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

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 L2L2 approximation error for the truncated inverse filters. The approach to determine the minimum inverse model-order subject to the maximum allowable L2L2 approximation error is also in demand. In this paper, we present two L2L2 approximation error measures and the two corresponding optimal finite-support approximates. Also, we derive the explicit L2L2 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 L2L2 approximation error. Our newly derived L2L2 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.