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It was previously shown that sigma-delta (/spl Sigma//spl Delta/) modulators of "asymptotic" type theoretically yield an equivalent feedforward system where the recursive nonlinear mechanisms are extracted from the feedback loop and reduced to a memoryless function. With time-varying inputs, we show in this paper, partially by mathematical derivations and partially by experiment, that this system is quasi-equivalent to the original modulator in a sense that we explain. This reduction of the nonlinear mechanisms should permit more refined modeling of the /spl Sigma//spl Delta/ errors in future research, with a better account of the original nonlinearities of asymptotic /spl Sigma//spl Delta/ modulation.

The theoretical error signal analysis of a sigma-delta (/spl Sigma//spl Delta/) modulator is a difficult problem due to the presence of a nonlinear operation (the amplitude quantization) in a feedback loop. In this paper, new deterministic knowledge on the transfer function of a /spl Sigma//spl Delta/ modulator is established, thanks to some recently observed properties of its state variables. For a large class of typical /spl Sigma//spl Delta/ modulators with constant inputs, the state variables appear to remain in a tile. We show what characteristics in a /spl Sigma//spl Delta/ modulator are specifically responsible for this property and give some initial proof of it. Under a constant input, the tiling phenomenon has as fundamental consequence that the output is a fixed and memoryless modulo function of n successive integrated versions of the input. This gives the theoretical knowledge that the modulator has an equivalent feedforward circuit expression. We give some immediate theoretical consequences on error analysis including the case of time-varying inputs.

The mean squared error (MSE) of an nth-order /spl Sigma//spl Delta/ modulator is classically understood to decay with the oversampling ratio M at the rate /spl alpha//M/sup 2n+1/. In this paper, we demonstrate by experimental evidence and basic analytical explanations that the accurate general MSE law is in fact in (/spl beta//M/sup 2n/)+(/spl gamma//M/sup 2n+1/) instead. The coefficient /spl beta/ is in general not zero, except when the input is rigorously constant, or when using the ideal standard nonoverloaded n-bit n-loop modulator. The term /spl beta//M/sup 2n/ is dominant starting from low values of M when using a one-bit quantizer.

We say that a /spl Sigma//spl Delta/ modulator is asymptotic when all the zeros of its noise transfer function are located at z = 1 (zero frequency). It is usually understood from the classical white-noise model that the output of a stable mth-order asymptotic modulator includes a baseband mean squared error (MSE) that decays with the oversampling ratio R in O(1/R/sup 2m+1/). While this is indeed always true with constant inputs, we show numerically that with time-varying inputs, this result is in general not true, except in the particular case of the standard nonoverloading multibit multiloop configuration. The general MSE decay rate is in fact in O (1/R/sup 2m/) instead. We give mixed analytical and experimental explanations for this phenomenon thanks to the notion of "centroid function" of a modulator. We show that, besides the standard multibit multiloop configuration, it is actually possible to force an mth-order /spl Sigma//spl Delta/ modulator to yield an MSE decay rate of O(1/R/sup 2m+1/) with time-varying inputs, but this necessitates the introduction of a new species of EA modulators that include additional nonlinear memoryless operations.

In this work we propose a systematic approach to the digital compensation for nonideal deterministic characteristics in a /spl Sigma//spl Delta/ modulator. It is based on inverting the complete nonlinear deterministic transfer function of the modulator. This operation leads us to a whole set of possible reconstruction estimates from which we systematically pick the geometric center as signal decoding scheme. This approach already proves effective in the particular case of static deviations in the built-in analog-to-digital converter and digital-to-analog converter of the modulator.

When considering a class of finite-dimensional input signals such as bandlimited signals in the discrete Fourier sense within a finite time window [O,T] of observation, we show that a single-loop /spl Sigma//spl Delta/ modulator behaves like the encoder of a vector quantizer. The intrinsic behavior of the modulator can be studied by analyzing the partition it generates in the input space. We show that this partition yields a particular structure that we call the "hyperplane wave structure". As one consequence to this analysis, it can be proved for the considered class of bandlimited signals that the mean squared error (MSE) of reconstruction cannot asymptotically decrease with the oversampling ratio R faster than O(R/sup -4/), regardless of the type of reconstruction used. We then generalize this analysis to n-loop /spl Sigma//spl Delta/ modulators and show that the MSE cannot decrease faster than O(R/sup -(2n+2)/).

The mean squared error (MSE) of an nth order /spl Sigma//spl Delta/ modulator is classically understood to decay with the oversampling ratio M at the rate /spl alpha//M/sup 2n+1/. In this paper, we show that the accurate general MSE law is in fact in /spl beta//M/sup 2n/ + /spl gamma//M/sup 2n+1/ instead. The coefficient /spl beta/ is in general not zero, except when the input is rigorously constant, or when using the ideal standard nonoverloaded n-bit n-loop modulator. The term /spl beta//M/sup 2n/ is dominant starting from low values of M when using a one-bit quantizer.

Research has been devoted to optimal signal reconstruction by /spl Sigma//spl Delta/ modulation. Using classical linear filtering, the reconstruction mean squared error (MSE) asymptotically decreases with the oversampling ratio R by R/sup -(2n+1)/, where n is the order of the modulator. Using a nonlinear reconstruction scheme, an improvement of the MSE asymptotic behavior by 3 dB per octave of R has been observed for time-varying, periodic and bandlimited input signals. This implies an MSE of /spl Oscr/(R/sup -(2n+2)/). We show for the multi-loop configuration of the /spl Sigma//spl Delta/ modulation and with the same kind of input signals that the asymptotic behavior of the reconstruction MSE cannot be less than /spl Oscr/(R/sup -(2n+2)/), including optimal reconstruction. >

We analyze the behavior of a single-loop /spl Sigma//spl Delta/ modulator using a vector quantization (VQ) approach. We extract the encoder-decoder structure typical of VQ, existing in a /spl Sigma//spl Delta/ modulator. The study of the encoding part gives the intrinsic behavior of the modulator. Bounds to the intrinsic performance of the /spl Sigma//spl Delta/ modulator can be derived through this approach in the oversampling situation, assuming periodicity of the bandlimited input signals. >

We give a general approach to the problem of digital inversion of nonlinear deterministic characteristics in a /spl Sigma//spl Delta/ modulator. This consists of going back to the elementary definition of a /spl Sigma//spl Delta/ modulator as a mapping. The difficulty of this approach is that this inversion leads to a whole set of solutions. However, it turns out that the output in an ideal /spl Sigma//spl Delta/ modulator coincides with the geometric center of this set. While this is no longer the case in the presence of nonlinear analog imperfections, a general scheme for their digital compensation can be defined by systematically providing as final output the true geometric center of the set. As a preliminary validation, we show the remarkable efficiency of this general correction scheme in the case of static ADC-DAC deviations.

The classical noise-shaping analysis provides a framework to invert any existing distortion in the transfer function of a /spl Sigma//spl Delta/ modulator, provided that the distortion has the form of a time-invariant and linear operation. We establish here a more general algebraic framework for the analysis of /spl Sigma//spl Delta/ modulation, which can explicitly take into account any nonlinear deterministic distortion. We show in this paper that this framework permits the algebraic inversion of nonlinear characteristics in the specific case of static ADC-DAC nonuniformity, which can be seen as a generalization of the linear noise-shaping inversion formula. From this algebraic inversion formula, the classical qualitative knowledge of the behavior of /spl Sigma//spl Delta/ modulation with regard to ADC-DAC nonuniformity can be reexplained in a concise and formal manner, with some new insights and results.

We present a synthesis of recent results obtained by various collaborators contributing to a new analysis framework for one-bit nth order /spl Sigma//spl Delta/ modulators. This framework leads to a better theoretical understanding of the intriguing MSE behavior of such modulators and new general principles for the design of nonlinear /spl Sigma//spl Delta/ modulators.