In many applications, physical systems are modeled as linear time invariant (LTI) systems. The dynamic behavior of an LTI system to deterministic inputs is described by linear differential equations. We are familiar with time and transform domain (such as Laplace transform and Fourier transform) techniques to solve these equations. In this lecture, we develop the technique to analyze the response of an LTI system to WSS random process.
The purpose of this study is two-folds:
Analysis of the response of a system
Finding an LTI system that can optionally estimate an unobserved random process from an observed process. The observed random process is statistically related to the unobserved random process. For example, we may have to find LTI system (also called a filter) to estimate the signal from the noisy observations.
Basics of Linear Time Invariant Systems: A system is modeled by a transformation T that maps an input signal to an output signal y(t). We can thus write,
The system is called linear if superposition applies: the weighted sum of inputs results in the weighted sum of the corresponding outputs. Thus for a linear system
Example : Consider the output of a linear differentiator, given by
Hence the linear differentiator is a linear system.
Linear time-invariant system Consider a linear system with y(t) = T x(t). The system is called time-invariant if
It is easy to check that that the differentiator in the above example is a linear time-invariant system.
The system is called causal if the output of the system at depends only on the present and past values of input. Thus for a causal system