For f(t) a real-valued signal band-limited to - \pi r \leq \omega \leq \pi r (0 < r < 1) and represented by its Fourier integral, upper bounds are established for the magnitude of the truncation error when f(t) is approximated at a generic time t by an appropriate selection of N_{1} + N_{2} + 1 terms from its Shannon sampling series expansion, the latter expansion being associated with the full band [-\pi, \pi] and thus involving samples of f taken at the integer points. Results are presented for two cases: 1) the Fourier transform F(\omega) is such that |F(\omega)|^{2} is integrable on [-\pi, \pi r] (finite energy case), and 2) |F(\omega)| is integrable on [-\pi r, \pi r] . In case 1) it is shown that the truncation error magnitude is bounded above by g(r, t) \cdot \sqrt{E} \cdot \left( \frac{1}{N_{1}} + \frac{1}{N_{2}} \right) where E denotes the signal energy and g is independent of N_{1}, N_{2} and the particular band-limited signal being approximated. Correspondingly, in case 2) the error is bounded above by h(r, t) \cdot M \cdot \left( \frac{1}{N_{1}} + \frac{1}{N_{2}} \right) where M is the maximum signal amplitude and h is independent of N_{1}, N_{2} and the signal. These estimates possess the same asymptotic behavior as those exhibited earlier by Yao and Thomas [2], but are derived here using only real variable methods in conjunction with the signal representation. In case 1), the estimate obtained represents a sharpening of the Yao-Thomas bound for values of r dose to unity.

	Year	Citations

References