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

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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.

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References

  • Aldroubi, A., & Gröchenig, K. (2001). Nonuniform sampling and reconstruction in shift-invariant spaces. SIAM Review, 43(4), 585–620.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Balakrishnan, A. V. (1962). On the problem of time jitter in sampling. IRE Transactions on Information Theory, 8(3), 226–236.

    Article  MATH  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    MathSciNet  Google Scholar 

  • Eng, F. (2007). Nonuniform sampling in statistical signal processing, Ph.D. Thesis, Linkoping University, Department of Electrical Engineering.

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • Feichtinger, H. G., & Gröchenig, K. (1995). Efficient numerical methods in non-uniform sampling theory. Numerische Mathmatik, 69(4), 423–440.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Huo, H. (2019a). Uncertainty principles for the offset linear canonical transform. Circuits, Systems and Signal Processing, 38(1), 395–406.

    Article  MathSciNet  Google Scholar 

  • Huo, H. (2019b). A new convolution theorem associated with the linear canonical transform. Signal, Image and Video Processing, 13(1), 127–133.

    Article  Google Scholar 

  • Huo, H., & Sun, W. (2015). Sampling theorems and error estimates for random signals in the linear canonical transform domain. Signal Processing, 111, 31–38.

    Article  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Leow, K. (2010). Reconstruction from non-uniform samples. Master Thesis, Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science.

  • 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.

    Article  MathSciNet  Google Scholar 

  • Marvasti, F. (2012). Nonuniform sampling: Theory and practice. Berlin: Springer.

    MATH  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Maymon, S., & Oppenheim, A. V. (2011). Sinc interpolation of nonuniform samples. IEEE Transactions on Signal Processing, 59(10), 4745–4758.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • Papoulis, A. (1966). Error analysis in sampling theory. Proceedings of IEEE, 54(7), 947–955.

    Article  Google Scholar 

  • Selva, J. (2009). Functionally weighted lagrange interpolation of band-limited signals from nonuniform samples. IEEE Transactions on Signal Processing, 57(1), 168–181.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • Stern, A. (2006a). Why is the linear canonical transform so little known? AIP Conference Proceedings, 860(1), 225–234.

    Article  Google Scholar 

  • Stern, A. (2006b). Sampling of linear canonical transformed signals. Signal Processing, 86(7), 1421–1425.

    Article  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • Tao, R., Zhang, F., & Wang, Y. (2011). Sampling random signals in a fractional Fourier domain. Signal Processing, 91(6), 1394–1400.

    Article  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • 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).

  • 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 

  • 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.

    Article  MathSciNet  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • Xiao, L., & Sun, W. (2013). Sampling theorems for signals periodic in the linear canonical transform domain. Optics Communications, 290, 14–18.

    Article  Google Scholar 

  • 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.

  • 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.

    Article  Google Scholar 

  • 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 

  • Xu, L., Zhang, F., & Tao, R. (2016). Randomized nonuniform sampling and reconstruction in fractional Fourier domain. Signal Processing, 120, 311–322.

    Article  Google Scholar 

  • Yao, K., & Thomas, J. (1967). On some stability and interpolatory properties of nonuniform sampling expansions. IEEE Transactions on Circuit Theory, 14(4), 404–408.

    Article  Google Scholar 

  • Yen, J. L. (1956). On nonuniform sampling of bandwidth-limited signals. IRE Transactions on Circuit Theory, 3(4), 251–257.

    Article  Google Scholar 

  • Zhang, Q. (2016). Zak transform and uncertainty principles associated with the linear canonical transform. IET Signal Processing, 10(7), 791–797.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

Download references

Acknowledgements

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

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Authors and Affiliations

  1. Department of Mathematics, School of Science, Nanchang University, Nanchang, 330031, Jiangxi, China

    Haiye Huo

  2. School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, China

    Wenchang Sun

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  1. Haiye Huo
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  2. Wenchang Sun
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Correspondence to Haiye Huo.

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This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 11801256, 11525104 and 11531013).

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Huo, H., Sun, W. Nonuniform sampling for random signals bandlimited in the linear canonical transform domain. Multidim Syst Sign Process 31, 927–950 (2020). https://doi.org/10.1007/s11045-019-00691-2

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  • DOI: https://doi.org/10.1007/s11045-019-00691-2

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