The problem of suppressing zero-input limit cycles in coupled-form-state-space digital filters when the quantization is performed just after the multiplication is here addressed. No proof has been presented guaranteeing that zeroinput limit cycles are not sustained in the filter output when the poles are anywhere inside the unit circle. Some authors have addressed this subject, but the results in the literature constrain the poles to stay in a bounded region inside the unit circle. To explore this topic further, this paper presents a proof of the stability of the second-order-coupled-form digital filter whose poles are inside the unit circle in the direction of 0, 45, 90, 135 or 180 degrees. This freedom of zero-input limit cycles is guaranteed when the quantizers placed implement the magnitude-truncation scheme, as it happens in the case of quantization applied to the state variables
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