Statistical properties of quantisation noise in analogue-to-digital converter with oversampling and decimation

Statistical properties of quantisation noise in analogue-to-digital converter with oversampling and decimation

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This study presents the analysis of statistical parameters of the output quantisation noise amplitude distribution in the analogue-to-digital converter (ADC) with oversampling and decimation. Hypothesis was tested that quantisation noise amplitudes on the output of the system have a Gaussian distribution. To test this hypothesis, Chi-squared statistical test was performed with parameter estimation based on grouped data. Simulation models were made for two types of ADCs: classical b-bit and sigma–delta. Various realisations of decimation filters were used as well as various values of the decimation factor. Based on the number of cases in which the hypothesis was rejected, the significance of the test was determined, i.e. the deviation from the ideal Gaussian distribution was quantified. Results of this research can prove to be useful in system design process, where accurate knowledge of the quantisation noise statistical model parameters is required. This study presents general procedure for analysing the amplitude distribution of the quantisation noise, which can easily be implemented on other modified models of the ADC with oversampling.


    1. 1)
      • 1. Hauser, M.W.: ‘Principles of oversampling A/D conversion’, AES: J. Audio Eng. Soc., 1991, 39, (1-2), pp. 326.
    2. 2)
      • 2. De La Rosa, J.M.: ‘Sigma-Delta modulators: tutorial overview, design guide, and state-of-the-art survey’, IEEE Trans. Circuits Syst. I, Regul. Pap., 2011, 58, (1), pp. 121.
    3. 3)
      • 3. Widrow, B., Kollár, I., Liu, M.-C.: ‘Statistical theory of quantisation’, IEEE Trans. Instrum. Meas., 1996, 45, (2), pp. 353361.
    4. 4)
      • 4. Candy, J.C., Benjamin, O.J.: ‘The structure of quantisation noise from Sigma-Delta modulation’, IEEE Trans. Commun., 1981, 29, (9), pp. 13161323..
    5. 5)
      • 5. Lopez, J.A., Caffarena, G., Carreras, C., et al: ‘Fast and accurate computation of the round-off noise of linear time-invariant systems’, IET Circuits Devices Syst., 2008, 2, (4), pp. 393408.
    6. 6)
      • 6. Caffarena, G., Carreras, C., Lopez, J.A., et al: ‘SQNR estimation of fixed-point DSP algorithms’, EURASIP J. Adv. Signal Process., 2010, pp. 112.
    7. 7)
      • 7. Naud, J.C., Menard, D., Caffarena, G., et al: ‘A discrete model for correlation between quantisation noises’, IEEE Trans. Circuits Syst. II, Express Briefs, 2012, 59, (11), pp. 800804.
    8. 8)
      • 8. Rocher, R., Menard, D., Scalart, P., et al: ‘Analytical approach for numerical accuracy estimation of fixed-point systems based on smooth operations’, IEEE Trans. Circuits Syst. I, Regul. Pap., 2012, 59, (10), pp. 23262339.
    9. 9)
      • 9. Rocher, R., Scalart, P.: ‘Noise probability density function in fixed-point systems based on smooth operators’. Proc Conf. Design and Architectures for Signal and Image Processing (DASIP), Karlsruhe, Germany, October 2012, pp. 139146.
    10. 10)
      • 10. Caffarena, G., Menard, D.: ‘Quantisation noise power estimation for floating-point DSP circuits’, IEEE Trans. Circuits Syst. II, Express Briefs, 2016, 63, (6), pp. 593597.
    11. 11)
      • 11. Wang, Z., Zhang, J., Verma, N.: ‘Reducing quantisation errors for inner-product operations in embedded digital signal processing systems [Tips&Tricks]’, IEEE Signal Process. Mag., 2016, 33, (6), pp. 141147.
    12. 12)
      • 12. Benkrid, A., Benkrid, K.: ‘A statistical framework to minimise and predict the range values of quantisation errors in fixed-point FIR filters architectures’, Digital Signal Process., Rev. J., 2013, 23, (1), pp. 453469.
    13. 13)
      • 13. Kipnis, A., Goldsmith, A.J., Eldar, Y.C.: ‘Optimal trade-off between sampling rate and quantisation precision in Sigma-Delta A/D conversion’. Proc. Int. Conf. Sampling Theory and Applications (SampTA), Washington, DC, USA, May 2015, pp. 627631.
    14. 14)
      • 14. Nehmeh, R., Menard, D., Banciu, A., et al: ‘Integer word-length optimization for fixed-point systems’. Proc. IEEE Int. Conf. Acoustics, Speech and Signal Processing (ICASSP), Florence, Italy, May 2014, pp. 83218325.
    15. 15)
      • 15. Parashar, K., Menard, D., Rocher, R., et al: ‘Shaping probability density function of quantisation noise in fixed point systems’. Proc. Conf. Record of the Forty Fourth Asilomar Conf. Signals, Systems and Computers, Pacific Grove, CA, USA, November 2010, pp. 16751679.
    16. 16)
      • 16. Oppenheim, A.V., Schafer, R.W.: ‘Discrete-time signal processing’ (Prentice-Hall, Englewood Cliffs, NJ, 1989).
    17. 17)
      • 17. Radonjić, A.D., Ćertić, J.D.: ‘Analysis of half-band approximately linear phase IIR filter realisation structure in MATLAB’, Facta Univer., Series: Electron. Energ., 2015, 28, (4), pp. 611623.
    18. 18)
      • 18. Constantinides, G.A., Cheung, P.Y.K., Luk, W.: ‘Truncation noise in fixed-point SFGs’, Electron. Lett., 1999, 35, (23), pp. 20132014.
    19. 19)
      • 19. Hogenauer, E.B.: ‘An economical class of digital filters for decimation and interpolation’, IEEE Trans. Acoustics Speech Signal Process., 1981, 29, (2), pp. 155162.
    20. 20)
      • 20. Lehtinen, V., Renfors, M.: ‘Truncation noise analysis of noise shaping DSP systems with application to CIC decimators’. Proc. 11th European Signal Processing Conf., Toulouse, France, September 2002, pp. 14.
    21. 21)
      • 21. Milić, Lj.: ‘Multirate filtering for digital signal processing: MATLAB applications’ (Info. Science Reference, Hershey, PA, 2009).
    22. 22)
      • 22. Presti, L.L.: ‘Efficient modified-sinc filters for sigma-delta A/D converters’, IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., 2000, 47, (11), pp. 12041213.
    23. 23)
      • 23. Jovanovic Dolecek, G., Laddomada, M.: ‘A novel two-stage non-recursive architecture for the design of generalized comb filters’, Digit. Signal Process., 2012, 22, (5), pp. 859868.
    24. 24)
      • 24. Jovanovic Dolecek, G., Mitra, S.K.: ‘On design of CIC decimation filter with improved response’. Proc. Int. Conf. ISCCSP, Malta, March 2008.
    25. 25)
      • 25. Renfors, M., Neuvo, Y.: ‘The maximum sampling rate of digital filters under hardware speed constraints’, IEEE Trans. Circuits Syst., 1981, 28, (3), pp. 196202.
    26. 26)
      • 26. Cochran, W.G.: ‘The χ2 test of goodness of fit’, Ann. Math. Stat., 1952, 23, pp. 315345.
    27. 27)
      • 27. Voinov, V., Nikulin, M., Balakrishnan, N.: ‘Chi-squared goodness of fit tests with applications’ (Elsevier, Amsterdam, 2013, 1st edn.), pp. 1126.
    28. 28)
      • 28. Widrow, B.: ‘A study of rough amplitude quantisation by means of Nyquist sampling theory’, IRE Trans. Circuit Theory, 1956, 3, (4), pp. 266276.
    29. 29)
      • 29. Montgomery, D.C.: ‘Applied statistics and probability for engineers’ (John Wiley & Sons Inc., 2003, 3rd edn.), Chapter 1.
    30. 30)
      • 30. Sturges, H.A.: ‘The choice of a class interval’, J. Am. Stat. Assoc., 1926, 21, (153), pp. 6566.
    31. 31)
      • 31. Scott, D.W.: ‘Sturges’ rule’, Wiley Interdiscip. Rev., Comput. Stat., 2009, 1, (3), pp. 303306.
    32. 32)
      • 32. Lilliefors, H.: ‘On the Kolmogorov–Smirnov test for normality with mean and variance unknown’, J. Am. Stat. Assoc., 1967, 62, pp. 399402.
    33. 33)
      • 33. Schreier, R., Temes, G.C.: ‘Understanding delta–sigma data converters’ (Wiley-IEEE Press, 2004, 1st edn.), Chapter 1.

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