Recovering band-limited signals under noise

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/n/spl tau/ yksin(/spl Omega/(t-k/spl tau/))//spl pi/(t-k/spl tau/),|t|/spl les/n/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)/n).

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