Waveform reconstruction for recurrent non-uniform discrete-time sampling scheme: Discussion of its instability

The main purpose of this paper is to analyze the instability in waveform reconstruction from non-uniformly sampled signals generated by recurrent non-uniform discrete-time sampling scheme. From the applied equation, the matrix of conversion, [A], associates the uniformly sampled signals to non-uniformly sampled ones, and plays a key role in the reconstruction. To recover the uniformly sampled signals, we need to compute the inversion of [A], [A]-1. Although [A]-1 is known to be numerically unstable in some occasions, the fundamental reason and premise for this instability have not been discussed. Thus, by obtaining the closed form of det[A], this paper shows that the root cause of the instability of [A]-1 is related to the singularity of [A], which has det[A] = 0 resulting from specific patterns of timing offsets in the recurrent nonuniform discrete-time sampling scheme. Numerical results confirm the analysis.