On the stability of coupled-form state-space digital filters with quantization before summation

The problem of suppressing zero-input limit cycles in coupled-form state-space digital filters when the quantization is performed after the multiplication is addressed. To the extent of the authors´ knowledge, no proof has been presented that the parasitic oscillations are suppressed for poles anywhere in the unit circle, under that condition. Some authors have addressed this subject, but their results constrain the poles to a bounded region inside the unit circle. With the objective of exploring this topic further, the authors present a proof that for poles whose angle is either 0, ±45, ±90, ±135 or 180 degrees and whose radius is lower than one, the second-order state-space coupled-form digital filter is free of zero-input limit cycles when quantizers placed just after the multipliers implement magnitude truncation