Abstract

Abstract—In this paper, an initial condition of strictly causal
rational interpolative sigma-delta modulators (SDMs) is
estimated based on quantizer output bit streams and an input
signal. A set of initial conditions generating bounded trajectories
is characterized. It is found that a set of initial conditions
generating bounded trajectories but not necessarily
corresponding to quantizer output bit streams is convex. Also, it is
found that a set of initial conditions corresponding to quantizer
output bit streams but not necessarily generating bounded
trajectories is convex too. Moreover, it is found that an initial
condition both corresponding to quantizer output bit streams and
generating bounded trajectories is uniquely defined if the loop
filter is unstable (Here, an unstable loop filter refers to that with
at least one of its poles being strictly outside the unit circle). To
estimate that unique initial condition, a projection onto convex set
approach is employed. Numerical computer simulations show that
the employed method can estimate the initial condition effectively.