The response of a particular nonlinear system with feedback to each of two random processes

A discrete time, nonlinear system composed of an integrator preceded by a binary quantizer with integrated negative feedback, which can model a tracking loop or a single integrating delta modulation communication system, is discussed with regard to the input-output statistics for two types of input processes: independent inputs and independent increments inputs. A recursion on time for the joint distribution of input and output is obtained for the independent inputs process and explicitly solved for the time asymptotic distribution, when it exists. The solution is examined in greater detail for the special case of IID normal inputs. When the system is excited by a process of independent increments, the asymptotic behavior of the input and output (they diverge) is of less interest than that of the difference between input and output, the tracking error. A recursion in time for the characteristic function of the error is developed and the time asymptotic solution found, The tracking error is interpreted by decomposition into static and dynamic parts, and an exponential bound to its distribution is provided. The particular case of normal increments input is discussed in additional detail.