An analysis of nonlinear behavior in delta - sigma modulators

Abstract

This paper introduces a new method of analysis for Delta-Sigma Modulators based on modeling the nonlinear quantizer with a linerized gain, obtained by minimizing a mean-square-error criterion[7] , followed by an additive noise source representing distortion components. In the paper, input-signal amplitude dependencies of Delta-Sigma Modulator stability and signal to noise ratio are analyzed. It is shown that due to the nonlinearity of the quantizer, the signal-to-noise ratio of the modulator may decrease as the input amplitude increases prior to saturation. Also, a stable third-order Delta-Sigma Modulator may become unstable by increasing the input amplitude beyond a certain threshold. Both of these phenomenon are explained by the nonlinear analysis of this paper. The analysis is carried out for both DC and sinusoidal excitations. 3 3.2 Sinusoidal Input and Nonlinearity Modeling (, 4. Noise Spectra and Signal to Noise Ratio 9 4.1 Exact Numerical Calculation of Signal to Noise Ratio 10 5.

1-Bit processing based model predictive control for fractionated satellite missions

“…In most digitalized control applications, the quantization noise is neglected due to the assumption that the error is filtered by the physical system (physical systems are generally not sensitive to signals in high-frequency range). However, as mentioned before, for ∆∑ modulators, the nonlinearity introduces spectral components which cover a wide bandwidth including the baseband (Ardalan and Paulos, 1987). Therefore it is hard to adequately filter the quantization noise if the noise power in the base band is too high.…”

Nonlinear Stability Prediction of Multibit Delta–Sigma Modulators for Sinusoidal Inputs

“…Another approach to simplify the analysis has been to assume a DC input to the Δ-Σ modulator [2]- [7]. In [8], separate signal and quantization noise nonlinear gains have been used for the stability analysis of 2 nd -and 3 rd -order Δ-Σ modulators for DC and sinusoidal inputs using the root locus approach. The nonlinear gains have been derived from the concept of modified nonlinearity in nonlinear control theory [9].…”

“…the signal and quantization noise. The linearized modeling approach using nonlinear gains in [8] did not previously provide useful stability predictions, until a new interpretation of the instability mechanism for Δ-Σ modulators based on the quantization noise amplification was given in [10]. However, this is restricted to DC inputs.…”

Rigorous Analysis of Delta–Sigma Modulators for Fractional-<tex>$N$</tex>PLL Frequency Synthesis

“…Despite the fact that the linearized model can provide reasonable results with respect to signal-to-noise ratio (SNR) performance, it completely fails to determine the exact spectral shaping of the modulator, which is highly likely to be discrete and colored. A more advanced linearized model for analyzing modulators is the so-called describing function method, in which the quantizer is modeled as a linear gain (chosen in a minimum mean-square-error sense) followed by an input dependent additive white noise source [5]. This improved model gives crude explanations on the input dependent stability and large amplitude limit cycle behavior of the modulators.…”

“…This improved model gives crude explanations on the input dependent stability and large amplitude limit cycle behavior of the modulators. However, it is still an approximation in the sense that, except the fundamental component (assuming a sinusoidal input), all other harmonics generated by the quantizer are perfectly removed by linear filters within the modulator [4], [5]. The exact quantization noise shaping performance of modulators is an important characteristic, as in certain applications the spectral spikes contained in the output spectrum may be strictly objectionable, for instance in the fractional-phase-locked-loop (PLL) frequency synthesis applications [6], [7].…”