Keywords: fuzzy adaptive filter, sound environment system, unknown structure, state estimation, psychological evaluation, loudness.

The internal physical mechanism of an actual sound environment system having a complicated relation to various factors is often difficult to recognize analytically, and it contains an unknown structure. Furthermore, the stochastic process observed in the actual phenomenon exhibits a complex fluctuation pattern and there are potentially various nonlinear correlations in addition to the linear correlation between the input and output time series [1].

On the other hand, it is necessary to pay our attention to the fact that the observation data in the sound environment system often contain fuzziness due to several causes. For example, it has been reported in psychological acoustics that the human psychological evaluation for loudness can be distinguished with up to 7 scores: 1. very calm, 2. calm, 3. mostly calm, 4. little noisy, 5. noisy, 6. fairly noisy, 7. very noisy [2]. However, each score is affected by the human subjectivity and the borders between two neighbouring scores are vague. In this situation, in order to evaluate the objective sound environment system, it is desirable to estimate the waveform fluctuation of the specific signal based on the observed data with fuzziness.

As a typical method in the state estimation problem, the Kalman filtering theory and its extended filter are well known [3,4,5]. These theories are originally based on the additive model of the specific signal and an external noise. The actual sound environment systems often contain unknown characteristics in the relationship between the state variable and the observation. Furthermore, the observation data often contain fuzziness.

In this study, a method for estimating adaptively the specific signal for sound environment systems with an unknown structure and fuzzy observations is theoretically proposed. More specifically, by regarding the loudness scores as observation data with fuzziness, an estimation method for the specific signal is proposed by introducing a probability measure of fuzzy events [6]. Furthermore, the effectiveness of the proposed theory is confirmed experimentally by applying it to the estimation of sound level based on the successive observation of the psychological evaluation for loudness.

1

A. Ikuta, H. Masuike, M. Ohta, "A Digital Filter for Stochastic Systems with Unknown Structure and Its Application to Psychological Evaluation of Sound Environment", IEICE Transactions on Information and Systems, E88-D, 1519-1525, 2005. doi:10.1093/ietisy/e88-d.7.1519

2

S. Namba, S. Kuwano, "Psychological Measurement Method for Sound Evaluation", Corona Press, 413-164, 1998.

3

R.E. Kalman, "A New Approach to Linear Filtering and Prediction Problem", Trans. ASME, Series D, J. Basic Engineering, 82, 35-45, 1960.

4

R.E. Kalman, R. Buch, "New Results in Linear Filtering and Prediction Theory", Trans. ASME, Series D, J. Basic Engineering, 83, 95-108, 1961.

5

H.J. Kushner, "Approximations to Optimal Nonlinear Filter", IEEE Trans. Autom. Control, AC-12, 546-556, 1967. doi:10.1109/TAC.1967.1098671

6

L.A. Zadeh, "Probability Measures of Fuzzy Events", J. Math. Anal. Appl., 23, 199-212, 1968. doi:10.1016/0022-247X(68)90078-4

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