Advertisement

Nonuniform sampling for random signals bandlimited in the linear canonical transform domain

  • Haiye HuoEmail author
  • Wenchang Sun
Article
  • 38 Downloads

Abstract

In this paper, we mainly investigate the nonuniform sampling for random signals which are bandlimited in the linear canonical transform (LCT) domain. We show that the nonuniform sampling for a random signal bandlimited in the LCT domain is equal to the uniform sampling in the sense of second order statistic characters after a pre-filter in the LCT domain. Moreover, we propose an approximate recovery approach for nonuniform sampling of random signals bandlimited in the LCT domain. Furthermore, we study the mean square error of the nonuniform sampling. Finally, we do some simulations to verify the correctness of our theoretical results.

Keywords

Nonuniform sampling Linear canonical transform Random signals Approximate reconstruction Sinc interpolation 

Notes

Acknowledgements

The authors thank the referees very much for carefully reading the paper and for elaborate and valuable suggestions.

References

  1. Aldroubi, A., & Gröchenig, K. (2001). Nonuniform sampling and reconstruction in shift-invariant spaces. SIAM Review, 43(4), 585–620.MathSciNetzbMATHCrossRefGoogle Scholar
  2. Amirtharajah, R., Collier, J., Siebert, J., Zhou, B., & Chandrakasan, A. (2005). DSPs for engery harvesting sensors: Applications and architectures. IEEE Pervasive Computing, 4(3), 72–79.CrossRefGoogle Scholar
  3. Balakrishnan, A. V. (1962). On the problem of time jitter in sampling. IRE Transactions on Information Theory, 8(3), 226–236.zbMATHCrossRefGoogle Scholar
  4. Bastiaans, M. J. (1979). Wigner distribution function and its application to first-order optics. Journal of the Optical Society of America, 69(12), 1710–1716.CrossRefGoogle Scholar
  5. Chen, Y., Goldsmith, A. J., & Eldar, Y. C. (2014). Channel capacity under sub-Nyquist nonuniform sampling. IEEE Transactions on Information Theory, 60(8), 4739–4756.MathSciNetzbMATHCrossRefGoogle Scholar
  6. Chen, X., Guan, J., Huang, Y., Liu, N., & He, Y. (2015). Radon-linear canonical ambiguity function-based detection and estimation method for marine target with micromotion. IEEE Transactions on Geoscience and Remote Sensing, 53(4), 2225–2240.CrossRefGoogle Scholar
  7. Deng, B., Tao, R., & Wang, Y. (2006). Convolution theorems for the linear canonical transform and their applications. Science in China Series F: Information Sciences, 49(5), 592–603.MathSciNetGoogle Scholar
  8. Eng, F. (2007). Nonuniform sampling in statistical signal processing, Ph.D. Thesis, Linkoping University, Department of Electrical Engineering.Google Scholar
  9. Feichtinger, H. G., & Gröchenig, K. (1992). Irregular sampling theorems and series expansions of band-limited functions. Journal of Mathematical Analysis and Applications, 167(2), 530–556.MathSciNetzbMATHCrossRefGoogle Scholar
  10. Feichtinger, H. G., & Gröchenig, K. (1995). Efficient numerical methods in non-uniform sampling theory. Numerische Mathmatik, 69(4), 423–440.MathSciNetzbMATHCrossRefGoogle Scholar
  11. Huang, X., Zhang, L., Li, S., & Zhao, Y. (2018). Radar high speed small target detection based on keystone transform and linear canonical transform. Digital Signal Processing, 82, 203–215.CrossRefGoogle Scholar
  12. Huo, H. (2019a). Uncertainty principles for the offset linear canonical transform. Circuits, Systems and Signal Processing, 38(1), 395–406.MathSciNetCrossRefGoogle Scholar
  13. Huo, H. (2019b). A new convolution theorem associated with the linear canonical transform. Signal, Image and Video Processing, 13(1), 127–133.CrossRefGoogle Scholar
  14. Huo, H., & Sun, W. (2015). Sampling theorems and error estimates for random signals in the linear canonical transform domain. Signal Processing, 111, 31–38.CrossRefGoogle Scholar
  15. Huo, H., Sun, W., & Xiao, L. (2019). Uncertainty principles associated with the offset linear canonical transform. Mathematical Methods in the Applied Sciences, 42(2), 466–474.MathSciNetzbMATHCrossRefGoogle Scholar
  16. Ignjatović, A., Wijenayake, C., & Keller, G. (2018). Chromatic derivatives and approximations in practice–part II: Nonuniform sampling, zero-crossings reconstruction, and denoising. IEEE Transactions on Signal Processing, 66(6), 1513–1525.MathSciNetzbMATHCrossRefGoogle Scholar
  17. Janik, J., & Bloyet, D. (2004). Timing uncertainties of A/D converters: Theoretical study and experiments. IEEE Transactions on Instrumentation and Measurement, 53(2), 561–565.CrossRefGoogle Scholar
  18. Leow, K. (2010). Reconstruction from non-uniform samples. Master Thesis, Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science.Google Scholar
  19. Liu, X., Shi, J., Xiang, W., et al. (2014). Sampling expansion for irregularly sampled signals in fractional Fourier transform domain. Digital Signal Processing, 34, 74–81.MathSciNetCrossRefGoogle Scholar
  20. Marvasti, F. (2012). Nonuniform sampling: Theory and practice. Berlin: Springer.zbMATHGoogle Scholar
  21. Marvasti, F., Analoui, M., & Gamshadzahi, M. (1991). Recovery of signals from nonuniform samples using iterative methods. IEEE Transactions on Signal Processing, 39(4), 872–878.CrossRefGoogle Scholar
  22. Maymon, S., & Oppenheim, A. V. (2011). Sinc interpolation of nonuniform samples. IEEE Transactions on Signal Processing, 59(10), 4745–4758.MathSciNetzbMATHCrossRefGoogle Scholar
  23. Nazarathy, M., & Shamir, J. (1982). First-order optics–a canonical operator representation: lossless systems. Journal of the Optical Society of America, 72(3), 356–364.MathSciNetCrossRefGoogle Scholar
  24. Papoulis, A. (1966). Error analysis in sampling theory. Proceedings of IEEE, 54(7), 947–955.CrossRefGoogle Scholar
  25. Selva, J. (2009). Functionally weighted lagrange interpolation of band-limited signals from nonuniform samples. IEEE Transactions on Signal Processing, 57(1), 168–181.MathSciNetzbMATHCrossRefGoogle Scholar
  26. Senay, S., Chaparro, L., & Durak, L. (2009). Reconstruction of nonuniformly sampled time-limited signals using prolate spheroidal wave functions. Signal Processing, 89(12), 2585–2595.zbMATHCrossRefGoogle Scholar
  27. Shi, J., Liu, X., He, L., Han, M., Li, Q., & Zhang, N. (2016). Sampling and reconstruction in arbitrary measurement and approximation spaces associated with linear canonical transform. IEEE Transactions on Signal Processing, 64(24), 6379–6391.MathSciNetzbMATHCrossRefGoogle Scholar
  28. Shi, J., Liu, X., Sha, X., & Zhang, N. (2012). Sampling and reconstruction of signals in function spaces associated with the linear canonical transform. IEEE Transactions on Signal Processing, 60(11), 6041–6047.MathSciNetzbMATHCrossRefGoogle Scholar
  29. Stern, A. (2006a). Why is the linear canonical transform so little known? AIP Conference Proceedings, 860(1), 225–234.CrossRefGoogle Scholar
  30. Stern, A. (2006b). Sampling of linear canonical transformed signals. Signal Processing, 86(7), 1421–1425.zbMATHCrossRefGoogle Scholar
  31. Tao, R., Li, B.-Z., Wang, Y., & Aggrey, G. K. (2008). On sampling of band-limited signals associated with the linear canonical transform. IEEE Transactions on Signal Processing, 56(11), 5454–5464.MathSciNetzbMATHCrossRefGoogle Scholar
  32. Tao, R., Zhang, F., & Wang, Y. (2011). Sampling random signals in a fractional Fourier domain. Signal Processing, 91(6), 1394–1400.zbMATHCrossRefGoogle Scholar
  33. Tuncer, T. E. (2007). Block-based methods for the reconstruction of finite-length signals from nonuniform samples. IEEE Transactions on Signal Processing, 55(2), 530–541.MathSciNetzbMATHCrossRefGoogle Scholar
  34. Venkataramani, R., & Bresler, Y. (2000). Perfect reconstruction formulas and bounds on aliasing error in sub-Nyquist nonuniform sampling of multiband signals. IEEE Transactions on Information Theory, 46(6), 2173–2183.MathSciNetzbMATHCrossRefGoogle Scholar
  35. Verbeyst, F., Rolain, Y., Schoukens, J., et al. (2006). System identification approach applied to jitter estimation. In Instrumentation and measurement technology conference (pp. 1752–1757).Google Scholar
  36. Wang, J., Ren, S., Chen, Z., et al. (2018a). Periodically nonuniform sampling and reconstruction of signals in function spaces associated with the linear canonical transform. IEEE Communications Letters, 22(4), 756–759.Google Scholar
  37. Wang, J., Wang, Y., Ren, S., et al. (2018b). Periodically nonuniform sampling and averaging of signals in multiresolution subspaces associated with the fractional wavelet transform. Digital Signal Processing, 80, 1–11.MathSciNetCrossRefGoogle Scholar
  38. Wei, D., & Li, Y. (2014). Reconstruction of multidimensional bandlimited signals from multichannel samples in the linear canonical transform domain. IET Signal Processing, 8(6), 647–657.CrossRefGoogle Scholar
  39. Wei, D., Ran, Q., & Li, Y. (2011). Multichannel sampling and reconstruction of bandlimited signals in the linear canonical transform domain. IET Signal Processing, 8(5), 717–727.MathSciNetCrossRefGoogle Scholar
  40. Xiao, L., & Sun, W. (2013). Sampling theorems for signals periodic in the linear canonical transform domain. Optics Communications, 290, 14–18.CrossRefGoogle Scholar
  41. Xu, L., Zhang, F., Lu, M., et al. (2016). Random signal analysis in the linear canonical transform domain. In URSI Asia-Pacific radio science conference (URSI AP-RASC) (pp. 1862–1865). IEEE.Google Scholar
  42. Xu, S., Feng, L., Chai, Y., et al. (2018a). Analysis of A-stationary random signals in the linear canonical transform domain. Signal Processing, 146, 126–132.CrossRefGoogle Scholar
  43. Xu, S., Jiang, C., Chai, Y., et al. (2018b). Nonuniform sampling theorems for random signals in the linear canonical transform domain. The International Journal of Electronics, 105(6), 1051–1062.Google Scholar
  44. Xu, L., Zhang, F., & Tao, R. (2016). Randomized nonuniform sampling and reconstruction in fractional Fourier domain. Signal Processing, 120, 311–322.CrossRefGoogle Scholar
  45. Yao, K., & Thomas, J. (1967). On some stability and interpolatory properties of nonuniform sampling expansions. IEEE Transactions on Circuit Theory, 14(4), 404–408.CrossRefGoogle Scholar
  46. Yen, J. L. (1956). On nonuniform sampling of bandwidth-limited signals. IRE Transactions on Circuit Theory, 3(4), 251–257.CrossRefGoogle Scholar
  47. Zhang, Q. (2016). Zak transform and uncertainty principles associated with the linear canonical transform. IET Signal Processing, 10(7), 791–797.CrossRefGoogle Scholar
  48. Zhang, Z. C. (2016). An approximating interpolation formula for bandlimited signals in the linear canonical transform domain associated with finite nonuniformly spaced samples. Optik, 127(17), 6927–6932. CrossRefGoogle Scholar
  49. Zhao, H., Ran, Q. W., Tan, L. Y., et al. (2009). Reconstruction of bandlimited signals in linear canonical transform domain from finite nonuniformly spaced samples. IEEE Signal Processing Letters, 16(12), 1047–1050.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics, School of ScienceNanchang UniversityNanchangChina
  2. 2.School of Mathematical Sciences and LPMCNankai UniversityTianjinChina

Personalised recommendations