Random Signals and Systems with Random Inputs

Abstract

In the problem of estimating a signal s(n) from the measurements z(n) = g(s(n), v(n), n), the noise term v(n) usually varies “randomly,” and thus modeling v(n) requires that we use a random signal formulation. The signal s(n) may also include some random variation, and thus it too must be modeled in general as a random signal. The random signal formulation is generated by taking v(n) and s(n) to be random variables for each value of the time index n. We begin by presenting the fundamentals of random variables in Section 2.1, and then in Section 2.2 we consider random discrete-time signals. In the last section of the chapter, we study linear time-varying and time-invariant discrete-time systems driven by random signal inputs. The treatment of random signals and systems with random inputs given in this chapter is presented in sufficient depth to allow the reader to then follow the development of optimal filtering given in this text.