An Optimal Family of Exponentially Accurate One-Bit Sigma-Delta Quantization Schemes

Sigma-Delta modulation is a popular method for analog-to-digital conversion of bandlimited signals that employs coarse quantization coupled with oversampling. The standard mathematical model for the error analysis of the method measures the performance of a given scheme by the rate at which the associated reconstruction error decays as a function of the oversampling ratio $\lambda$. It was recently shown that exponential accuracy of the form $O(2^{-r\lambda})$ can be achieved by appropriate one-bit Sigma-Delta modulation schemes. By general information-entropy arguments $r$ must be less than 1. The current best known value for $r$ is approximately 0.088. The schemes that were designed to achieve this accuracy employ the "greedy" quantization rule coupled with feedback filters that fall into a class we call "minimally supported". In this paper, we study the minimization problem that corresponds to optimizing the error decay rate for this class of feedback filters. We solve a relaxed version of this problem exactly and provide explicit asymptotics of the solutions. From these relaxed solutions, we find asymptotically optimal solutions of the original problem, which improve the best known exponential error decay rate to $r \approx 0.102$. Our method draws from the theory of orthogonal polynomials; in particular, it relates the optimal filters to the zero sets of Chebyshev polynomials of the second kind.Comment: 35 pages, 3 figure

Uniform Approximation by Polynomials with Integer Coefficients via the Bernstein Lattice

Let $\mathscr{C}_\mathbb{Z}([0,1])$ be the metric space of real-valued continuous functions on $[0,1]$ with integer values at $0$ and $1$, equipped with the uniform (supremum) metric $d_\infty$. It is a classical theorem in approximation theory that the ring $\mathbb{Z}[X]$ of polynomials with integer coefficients, when considered as a set of functions on $[0,1]$, is dense in $\mathscr{C}_\mathbb{Z}([0,1])$. In this paper, we offer a strengthening of this result by identifying a substantially small subset $\bigcup_n \mathscr{B}_n$ of $\mathbb{Z}[X]$ which is still dense in $\mathscr{C}_\mathbb{Z}([0,1])$. Here $\mathscr{B}_n$, which we call the ``Bernstein lattice,'' is the lattice generated by the polynomials $$p_{n,k}(x) := \binom{n}{k} x^k(1-x)^{n-k}, ~~k=0,\dots,n.$$ Quantitatively, we show that for any $f \in \mathscr{C}_\mathbb{Z}([0,1])$, $$d_\infty(f, \mathscr{B}_n) \leq \frac{9}{4} \omega_f(n^{-1/3}) + 2 n^{-1/3}, ~~n \geq 1,$$ where $\omega_f$ stands for the modulus of continuity of $f$. We also offer a more general bound which can be optimized to yield better decay of approximation error for specific classes of continuous functions.Comment: 10 pages; presented at CANT 23 (Combinatorial and Additive Number Theory 2023

Ergodic dynamics in sigma–delta quantization: tiling invariant sets and spectral analysis of error

AbstractThis paper has two themes that are intertwined. The first is the dynamics of certain piecewise affine maps on Rm that arise from a class of analog-to-digital conversion methods called ΣΔ (sigma–delta) quantization. The second is the analysis of reconstruction error associated with each such method.ΣΔ quantization generates approximate representations of functions by sequences that lie in a restricted set of discrete values. These are special sequences in that their local averages track the function values closely, thus enabling simple convolutional reconstruction. In this paper, we are concerned with the approximation of constant functions only, a basic case that presents surprisingly complex behavior. An mth order ΣΔ scheme with input x can be translated into a dynamical system that produces a discrete-valued sequence (in particular, a 0–1 sequence) q as its output. When the schemes are stable, we show that the underlying piecewise affine maps possess invariant sets that tile Rm up to a finite multiplicity. When this multiplicity is one (the single-tile case), the dynamics within the tile is isomorphic to that of a generalized skew translation on Tm.The value of x can be approximated using any consecutive M elements in q with increasing accuracy in M. We show that the asymptotical behavior of reconstruction error depends on the regularity of the invariant sets, the order m, and some arithmetic properties of x. We determine the behavior in a number of cases of practical interest and provide good upper bounds in some other cases when exact analysis is not yet available

Sigma Delta Quantization for Compressed Sensing

Abstract—Recent results make it clear that the compressed sensing paradigm can be used effectively for dimension reduction. On the other hand, the literature on quantization of compressed sensing measurements is relatively sparse, and mainly focuses on pulse-code-modulation (PCM) type schemes where each measurement is quantized independently using a uniform quantizer, say, of step size δ. The robust recovery result of Candès et al. and Donoho guarantees that in this case, under certain generic conditions on the measurement matrix such as the restricted isometry property, ℓ 1 recovery yields an approximation of the original sparse signal with an accuracy of O(δ). In this paper, we propose sigma-delta quantization as a more effective alternative to PCM in the compressed sensing setting. We show that if we use an rth order sigma-delta scheme to quantize m compressed sensing measurements of a k-sparse signal in R N, the reconstruction accuracy can be improved by a factor of (m/k) (r−1/2)α for any 0 < α < 1 if m �r k(log N) 1/(1−α) (with high probability on the measurement matrix). This is achieved by employing an alternative recovery method via rth-order Sobolev dual frames. I