Controlled rounding arithmetics, for second-order direct-form digital filters, that eliminate all self-sustained oscillations

Quantization often allows a recursive second-order filter to oscillate even when the underlying linear model is absolutely stable and there is no external signal present to excite the filter. One of the significant innovative ideas recently introduced to combat this condition is "controlled rounding" which utilizes memory in the rounding decision process. Specifically, the selection at time
(n+ 1)
of the state variable of filter,
x_(n+1)
depends not only on the quantity to be rounded, but also on the two previous outputs of the rounding operation,
x_n
and
x_{n-1}
; the latter two quantities are anyhow readily available at the time of decision. This paper gives controlled rounding arithmetics that suppress all selfsustained oscillations in all direct-form second-order filters for which the underlying ideal linear model is stable. Not just rounding but also overflow, and hence all the features that make a digital filter a finite state machine, are taken into account. The incremental cost of the hardware which is mostly on account of the additional logic is reckoned to be slight.