Quantization noise in single-loop sigma-delta modulation with sinusoidal inputs

An exact nonlinear difference equation is derived and solved for a simple sigma-delta modulator consisting of a discrete-time integrator and a binary quantizer inside a single feedback loop and an arbitrary input signal. It is shown that the system can be represented as an affine operation (discrete-time integration of a biased input) followed by a memoryless nonlinearity. An extension of the transform method for the analysis of nonlinear systems is applied to obtain formulas for first- and second-order time-average moments of the binary quantization noise, including the sample mean, energy, and autocorrelation. The results are applied to the special case of a sinusoidal input signal to evaluate these time averages and the power spectrum. In the limit of large oversampling ratios, the marginal moments behave as if the quantization noise had a uniform distribution. The spectrum is neither white nor continuous, however, even in the limit of large oversampling ratios. >

[1]  Robert M. Gray,et al.  Two-stage sigma-delta modulation , 1990, IEEE Trans. Acoust. Speech Signal Process..

[2]  Karl Petersen.  Ergodic Theory , 1983 .

[3]  Y. Katznelson.  An Introduction to Harmonic Analysis: Interpolation of Linear Operators , 1968 .

[4]  Atsushi Iwata,et al.  VLSI- A to D and D to A converters with multi-stage noise shaping modulators , 1986, ICASSP '86. IEEE International Conference on Acoustics, Speech, and Signal Processing.

[5]  A. R. Bergen,et al.  Justification of the Describing Function Method , 1971 .

[6]  J. Candy,et al.  The Structure of Quantization Noise from Sigma-Delta Modulation , 1981 .

[7]  Salomon Bochner,et al.  Almost periodic functions in groups. II , 1935 .

[8]  Robert M. Gray,et al.  Spectral analysis of quantization noise in a single-loop sigma-delta modulator with DC input , 1989, IEEE Trans. Commun..

[9]  W. Root,et al.  An introduction to the theory of random signals and noise , 1958 .

[10]  James C. Candy,et al.  A Use of Double Integration in Sigma Delta Modulation , 1985, IEEE Trans. Commun..

[11]  Robert M. Gray,et al.  Oversampled Sigma-Delta Modulation , 1987, IEEE Trans. Commun..

[12]  James C. Candy,et al.  Using Triangularly Weighted Interpolation to Get 13-Bit PCM from a Sigma-Delta Modulator , 1976, IEEE Trans. Commun..

[13]  Hiroshi Inose,et al.  A unity bit coding method by negative feedback , 1963 .

[14]  S. Rice.  Mathematical analysis of random noise , 1944 .

[15]  James C. Candy,et al.  A Use of Limit Cycle Oscillations to Obtain Robust Analog-to-Digital Converters , 1974, IEEE Trans. Commun..

[16]  Derek P. Atherton,et al.  Stability of nonlinear systems , 1981 .

[17]  John J. Paulos,et al.  An analysis of nonlinear behavior in delta-sigma modulators , 1987 .

[18]  James C. Candy,et al.  Decimation for Sigma Delta Modulation , 1986, IEEE Trans. Commun..

[19]  S. Bochner,et al.  Beiträge zur Theorie der fastperiodischen Funktionen , 1927 .

[20]  G. A. Watson.  A treatise on the theory of Bessel functions , 1944 .

[21]  Harvey Cohn,et al.  Almost Periodic Functions , 1947 .

[22]  J. Iwersen.  Calculated quantizing noise of single-integration delta-modulation coders , 1969 .

[23]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[24]  Austin Blaquière,et al.  Nonlinear System Analysis , 1966 .

[25]  G. Siouris,et al.  Nonlinear Control Engineering , 1977, IEEE Transactions on Systems, Man, and Cybernetics.

[26]  Russell Lyons,et al.  Fourier-Stieltjes Coefficients and Asymptotic Distribution Modulo 1 , 1985 .

[27]  H. S. Carslaw,et al.  Introduction to the Theory of Fourier's Series and Integrals , 1921, Nature.