带宽有限随机过程从过采样点的指数阶逼近

Exponential approximation of bandlimited random processes from oversampling

带宽有限的宽平稳随机过程的Shannon采样定理在1957年被建立起来.从那以后,关于它在其他随机过程的推广有广泛的研究.然而,直接截断Shannon级数收敛较慢.特别地,我们知道利用在Nyquist采样率下得到的n个采样点的截断级数的均方逼近误差的收敛速率是O(1/n^(1/2)).本文我们n考虑用有限的过采样点来重构带宽有限宽平稳随机过程,其中过采样点是指连续两个采样点之间的距离小于Nyquist采样率.我们研究了最优的线性重构算法和与其相关的本性逼近误差阶.通过过采样,我们发现线性重构算法可以达到指数阶衰减逼近,并且我们还证明线性重构算法不可能有快于指数阶的衰减速率.另外,我们还构造了两个具体的指数阶衰减的重构算法.

The Shannon sampling theorem for bandlimited wide sense stationary random processes was es- tablished in 1957, which and its extensions to various random processes have been widely studied since then. However, truncation of the Shannon series suffers the drawback of slow convergence. Specifically, it is well-known that the mean-square approximation error of the truncated series at n points sampled at the exact Nyquist rate 1 is of the order O（1/n√（1/2））. We consider the reconstruction of bandlimited random processes from finite oversampling points, namely, the distance between consecutive points is smaller than the Nyquist sampling rate. The optimal deterministic linear reconstruction method and the associated intrinsic approximation error are studied. It is found that one can achieve exponentially-decaying （but not faster） approximation errors from oversampling. Two practical reconstruction methods with exponential approximation ability are also presented.