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A describing function study of saturated quantization and its application to the stability analysis of multi-bit sigma delta modulators

Pieter Rombouts (UGent) , Maarten De Bock (UGent) , Jeroen De Maeyer (UGent) and Ludo Weyten (UGent)
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Abstract
Just as their single-bit counterparts, multi-bit sigma delta modulators exhibit nonlinear behavior due to the presence of the quantizer in the loop. In the multi-bit case this is caused by the fact that any quantizer has a limited output range and hence gives an implicit saturation effect. Due to this, any multi-bit modulator is prone to modulator overloading. Unfortunately, until now, designers had to rely on extensive time-domain simulations to predict the overloading level, because there is no adequate analytical theory to model this effect. In this work, we have developed such an analytical theory based on multiple input describing function analysis. This way, we obtained expressions for the signal gain, the noise gain and the variance of the quantization noise. Here, both the case of DC as well as sinusoidal signals was considered. These results were used for the stability analysis of multi-bit Sigma Delta modulators, which allows to predict the overloading level. Code implementing the proposed expressions is available for download at http://cas1.elis.ugent. be/cas/en/download/
Keywords
sigma-delta modulation, quantization, DELAY, EXCESS, NOISE, CONVERTERS, SINUSOIDAL INPUTS, A/D-conversion, describing function theory

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MLA
Rombouts, Pieter et al. “A Describing Function Study of Saturated Quantization and Its Application to the Stability Analysis of Multi-bit Sigma Delta Modulators.” IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I-REGULAR PAPERS 60.7 (2013): 1740–1752. Print.
APA
Rombouts, P., De Bock, M., De Maeyer, J., & Weyten, L. (2013). A describing function study of saturated quantization and its application to the stability analysis of multi-bit sigma delta modulators. IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I-REGULAR PAPERS, 60(7), 1740–1752.
Chicago author-date
Rombouts, Pieter, Maarten De Bock, Jeroen De Maeyer, and Ludo Weyten. 2013. “A Describing Function Study of Saturated Quantization and Its Application to the Stability Analysis of Multi-bit Sigma Delta Modulators.” Ieee Transactions on Circuits and Systems I-regular Papers 60 (7): 1740–1752.
Chicago author-date (all authors)
Rombouts, Pieter, Maarten De Bock, Jeroen De Maeyer, and Ludo Weyten. 2013. “A Describing Function Study of Saturated Quantization and Its Application to the Stability Analysis of Multi-bit Sigma Delta Modulators.” Ieee Transactions on Circuits and Systems I-regular Papers 60 (7): 1740–1752.
Vancouver
1.
Rombouts P, De Bock M, De Maeyer J, Weyten L. A describing function study of saturated quantization and its application to the stability analysis of multi-bit sigma delta modulators. IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I-REGULAR PAPERS. 2013;60(7):1740–52.
IEEE
[1]
P. Rombouts, M. De Bock, J. De Maeyer, and L. Weyten, “A describing function study of saturated quantization and its application to the stability analysis of multi-bit sigma delta modulators,” IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I-REGULAR PAPERS, vol. 60, no. 7, pp. 1740–1752, 2013.
@article{4170389,
  abstract     = {Just as their single-bit counterparts, multi-bit sigma delta modulators exhibit nonlinear behavior due to the presence of the quantizer in the loop. In the multi-bit case this is caused by the fact that any quantizer has a limited output range and hence gives an implicit saturation effect. Due to this, any multi-bit modulator is prone to modulator overloading. Unfortunately, until now, designers had to rely on extensive time-domain simulations to predict the overloading level, because there is no adequate analytical theory to model this effect. In this work, we have developed such an analytical theory based on multiple input describing function analysis. This way, we obtained expressions for the signal gain, the noise gain and the variance of the quantization noise. Here, both the case of DC as well as sinusoidal signals was considered. These results were used for the stability analysis of multi-bit Sigma Delta modulators, which allows to predict the overloading level. Code implementing the proposed expressions is available for download at http://cas1.elis.ugent. be/cas/en/download/},
  author       = {Rombouts, Pieter and De Bock, Maarten and De Maeyer, Jeroen and Weyten, Ludo},
  issn         = {1549-8328},
  journal      = {IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I-REGULAR PAPERS},
  keywords     = {sigma-delta modulation,quantization,DELAY,EXCESS,NOISE,CONVERTERS,SINUSOIDAL INPUTS,A/D-conversion,describing function theory},
  language     = {eng},
  number       = {7},
  pages        = {1740--1752},
  title        = {A describing function study of saturated quantization and its application to the stability analysis of multi-bit sigma delta modulators},
  url          = {http://dx.doi.org/10.1109/TCSI.2012.2230585},
  volume       = {60},
  year         = {2013},
}

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