Bounds for truncation error in sampling expansions of band-limited signals
- 1 July 1969
- journal article
- Published by Institute of Electrical and Electronics Engineers (IEEE) in IEEE Transactions on Information Theory
- Vol. 15 (4) , 440-444
- https://doi.org/10.1109/tit.1969.1054335
Abstract
Forf(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 whenf(t)is approximated at a generic timetby an appropriate selection ofN_{1} + N_{2} + 1terms from its Shannon sampling series expansion, the latter expansion being associated with the full band[-pi, pi]and thus involving samples offtaken at the integer points. Results are presented for two cases: 1) the Fourier transformF(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 byg(r, t) cdot sqrt{E} cdot left( frac{1}{N_{1}} + frac{1}{N_{2}} ight)whereEdenotes the signal energy andgis independent ofN_{1}, N_{2}and the particular band-limited signal being approximated. Correspondingly, in case 2) the error is bounded above byh(r, t) cdot M cdot left( frac{1}{N_{1}} + frac{1}{N_{2}} ight)whereMis the maximum signal amplitude andhis independent ofN_{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 ofrdose to unity.Keywords
This publication has 4 references indexed in Scilit:
- Truncated sampling expansionsIEEE Transactions on Automatic Control, 1967
- On Truncation Error Bounds for Sampling Representations of Band-Limited SignalsIEEE Transactions on Aerospace and Electronic Systems, 1966
- Bounds for Truncation Error of the Sampling ExpansionSIAM Journal on Applied Mathematics, 1966
- Truncation Error of Sampling-Theorem ExpansionsProceedings of the IRE, 1962