Characterizing the impact of time error on general systems

Time error x(t) impacts the performance of general systems principally by generating multiplicative signal error (ME) or its short term manifestation multiplicative noise (MN). In this paper, we examine how to characterize x(t) induced ME and MN, especially in the presence of negative power law (negp) x(t) noise, that is, noise with a power spectral density (PSD) Lx(f) prop fp for p<0. In the paper, it is shown that one can spectrally characterize the MN as Ldeltav(fg,f) the dual frequency Loeve spectrum of the x(t) induced ME deltav(t) = v(t+x(t)) - v(t), where v(t) is the ideal generated or processed signal. It is also shown that Ldeltav(fg,f) can be written as the convolution of Lv(fg,f) the Loeve spectrum of v(t) with the spectrum of the x(t) generating the MN. When v(t) is broadband, it is further shown that the evaluation of Ldeltav(fg,f) is problematic in the presence of neg-p x(t) noise, if one naively interprets the x-spectrum in the convolution merely as LTB(f) the raw x-PSD of a time source or timebase (TBs) used to generate or process the signal. The paper then demonstrates that this problem is mitigated in most systems because such a neg-p LTB(f) is naturally high-pass (HP) filtered in the convolution due to two effects. The first of these HP effects is generated by topological structures such as phase lock loops and delay mismatches. The second HP effect arises when one properly defines the x-PSD for the MN convolution as that of the residual TB error after removing an estimate of the long term causal x(t) behavior, not that of the total TB error. It is finally shown that such causal removal guarantees the convergence of the MN convolution for any order of neg-p TB noise, if an appropriate causal estimation model is chosen.