Theoretical analysis on the finite-support approximation for the mixing-phase FIR systems

Abstract

The inverse system approximation using the finite impulse responses (FIR) and the corresponding model-order determination are essential to a broad area of science and technology utilizing signal processing. To the best of our knowledge, there exists no explicit formulation of the exact L 2 approximation error for the truncated inverse filters. The approach to determine the minimum inverse model-order subject to the maximum allowable L 2 approximation error is also in demand. In this paper, we present two L 2 approximation error measures and the two corresponding optimal finite-support approximates. Also, we derive the explicit L 2 approximation error functions with respect to roots, multiplicities and model orders for these two kinds of approximates. Then, we propose a new algorithm to determine the minimum total model order of the appropriate truncated inverse filter to achieve a specified L 2 approximation error. Our newly derived L 2 approximation error evaluation method can be employed for signal processing, telecommunication, control systems involving the inverse filtering in the future. Besides, our novel model-order determination algorithm can be utilized for efficient dynamic memory allocation in a wide variety of applications since such a minimum total model order is proportional to the memory usage for any inverse filter implementation.