Nonconcurrent Error Correction in the Presence of Roundoff Noise

This paper analyzes the effects of roundoff noise on our ability to nonconcurrently detect and identify transient faults that corrupt state variables during the operation of a fault-tolerant discrete-time linear time-invariant (LTI) dynamic system. Our analysis leads to two decoding algorithms, i.e., one based on the Peterson-Gorenstein-Zierler algorithm and the other based on singular-value decomposition techniques. We analytically obtain bounds on the roundoff noise level (equivalently, the precision) at which both algorithms can guarantee the correct determination of the number of errors. Our simulations verify our analysis and suggest that our bounds can be very tight for certain choices of design parameters. The developments in this paper can be used to provide guidance about the design of fault-tolerant systems and have immediate implications for digital implementations of LTI dynamic systems (e.g., digital filters) because such implementations unavoidably have to deal with finite-precision effects.