##
**Approximating a bandlimited function using very coarsely quantized data: a family of stable sigma-delta modulators of arbitrary order.**
*(English)*
Zbl 1058.94004

In order to perform digital signal processing, an analog signal such as an audio signal is first converted into a digital signal by an analog/digital (A/D) conversion. The digitization of an audio signal rests on two pillars: sampling and quantization. It is standard to model audio signals by bandlimited functions, i.e. functions \(f\in L^2(\mathbb{R})\) whose Fourier transform vanishes outside an interval \([-\Omega, \Omega]\) with \(\Omega\) being the bandwidth. For simplicity of discussion, one in general normalizes \(\Omega=\pi\). By sampling a bandlimited function \(f\) with bandwidth \(\pi\), for \(\lambda>1\), one gets the samples \(\{ f(\frac{n}{\lambda})\}_{n\in \mathbb Z}\). Let \(g\) be a function such that \(\widehat g\in C^\infty(\mathbb{R})\), \(\widehat g(\xi)=\frac{1}{\sqrt{2\pi}}\) for \(| \xi| \leq \pi\), and \(\widehat g(\xi)=0\) for \(| \xi| > \lambda \pi\). Then one can exactly recover the original function \(f\) from its samples \(\{ f(\frac{n}{\lambda})\}_{n\in \mathbb Z}\) by the following sampling formula:
\[
f(t)=\frac{1}{\lambda} \sum_{n\in \mathbb{Z}} f\left(\frac{n}{\lambda}\right) g\left(t-\frac{n}{\lambda}\right).
\]
In the quantization step, the sample \(f(\frac{n}{\lambda})\), which is a real number, is replaced by a discrete representation. As discussed in this paper, the first order sigma-delta quantization scheme is a robust scheme to achieve this goal:
\[
u_n=u_{n-1}+f\biggl(\frac{n}{\lambda}\biggr)-q_n^\lambda
\]

\[ q_n^\lambda=\text{sign} \Biggl(u_{n-1}+f\biggl (\frac{n}{\lambda}\biggr)\Biggr), \] where \(u_n\) is the internal state variable and the sample \(f(\frac{n}{\lambda})\) is replaced by the quantized 1-bit number \(q_n^\lambda \in \{-1,1\}\). In this paper, the authors establish the fundamental result on the approximation error of the first order sigma-delta quantization scheme by showing that \[ | e(f, f_{q_n^\lambda})(t)| \leq \frac{1}{\lambda}\| g'\| _{L^1} \quad \text{with} \quad e(f, f_{q_n^\lambda})(t)=f(t)-\frac{1}{\lambda} \sum_{n\in \mathbb Z} q_n^\lambda g \biggl(t-\frac{n}{\lambda}\biggr). \] Building on their study of the first order sigma-delta quantization scheme, the authors propose a family of innovative higher order sigma-delta quantization schemes in this paper. For the \(k\)th order sigma-delta quantization scheme proposed in this paper, the authors show that their scheme is stable (i.e. the internal stable variables are uniformly bounded) and moreover, they prove that the approximation error \(e(f, f_{q_n^\lambda})\) decays like \(O(\lambda^{-k})\), as \(\lambda\to\infty\). This paper not only lays down a rigorous mathematical foundation for analyzing sigma-delta quantization schemes, but it also proposes a family of new higher order sigma-delta quantization schemes, which are of interest in both mathematics and electrical engineering. Many advantages of sigma-delta quantization schemes such as robustness to imperfection in circuits are discussed in this paper. Many challenging open problems, arising from their study of stable sigma-delta quantization schemes, are also given herein.

\[ q_n^\lambda=\text{sign} \Biggl(u_{n-1}+f\biggl (\frac{n}{\lambda}\biggr)\Biggr), \] where \(u_n\) is the internal state variable and the sample \(f(\frac{n}{\lambda})\) is replaced by the quantized 1-bit number \(q_n^\lambda \in \{-1,1\}\). In this paper, the authors establish the fundamental result on the approximation error of the first order sigma-delta quantization scheme by showing that \[ | e(f, f_{q_n^\lambda})(t)| \leq \frac{1}{\lambda}\| g'\| _{L^1} \quad \text{with} \quad e(f, f_{q_n^\lambda})(t)=f(t)-\frac{1}{\lambda} \sum_{n\in \mathbb Z} q_n^\lambda g \biggl(t-\frac{n}{\lambda}\biggr). \] Building on their study of the first order sigma-delta quantization scheme, the authors propose a family of innovative higher order sigma-delta quantization schemes in this paper. For the \(k\)th order sigma-delta quantization scheme proposed in this paper, the authors show that their scheme is stable (i.e. the internal stable variables are uniformly bounded) and moreover, they prove that the approximation error \(e(f, f_{q_n^\lambda})\) decays like \(O(\lambda^{-k})\), as \(\lambda\to\infty\). This paper not only lays down a rigorous mathematical foundation for analyzing sigma-delta quantization schemes, but it also proposes a family of new higher order sigma-delta quantization schemes, which are of interest in both mathematics and electrical engineering. Many advantages of sigma-delta quantization schemes such as robustness to imperfection in circuits are discussed in this paper. Many challenging open problems, arising from their study of stable sigma-delta quantization schemes, are also given herein.

Reviewer: Bin Han (Edmonton)