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
Equation 8.5
Equation 8.6
The a posteriori probabilities
of the transmitted symbols can then be calculated from the joint posterior
distribution (8.6) according to
Equation 8.7
Equation 8.8
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
Equation 8.9
An online symbol estimate can then be obtained from the
marginal posterior distribution
Equation 8.10
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.
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