Abstract
The random equivalent sampling (RES) is a well-known sampling technique that can be used to capture a high-speed repetitive waveform with low sampling rate. In this paper, the feasibility of spectrum-blind multiband signal reconstruction for data sampled from RES is investigated. We propose a RES sampling pattern and its corresponding mathematical model that guarantees well-conditioned reconstruction of multiband signal with unknown spectral support. We give the minimum number of RES acquisitions that hold overwhelming probability to successfully reconstruct original signal. We demonstrate that for signal with specific spectral occupation, the minimum number of RES acquisitions and the minimum sampling rate could be approached. The signal reconstruction is studied in the framework of compressive sampling theory. The eigen-decomposition and minimum description length criteria are adopted to adaptively estimate the dimension of signal, and the number of unknowns of reconstruction problem is reduced. Experimental results are reported to indicate that, for a spectrum-blind sparse multiband signal, the proposed reconstruction algorithm for RES is feasible and robust.
Similar content being viewed by others
References
Y. Bresler, Spectrum-blind sampling and compressive sensing for continuous-index signals. IEEE Info. Theory Appl. Workshop, 547–554 (2008). doi:10.1109/ITA.2008.4601017
E. Candès, J. Romberg, T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 52(2), 489–509 (2006)
E. Candès, M. Wakin, An introduction to compressive sampling [a sensing/sampling paradigm that goes against the common knowledge in data acquisition]. IEEE Sig. Proc. Mag. 52(2), 21–30 (2006)
D. Donoho, Compressed sensing. IEEE Trans. Inform. Theory 52(4), 1289–1306 (2006)
P. Feng, Y. Bresler, Spectrum-blind minimum-rate sampling and reconstruction of multi-band signals. Proc. IEEE Int. Conf. Acoust. Speech Signal Process. 3, 1688–1691 (1996)
V.G. Ivchenko, A.N. Kalashnikov, R.E. Challis, B.R. Hayes-Gill, High-speed digitizing of repetitive waveforms using accurate interleaved sampling. IEEE Trans. Instrum. Meas. 56(4), 1322–1328 (2007)
H.J. Landau, Necessary density conditions for sampling and interpolation of certain entire functions. Acta Math. 117, 37–52 (1967)
M. Lexa, M. Davies, J. Thompson, Multi-coset sampling and recovery of sparse multiband signals. [Technical report], http://www.see.ed.ac.uk/~lexa/supportingdocs/mlexa_techreport_mc.pdf. Accessed 10 Aug 2014
M. Mishali, Y.C. Eldar, Blind multiband signal reconstruction: compressed sensing for analog signals. IEEE Trans. Signal Process. 57(3), 993–1009 (2009)
A.V. Oppenheim, A.S. Willsky, S. Hamid, Signal and Systems, 2nd edn. (Prentice Hall, Englewood Cliffs, 1996)
B. Provost, E. Sanchez-Sinencio, A practical self-calibration scheme implementation for pipeline ADC. IEEE Trans. Instrum. Meas. 53(2), 448–456 (2004)
P.J. Pupalaikis, Random Interleaved Sampling, http://www.lecroy.com/files/WhitePapers/WP_Ris_102203.pdf. Accessed 10 Aug 2014
C.E. Shannon, A Mathematical Theory of Communication (University of Illinois Press, Urbana, 1949)
D.E. Toeppen, Acquisition clock dithering in a digital oscilloscope. Hewlett-Packward J. 48(2), 1–4 (1997)
J.A. Tropp, Algorithms for simultaneous sparse approximation. Part I: Greedy pursuit. Signal Process (Special Issue on Sparse Approximations in Signal and Image Processing) 86, 572–588 (2006)
R. Venkataramani, Y. Bresler, Perfect reconstruction formulas and bounds on aliasing error in sub-Nyquist nonuniform sampling of multiband signals. IEEE Trans. Info. Theory 46(6), 2173–2183 (2000)
M. Wax, T. Kailath, Detection of signals by information theoretic criteria. IEEE Trans. Acoust. Speech Signal Process. 33(2), 387–392 (1985)
D.Y. Wei, Y.M. Li, Reconstruction of multidimensional bandlimited signals from multichannel samples in the linear canonical transform domain. IET Signal Process. 8(6), 647–657 (2014)
R.A. Witte, Sample rate and display rate in digitizing oscilloscopes. Hewlett-Packward J. 43, 18–19 (1992)
Y.J. Zhao, X.Y. Zhuang, L. Wang, The research and application of random sampling in digital storage oscilloscope, in Proceedings of IEEE Circuits, Systems International Conference (Chengdu, China, 2009), pp. 1–3
Y.J. Zhao, Y.H. Hu, H.J. Wang, Enhanced random equivalent sampling based on compressed sensing. IEEE Trans. Instrum. Meas. 61(3), 579–586 (2012)
Y.J. Zhao, X.Y. Zhuang, H.J. Wang, Z.J. Dai, Ultrasonic signal compressive detection using improved random equivalent sampling. IET Sci. Meas. Technol. 6(4), 261–266 (2012)
Acknowledgments
This work is supported in part by the National Natural Science Foundation of China (Grant No. 61301264), in part by the Ph.D. Programs Foundation of Ministry of Education of China (Grant No. 20130185120019), and in part by the Fundamental Research Funds for the Central Universities (Grant No. ZYGX2013J089).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhao, Y., Wang, L., Wang, H. et al. Minimum Rate Sampling and Spectrum-Blind Reconstruction in Random Equivalent Sampling. Circuits Syst Signal Process 34, 2667–2680 (2015). https://doi.org/10.1007/s00034-015-9989-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00034-015-9989-4