Theories, Analysis, and Bounds of the Finite-Support Approximation for the Inverses of Mixing-Phase FIR Systems

Inverse-system approximation using finite-impulse responses (FIR) is essential to a broad area of signal-processing applications. The conventional Wiener filtering techniques based on the least-square approach cannot provide an analytical framework simultaneously governing two crucial problems, namely, the selection of model order and the evaluation of asymptotical error bounds. In fact, the square approximation error induced from the FIR realization of a linear time-invariant system is quite complicated, specifically for those system transfer functions possessing repeated zeros with large multiplicities. Therefore, in this paper, we establish an isomorphism to characterize the
z
-transform pairs. In this mathematical paradigm, we elaborate the problem of approximating an inverse system or filter with an infinite number of coefficients by an FIR filter and derive the new
L

₁
and
L

₂
approximation-error bounds between the actual inverse filter and the corresponding approximated FIR. Our new theories, analysis, and bounds can be utilized to quantify the appropriate model order for the inverse-system approximation that is often needed for signal processing, control, communications, etc.