On the effect of the integrator initial state in a non-ideal sigma–delta-modulator

This paper examines the dynamics of an electronic system—a single-loop one-bit sigma–delta modulator in the presence of integrator leak which is not necessarily a constant. We prove that the difference of the moving averages of any two output sequences of the modulator (arising from different initial conditions of the integrator) is inversely proportional to the averaging length. The result demonstrates the insensitivity of the time average of the modulator output to the initial condition of the integrator.