Stability tests for single-bit sigma-delta modulators with second-order FIR noise transfer functions

The authors study the stability of sigma-delta modulators with second-order FIR (finite impulse response) noise transfer functions. At present, there is no analytical criterion which is both necessary and sufficient for the stability of such systems, even when the input is known. The authors present a variety of tests, culminating with the results of a simple computer algorithm which proves the zero-input stability of 94% of the modulators of the foregoing type which are, according to brute-force simulations, believed to be stable. The algorithm may be extended to higher-order FIR, or even IIR (infinite impulse response) transfer functions and may also be modified to incorporate bounded inputs