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On mean-square aliasing error in the cardinal series expansion of random processes (Corresp.)

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An upper bound is derived for the mean-square error involved when a non-band-limited, wide-sense stationary random process x(t) (possessing an integrable power spectral density) is approximated by a cardinal series expansion of the form \sum ^{\infty }_{-\infty }x(n/2W) sinc 2W(t-n/2W) , a sampling expansion based on the choice of some nominal bandwidth W > 0 . It is proved that \lim_{N \rightarrow \infty } E {|x(t) - x_{N}(t)|^{2}} \leq frac{2}{\pi}\int_{| \omega | > 2 \pi W}S_{x}( \omega ) d \omega , where x_{N}(t) = \sum _{-N}^{N}x(n/2W) sinc 2W(t-n/2W) , and S_{x}(\omega ) is the power spectral density for x(t) . Further, the constant 2/ \pi is shown to be the best possible one if a bound of this type (involving the power contained in the frequency region lying outside the arbitrarily chosen band) is to hold uniformly in t . Possible reductions of the multiplicative constant as a function of t are also discussed, and a formula is given for the optimal value of this constant.

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Information Theory, IEEE Transactions on  (Volume:24 ,  Issue: 2 )