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

The Continuous Skolem-Pisot Problem: On the Complexity of Reachability for Linear Ordinary Differential Equations

We study decidability and complexity questions related to a continuous analogue of the Skolem-Pisot problem concerning the zeros and nonnegativity of a linear recurrent sequence. In particular, we show that the continuous version of the nonnegativity problem is NP-hard in general and we show that the presence of a zero is decidable for several subcases, including instances of depth two or less, although the decidability in general is left open. The problems may also be stated as reachability problems related to real zeros of exponential polynomials or solutions to initial value problems of linear differential equations, which are interesting problems in their own right.Comment: 14 pages, no figur

A Spectral Method for Elliptic Equations: The Neumann Problem

Let $\Omega$ be an open, simply connected, and bounded region in $\mathbb{R}^{d}$, $d\geq2$, and assume its boundary $\partial\Omega$ is smooth. Consider solving an elliptic partial differential equation $-\Delta u+\gamma u=f$ over $\Omega$ with a Neumann boundary condition. The problem is converted to an equivalent elliptic problem over the unit ball $B$, and then a spectral Galerkin method is used to create a convergent sequence of multivariate polynomials $u_{n}$ of degree $\leq n$ that is convergent to $u$. The transformation from $\Omega$ to $B$ requires a special analytical calculation for its implementation. With sufficiently smooth problem parameters, the method is shown to be rapidly convergent. For $u\in C^{\infty}(\overline{\Omega})$ and assuming $\partial\Omega$ is a $C^{\infty}$ boundary, the convergence of $\Vert u-u_{n}\Vert_{H^{1}}$ to zero is faster than any power of $1/n$. Numerical examples in $\mathbb{R}^{2}$ and $\mathbb{R}^{3}$ show experimentally an exponential rate of convergence.Comment: 23 pages, 11 figure

Approximation with sums of exponentials in Lp[0, ∞)

AbstractWe consider the problem of approximating a given f from Lp [0, ∞) by means of the family Vn(S) of exponential sums; Vn(S) denotes the set of all possible solutions of all possible nth order linear homogeneous differential equations with constant coefficients for which the roots of the corresponding characteristic polynomials all lie in the set S. We establish the existence of best approximations, show that the distance from a given f to Vn(S) decreases to zero as n becomes infinite, and characterize such best approximations with a first-order necessary condition. In so doing we extend previously known results that apply in Lp[0, 1]

Sparsity, Randomness and Convexity in Applied Algebraic Geometry

In this dissertation we study three problems in applied algebraic geometry. The first problem is to construct an algorithmically efficient approximation to the real part of the zero set of an exponential sum. We construct such a polyhedral approximation using techniques from tropical geometry. We prove precise distance bounds between our polyhedral approximation and the real part of the zero set. Our bounds depend on the number of terms of the exponential sum and the minimal distance between the exponents. Despite the computational hardness of the membership problem for the real part of the zero set, we prove that our polyhedral approximation can be computed by linear programing on the real BSS machine. The second problem is to study the ratio of sums of squares polynomials inside the set of nonnegative polynomials. Our focus is on the effect of fixed monomial structure to the ratio of these two sets. We study this problem quantitatively by combining convex geometry and algebra. Some of our methods work for arbitrary Newton polytopes; however our main theorem is stated for multi-homogenous polynomials. Our main theorem provides quantitative versions of some known algebraic facts, and also refines earlier quantitative results. The third problem is to study the condition number of polynomial systems ‘on average’. Condition number is a vital invariant of polynomial systems which controls their computational complexity. We analyze the condition number of random polynomial systems for a broad family of distributions. Our work shows that earlier results derived for the polynomial systems with real Gaussian independent random coefficients can be extended to the broader family of sub-Gaussian random variables allowing dependencies. Our results are near optimal for overdetermined systems but there is room for improvement in the case of square systems of random polynomials. The main idea binding our three problems is to observe structure and randomness phenomenon in the space of polynomials. We used combinatorial algebraic geometry to observe the ‘structure’ and convex geometric analysis to understand the ‘randomness’. We believe results presented in this dissertation are just the first steps of the interaction between these two fields