Digital Signal Processing with Optimal Analog Performance via Sampled-data Control Theory

本研究项目旨在发展一种新的信号处理理论和方法，利用现代采样数据控制理论优化模拟性能。这只能通过采样数据控制理论实现，因为该理论可以考虑采样间行为。这与信号处理界流行的香农采样定理形成了鲜明的对比。在这种传统方法中，人们通常会假设原始模拟信号必须是完全带限的。这种人为的假设在现实中永远不会实现。此外，它还会在许多信号中产生不必要的超调。目前的研究已经成功地推导出了一种导致次优模拟性能的滤波器，并保证了它在声音处理方面的优越结果。这在实验室实验中也得到了保证，我们在带有DSP板的PC上实现了最佳数字滤波器。在这种DSP电路中实现的构造的数模转换器轻松优于目前可用的高端CD播放机。我们还获得了一种新的最优采样率转换器的设计方法，以及一种数字通信系统的最优滤波器的设计方法。由于这些应用中的设计问题往往是高维的，因此也发展了一种新的数值设计方法。研究表明，一种凸优化问题，即所谓的切面法，在这类问题中是非常有效的。最后指出，将这种滤波器应用于图像处理是未来研究的一个前景方向。

This research project intends to develop a new theory and method of signal processing that optimize analog performance via modern sampled-data control theory. This is made possible only via sampled-data control theory in that this theory can take intersampling behavior into account. This is in marked contrast to the Shannon sampling theorem which is so popular in signal processing community. In such a conventional approach, one usually resorts to the assumption that the original analog signals must be fully band-limited. This artificial assumption is never fulfilled in reality. Furthermore, it has an added artifact that it produces an unnecessary overshoot in many signals. The present research has successfully derived a filter that gives rise to a suboptimal analog performance, and it has been guaranteed that it gives a superior result in sound processing. This has been also assured in a lab experiment in which we implemented an optimal digital filter to a PC with DSP board. The constructed DA converter realized in such a DSP circuit easily outperformed the current high-end CD players available at the moment.We have also obtained a new design method for optimal sample-rate converters, and also a design method for optimal filters for digital communication systems. A new numerical design method has also been developed since the design problem in these applications tend to be high-dimensional. It has been shown that a convex optimization problem, the so-called cutting-plane method, is very effective in such problems.Finally, it has been observed that application to image processing of such filters can be a prospective direction for future research.