Skip to main navigation menu Skip to main content Skip to site footer

Positive B-splines used as Mappings in the Probabilistic Quantizer

Abstract

A Dithered quantizer consists of an external signal called Dither added to the input signal prior to quantization to control the statistical properties of the quantization error. In the framework known as Quantum Signal Processing (QSP) an equivalent quantizer was developed, called probabilistic quantizer, which is able to generate a Dither signal with an arbitrary joint probability distribution. This paper demonstrates how positive B-spline functions can be used as a mapping in the probabilistic quantizer and the mathematical advantages to perform their analysis. In addition, we established a relation between the order of the B-spline and the rendering of conditional moments of the error. Experimental results show that the proposed approach performs on par with Dither quantizer, and its implementation is easier.

Keywords

B-spline, conditional moments, Dither quantizer, probabilistic quantizer, quantum signal processing

PDF

References

  • R. M. Gray, D. L. Neuhoff, “Quantization,” IEEE Transactions Information Theory, vol. 44, no. 6, pp. 2325-2383, 1998. https://doi.org/10.1109/18.720541
  • B. Widrow, I. Kollar, Quantization Noise: Roundoff Error in Digital Computation, Signal Processing, Control, and Communications. Cambridge University Press, 2008. https://doi.org/10.1017/cbo9780511754661
  • S. P. Lipshitz, R. A. Wannamaker, J. Vanderkooy, “Quantization and Dither: A theoretical survey,” Journal of the Audio Engineering Society, vol. 40, no. 5, pp. 355-375, 1992.
  • E. Akyol, K. Rose, “Nonuniform Dithered Quantization,” in Data Compression Conference, 2009, pp. 435-435. https://doi.org/10.1109/dcc.2009.78
  • E. Akyol, K. Rose, “On Constrained Randomized Quantization,” IEEE Transactions Signal Processing, vol. 61, no. 13, pp. 3291-3302, 2013. https://doi.org/10.1109/tsp.2013.2261296
  • R. A. Wannamaker, S. P. Lipshitz, J. Vanderkooy, J. N. Wright, “A theory of nonsubtractive Dither,” IEEE Transactions Signal Processing, vol. 48, no. 2, pp. 499-516, 2000. https://doi.org/10.1109/78.823976
  • R. A. Wannamaker, “The theory of Dithered quantization,” Doctoral Dissertation, University of Waterloo, Ontaria, Canada, 1997.
  • L. Yue, P. Ganesan, B. S. Sathish, C. Manikandan, A. Niranjan, V. Elamaran, A. F. Hussein, “The importance of Dithering technique revisited with biomedical images-a survey,” IEEE Access, vol. 7, pp. 3627-3634, 2019. https://doi.org/10.1109/access.2018.2888503
  • H. Pan, A. Abidi, “Spectral spurs due to quantization in nyquist adcs,” IEEE Transactions Circuits Systems I, vol. 51, no. 8, pp. 1422-1439, 2004. https://doi.org/10.1109/tcsi.2004.832755
  • L. He, L. Jin, J. Yang, F. Lin, L. Yao, X. Jiang, “Self-Dithering technique for high-resolution sar adc design,” IEEE Transactions Circuits Systems II, vol. 62, no. 12, pp. 1124-1128, 2015. https://doi.org/10.1109/tcsii.2015.2468921
  • T. Miki, N. Miura, H. Sonoda, K. Mizuta, M. Nagata, “A random interrupt Dithering sar technique for secure adc against reference-charge side-channel attack,” IEEE Transactions Circuits Systems II, vol. 67, no. 1, pp. 14-18, 2020. https://doi.org/10.1109/tcsii.2019.2901534
  • H. Mo, X. Tan, M. P. Kennedy, “Maximizing the fundamental period of a Dithered digital delta-sigma modulator with constant input,” in Proceeddings IEEE ICECS, 2016, pp. 472-475. https://doi.org/10.1109/icecs.2016.7841241
  • H. Mo, M. P. Kennedy, “Masked Dithering of MASH Digital Delta-Sigma Modulators With Constant Inputs Using Multiple Linear Feedback Shift Registers,” IEEE Transactions Circuits Systems I, vol. 64, no. 6, pp. 1390-1399, 2017. https://doi.org/10.1109/tcsi.2017.2670365
  • Y. Liao, X. Fan, Z. Hua, “Influence of lfsr Dither on the periods of a mash digital delta–sigma modulator,” IEEE Transactions Circuits Systems II, vol. 66, no. 1, pp. 66-70, 2019. https://doi.org/10.1109/tcsii.2018.2828600
  • M. S. Fu, O. C. Au, “Data hiding in ordered Dithered halftone images,” Circuits Systems Signal Process, vol. 20, pp. 209-232, 2001. https://doi.org/10.1007/bf01201139
  • J. Rapp, R. M. A. Dawson, V. K. Goyal, “Improving Lidar Depth Resolution with Dither,” in Proceeddings IEEE ICIP, 2018, pp. 1553-1557. https://doi.org/10.1109/icip.2018.8451528
  • E. T. Mbitu, S.-C. Chen, “Designing limit-cycle suppressor using Dithering and dual-input describing function methods,” Mathematics, vol. 8, no. 11, e1978, 2020. https://doi.org/10.3390/math8111978
  • V. K. Goyal, J. Kovacevic, J. A. Kelner, “Quantized Frame Expan-´ sions with Erasures,” Applied and Computational Harmonic Analysis, vol. 10, no. 3, pp. 203-233, 2001. https://doi.org/10.1006/acha.2000.0340
  • S. Rangan, V. K. Goyal, “Recursive consistent estimation with bounded noise,” IEEE Transaction Information Theory, vol. 47, no. 1, pp. 457-464, 2001. https://doi.org/10.1109/18.904562
  • B. G. Bodmann, S. P. Lipshitz, “Randomly Dithered quantization and sigma–delta noise shaping for finite frames,” Applied and Computational Harmonic Analysis, vol. 25, no. 3, pp. 367-380, 2008. https://doi.org/10.1016/j.acha.2007.12.003
  • P. T. Boufounos, “Universal Rate-Efficient Scalar Quantization,” IEEE Transactions Information Theory, vol. 58, no. 3, pp. 1861-1872, 2012. https://doi.org/10.1109/tit.2011.2173899
  • C. Xu, V. Schellekens, L. Jacques, “Taking the Edge off Quantization: Projected Back Projection in Dithered Compressive Sensing,” in Proceeddings IEEE SSP, 2018, pp. 203-207. https://doi.org/10.1109/ssp.2018.8450784
  • L. Jacques, V. Cambareri, “Time for Dithering: fast and quantized random embeddings via the restricted isometry property,” Information and Inference: A Journal of the IMA, vol. 6, no. 4, pp. 441-476, 2017. https://doi.org/10.1093/imaiai/iax004
  • A. Parada-Mayorga, D. L. Lau, J. H. Giraldo, G. R. Arce, “Blue-Noise sampling on graphs,” IEEE Transactions Signal Information Processing, vol. 5, no. 3, pp. 554-569, 2019. https://doi.org/10.1109/tsipn.2019.2922852
  • A. Sanyal, N. Sun, “A simple and efficient Dithering method for vector quantizer based mismatch-shaped DACs,” in Proceedings IEEE ISCAS, 2012, pp. 528-531. https://doi.org/10.1109/iscas.2012.6272082
  • N. West, G. Scheets, “Increasing the resolution of a uniform quantizer using a deterministic Dithering signal,” in Proceedings IEEE AUTOTESTCON, 2012, pp. 54-57. https://doi.org/10.1109/autest.2012.6334521
  • R. Hadad, U. Erez, “Dithered Quantization via Orthogonal Transformations,” IEEE Transactions Signal Processing, vol. 64, no. 22, pp. 5887-5900, 2016. https://doi.org/10.1109/tsp.2016.2599482
  • Y. C. Eldar, “Quantum signal processing,” Doctora Dissertation, Massachusetts Institute of Technology, 2001.
  • M. Unser, T. Blu, “Wavelet theory demystified,” IEEE Transactions Signal Processing, vol. 51, no. 2, pp. 470-483, 2003. https://doi.org/10.1109/tsp.2002.807000
  • M. Unser, A. Aldroubi, M. Eden, “A family of polynomial spline wavelet transforms,” Signal Processing, vol. 30, no. 2, pp. 141-162, 1993. https://doi.org/10.1016/0165-1684(93)90144-y
  • G. Makkena, M. B. Srinivas, “Nonlinear sequence transformation based continuous-time wavelet filter approximation,” Circuits, Systems, and Signal Processing, vol. 37, no. 3, p. 965-983, 2018. https://doi.org/10.1007/s00034-017-0591-9
  • P. Noras, N. Aghazadeh, “Directional schemes for edge detection based on b-spline wavelets,” Circuits, Systems, and Signal Processing, vol. 37, pp. 3973-3994, 2018. https://doi.org/10.1007/s00034-018-0753-4
  • M. A. Unser, “Ten good reasons for using spline wavelets,” in Wavelet Applications in Signal and Image Processing V, A. Aldroubi, A. F. Laine, M. A. Unser, Eds., vol. 3169, International Society for Optics and Photonics. SPIE, 1997, pp. 422-431. https://doi.org/10.1117/12.292801
  • M. Unser, T. Blu, “Fractional splines and wavelets,” SIAM Review, vol. 42, no. 1, pp. 43-67, 2000. https://doi.org/10.1137/s0036144598349435
  • L. Devroye, Non-Uniform Random Variate Generation, Springer New York, 1986. https://doi.org/10.1007/978-1-4613-8643-8

Downloads

Download data is not yet available.

Similar Articles

You may also start an advanced similarity search for this article.