Iterative Design of L_p Digital Filters

The design of digital filters is a fundamental process in the context of digital signal processing. The purpose of this paper is to study the use of $\lp$ norms (for $2 < p < \infty$) as design criteria for digital filters, and to introduce a set of algorithms for the design of Finite (FIR) and Infinite (IIR) Impulse Response digital filters based on the Iterative Reweighted Least Squares (IRLS) algorithm. The proposed algorithms rely on the idea of breaking the $\lp$ filter design problem into a sequence of approximations rather than solving the original $\lp$ problem directly. It is shown that one can efficiently design filters that arbitrarily approximate a desired $\lp$ solution (for $2 < p < \infty$) including the commonly used $l_\infty$ (or minimax) design problem. A method to design filters with different norms in different bands is presented (allowing the user for better control of the signal and noise behavior per band). Among the main contributions of this work is a method for the design of {\it magnitude} $\lp$ IIR filters. Experimental results show that the algorithms in this work are robust and efficient, improving over traditional off-the-shelf optimization tools. The group of proposed algorithms form a flexible collection that offers robustness and efficiency for a wide variety of digital filter design applications.