مرکز منطقه ای اطلاع رساني علوم و فناوري - Bounds for truncation error in sampling expansions of band-limited signals

Abstract :

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.