Linear System Response to Random Inputs

In system analysis, we are interested in determining the response of a system for given initial conditions, input, and system dynamics, to characterize the system behavior. For deterministic systems, we use time domain or transform-domain methods to solve for the system response and can completely determine the output. For random input, the response to a particular realization of the input process does not characterize the system response because the response can be different for different realizations. We must therefore characterize the response using averaged measures. Another problem that we face with random inputs is that they typically do not satisfy the assumptions required for the mathematical machinery of calculus. Fortunately, using modified definitions of limits, derivatives and integrals, a calculus can be developed for random signals. We begin this chapter with a brief overview of the calculus for random signals, then discuss the behavior of linear systems with random inputs. We discuss both continuous-time and discrete-time systems. We consider two cases, stationary analysis for stable systems in the steady state, and nonstationary analysis where the system need not be stable and where the transient response is considered. For stationary analysis, the response is due to a random input because the response due to the initial conditions is zero in the steady state for stable systems. As with deterministic analysis of linear systems, the nonstationary response comprises the zero-input response due to the initial conditions, and the zero-state response due to the random input.