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

Jaswinder Lota, Mohammed Al-Janabi, Izzet Kale
2014 IEEE Transactions on Instrumentation and Measurement  
This paper proposes a novel algorithm that can be integrated with various design and evaluation tools, to more accurately and rapidly predict stability in multi-bit delta-sigma (Δ-Σ) modulators. Analytical expressions using the nonlinear gains from the concept of modified nonlinearity in control theory are incorporated into the mathematical model of multi-bit Δ-Σ modulators to predict the stable amplitude limits for sinusoidal input signals. The nonlinear gains lead to a set of equations which
more » ... an numerically estimate the quantizer gain as a function of the input sinusoidal signal amplitude. This method is shown to accurately predict the stable amplitude limits of sinusoids for 2 nd -, 3 rd -, 4 th -, 5 th -and 6 th -order 3-and 5-level mid-tread quantizer based Δ-Σ modulators. The algorithm is simple to apply and can be extended to midrise quantizers or to any number of quantizer levels. The only required input parameters for this algorithm are the number of quantizer levels and the coefficients of the noise transfer function. Index Terms-delta-sigma, multi-bit quantizer, nonlinear gains, stability I. INTRODUCTION A. Literature review-limitations of existing approaches The stable input amplitude limits for Delta-Sigma (Δ-Σ) modulators are complicated to predict due to the non-linearity of the quantizer. The stable amplitude limit is defined as the amplitude beyond which the quantizer input exhibits large oscillations before eventually increasing to an exponentially large value. This stable amplitude limit decreases as the order of the Δ-Σ modulator increases. Various techniques have been proposed for predicting the stability of one-bit quantizer based Δ-Σ modulators. One technique is to model the quantizer as a threshold function in the state equations, which gets complicated for higher-order Δ-Σ modulators and is limited to 1 st -and 2 ndorder Δ-Σ modulators [1] . 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] . This approach of using a quasi-linear technique allows the nonlinear quantizer to be replaced by an equivalent gain, for each of the inputs, i.e. 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. A combined approach of deploying the separate signal, quantization noise gains in [8] , and of the quantization noise amplification in [10] is given in [11] , where stability has been predicted for a single-sinusoidal input. In [12] , the analysis is extended for predicting stability for dualsinusoidal inputs. An in-depth analysis of the approach in [11], [12] with detailed simulation results is given in [13] . As the approaches in [11]-[13] are applied to quantify the stability limits of low-pass ∆-Σ modulators, the analysis and results for predicting stability in band-pass ∆-Σ modulators are detailed in [14] . A novel method based on this approach is given in [15] . It quantifies the maximum stability limits in higher-order ∆-Σ modulators for multiple-sinusoidal inputs. It can be observed that all
doi:10.1109/tim.2013.2273597 fatcat:aqagkwzimfdmvmdf6ezsu2wc5e