Introduction to Random Signals and Noise

7

Detection and Optimal Filtering

Thus far the treatment has focused on the description of random signals and their analyses, and how these signals are transformed by linear time-invariant systems. In this chapter we take a somewhat different approach; namely starting with what is known about input processes and of system requirements we look for an optimum system. This means that we are going to perform system synthesis. The approach achieves an optimal reception of information signals that are corrupted by noise. In this case the input process consists of two parts, the information bearing or data signal and noise, and we may wonder what the optimal receiver or processing looks like, subject to some criterion.

When designing an optimal system three items play a crucial role. These are:

1. A description of the input noise process and the information bearing signal;
2. Conditions to be imposed on the system;
3. A criterion that defines optimality.

In the following we briefly comment on these items:

1. It is important to know the properties of the system inputs, e.g. the power spectral density of the input noise, whether it is wide-sense stationary, etc. What does the information signal look like? Are information signal and noise additive or not?
2. The conditions to be imposed on the system may influence performance of the receiver or the processing. We may require the system to be linear, time-invariant, realizable, etc. To start with and to simplify matters we will not bother about realizability. ...