A Novel Approach for Blind Estimation of Reverberation Time using Rayleigh Distribution Model

In this paper a blind estimation approach is proposed which directly utilizes the reverberant signal for estimating the RT (Reverberation Time).For estimation a very well-known method is used; MLE (Maximum Likelihood Estimation). Distribution of the decay rate is the core of the proposed method and can be achieved from the analysis of decay curve of the energy of the sound or from enclosure impulse response. In a pre-existing state of the art method Laplace distribution is used to model reverberation decay. The method proposed in this paper make use of the Rayleigh distribution and a spotting approach for modelling decay rate and identifying region of free decay in reverberant signal respectively. Motivation for the paper was deduced from the fact, when the reverberant speech RT falls in specific range then the signals decay rate impersonate Rayleigh distribution. On the basis of results of the experiments carried out for numerous reverberant signal it is clear that the performance and accuracy of the proposed method is better than other pre-existing methods.

and its geometry (volume and surface area) are the major ingredients of the proposed formula.
However, to use these empirical formulae the information of the absorptive characteristic and the geometry of the target enclosure were essential requirements.The extraction of these information for an enclosure is not easy task, hence a new method were required to estimate RT using on the impulse response of an enclosure.
In [4] similar approach is used in which the energy decay curves are used in measuring approaches of RT.
The desire decay curve can be obtained by turning off a noise producing source after achieving its steady state.Slope of the decay curve is used in the estimation of the RT.The noise source fluctuation leads to multiple decay curves.The reliable value of the RT is achieved by averaging technique.In some other approaches the reverberant signal partitioning approach is used for its gap detection within that signal [5][6] resulting decay curves tracking.This issue is rectified in 1965 by introducing an impulse response based approach [7].This approach prevailed the field of signal processing (speech) for several decades however, on the other hand a technique was required having the capability of estimating the value of the RT of a reverberant signal directly.In other words a blind estimation approach were required which should takes only the reverberant signal as an input.Blind estimation of RT for a reverberant signal can be obtained by numerous approaches [2,[8][9][10][11].In these approaches statistical models and Maximum likelihood estimation are the base element for the optimized RT value determination.Some other methods are also developed which make use of the neural networks in order to obtain the enclosure characteristics and are called semi-blind approaches [12][13][14].Ratnam,et. al. [2] modelled the characteristics of reverberation of a room by utilizing an approach which is based on the decay curve of the noise, in order to develop an algorithm for estimating the value of RT blindly.In [2] the detection of the sound decay is achieved by utilizing a repetitive process and this repetition process made the algorithm expensive in terms of computation.Ratnam, et.al. [10] increased the computational efficiency of the algorithm by making some changes in it and presented the new version.
Recently Lollmann, et.al. [8] presented a blind approach RT estimation algorithm based on the sounds statistical decay model.A method of pre selection is utilized in order to detect the possible decay in sound signal.This pre selection process made the algorithm more robust in terms of estimation and more efficient computationally.
In this paper a Lollmann, et.al. [8] method based approach is proposed for estimating reverberation time blindly along with Rayleigh distribution model for decay rate modelling.

PROPOSED MODEL AND MLE
In this paper Rayleigh distribution based statistical model is used for modelling the energy decay of the target reverberant signal.For a specific RT range the speech signal amplitude distribution is then better modelled by Rayleigh distribution [15].The method presented in [15] is the motivation for the proposed work.Fig. 1 derived from [15] also shows the motivation.
Small enclosure such as ATM booth, Telephone booth etc. are mainly focused in this work in order to estimate the RT value for improving the quality of the reverberant speech signal.In such enclosure RT falls in the range 0-150msand therefore Rayleigh distribution would be utilized for decay curve modelling of the target speech.

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Hence, Rayleigh distribution G(x,) accompanied by a random variable sequence is used to model the tail of the decaying reverberant sound.In G(x,) random variable is represented by x and variance by .Mathematically the Rayleigh distribution is expressed as [16]: The model evolution is based on an assumption.In this assumption the tail of the decaying reverberant signal denoted by y is expressed by the expression below: Where the observation of y is used to estimate the unknown values of the parameters (b,).For simplifying the derived likelihood function in Equation ( 4) a supposition is done.In this supposition  (unity decay rate) is used for the damping of the target reverberant sound signal energy envelop, resulting in an expression for b(n) as given in [2].
A single scalar argument band  can be utilized to express the multi-dimensional argument b(n).The parameter b is given as: By making use of Equations (5-6) we get: After integrating Equation (7) with Equation (4) yields:  results.In [7] RIR is utilized for RT value calculation and in [19] Laplace distribution model is used.For the evaluation of the algorithm the RT estimated value is compared with the results of the above mention algorithms.
Results for the shorter distance and comparatively larger distance are shown in Figs.1-2 respectively.By observing the results it is revealed that for small distance the results of the proposed methods are closed to the results of the RIRs (0.49 and 0.429sec).Over all the results produced by the proposed method is in close agreement to the baseline (Schroeder) methods.

CONCLUSION
The proposed method is a novel approach for estimating RT blindly.Statistical model based on Raleigh distribution and ML scheme is utilized for representing echoic signal decay and arguments estimation respectively.Experiment exhibited that the results from the proposed arrangements are in close concession with the results acquired by the up-to-the-minute approach.The current statistical model produce good results when the RT is in lower range (0-300ms) in future we would try to modify it so that it could be used for wide range of TR values.[19].ORANGE LINES= STANDARD DEVIATIONS FIG. 3. BLUE BARS = BY SCHROEDER'S METHOD [7] (RIRS RESULTS), GREEN BARS =PROPOSED METHOD, BROWN BARS = LAPLACE DISTRIBUTION [19].ORANGE LINES= STANDARD DEVIATIONS

FIG. 2..BLUE BARS = BY SCHROEDER'S METHOD [7] (RIRS RESULTS),GREEN BARS =PROPOSED METHOD, BROWN BARS =LAPLACE DISTRIBUTION
Here b and x represents deterministic envelop and the fine structure of the random process respectively.The y(n) is not similarly distributed and have pdf G(b(n)) due to time variant x(n) although having an independent y(n).For estimating target (reverberant) signal decay rate a limited sequence, n = 0,1,2,..., M"1, is selected.For convenience in terms of notation and simplicity b and y are introduced to represent the b and y vector (Mdimensional).The joint probability density or the function of likelihood for y having the parameters  and b is given as, rearrangements and use of mathematical induction made Equation (8) simplify and becomes:

FIG. 1 .
FIG. 1. NUMEROUS DISTRIBUTIONTHEORETICAL PDF (LAPLACE DISTRIBUTION BLUE LINE, GAMMA DISTRIBUTION BLACK LINE, GAUSSIAN DISTRIBUTION PINK LINE) WITH ZERO MEAN ALONG WITH DISTRIBUTION OF ECHOIC SIGNAL