This paper proposes a novel Nyquist-rate analog-to-digital (A/D) conversion algorithm which achieves exponential accuracy in the bit-rate despite using imperfect components. The proposed algorithm is based on a robust implementation of a beta-encoder with β = φ = (1 + √5)/2, the golden ratio. It was previously shown that beta-encoders can be implemented in such a way that their exponential accuracy is robust against threshold offsets in the quantizer element. This paper extends this result by allowing for imperfect analog multipliers with imprecise gain values as well. Furthermore, a formal computational model for algorithmic encoders and a general test bed for evaluating their robustness is proposed.

The quantization operation has been traditionally analyzed in signal processing and data conversion as a source of white noise although it is of deterministic nature. This model has become particularly inaccurate in ΣΔ modulation where the quantization resolution can be as low as 1-bit. For an important class of ΣΔ modulators with constant inputs, we build rigorous foundations to the quantization error analysis based on the spectral theory of unitary operators.

This paper analyzes mathematically the effect of quantizer threshold imperfection commonly encountered in the circuit implementation of analog-to-digital (A/D) converters such as pulse code modulation (PCM) and sigma-delta (SigmaDelta) modulation. SigmaDelta modulation, which is based on coarse quantization of oversampled (redundant) samples of a signal, enjoys a type of self-correction property for quantizer threshold errors (bias) that is not shared by PCM. Although "classical" SigmaDelta modulation is inferior to PCM in the rate-distortion sense, this robustness feature is believed to be one of the reasons why SigmaDelta modulation is preferred over PCM in A/D converters with imperfect quantizers. Motivated by these facts, other encoders are constructed in this paper that use redundancy to obtain a similar self-correction property, but that achieve higher order accuracy relative to bit rate compared to classical SigmaDelta. More precisely, two different types of encoders are introduced that exhibit exponential accuracy in the bit rate (in contrast to the polynomial-type accuracy of classical SigmaDelta) while possessing the self-correction property

Although the technique of sigma-delta (/spl Sigma//spl Delta/) modulation is well established in practice for performing high-resolution analog-to-digital (A/D) conversion, theoretical analysis of the error between the input signal and the reconstructed signal has remained partial. For modulators of order higher than 1, the only rigorous error analysis currently available that matches practical and numerical simulation results is only applicable to a very special configuration, namely, the standard and ideal k-bit k-loop /spl Sigma//spl Delta/ modulator. Moreover, the error measure involves averaging over time as well as possibly over the input value. At the second order, it is known in practice that the mean-squared error decays with the oversampling ratio /spl lambda/ at the rate O(/spl lambda//sup -5/). In this paper, we introduce two new fundamental results in the case of constant input signals. We first establish a framework of analysis that is applicable to all second-order modulators provided that the built-in quantizer has uniformly spaced output levels, and that the noise transfer function has its two zeros at the zero frequency. In particular, this includes the one-bit case, a rigorous and deterministic analysis of which is still not available. This generalization has been possible thanks to the discovery of the mathematical tiling property of the state variables of such modulators. The second aspect of our contribution is to perform an instantaneous error analysis that avoids infinite time averaging. Until now, only an O(/spl lambda//sup -4/) type error bound was known to hold in this setting. Under our generalized framework, we provide two types of squared-error estimates; one that is statistically averaged over the input and another that is valid for almost every input (in the sense of Lebesgue measure). In both cases, we improve the error bound to O(/spl lambda//sup -4.5/), up to a logarithmic factor, for a general class of modulators including some specific ones that are covered in this paper in detail. In the particular case of the standard and ideal two-bit double-loop configuration, our methods provide a (previously unavailable) instantaneous error bound of O(/spl lambda//sup -5/), again up to a logarithmic factor.

Sigma-delta modulation, a widely used method of analog-to-digital (A/D) signal conversion, is known to be robust to hardware imperfections, i.e., bit streams generated by slightly imprecise hardware components can be decoded comparably well. We formulate a model for robustness and give a rigorous analysis for single-loop sigma-delta modulation applied to constant signals (DC inputs) for N time cycles, with an arbitrary (small enough) initial condition u/sub o/, and a quantizer that may contain an offset error. The mean-square error (MSE) of any decoding scheme for this quantizer (with u/sub o/ and the offset error known) is bounded below by 1/96N/sup -3/. We also determine the asymptotically best possible MSE as N/spl rarr//spl infin/ for perfect decoding when u/sub o/=0 and u/sub o/= 1/2 . The robustness result is the upper bound that a triangular linear filter decoder (with both u/sub o/ and the offset error unknown) achieves an MSE of 40/3N/sup -3/. These results establish the known result that the O(1/N/sup 3/) decay of the MSE with N is optimal in the single-loop case, under weaker assumptions than previous analyses, and show that a suitable linear decoder is robust against offset error. These results are obtained using methods from number theory and Fourier analysis.

Two new design techniques for adaptive orthogonal block transforms based on vector quantization (VQ) codebooks are presented. Both techniques start from reference vectors that are adapted to the characteristics of the signal to be coded, while using different methods to create orthogonal bases. The resulting transforms represent a signal coding tool that stands between a pure VQ scheme on one extreme and signal-independent, fixed block transformation-like discrete cosine transform (DCT) on the other. The proposed technique has superior compaction performance as compared to DCT both in the rendition of details of the image and in the peak signal-to-noise ratio (PSNR) figures.