implementation of the optimal
sequential Bayesian equalization above involves 
multidimensional integrals at time i, which is
increasing exponentially in time.
It is seen from the discussions above that although the optimal (i.e., Bayesian) signal processing procedures achieve the best performance (i.e., the Bayesian solutions achieve the minimum probability of error on symbol detection), they exhibit prohibitively high computational complexity and thus are not generally implementable in practice. The recently developed Monte Carlo methods for Bayesian computation have provided a viable approach to solving many such optimal signal processing problems with reasonable computational cost.