Iterative Reweighted Least Squares Design of Digital Filters

这项研究正在开发一种新的、明显更好的方法来设计各种数字滤波器。新方法基于一种称为迭代重加权最小二乘(IRLS)的逐次逼近算法。IRLS的一种形式，劳森算法，以前曾被使用过，但由于收敛速度慢且不一致而没有得到广泛应用。新方法的初步结果是，采用二阶IRLS和路径跟踪或同伦修改的形式，产生了一种稳健的、二次收敛的、高效的、通用的滤波器设计技术。该方法得到了具有1=p的广义L^p逼近，并允许在不同的频带内采用不同的准则来设计混合准则最优滤波器。新方法不仅解决了标准的滤波器设计问题，而且解决了许多Parks-McClellan Remez算法或加窗最小二乘法所不能解决的问题。在约束L p准则、复近似和二维设计上的初步结果显示了很好的前景。稳健收敛对于任何迭代算法的实际成功都是至关重要的。我们认为，我们在这个问题上有潜在的突破。本研究对更一般的优化准则的收敛进行了理论分析，开发了一个实用的自适应滤波器设计程序，并以实际工程师可以理解的形式展示了它。该算法将被应用于三个滤波器设计问题：a)约束最小二乘问题，以及更一般的约束L^p逼近问题。当人们使用切比雪夫或窗户设计时，这往往是他们真正想要的。B)复逼近的重要问题，它是均衡自适应滤波器设计中的重要问题。C)多维滤波器的设计和图像处理问题。这一新方法显示出极好的前景，并可能对现有技术进行重大改进。

This research is developing a new and significantly better method for the design of a wide variety of digital filters. The new method is based on a successive approximation algorithm called Iteratively Reweighted Least Squares (IRLS). One form of IRLS, Lawson's algorithm, has been used before but not extensively because of slow and inconsistent convergence. Initial results with the new method using a second order IRLS with a form of path- following or homotopy modification have produced a robust, quadratically converging, efficient, versatile filter design technique. The method finds general L^p approximations with 1=p. It also allows different criteria in different bands for the design of mixed criteria optimal filters. The new method solves, not only the standard filter design problems, but many that the Parks- McClellan Remez algorithm or windowed least wquares will not. Initial results on designing with constrained L^p criteria, complex approximation, and two dimensions show great promise. Robust convergence is crucial for the practical success of any iterative algorithm. We feel that we have a potential breakthrough on that issue. This research is doing a theoretical analysis of the convergence for the more general optimization criteria, developing a practical adaptive filter design program, and presenting it in a form accessible to the practicing engineer. The algorithm will be applied to three filter design problems: a) the constrained least-squares and, more generally, the constrained L^p approximation problem. This is often what people really want when they use a straight Chebyshev or window design. b) The important problems of complex approximation which is important in equalization adaptive filter design. c) The multidimensional filter design and image processing problems. This new approach shows excellent promise and a possible great improvement over present techniques.