Keywords: active noise control, simultaneous equations method, frequency domain adaptive algorithm, block implementation, linear convolution, circular convolution.

The filtered-x algorithm is widely applied to feedforward type active noise control (ANC) systems. This algorithm requires a filter, called the secondary path filter, exactly modeled on a secondary path from a loudspeaker to an error microphone, whereas the path in practical systems continuously changes. This path change inevitably increases the modeling error, and at worst, the ANC system thereby falls into an uncontrollable state.

To solve this problem, we have proposed a simultaneous equations method [1,2,3]. In this method, a block implementation type of frequency domain adaptive algorithm is used for identifying the overall path from a noise detection microphone, through a primary path, a noise control filter and the secondary path, to an error microphone. However, the performance of this algorithm is not yet adequately analyzed. In this paper, we derive the minimum block length maximizing the convergence speed.

Frequency domain adaptive algorithms require transforming the circular convolution to the linear convolution, which can be accomplished by replacing a part of elements of signal vectors with zeros. However, this replacement generates an auto-correlation matrix whose non-diagonal elements are non-zero, which delays the convergence of the coefficient vector of adaptive filter. Applying the block implementation to the frequency domain adaptive algorithm decreases the non-diagonal elements. However, the excess block length delays the convergence.

This paper derives the minimum block length making the non-diagonal elements negligible in comparison to the diagonal those, from the results of the computer simulations using three kinds of noise. One is a white noise synthesized by a personal computer. Another is the output signal obtained by feeding the white noise to a band pass filter whose resonant frequency is 1 kHz. The third is a recorded noise of a diesel engine (DS3512, Caterpillar US). The results show that the minimum block length is about five in the case of the white noise and about twelve in other cases. In this simulation, the filter output is stationary, and its spectrum is monotonous. On the other hand, the diesel engine noise is unstationary and involves the periodic components. In addition, its spectrum is complicated. These two noises, however, indicate that the minimum block length is twelve as a tentative value. We have only to adjust it slightly if necessary. In this paper, we finally verify it by using an experimental system.

1

K. Fujii, M. Muneyasu and J. Ohga, "Active noise control systems by using the simultaneous equations method without estimation of the error path filter coefficients," IEICE Trans. Fundamentals, vol. J82-A, no. 3, pp. 299-305, March 1999.

2

K. Fujii, S. Hashimoto and M. Muneyasu, "Application of a frequency domain processing technique to the simultaneous equation method," IEICE Trans. Fundamentals, EA86-A, pp. 2020-2027, Aug. 2003.

3

K. Fujii, K. Yamaguchi, S. Hashimoto, Y. Fujita and M. Muneyasu, "Verification of simultaneous equation method by an experimental active noise control system," Acoust. Sci. & Tech., vol. 27, no. 5, pp. 270-277, Sept. 2006. doi:10.1250/ast.27.270

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