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

本研究计划旨在发展一种新的信号处理理论和方法，通过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来实现，因为该理论可以考虑互采样行为。这与在信号处理领域非常流行的Shannon采样定理形成了鲜明的对比。在这样的传统方法中，人们通常诉诸于原始模拟信号必须是完全带限的假设。这种人为的假设在现实中永远不会实现。此外，它还有一个附加的伪像，即在许多信号中产生不必要的过冲。目前的研究已经成功地得出了一个滤波器，产生了一个次优的模拟性能，它已经保证，它给出了一个上级的结果，在声音处理。这一点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。用这种DSP电路实现的DA转换器的性能明显优于目前的高端CD播放器，同时也为最佳采样率转换器的设计提供了一种新的方法，为数字通信系统中最佳滤波器的设计提供了一种新的方法。由于这些应用中的设计问题往往是高维的，一种新的数值设计方法也被开发出来。结果表明，凸优化问题，即所谓的割平面方法，在这类问题中是非常有效的。最后指出，将这类滤波器应用于图像处理是一个有前景的研究方向。

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.