Recovery of randomly sampled signals by simple interpolators

Abstract

The mean square error performance of simple polynomial interpolators is analyzed for wide-sense stationary signals subjected to randomly timed sampling represented by stationary point processes. This performance is expressed in dimensionsless parametric terms, with emphasis on asymptotic error behavior at high dimensionless sampling rates γ . The form of the asymptotic error expression, and particularly its dependence on γ , is shown to vary according to the number of points utilized, together with the differentiability properties of the signal. One point extrapolation yields a mean square error varying with γ −2 if the signal is differentiable, and as γ −1 if the signal is not. Similarly, two-point (polygonal) interpolation error exhibits linearity in γ −4 , γ −3 or γ −2 , according as the signal is twice, exactly once, or nondifferentiable. Specific examples are offered to furnish insight into actual error magnitudes. It is shown, for instance, that introduction of jitter in the sampling sequence increases the error by only a negligible amount. Exponential decay of the sample values is compared with stepwise holding; little is gained for a nondifferentiable signal, while for a differentiable signal the error performance deteriorates from γ −2 to γ −1 at high sampling rates. When more than two points are used in a polynomial fitting recovery scheme, specific computations or error become excessively difficult. However, it is proved that the asymptotic mean square error varies with γ −2 n when n points are utilized, and the signal is continuously differentiable at least n times. Finally, we compare the mean square errors of one and two sample schemes as described above with those attained by causal (extrapolating) and noncausal (interpolating) Wiener-Kolmogorov optimal filters. We demonstrate nontrivial instances in which the Wiener-Kolmogorov mean square error varies as γ −1/2 , so that any of the simple recovery schemes considered exhibits superior performance at high sampling rates. This is explained by noting that the latter represent time-varying filters, whereas the Wiener-Kolmogorov filter is timeinvariant.