Internal physical mechanism of actual sound environment system is often difficult to recognize analytically, and it contains unknown structural characteristics. Furthermore, the observation data often contain fuzziness due to several causes and exhibit level saturation owing to the existence of a finite dynamic range. Therefore, it is necessary to propose a new state estimation method by considering fuzziness and finite amplitude fluctuation of observation data. In this paper, a method for estimating the specific signal for sound environment system with unknown structure is proposed in an appropriate form for the finite level range of the measured fuzzy observation data by introducing an expansion expression of probability distribution with Bata distribution in the first term and new type of membership function. The effectiveness of the proposed theoretical method is confirmed by applying it to the actual problem in the sound environment.
The internal physical mechanism of actual sound environment system having a complicated relation to various factors is often difficult to recognize analytically, and it contains unknown structure. Furthermore, the stochastic process observed in the actual phenomenon exhibits complex fluctuation pattern and there are potentially various nonlinear correlations in addition to the linear correlation between input and output time series [
Furthermore, it is necessary to pay our attention on the fact that the observation data in the sound environment system often contain fuzziness due to several causes, for example, the permissible error of the accuracy in measurements, the quantized error in the digitization of observation data, and the existence of confidence limitation in measuring instruments. In our previous studies, state estimation methods for sound environment system based on fuzzy observation have been proposed by considering the standard Gaussian type membership function [
As a typical method in the state estimation problem, the Kalman filtering theory and its extended filter are well known [
In this study, based on the fuzzy observations, a Bayesian filter for estimating the specific signal of sound environment systems with unknown structural characteristic is theoretically proposed in an appropriate form for the finite amplitude range of the measured data. More specifically, complex sound environment systems which have to be treated as the systems with unknown characteristics are paid our attention. By introducing the orthogonal expansion expression of the probability distribution with a Beta distribution as the first term, and a new type of membership function, which are suitable for the finite amplitude fluctuation range of the signal and observation, a method to estimate the waveform fluctuation of the specific signal based on the fuzzy observation data is proposed. After adopting a previously reported expansion expression of the conditional probability distribution as the correlation information between the specific signal and observation [
Finally, the effectiveness of the proposed Bayesian filter focusing on the relationship between variables is confirmed experimentally too by applying the theory to the estimation of sound level based on the observation data containing fuzziness and amplitude saturation.
Let
with
where
These orthonormal polynomials can be decomposed by using Schmidt’s orthogonalization [
Though Equation (5) is originally infinite series expansion, finite expansion series with
Furthermore, let
Since the objective system contains an unknown specific signal and unknown structure, the expansion coefficients
the following simple dynamical models are introduced for the simultaneous estimation of the parameters with the specific signal
where
On the other hand, the following time transition model for the input signal is generally established.
where
A method to estimate
In order to derive an estimation algorithm for a specific signal
where
with
The functions
Based on Equation (11), the recurrence algorithm for estimating an arbitrary
where
In order to make the general theory for estimation algorithm more concrete, the well-known Gaussian distribution is adopted as
with
, (20)
Furthermore, the Bata distribution [
with
, (22)
where
where
where s is a parameter.
Accordingly, Equation (12) can be given by
with
, (28)
The fuzzy data
By considering the orthonormal condition of Jacobi polynomial [
where a few concrete expressions of
In two special cases when
with
Using the property of conditional expectation and Equation (5), the two variables
with
where T denotes the transpose of a matrix. The coefficients
, (38)
Furthermore, using Equation (5) and the orthonormal condition of Equation (4), each expansion coefficient
(: appropriate coefficients). (39)
In the above, the expansion coefficient
Finally, by considering Equation (9), the prediction step which is essential to perform the recurrence estimation can be given by
By replacing
In order to confirm the effectiveness of the proposed method, it was applied to real data observed in a sound environment system. Acoustic signals observed by two microphones in indoors and outdoors for a house were adopted as input and output data for the sound insulation system shown by the frame in
After generating the music sound inside the house, the indoor and outdoor sound pressure levels were regarded as the input signal
Figures 2-4 show the estimation results of the fluctuation wave form of the input signal for Data 1 in a typical case of widely fluctuating signals, by applying the proposed algorithm to the quantized observation data with 1 dB, 2 dB and 3 dB widths. Furthermore, Figures 5-7 show the estimation results for Data 2 in a typical case of narrowly fluctuating signals. In these estimations, the finite number of expansion coefficients
where
The results estimated by the proposed method show good agreement with the
Data Number | Data 1 | Data 2 | Data 3 | Data 4 | Data 5 |
---|---|---|---|---|---|
Mean Value | 80.8 | 79.6 | 80.4 | 82.4 | 80.8 |
Standard Deviation | 2.25 | 1.31 | 2.38 | 2.26 | 2.50 |
Data Number | Data 1 | Data 2 | Data 3 | Data 4 | Data 5 |
---|---|---|---|---|---|
Mean Value | 60.4 | 58.2 | 59.7 | 60.8 | 60.0 |
Standard Deviation | 1.92 | 1.45 | 1.74 | 1.85 | 1.75 |
true values. On the other hand, there are great discrepancies between the esti- mates based on the standard type dynamical estimation method (i.e., extended Kalman filter). For these differences on estimated results, the following reasons can be considered: 1) The standard method assumes the simple observation model in Equation (42). On the other hand, the proposed method introduces the conditional probability distribution in Equation (5) as the observation model, which can be considered the whole fluctuation of input and output signals; 2) The standard method is not considered the finite amplitude fluctuation of the observation data; 3) To consider the quantized observation by introducing fuzzy theory is more useful than the standard method by introducing the quantized noise.
The squared sums of the estimation error are shown in Tables 3-5. It can be found numerically that the proposed method is more useful than the extended Kalman filter.
In this study, based on the observed data with fuzziness and the finite level range, a new adaptive method for estimating the input signal for sound environment systems with unknown structure has been proposed. The proposed estimation method has been realized by introducing a system model of conditional probability type and the probability measure of fuzzy events. The proposed method has been applied to the estimation for the input signal of an actual sound environment system, and it has been experimentally verified that better results are obtained as compared with the standard estimation method without considering fuzzy theory.
The proposed approach is quite different from the traditional standard techniques. However, we are still in an early stage of development, and a number of practical problems are yet to be investigated in the future. These include: 1) application to a diverse range of sound signals in actual noise environment; 2) extension to cases with multi-noise sources, and 3) finding an optimal number of expansion terms for the expansion-based probability expressions adopted.
Data Number | Data 1 | Data 2 | Data 3 | Data 4 | Data 5 |
---|---|---|---|---|---|
Proposed Methods | 1.61 | 1.39 | 1.32 | 1.25 | 1.45 |
Extended Kalman Filter | 2.93 | 3.18 | 2.01 | 1.55 | 2.60 |
Data Number | Data 1 | Data 2 | Data 3 | Data 4 | Data 5 |
---|---|---|---|---|---|
Proposed Methods | 1.63 | 1.47 | 1.34 | 1.38 | 1.50 |
Extended Kalman Filter | 2.93 | 3.80 | 2.01 | 1.40 | 2.58 |
Data Number | Data 1 | Data 2 | Data 3 | Data 4 | Data 5 |
---|---|---|---|---|---|
Proposed Methods | 1.63 | 1.53 | 1.39 | 1.53 | 1.51 |
Extended Kalman Filter | 2.83 | 4.38 | 2.09 | 1.78 | 2.80 |
The authors are grateful to Emeritus Prof. Mitsuo Ohta of Hiroshima University for his advice during this study. This work was supported in part by fund from the Grant-in-Aid for Scientific Research No.15K06116 from the Ministry of Education, Culture, Sports, Science and Technology-Japan.
Ikuta, A., Orimoto, H. and Gallagher, G. (2017) State Estimation for Fuzzy Sound Environment System with Finite Amplitude Fluctuation. Journal of Software Engineering and Applications, 10, 625-638. https://doi.org/10.4236/jsea.2017.107034