An analysis of nonlinear behavior in delta - sigma modulators

TL;DR: This paper introduces a new method of analysis for deltasigma modulators based on modeling the nonlinear quantizer with a linearized gain, obtained by minimizing a mean-square-error criterion, followed by an additive noise source representing distortion components.

Abstract: This paper introduces a new method of analysis for deltasigma modulators based on modeling the nonlinear quantizer with a linearized 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 phenomena are explained by the nonlinear analysis of this paper. The analysis is carried out for both dc and sinusoidal excitations.

Summary

1.2 Background on Methodology

• Since the modulator response is now seen to consist of both a signal related to the input to the modulator and random distortion components, the response of the nonlinearity to multiple inputs must be considered.
• This technique has been extensively studied in [7] .
• Using this method, a nonlinear feedback system can be represented as two interlocked linear systems.
• Thus, the nonlinearity is modeled as two linearized gains, one for the input signal to the system and one for the random distortion components which are assumed to have a Gaussian pdf.

2.Delta-Sigma Modulation

• This circuit can be implemented with a differential integrator, a comparator, and a flip-flop or sampleand-hold amplifier.
• The output of this system is a bit stream whose pulse density is proportional to the applied input signal amplitude.
• In previous work [4, 5, 3] , this system has been modeled by replacing the nonlinear element with a unity-gain linear element followed by an additive noise process.
• This simple model has proven to be inadequate for the accurate analysis of higher-order modulator stability since it does not reflect the dependence of the nonlinear system on the input signal to the nonlinearity.
• Since the sample-and-hold circuit and the integrator in the inner loop are cascaded with no sampler in between, they must be combined prior to obtaining the z-Transform.

2) z-l

• Thez-domain representation, however, is an approximation to the actual behavior of the continuous system.
• The identification problem is to determine K x and K y such that the mean square error in modeling the nonlinear element using the linearized gains is minimized.
• An important consequence of using the above linear gains is that the error, e(t), becomes uncorrelated with y(t).

E{y(t)e(t)}

• In the application of the above modeling technique to the field of nonlinear control, the error e (t) is usually neglected.
• This is based on the assumption that the error is filtered by the plant after feedback and forms a negligible part of the input signal to the nonlinearity [7] .
• Thus the authors replace the nonlinear quantizer in the modulator by the two linearized gains followed by an additive noise source representing the error.
• Also" the linear gains K; and K n were calculated from the simulation using time averages based on (3.6) and (3.7).

3.2 Sinusoidal Input and Nonlinearity Modeling

• Consider the case where the input to the nonlinear quantizer consists of a sinusoid, x(t), and a Gaussian signal, y(t).
• Based on the previous discussion, the authors can associate the linear gains K x and K y with the sinusoidal and Gaussian inputs.
• By replacing the nonlinear quantizer by the linearized gains followed by an additive noise source, rut), the Delta-Sigma Modulator can be separated into two interlocked linear systems as illustrated in Fig. 7 and 8.
• In one system, the input forcing function is the sinusoid x(k).
• A few words are in order about the Confluent Hypergeometric Functions.

Exact Numerical Calculation of Signal to Noise Ratio

• Based on the analytical results presented in section 3, the authors can compute the signal-to-noise ratio as a function of input-signal amplitude for both DC and sinusoidal inputs.
• Once K; and (1; are known, then the noise spectra Snn (f) can be numerically integrated over the baseband to yield the in-band noise component (J' ;8.

2(J";B

• Figure 16 shows the calculated SNR of second and third-order modulators plotted against the DC amplitude.
• The SNR decreases although based on (3.21) the additive noise variance decreases with increasing amplitude!.
• As was pointed out in section 3, the linear gain K; decreases as the input amplitude increases.
• Also, the variance of the sinusoidal component at the quantizer input also increases, surpassing the noise variance (J' ' 1.
• This analytical observation, which has been observed in experimental circuits in [5] , has important consequences on the design of actual circuits where the dynamic range of signals is limited.

5. Stability Analysis

• One of the major problems associated with higher-order Delta-Sigma Modulators is their stability.
• In this section the authors present a stability analysis of the second and third-order modulators.
• Expressions are derived which give bounds on the loop gains for stable operation.
• Furthermore, the authors show that a stable third-order modulator will become unstable as the input-signal amplitude is increased beyond a certain threshold.

5.1 Double Loop System

• This is not the case for the third-order system as will be shown below.
• Therefore, the only degree of freedom is the choice of (Xl.
• The frequency of oscillation for the second solution is determined by the following equation, We observe that, as the hneanzed gain K decreases, the authors approach the second solution given by (5.9) ~d (5:10).the authors.
• There is an amplitude region, however, for which the modulator IS stable.

6. Conclusions

• A new method of analysis for Delta-Sigma Modulators based on modeling the nonlinear quantizer with minimum-mean-square error linearized gains followed by an additive noise source representing distortion components is described.
• The effects of increasing input-signal amplitude on the shaping of the noise spectra and signal-to-noise ratio have been presented.
• The signal-to-noise ratio of the modulator is calculated directly as a function of input-signal amplitude using analytical methods.
• The analysis is carried out for both DC and sinusoidal excitations.
• Moreover, regions of stability for second and third-order loops have been obtained, including bounds on the loop gains for stable operation.

Figures

Fig. 6. Minimum Mean Square Error Linearized Gain Modeling of Nonlinearity

Fig. 23 Nyquist Plot 3rd Order Modulator «11=0.1, (12=0.1, (13=1.0)

Fig. 16. Calculated SNR for 2nd and 3rd Order Modulator, ~: =256

Fig. 21. Calculated Variances at Quantizer Input, 2nd Order Sinusoidal Input

Fig. 22 Nyquist Plot 2nd Order Modulator (Q(=O.5, Q2=1.0)