We consider the problem of recovering a band-limited signal f(t) from noisy data yk=f(k/spl tau/)+/spl Gt/epsilon//sub k/, where /spl tau/ is the sampling rate. Starting from the truncated Whittaker-Shannon cardinal expansion with or without sampling windows (both cases yield inconsistent estimates of f(t)) we propose estimators that are convergent to f(t) in the pointwise and uniform sense. The basic idea is to cut down high frequencies in the data and to use suitable oversampling /spl tau//spl les//spl pi///spl Omega/, /spl Omega/ being the bandwidth (maximum frequency) of f(t). The simplest estimator we propose is given by f/spl circ//sub n/(t)=/spl tau/ /spl Sigma//|t-k/spl tau/|/spl les/spl tau/ yksin(/spl Omega/(t-k/spl tau/))//spl pi/(t-k/spl tau/),|t|/spl les/spl tau/. Generalizations of f/spl circ//sub n/ including sampling windows are also examined. The main aim is to examine the mean squared error (MSE) properties of such estimators in order to determine the optimal choice of the sampling rate /spl tau/ yielding the fastest possible rate of convergence. The best rate for the MSE we obtain is O(In(n)).
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机译：我们考虑从噪声数据yk = f（k / spl tau /）+ / spl Gt / epsilon // sub k /中恢复带限信号f（t）的问题，其中/ spl tau /是采样率。从有或没有采样窗口的两种截短的Whittaker-Shannon基数展开开始（这两种情况均得出f（t）的估计不一致），我们提出了在点和统一意义上收敛于f（t）的估计。基本思想是减少数据中的高频并使用适当的过采样/ spl tau // spl les // spl pi /// spl Omega /，/ spl Omega /是f（t的带宽（最大频率） ）。我们建议的最简单的估计量由f / spl circ // sub n /（t）= / spl tau / / spl Sigma // || tk / spl tau / | / spl les / n / spl tau / yksin（/ spl Ω/（tk / spl tau /））// spl pi /（tk / spl tau /），| t | / spl les / n / spl tau /。还检查了f / spl circ // sub n /的一般性，包括采样窗口。主要目的是检查此类估计器的均方误差（MSE）属性，以便确定采样率/ sp tau /的最佳选择，以产生最快的收敛速度。我们获得的MSE的最佳速率为O（In（n）/ n）。
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