ITR: Collaborative Research: Accurate Representations of Signals in a Coarse-Grained Environment

其目的是为高过采样环境下信号的粗量化提供更好的数学框架和替代设计方法。中心主题是音频信号模数转换中的“Sigma-Delta量化”，以及图像数字半色调中的“误差扩散”。在这两种情况下，给定的目标模拟信号由明智地选择的一位(或几位)序列表示，该序列在合适的低通子空间中近似于信号。为此，存在多种不同的方案，每种方案对应于不同的实现和近似顺序。作为我们的主要数学目标，我们期望更好地理解现有方案的误差衰减规律，并通过新的设计来改进这些方案。在算法方面，我们的目标是将这种分析扩展到具有计算和其他实现限制的设置。数字信号和数据集可以以极高的精度和速度被处理、存储和检索这一事实对从模拟世界到模拟世界的转换过程提出了很高的要求。然而，在转换过程中使用的设备(例如音频应用中的模数和数模转换电路以及图像再现中的打印机)是必需的模拟设备，其物理限制乍看起来与这些精度要求相冲突。为了解决这个问题，工程师们根据经验开发了特殊的信号处理技术，导致了与标准十进制或二进制表示法截然不同的替代信号和数字表示法。典型的技术利用模拟设备在时间或空间上非常密集地采样的高精度性能来补偿这些设备的幅度精度的不足。有趣的是，虽然这些经验方案已经被证明是有效的，并且已经在消费产品中使用了很长一段时间，但相应的信号处理框架几乎没有数学支持。我们的目标是更详细地研究这些方案，目的是改进数学理论，提出比目前使用的方案更好的变体，并确定更广泛的应用范围。

The aim is to provide a better mathematical framework andalternative design methodologies for coarse quantization of signals ina highly oversampled setting. The central themes is "sigma-deltaquantization" in analog-to-digital conversion of audio signals and itscounterpart "error-diffusion" in digital halftoning of images. In bothcases, a given target analog signal is represented by a judiciouslychosen one-bit (or few-bit) sequence which approximates the signal in asuitable low-pass subspace. A wide range of different schemes exist forthis purpose, each corresponding to a different implementation andapproximation order. As our primary mathematical objective, we expectto better understand the law of error decay of the existing schemes aswell as to improve on these by new designs. On the algorithmic side, weaim to extend this analysis to settings with computational and otherimplementational constraints.The fact that digital signals and data sets can be processed, storedand retrieved with great precision and speed places high demands ofaccuracy on the conversion process from and to the analog world.However, the devices used in the translation process (such as in thecase of analog-to-digital and digital-to-analog conversion circuits inaudio applications, and printers in image reproduction) are ofnecessity analog devices, which have physical limitations that, atfirst sight, conflict with those accuracy demands. To cope with thisproblem, engineers have empirically developed special signal processingtechniques leading to alternative signal and number representationsthat are quite different from standard decimal or binaryrepresentations. Typical techniques take advantage of the highlyaccurate performance of the analog devices in sampling very densely intime or space to compensate for the lack of amplitude precision ofthose devices. Interestingly, while these empirical schemes have provedefficient and have been used in consumer products for a long time, thecorresponding framework of signal processing has little mathematicalsupport. We aim to study these schemes in more detail, with thegoals of improving the mathematical theory as well as proposingvariants that outperform those used presently, and to identify a widerrange of applications.