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Second-order direct-form digital filters with two two's-complement truncation quantizers are analyzed. The regions of the parameter plane where limit cycles of period one and two exist are first determined, and their locations are related to the possible cycle values. The global asymptotic stability (GAS) of the filter is then investigated in the region where no such limit cycles are possible within the stability triangle of the associated linear filter. The GAS is proved for a fairly large area by an analytical procedure. The GAS area is then extended using an exact exhaustive search.< >