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In the active control of noise in ducts, it is common practice to locate an error microphone far from the control source to avoid the near-field effects by evanescent waves. Such a distance between the control source and the error microphone makes a certain level of time delay inevitable and, hence, may yield undesirable effects on the convergence properties of control algorithms such as filtered-x and filtered-u least-mean-square (LMS). This paper discusses the dependence of the convergence rate on the acoustic error path in these popular algorithms and introduces new algorithms "constraint filtered-x and constraint filtered-u LMS," which increase the convergence region regardless of the time delay in the acoustic error path. In the algorithms, coefficients of the controller are adapted using modified residuals that are defined in such a way that the control process become stationary. Performances of the new LMS algorithms are presented in comparison with those by the conventional algorithms based on computer simulations.

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