In acoustic multi-channel equalization techniques, such as complete multi-channel equalization based on the\nmultiple-input/output inverse theorem (MINT), relaxed multi-channel least-squares (RMCLS), and partial multi-channel\nequalization based on MINT (PMINT), the length of the reshaping filters is generally chosen such that perfect\ndereverberation can be achieved for perfectly estimated room impulse responses (RIRs). However, since in practice\nthe available RIRs typically differ from the true RIRs, this reshaping filter length may not be optimal. This paper\nprovides a mathematical analysis of the robustness increase of equalization techniques against RIR perturbations\nwhen using a shorter reshaping filter length than conventionally used. Based on the condition number of the\n(weighted) convolution matrix of the RIRs, a mathematical relationship between the reshaping filter length and the\nrobustness against RIR perturbations is established. It is shown that shorter reshaping filters than conventionally used\nyield a smaller condition number, i.e., a higher robustness against RIR perturbations. In addition, we propose an\nautomatic non-intrusive procedure for determining the reshaping filter length based on the L-curve. Simulation\nresults confirm that using a shorter reshaping filter length than conventionally used yields a significant increase in\nrobustness against RIR perturbations for MINT, RMCLS, and PMINT. Furthermore, it is shown that PMINT using an\noptimal intrusively determined reshaping filter length outperforms all other considered techniques. Finally, it is shown\nthat the automatic non-intrusively determined reshaping filter length in PMINT yields a similar performance as the\noptimal intrusively determined reshaping filter length.
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