# Patent

## Title
Noise estimation apparatus, noise estimation method, noise estimation program, and recording medium

## Abstract

A noise estimation apparatus which estimates a non-stationary noise component on the basis of the likelihood maximization criterion is provided. The noise estimation apparatus obtains the variance of a noise signal that causes a large value to be obtained by weighted addition of the sums each of which is obtained by adding the product of the log likelihood of a model of an observed signal expressed by a Gaussian distribution in a speech segment and a speech posterior probability in each frame, and the product of the log likelihood of a model of an observed signal expressed by a Gaussian distribution in a non-speech segment and a non-speech posterior probability in each frame, by using complex spectra of a plurality of observed signals up to the current frame.

## Background
TECHNICAL FIELD
The present invention relates to a technology for estimating a noise component included in an acoustic signal observed in the presence of noise (hereinafter also referred to as an “observed acoustic signal”) by using only information included in the observed acoustic signal.
BACKGROUND ART
In the subsequent description, symbols such as “ ̃” should be printed above a letter but will be placed after the letter because of the limitation of text notation. These symbols are printed in the correct positions in formulae, however. If an acoustic signal is picked up in a noisy environment, that acoustic signal includes the sound to be picked up (hereinafter also referred to as “desired sound”) on which noise is superimposed. If the desired sound is speech, the clarity of speech contained in the observed acoustic signal would be lowered greatly because of the superimposed noise. This would make it difficult to extract the properties of the desired sound, significantly lowering the recognition rate of automatic speech recognition (hereinafter also referred to simply as “speech recognition”) systems. If a noise estimation technology is used to estimate noise, and the estimated noise is eliminated by some method, the clarity of speech and the speech recognition rate can be improved. Improved minima-controlled recursive averaging (IMCRA hereinafter) in Non-patent literature 1 is a known conventional noise estimation technology.
Prior to a description of IMCRA, an observed acoustic signal model used in the noise estimation technology will be described. In general speech enhancement, an observed acoustic signal (hereinafter referred to briefly as “observed signal”) yn observed at time n includes a desired sound component and a noise component. Signals corresponding to the desired sound component and the noise component are respectively referred to as a desired signal and a noise signal and are respectively denoted by xn and vn. One purpose of speech enhancement processing is to restore the desired signal xn on the basis of the observed signal yn. Letting signals after short-term Fourier transformation of signals yn, xn, and be Yk,t, Xk,t, and Vk,t, where k is a frequency index having values of 1, 2, . . . , K (K is the total number of frequency bands), the observed signal in the current frame t is expressed as follows.

Yk,t=Xk,t+Vk,t  (1)

In the subsequent description, it is assumed that this processing is performed in each frequency band, and for simplicity, the frequency index k will be omitted. The desired signal and the noise signal are assumed to follow zero-mean complex Gaussian distributions with variance σx2 and variance σv2 respectively.
The observed signal has a segment where the desired sound is present (“speech segment” hereinafter) and a segment where the desired sound is absent (“non-speech segment” hereinafter), and the segments can be expressed as follows with a latent variable H having two values H1 and H0.




  
    
      
        
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The conventional method will be explained next with the variables described above.
IMCRA will be described with reference to FIG. 1. In a conventional noise estimation apparatus 90, first a minimum tracking noise estimation unit 91 obtains a minimum value in a given time segment of the power spectrum of the observed signal to estimate a characteristic (power spectrum) of the noise signal (refer to Non-patent literature 2).
Then, a non-speech prior probability estimation unit 92 obtains the ratio of the power spectrum of the estimated noise signal to the power spectrum of the observed signal and calculates a non-speech prior probability by determining that the segment is a non-speech segment if the ratio is smaller than a given threshold.
A non-speech posterior probability estimation unit 93 next calculates a non-speech posterior probability p(H0|Yi;θi ̃IMCRA) (1 or 0), assuming that the complex spectra of the observed signal and the noise signal after short-term Fourier transformation follow Gaussian distributions. The non-speech posterior probability estimation unit 93 further obtains a corrected non-speech posterior probability β0,iIMCRA from the calculated non-speech posterior probability p(H0|Yi;θi ̃IMCRA) and an appropriately predetermined weighting factor α.

β0,iIMCRA=(1−α)p(H0|Yi;{tilde over (θ)}iIMCRA)  (3)

A noise estimation unit 94 then estimates a variance σv,i2 of the noise signal in the current frame i by using the obtained non-speech posterior probability β0,iIMCRA, the power spectrum |Yi|2 of the observed signal in the current frame, and the estimated variance σv,i-12 of the noise signal in the frame i−1 immediately preceding the current frame i.

σv,i2=(1−β0,iIMCRA)σv,i-12+β0,iIMCRA|Yi|2  (4)

By successively updating the estimated variance σv,i2 of the noise signal, varying characteristics of non-stationary noise can be followed and estimated.
PRIOR ART LITERATURE
Non-Patent Literature


    Non-patent literature 1: I. Cohen, “Noise spectrum estimation in adverse environments: improved minima controlled recursive averaging”, IEEE Trans. Speech Audio Process., September 2003, vol. 11, pp. 466-475
    Non-patent literature 2: R. Martin, “Noise power spectral density estimation based on optimal smoothing and minimum statistics”, IEEE Trans. Speech Audio Process., July 2001, vol. 9, pp. 504-512.


SUMMARY OF THE INVENTION
Problems to be Solved by the Invention
In the conventional technology, the non-speech prior probability, the non-speech posterior probability, and the estimated variance of the noise signal are not calculated on the basis of the likelihood maximization criterion, which is generally used as an optimization criterion, but are determined by a combination of parameters adjusted by using a rule of thumb. This has caused a problem that the finally estimated variance of the noise signal is not always optimum but is quasi-optimum based on the rule of thumb. If the successively estimated variance of the noise signal is quasi-optimum, the varying characteristics of non-stationary noise cannot be estimated while being followed appropriately. Consequently, it has been difficult to achieve a high noise cancellation performance in the end.
An object of the present invention is to provide a noise estimation apparatus, a noise estimation method, and a noise estimation program that can estimate a non-stationary noise component by using the likelihood maximization criterion.
Means to Solve the Problems
To solve the problems, a noise estimation apparatus in a first aspect of the present invention obtains a variance of a noise signal that causes a large value to be obtained by weighted addition of the sums each of which is obtained by adding the product of the log likelihood of a model of an observed signal expressed by a Gaussian distribution in a speech segment and a speech posterior probability in each frame, and the product of the log likelihood of a model of an observed signal expressed by a Gaussian distribution in a non-speech segment and a non-speech posterior probability in each frame, by using complex spectra of a plurality of observed signals up to the current frame.
To solve the problems, a noise estimation method in a second aspect of the present invention obtains a variance of a noise signal that causes a large value to be obtained by weighted addition of the sums each of which is obtained by adding the product of the log likelihood of a model of an observed signal expressed by a Gaussian distribution in a speech segment and a speech posterior probability in each frame, and the product of the log likelihood of a model of an observed signal expressed by a Gaussian distribution in a non-speech segment and a non-speech posterior probability in each frame, by using complex spectra of a plurality of observed signals up to the current frame.
Effects of the Invention
According to the present invention, a non-stationary noise component can be estimated on the basis of the likelihood maximization criterion.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a functional block diagram of a conventional noise estimation apparatus;
FIG. 2 is a functional block diagram of a noise estimation apparatus according to a first embodiment;
FIG. 3 is a view showing a processing flow in the noise estimation apparatus according to the first embodiment;
FIG. 4 is a functional block diagram of a likelihood maximization unit according to the first embodiment;
FIG. 5 is a view showing a processing flow in the likelihood maximization unit according to the first embodiment;
FIG. 6 is a view showing successive noise estimation characteristics of the noise estimation apparatus of the first embodiment and the conventional noise estimation apparatus;
FIG. 7 is a view showing speech waveforms obtained by estimating noise and cancelling noise on the basis of the estimated variance of a noise signal in the noise estimation apparatus of the first embodiment and the conventional noise estimation apparatus;
FIG. 8 is a view showing results of evaluation of the noise estimation apparatus of the first embodiment and the conventional noise estimation apparatus compared in a modulated white-noise environment;
FIG. 9 is a view showing results of evaluation of the noise estimation apparatus of the first embodiment and the conventional noise estimation apparatus compared in a bubble noise environment;
FIG. 10 is a functional block diagram of a noise estimation apparatus according to a modification of the first embodiment; and
FIG. 11 is a view showing a processing flow in the noise estimation apparatus according to the modification of the first embodiment.



DETAILED DESCRIPTION OF THE EMBODIMENTS
Now, an embodiment of the present invention will be described. In the drawings used in the following description, components having identical functions and steps of performing identical processes will be indicated by identical reference characters, and their descriptions will not be repeated. A process performed in units of elements of a vector or a matrix is applied to all the elements of the vector or the matrix unless otherwise specified.
Noise Estimation Apparatus 10 According to First Embodiment
FIG. 2 shows a functional block diagram of a noise estimation apparatus 10, and FIG. 3 shows a processing flow of the apparatus. The noise estimation apparatus 10 includes a likelihood maximization unit 110 and a storage unit 120.
When reception of the complex spectrum Yi of the observed signal in the first frame begins (s1), the likelihood maximization unit 110 initializes parameters in the following way (s2).

σv,i-12=|Yi|2

σy,i-12=|Yi|2

β1,i-1=1−λ

α0,i-1=κ

α1,i-1=1−α0,i-1 

c0,i-1=α0,i-1 

c0,i-1=α1,i-1  (A)

Here, λ and κ are set beforehand to a given value in the range of 0 to 1. The other parameters will be described later in detail.
When the likelihood maximization unit 110 receives the complex spectrum Yi of the observed signal in the current frame i, the likelihood maximization unit 110 takes from the storage unit 120 the non-speech posterior probability η0,i-1, the speech posterior probability η1,i-1, the non-speech prior probability α0,i-1, the speech prior probability α1,i-1, the variance σy,i-12 of the observed signal, and the variance σv,i-12 of the noise signal, estimated in the frame i−1 immediately preceding the current frame i, for successive estimation of the variance σv,i2 of the noise signal in the current frame i (s3). On the basis of those values (or on the basis of the initial values (A), instead of the values taken from the storage unit 120, when the complex spectrum Yi of the observed signal in the first frame is received), by using the complex spectra Y0, Y1, . . . Yi of the observed signal up to the current frame i, the likelihood maximization unit 110 obtains the speech prior probability α1,i, the non-speech prior probability α0,i, the non-speech posterior probability η0,i, the speech posterior probability η1,i, the variance σv,i2 of the noise signal, and the variance σx,i2 of the desired signal in the current frame i such that the value obtained by weighted addition of the sums each of which is obtained by adding the product of the log likelihood log [α1p(Yt|H1;θ)] of a model of an observed signal expressed by a Gaussian distribution in a speech segment and the speech posterior probability η1,t(α′0,θ′) in each frame t (t=0, 1, . . . , i), and the product of the log likelihood log [α1p(Yt|H0;θ)] of a model of an observed signal expressed by a Gaussian distribution in a non-speech segment and the non-speech posterior probability η0,t(α′0,θ′) in each frame t, as given below, is maximized (s4), and stores them in the storage unit 120 (s5).




  
    
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The noise estimation apparatus 10 outputs the variance σv,i2 of the noise signal. Here, λ is a forgetting factor and a parameter set in advance in the range 0<λ<1. Accordingly, the weighting factor λi-t decreases as the difference between the current frame i and the past frame t increases. In other words, a frame closer to the current frame is assigned a greater weight in the weighted addition. Steps s3 to s5 are repeated (s6, s7) up to the observed signal in the last frame. The likelihood maximization unit 110 will be described below in detail.
Parameter Estimation Method Based on Likelihood Maximization Criterion
An algorithm for estimating the above-described parameters on the basis of the likelihood maximization criterion will now be derived. First, the speech prior probability and the non-speech prior probability are defined respectively as α1=P(H1) and α0=P(H0)=1−α1, and the parameter vector is defined as θ=[σv2, σx2]T. It is noted that σy2, σx2, and σv2 represent the variances of the observed signal, the desired signal, and the noise signal, respectively, and also their power spectra.
It is assumed as follows that the complex spectrum Yt of the observed signal follows a Gaussian distribution both in the speech segment and in the non-speech segment.




  
    
      
        
          
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With the above-described models, the non-speech prior probability α0, and the speech prior probability α1, the likelihood of the observed signal in the time frame t can be expressed as follows.

p(Yt;α0,θ)=α0p(Yt|H0;σv2)+α1p(Yt|H1;σv2,σx2)  (6)

According to the Bayes' theorem, the speech posterior probability η1,t(α0,θ)=p(H1|Yt;α0,θ) and the non-speech posterior probability η0,t(α0,θ)=p(H0|Yt;α0,θ) can be defined as follows.




  
    
      
        
          
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Here, s is a variable that has a value of either 0 or 1. With those models, parameters α0 and θ that maximize the likelihood defined by formula (6) can be estimated by repeatedly maximizing an auxiliary function. Specifically, by repeatedly estimating values α′0 and θ′ of unknown optimum values of the parameters that maximize the auxiliary function Q(α0,θ)=E{log [p(Yt,H;α0,θ)]|Yt;α′0,θ′}, the (local) optimum values (estimated maximum likelihood) of the parameters can be obtained. Here, E{•} is an expectation calculation function. In this embodiment, since the variance of a non-stationary noise signal is estimated, the parameters α0 and θ to be estimated (latent variables of the expectation maximization algorithm) could vary with time. Therefore, instead of the usual expectation maximization (EM) algorithm, a recursive EM algorithm (reference 1) is used.

    (Reference 1) L. Deng J. Droppo, and A. Acero, “Recursive estimation of nonstationary noise using iterative stochastic approximation for robust speech recognition”, IEEE Trans. Speech, Audio Process, November 2003, vol. 11, pp. 568-580


For the recursive EM algorithm, the following auxiliary function Qi(α0, θ) obtained by transforming the auxiliary function given above is introduced.




  
    
      
        
          
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By maximizing the auxiliary function Qi(α0, θ), the optimum parameter values α0,i, αi,1, θi={σv,i2, σx,i2} in the time frame i can be obtained. If the optimum estimates in the immediately preceding frame i−1 have always been obtained (α′s=αs,i-1, and θ′=θi-1 are assumed), the optimum parameter value α0,i can be obtained by partially differentiating the function L(α0, θ)=Qi(α0, θ)+μ(α1+α0−1) with respect to α1 and α0 and zeroing the result. Here, μ represents the Lagrange undetermined multiplier (adopted for optimization under the constraint α1+α0=1).
Through this operation, the following updating formula can be obtained.

αs,ici=csi  (9)

The variables in the formula are defined as follows.




  
    
      
        
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Formula (10) can be expanded as follows.

csi=λcs,i-1+ηs,i(α0,i-1,θi-1)  (12)

By partially differentiating the auxiliary function Q(α0,θ) with respect to σv2 and σx2 and zeroing the result, the following formula can be obtained for s=1.




  
    
      
        
          
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As for s=0, the following formula can be obtained.




  
    
      
        
          
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By inserting formula (10) into the first term on the left side of formula (14) and expanding the right side, the following formula can be obtained.

c0,iσv,i2=λc0,i-1σv,i-12+η0,i(α0,i-1,θi-1)|Yi|2  (15)

With formulae (12) and (15), a formula for successively estimating the variance σv,i2 of the noise signal can be derived as follows.

σv,12=(1−β0,i)σv,i-12+β0,i|Yi|2  (16)

Here, β0,i is defined as a time-varying forgetting factor, as given below.




  
    
      
        
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With formulae (12) and (13), a formula for updating the variance σy,i2 of the observed signal can also be obtained.

σy,i2=(1−β1,i)σy,i-12+β1,i|Yi|2  (18)

Here, β1,i is defined as a time-varying forgetting factor, as given below.




  
    
      
        
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When σy,i2 and σv,i2 are estimated, σx,i2 is estimated naturally (σy,i2=σv,i2+σx,i2). Therefore, the estimation of and σy,i2 is synonymous with the estimation of σx,i2.
Likelihood Maximization Unit 110
FIG. 4 shows a functional block diagram of the likelihood maximization unit 110, and FIG. 5 shows its processing flow. The likelihood maximization unit 110 includes an observed signal variance estimation unit 111, a posterior probability estimation unit 113, a prior probability estimation unit 115, and a noise signal variance estimation unit 117.
Observed Signal Variance Estimation Unit 111
The observed signal variance estimation unit 111 estimates a first variance σy,i,12 of the observed signal in the current frame i on the basis of the speech posterior probability η1,i-1(α0,i-2,θ1-2) estimated in the immediately preceding frame i−1, by weighted addition of the complex spectrum Yi of the observed signal in the current frame i and a second variance σy,i-1,22 of the observed signal estimated in the frame i−1 immediately preceding the current frame i. For example, the observed signal variance estimation unit 111 receives the complex spectrum Yi of the observed signal in the current frame i, and the speech posterior probability η1,i-1(α0,i-2,θ1-2) and the second variance σy,i-1,22 of the observed signal estimated in the immediately preceding frame i−1,
uses those values to estimate the first variance σy,i,12 of the observed signal in the current frame i, as given below, (s41) (see formulae (18), (19), and (12)), and outputs the first variance to the posterior probability estimation unit 113.




  
    
      
        
          σ
          
            y
            ,
            i
            ,
            1
          
          2
        
        =
        
          
            
              (
              
                1
                -
                
                  β
                  
                    1
                    ,
                    
                      i
                      -
                      1
                    
                  
                
              
              )
            
            ⁢
            
              σ
              
                y
                ,
                
                  i
                  -
                  1
                
                ,
                2
              
              2
            
          
          +
          
            
              β
              
                1
                ,
                
                  i
                  -
                  1
                
              
            
            ⁢
            
              
                
                
                  Y
                  i
                
                
              
              2
            
          
        
      
    
  
  
    
      
        
          
            β
            
              1
              ,
              
                i
                -
                1
              
            
          
          =
          
            
              
                n
                
                  1
                  ,
                  
                    i
                    -
                    1
                  
                
              
              ⁡
              
                (
                
                  
                    α
                    
                      0
                      ,
                      
                        i
                        -
                        2
                      
                    
                  
                  ,
                  
                    θ
                    
                      i
                      -
                      2
                    
                  
                
                )
              
            
            
              c
              
                1
                ,
                
                  i
                  -
                  1
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            c
            
              1
              ,
              
                i
                -
                1
              
            
          
          =
          
            
              λ
              ⁢
              
                
              
              ⁢
              
                c
                
                  1
                  ,
                  
                    i
                    -
                    2
                  
                
              
            
            +
            
              
                η
                
                  1
                  ,
                  
                    i
                    -
                    1
                  
                
              
              ⁡
              
                (
                
                  
                    α
                    
                      0
                      ,
                      
                        i
                        -
                        2
                      
                    
                  
                  ,
                  
                    θ
                    
                      i
                      -
                      2
                    
                  
                
                )
              
            
          
        
      
    
  




When the complex spectrum Yi of the observed signal in the first frame is received, the first variance σy,i,12 is obtained from the initial values β1,i-1=1−λ and σy,i-12=|Yi|2 in (A) above, instead of using η1,i-1(α0,i-2,θi-2) and σy,i-1,22.
The observed signal variance estimation unit 111 further estimates the second variance σy,i,22 of the observed signal in the current frame i on the basis of the speech posterior probability η1,i(α0,i-1,θi-1) estimated in the current frame I, by weighted addition of the complex spectrum Yi of the observed signal in the current frame i and the second variance σy,i-1,22 of the observed signal estimated in the frame i−1 immediately preceding the current frame i. For example, the observed signal variance estimation unit 111 receives the speech posterior probability η1,i(α0,i-1,θi-1) estimated in the current frame i, estimates the second variance σy,i,22 of the observed signal in the current frame i, as given below, (s45) (see formulae (18), (19), and (12)), and stores the second variance σy,i,22 as the variance σy,i2 of the observed signal in the current frame i in the storage unit 120.




  
    
      
        
          σ
          
            y
            ,
            i
            ,
            2
          
          2
        
        =
        
          
            
              (
              
                1
                -
                
                  β
                  
                    1
                    ,
                    i
                  
                
              
              )
            
            ⁢
            
              σ
              
                y
                ,
                
                  i
                  -
                  1
                
                ,
                2
              
              2
            
          
          +
          
            
              β
              
                1
                ,
                i
              
            
            ⁢
            
              
                
                
                  Y
                  i
                
                
              
              2
            
          
        
      
    
  
  
    
      
        
          
            β
            
              1
              ,
              i
            
          
          =
          
            
              
                n
                
                  1
                  ,
                  i
                
              
              ⁡
              
                (
                
                  
                    α
                    
                      0
                      ,
                      
                        i
                        -
                        1
                      
                    
                  
                  ,
                  
                    θ
                    
                      i
                      -
                      1
                    
                  
                
                )
              
            
            
              c
              
                1
                ,
                i
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            c
            
              1
              ,
              i
            
          
          =
          
            
              λ
              ⁢
              
                
              
              ⁢
              
                c
                
                  1
                  ,
                  
                    i
                    -
                    1
                  
                
              
            
            +
            
              
                η
                
                  1
                  ,
                  i
                
              
              ⁡
              
                (
                
                  
                    α
                    
                      0
                      ,
                      
                        i
                        -
                        1
                      
                    
                  
                  ,
                  
                    θ
                    
                      i
                      -
                      1
                    
                  
                
                )
              
            
          
        
      
    
  




In the first frame, the initial value c1,i-1=α0,i-1=κ in (A) above is used to obtain c1,i.
In other words, the observed signal variance estimation unit 111 estimates the first variance σy,i,12 by using the speech posterior probability η1,i-1(α0,i-2,θi-2) estimated in the immediately preceding frame i−1 and estimates the second variance σy,i,22 by using the speech posterior probability η1,i(α0,i-1,θi-1) estimated in the current frame i.
The observed signal variance estimation unit 111 stores the second variance σy,i,22 as the variance σy,i2 in the current frame i in the storage unit 120.
Posterior Probability Estimation Unit 113
It is assumed that the complex spectrum Yi of the observed signal in a non-speech segment follows a Gaussian distribution determined by the variance σv,i-12 of the noise signal (see formula (5)) and that the complex spectrum Yi of the observed signal in a speech segment follows a Gaussian distribution determined by the variance σv,i-12 of the noise signal and the first variance σy,i,12 of the observed signal (see formula (5) where σy,i,12=σx,i-12). The posterior probability estimation unit 113 estimates the speech posterior probability η1,i(α0,i-1,θi-1) and the non-speech posterior probability η0,i(α0,i-1,θi-1) for the current frame i by using the complex spectrum Yi of the observed signal and the first variance σy,i,12 of the observed signal in the current frame i and the speech prior probability α1,i-1 and the non-speech prior probability α0,i-1 estimated in the immediately preceding frame i−1. For example, the posterior probability estimation unit 113 receives the complex spectrum Yi of the observed signal and the first variance σy,i,12 of the observed signal in the current frame i, the speech prior probability α1,i-1 and the non-speech prior probability α0,i-1, and the variance σv,i-12 of the noise signal estimated in the immediately preceding frame i−1, uses those values to estimate the speech posterior probability η1,i(α0,i-1,θi-1) and the non-speech posterior probability η0,i(α0,i-1,θi-1) for the current frame i, as given below, (s42) (see formulae (7) and (5)), and outputs the speech posterior probability η1,i(α0,i-1,θi-1) to the observed signal variance estimation unit 111, the non-speech posterior probability η0,i(α0,i-1,θi-1) to the noise signal variance estimation unit 117, and the speech posterior probability η1,i(α0,i-1,θi-1) and the non-speech posterior probability η0,i(α0,i-1,θi-1) to the prior probability estimation unit 115.




  
    
      
        
          
            η
            
              s
              ,
              i
            
          
          ⁡
          
            (
            
              
                α
                
                  0
                  ,
                  
                    i
                    -
                    1
                  
                
              
              ,
              
                θ
                
                  i
                  -
                  1
                
              
            
            )
          
        
        =
        
          
            
              α
              
                s
                ,
                
                  i
                  -
                  1
                
              
            
            ⁢
            
              p
              ⁡
              
                (
                
                  
                    
                      Y
                      i
                    
                    ❘
                    
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                      s
                    
                  
                  ;
                  
                    θ
                    
                      i
                      -
                      1
                    
                  
                
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              ∑
              
                
                  s
                  ′
                
                =
                0
              
              1
            
            ⁢
            
              
            
            ⁢
            
              
                α
                
                  
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                    i
                    -
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          p
          ⁡
          
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                  i
                
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                v
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              2
            
          
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              -
              
                
                  
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          ⁡
          
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              ;
              
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        =
        
          
            1
            
              π
              ⁡
              
                (
                
                  
                    σ
                    
                      v
                      ,
                      
                        i
                        -
                        1
                      
                    
                    2
                  
                  +
                  
                    σ
                    
                      x
                      ,
                      
                        i
                        -
                        1
                      
                    
                    2
                  
                
                )
              
            
          
          ⁢
          
            e
            
              -
              
                
                  
                    
                    
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                      i
                    
                    
                  
                  2
                
                
                  
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                      v
                      ,
                      
                        i
                        -
                        1
                      
                    
                    2
                  
                  +
                  
                    σ
                    
                      x
                      ,
                      
                        i
                        -
                        1
                      
                    
                    2
                  
                
              
            
          
        
      
    
  
  
    
      
        
          σ
          
            x
            ,
            
              i
              -
              1
            
          
          2
        
        =
        
          
            σ
            
              y
              ,
              i
              ,
              1
            
            2
          
          -
          
            σ
            
              v
              ,
              
                i
                -
                1
              
            
            2
          
        
      
    
  




In addition, the speech posterior probability η1,i(α0,i-1,θi-1) and the non-speech posterior probability η0,i(α0,i-1,θi-1) are stored in the storage unit 120. When the complex spectrum Yi of the observed signal in the first frame i is received, the initial value σv,i-12=|Yi|2 in (A) above is used to obtain σx,i-12, and the initial values α0,i-1=κ and α1,i-1=1−α0,i-1=1−κ are used to obtain η1,i(α0,i-1,θi-1) and η0,i(α0,i-1,θi-1).
Prior Probability Estimation Unit 115
The prior probability estimation unit 115 estimates values obtained by weighted addition of the speech posterior probabilities and the non-speech posterior probabilities estimated up to the current frame i (see formula (10)), respectively, as the speech prior probability α1,i and the non-speech prior probability α0,i. For example, the prior probability estimation unit 115 receives the speech posterior probability η1,i(α0,i-1,θi-1) and the non-speech posterior probability η0,i(α0,i-1,θi-1) estimated in the current frame i, uses the values to estimate the speech prior probability α1,i and the non-speech prior probability α0,I, as given below, (s43) (see formulae (9), (12), and (11)), and stores them in the storage unit 120.



  
    
      α
      
        s
        ,
        i
      
    
    =
    
      
        c
        
          s
          ,
          i
        
      
      
        c
        i
      
    
  




  
    
      c
      
        s
        ,
        i
      
    
    =
    
      
        λ
        ⁢
        
          
        
        ⁢
        
          c
          
            s
            ,
            
              i
              -
              1
            
          
        
      
      +
      
        
          η
          
            s
            ,
            i
          
        
        ⁡
        
          (
          
            
              α
              
                0
                ,
                
                  i
                  -
                  1
                
              
            
            ,
            
              θ
              
                i
                -
                1
              
            
          
          )
        
      
    
  




  
    
      c
      i
    
    =
    
      
        c
        
          0
          ,
          i
        
      
      +
      
        c
        
          1
          ,
          i
        
      
    
  



As for cs,i-1, values obtained in the frame i−1 should be stored. For the initial frame i, the initial values c0,i-1=α0,i-1=κ and c1,i-1==1−α0,i-1=1−κ in (A) above are used to obtain cs,i-1.
cs,i-1 may be obtained from formula (10), but in that case, all of the speech posterior probabilities η1,0, η1,1, . . . , η1,i and non-speech posterior probabilities η0,0, η0,1, . . . , η0,1 up to the current frame must be weighted with λ1-t and added up, which will increase the amount of calculation.
(Noise Signal Variance Estimation Unit 117)
The noise signal variance estimation unit 117 estimates the variance σv,i2 of the noise signal in the current frame i on the basis of the non-speech posterior probability estimated in the current frame i, by weighted addition of the complex spectrum Yi of the observed signal in the current frame i and the variance σv,i-12 of the noise signal estimated in the frame i−1 immediately preceding the current frame i. For example, the noise signal variance estimation unit 117 receives the complex spectrum Yi of the observed signal, the non-speech posterior probability η0,i(α0,i-1,θi-1) estimated in the current frame i, and the variance σv,i-12 of the noise signal estimated in the immediately preceding frame i−1, uses these values to estimate the variance σv,i2 of the noise signal in the current frame i, as given below, (s44) (see formulae (16), (17)), and stores it in the storage unit 120.




  
    
      
        
          σ
          
            v
            ,
            i
          
          2
        
        =
        
          
            
              (
              
                1
                -
                
                  β
                  
                    0
                    ,
                    i
                  
                
              
              )
            
            ⁢
            
              σ
              
                v
                ,
                
                  i
                  -
                  1
                
              
              2
            
          
          +
          
            
              β
              
                0
                ,
                i
              
            
            ⁢
            
              
                
                
                  Y
                  i
                
                
              
              2
            
          
        
      
    
  
  
    
      
        
          β
          
            0
            ,
            i
          
        
        =
        
          
            
              η
              
                0
                ,
                i
              
            
            ⁡
            
              (
              
                
                  α
                  
                    0
                    ,
                    
                      i
                      -
                      1
                    
                  
                
                ,
                
                  θ
                  
                    i
                    -
                    1
                  
                
              
              )
            
          
          
            c
            
              0
              ,
              i
            
          
        
      
    
  
  
    
      
        
          c
          
            0
            ,
            i
          
        
        =
        
          
            λ
            ⁢
            
              
            
            ⁢
            
              c
              
                0
                ,
                
                  i
                  -
                  1
                
              
            
          
          +
          
            
              η
              
                0
                ,
                i
              
            
            ⁡
            
              (
              
                
                  α
                  
                    0
                    ,
                    
                      i
                      -
                      1
                    
                  
                
                ,
                
                  θ
                  
                    i
                    ,
                    1
                  
                
              
              )
            
          
        
      
    
  




The observed signal variance estimation unit 111 performs step s45 described above by using the speech posterior probability η1,i(α0,i-1,θi-1) estimated in the current frame i after the process performed by the posterior probability estimation unit 113.
Effects
According to this embodiment, the non-stationary noise component can be estimated successively on the basis of the likelihood maximization criterion. As a result, it is expected that the trackability to time-varying noise is improved, and noise can be cancelled with high precision.
Simulated Results
The capability to estimate the noise signal successively and the capability to cancel noise on the basis of the estimated noise component were compared with those of the conventional technology and evaluated to verify the effects of this embodiment.
Parameters λ and κ required to initialize the process were set to 0.96 and 0.99, respectively.
To simulate a noise environment, two types of noise, namely, artificially modulated white noise and bubble noise (crowd noise), were prepared. Modulated white noise is highly time-varying noise whose characteristics change greatly in time, and bubble noise is slightly time-varying noise whose characteristics change relatively slowly. These types of noise were mixed with clean speech at different SNRs, and the noise estimation performance and noise cancellation performance were tested. The noise cancellation method used here was the spectrum subtraction method (reference 2), which obtains a noise-cancelled power spectrum by subtracting the power spectrum of a noise signal estimated according to the first embodiment from the power spectrum of the observed signal. A noise cancellation method that requires an estimated power spectrum of a noise signal for cancelling noise (reference 3) can also be combined, in addition to the spectrum subtraction method, with the noise estimation method according to the embodiment.

    (Reference 2) P. Loizou, “Speech Enhancement Theory and Practice”, CRC Press, Boca Raton, 2007
    (Reference 3) Y. Ephraim, D. Malah, “Speech enhancement using a minimum mean square error short-time spectral amplitude estimator”, IEEE Trans. Acoust. Speech Sig. Process., December 1984, vol. ASSP-32, pp. 1109-1121


FIG. 6 shows successive noise estimation characteristics of the noise estimation apparatus 10 according to the first embodiment and the conventional noise estimation apparatus 90. The SNR was 10 dB at that time. FIG. 6 indicates that the noise estimation apparatus 10 successively estimated non-stationary noise effectively, whereas the noise estimation apparatus 90 could not follow sharp changes in noise and made big estimation errors.
FIG. 7 shows speech waveforms obtained by estimating noise with the noise estimation apparatus 10 and the noise estimation apparatus 90 and cancelling noise on the basis of the estimated variance of the noise signal. The waveform (a) represents clean speech; the waveform (b) represents speech with modulated white noise; the waveform (c) represents speech after noise is cancelled on the basis of noise estimation by the noise estimation apparatus 10; the waveform (d) represents speech after noise is cancelled on the basis of noise estimation by the noise estimation apparatus 90. In comparison with (d), (c) contains less residual noise. FIGS. 8 and 9 show the results of evaluation of the noise estimation apparatus 10 and the noise estimation apparatus 90 when compared in a modulated-white-noise environment and a bubble-noise environment. Here, the segmental SNR and PESQ value (reference 4) were used as evaluation criteria.

    (Reference 4) P. Loizou, “Speech Enhancement Theory and Practice”, CRC Press, Boca Raton, 2007


In the modulated-white-noise environment (see FIG. 8), the noise estimation apparatus 10 showed a great advantage over the noise estimation apparatus 90. In the bubble-noise environment (see FIG. 9), the noise estimation apparatus 10 showed slightly better performance than the noise estimation apparatus 90.
Modifications
Although β1,i-1 is calculated in the step (s41) of obtaining the first variance σy,i,12 in this embodiment, β1,i-1 calculated in the step (s45) of obtaining the second variance σy,i-1,22 in the immediately preceding frame i−1 may be stored and used. In that case, there is no need to store the speech posterior probability η1,i(α0,i-1,θi-1) and the non-speech posterior probability η0,i(α0,i-1,θi-1) in the storage unit 120.
Although c0,i is calculated in the step (s44) of obtaining the variance σv,i2 in this embodiment, c0,i calculated in the step (s43) of obtaining prior probabilities in the prior probability estimation unit 115 may be received and used. Likewise, although c1,i is calculated in the step (s45) of obtaining the second variance σy,i,22, c1,i calculated in the step (s43) of obtaining prior probabilities in the prior probability estimation unit 115 may be received and used.
Although the first variance σy,i,12 and the second variance σy,i,22 are estimated by the observed signal variance estimation unit 111 in this embodiment, a first observed signal variance estimation unit and a second observed signal variance estimation unit may be provided instead of the observed signal variance estimation unit 111, and the first variance σy,i,12 and the second variance σy,i,22 may be estimated respectively by the first observed signal variance estimation unit and the second observed signal variance estimation unit. The observed signal variance estimation unit 111 in this embodiment includes the first observed signal variance estimation unit and the second observed signal variance estimation unit.
The first variance σy,i,12 need not be estimated (s41). The functional block diagram and the processing flow of the likelihood maximization unit 110 in that case are shown in FIG. 10 and FIG. 11 respectively. Let the variance of the observed signal in the current frame i be σy,i2. The posterior probability estimation unit 113 performs estimation by using the variance σy,i-12 in the immediately preceding frame i−1 instead of the first variance σy,i,12. In that case, there is no need to store the speech posterior probability η1,i(α0,i-1,θi-1) and the non-speech posterior probability η0,i(α0,i-1,θi-1) in the storage unit 120. However, a higher noise estimation precision can be achieved through obtaining the first variance σy,i,12 by using βi-1, calculating βi, and then making an adjustment to obtain the second variance σy,i,22. This is because all the parameters are estimated in a form matching the current observation by using the first variance, in which the complex spectrum of the observed signal in the current frame is reflected, rather than by using the variance of the immediately preceding frame. Not estimating the first variance σy,i,12 has the advantage of reducing the amount of calculation in comparison with the first embodiment and has the disadvantage of a low noise estimation precision.
In step s4 in this embodiment, the likelihood maximization unit 110 obtains the speech prior probability α1,i, the non-speech prior probability α0,i, the non-speech posterior probability η0,1, the speech posterior probability η1,i, and the variance σx,i2 of the desired signal in the current frame i in order to perform successive estimation of the variance σv,i2 of the noise signal in the current frame i (to estimate the variance σv,i2 of the noise signal in the subsequent frame i+1 as well). If just the variance σv,i2 of the noise signal in the current frame i should be estimated, there is no need to obtain the speech prior probability α1,i, the non-speech prior probability α0,i, the non-speech posterior probability η0,i, the speech posterior probability η1,i, and the variance σx,i2 of the desired signal in the current frame i.
Although the parameters estimated in the frame i−1 immediately preceding the current frame i are taken from the storage unit 120 in step s4 in this embodiment, the parameters do not always have to pertain to the immediately preceding frame i−1, and parameters estimated in a given past frame i−τ may be taken from the storage unit 120, where τ is an integer not smaller than 1.
Although the observed signal variance estimation unit 111 estimates the first variance σy,i,12 of the observed signal in the current frame i on the basis of the speech posterior probability η1,i-1(α0,i-2,θi-2) estimated in the immediately preceding frame i−1 by using parameters α0,i-2 and θi-2 estimated in the second preceding frame i−2, the first variance σy,i,12 of the observed signal in the current frame i may be estimated on the basis of the speech posterior probability estimated in an earlier frame i−τ by using parameters α0,i-τ′ and θi-τ′ estimated in a frame i−τ′ before the frame i−τ. Here, τ′ is an integer larger than τ.
In step s4 in this embodiment, when the complex spectrum Yi of the observed signal in the current frame i is received, the parameters are obtained by using the complex spectra Y0, Y1, . . . , Yi of the observed signal up to the current frame i, such that the following is maximized.




  
    
      Q
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    ⁡
    
      (
      
        
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                  ′
                
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          ⁢
          
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            ⁡
            
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                  ⁡
                  
                    (
                    
                      
                        
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                    )
                  
                
              
              ]
            
          
        
      
    
  




Here, Q(α0, θ) may be obtained by using all values of the complex spectra Y0, Y1, . . . , Yi of the observed signal up to the current frame i. Alternatively, the parameters may also be obtained by using Qi-1 obtained in the immediately preceding frame i−1 and the complex spectrum Yi of the observed signal in the current frame i (by indirectly using the complex spectra Y0, Y1, . . . , Yi-1 of the observed signal up to the immediately preceding frame i−1) such that the following is maximized.




  
    
      Q
      i
    
    ⁡
    
      (
      
        
          α
          0
        
        ,
        θ
      
      )
    
  
  =
  
    
      
        Q
        
          i
          -
          1
        
      
      ⁡
      
        (
        
          
            α
            0
            ′
          
          ,
          
            θ
            ′
          
        
        )
      
    
    +
    
      
        ∑
        
          s
          =
          0
        
        1
      
      ⁢
      
        
      
      ⁢
      
        
          
            η
            
              s
              ,
              t
            
          
          ⁡
          
            (
            
              
                α
                0
                ′
              
              ,
              
                θ
                ′
              
            
            )
          
        
        ⁢
        
          log
          ⁡
          
            [
            
              
                α
                s
              
              ⁢
              
                p
                ⁡
                
                  (
                  
                    
                      
                        Y
                        i
                      
                      ❘
                      
                        H
                        s
                      
                    
                    ;
                    θ
                  
                  )
                
              
            
            ]
          
        
      
    
  




Therefore, Qi(α0, θ) should be obtained by using at least the complex spectrum Yi of the observed signal of the current frame.
In step s4 in this embodiment, the parameters are determined to maximize Qi(α0, θ). This value should not always be maximized at once. Parameter estimation on the likelihood maximization criterion can be performed by repeating several times the step of determining the parameters such that the value Qi(α0, θ) based on the log likelihood log [αsp(Yi|Hs;θ)] after the update is larger than the value Qi(α0, θ) based on the log likelihood log [αsp(Yi|Hs;θ)] before the update.
The present invention is not limited to the embodiment and the modifications described above. For example, each type of processing described above may be executed not only time sequentially according to the order of description but also in parallel or individually when necessary or according to the processing capabilities of the apparatus executing the processing. Appropriate changes can be made without departing from the scope of the present invention.
Program and Recording Medium
The noise estimation apparatus described above can also be implemented by a computer. A program for making the computer function as the target apparatus (apparatus having the functions indicated in the drawings in each embodiment) or a program for making the computer carry out the steps of procedures (described in each embodiment) should be loaded into the computer from a recording medium such as a CD-ROM, a magnetic disc, or a semiconductor storage or through a communication channel, and the program should be executed.
INDUSTRIAL APPLICABILITY
The present invention can be used as an elemental technology of a variety of acoustic signal processing systems. Use of the technology of the present invention will help improve the overall performance of the systems. Systems in which the process of estimating a noise component included in a generated speech signal can be an elemental technology that can contribute to the improvement of the performance include the following. Speech recorded in actual environments always includes noise, and the following systems are assumed to be used in those environments.
1. Speech recognition system used in actual environments
2. Machine control interface that gives a command to a machine in response to human speech and man-machine dialog apparatus
3. Music information processing system that searches for or transcripts a piece of music by eliminating noise from a song sung by a person, music played on an instrument, or music output from a speaker
4. Voice communication system which collects a voice by using a microphone, eliminates noise from the collected voice, and allows the voice to be reproduced by a remote speaker.

## Claims

1. A noise estimation apparatus comprising:
circuitry configured to
receive, as an input, complex spectra of inputted observed waveform signals, which are acoustic signals that include clean speech mixed with a noise signal, up to a current frame;
obtain a variance of the noise signal, where the noise signal follows a complex Gaussian distribution, such that a value of weighted addition of sums becomes large, wherein:
each of the sums is obtained by adding a first product and a second product; the first product in each frame is a product of a log likelihood of a model of an observed signal expressed by a Gaussian distribution in a speech segment and a speech posterior probability; and the second product in each frame is a product of a log likelihood of a model of an observed signal expressed by a Gaussian distribution in a non-speech segment and a non-speech posterior probability; and
the circuitry is further configured to estimate a variance σv,i2 of the noise signal in the current frame i by weighted addition of a complex spectrum Yi of an observed signal in the current frame i and a variance σv,i-τ2 of the noise signal estimated in a past frame i−τ, where τ is an integer greater than 1, on the basis of a non-speech posterior probability estimated in the current frame i,

wherein the circuitry is configured to output the variance σv,i2 of the noise signal for cancellation of the noise signal from the acoustic signals, wherein the cancellation of the noise signal includes subtracting a power spectrum of the noise signal, which is estimated based on the outputted variance σv,i2, from a power spectrum of the observed waveform signals.

2. The noise estimation apparatus according to claim 1, wherein the observed waveform signals include an observed signal in the current frame, and the circuitry is configured to obtain the variance of the noise signal, a speech prior probability, a non-speech prior probability, and a variance of a desired signal such that the value of the weighted addition of the sums becomes large.

3. The noise estimation apparatus according to claim 1, wherein a greater weight in the weighted addition is assigned to a frame closer to the current frame.

4. The noise estimation apparatus according to claim 2, wherein a greater weight in the weighted addition is assigned to a frame closer to the current frame.

5. The noise estimation apparatus according to one of claims 1 to 3 and 4, wherein the circuitry is further configured to estimate a first variance σy,i,12 of the observed signal in the current frame i by weighted addition of the complex spectrum Yi of the observed signal in the current frame i and a second variance σy,i-τ,22 of the observed signal estimated in the past frame i−τ, on the basis of the speech posterior probability estimated in the past frame i−τ;
estimate a speech posterior probability η1,i(α0,i-τ,θi-τ) and a non-speech posterior probability η0,i(α0,i-τ,θi-τ) for the current frame i by using the complex spectrum Yi of the observed signal and the first variance σy,i,12 of the observed signal in the current frame and a speech prior probability α1,i-τ and a non-speech prior probability α0,i-τ estimated in the past frame i−τ, assuming that the complex spectrum Yi of the observed signal in the non-speech segment follows a Gaussian distribution determined by the variance σv,i-τ2 of the noise signal and assuming that the complex spectrum Yi of the observed signal in the speech segment follows a Gaussian distribution determined by the variance σv,i-τ2 of the noise signal and the first variance σy,i,12 of the observed signal;
estimate values obtained by weighted addition of speech posterior probabilities and weighted addition of non-speech posterior probabilities estimated up to the current frame i as a speech prior probability α1,i and a non-speech prior probability α0,i, respectively; and
estimate a second variance σy,i,22 of the observed signal in the current frame i by weighted addition of the complex spectrum Yi of the observed signal in the current frame i and the second variance σy,i-τ,22 of the observed signal estimated in the past frame i−τ, on the basis of the speech posterior probability estimated in the current frame i.

6. The noise estimation apparatus according to one of claims 1 to 3 and 4, wherein the circuitry is further configured to
estimate a speech posterior probability η1,i(α0,i-τ,θ1-τ) and a non-speech posterior probability η0,i(α0,i-τ,θi-τ) for the current frame i by using the complex spectrum Yi of the observed signal in the current frame i and a variance σy,i-τ2 of the observed signal, a speech prior probability α1,i-τ, and a non-speech prior probability α0,i-τ estimated in the past frame i−τ, assuming that the complex spectrum Yi of the observed signal in the non-speech segment follows a Gaussian distribution determined by the variance of the noise signal and assuming that the complex spectrum Yi of the observed signal in the speech segment follows a Gaussian distribution determined by the variance σv,i-τ2 of the noise signal and a variance σy,i2 of the observed signal;
estimate values obtained by weighted addition of speech posterior probabilities and weighted addition of non-speech posterior probabilities estimated up to the current frame i as a speech prior probability α1,i and a non-speech prior probability α0,i, respectively; and
estimate the variance σy,i2 of the observed signal in the current frame i by weighted addition of the complex spectrum Yi of the observed signal in the current frame i and the variance σy,i-τ2 of the observed signal estimated in the past frame i−τ, on the basis of the speech posterior probability estimated in the current frame i.

7. The noise estimation apparatus according to claim 5, wherein the circuitry is further configured to
estimate the first variance σy,i,12 of the observed signal in the current frame i, as given below, by using the complex spectrum Yi of the observed signal in the current frame i and the second variance σy,i-τ,22 of the observed signal estimated in the past frame i−τ, where 0<λ<1 and is an integer larger than τ




  
    
      θ
      
        i
        -
        
          τ
          ′
        
      
    
    =
    
      
        [
        
          
            σ
            
              v
              ,
              
                i
                -
                
                  τ
                  ′
                
              
            
            2
          
          ,
          
            σ
            
              x
              ,
              
                i
                -
                
                  τ
                  ′
                
              
            
            2
          
        
        ]
      
      T
    
  




  
    
      c
      
        1
        ,
        
          i
          -
          τ
        
      
    
    =
    
      
        λ
        ⁢
        
          
        
        ⁢
        
          c
          
            1
            ,
            
              i
              -
              
                τ
                ′
              
            
          
        
      
      +
      
        
          η
          
            1
            ,
            
              i
              -
              τ
            
          
        
        ⁡
        
          (
          
            
              α
              
                0
                ,
                
                  i
                  -
                  
                    τ
                    ′
                  
                
              
            
            ,
            
              θ
              
                i
                -
                
                  τ
                  ′
                
              
            
          
          )
        
      
    
  




  
    
      β
      
        1
        ,
        
          i
          -
          τ
        
      
    
    =
    
      
        
          n
          
            1
            ,
            
              i
              -
              τ
            
          
        
        ⁡
        
          (
          
            
              α
              
                0
                ,
                
                  i
                  -
                  
                    τ
                    ′
                  
                
              
            
            ,
            
              θ
              
                i
                -
                
                  τ
                  ′
                
              
            
          
          )
        
      
      
        c
        
          1
          ,
          
            i
            -
            τ
          
        
      
    
  




  
    
      
        σ
        
          y
          ,
          i
          ,
          1
        
        2
      
      =
      
        
          
            (
            
              1
              -
              
                β
                
                  1
                  ,
                  
                    i
                    -
                    τ
                  
                
              
            
            )
          
          ⁢
          
            σ
            
              y
              ,
              
                i
                -
                τ
              
              ,
              2
            
            2
          
        
        +
        
          
            β
            
              1
              ,
              
                i
                -
                τ
              
            
          
          ⁢
          
            
              
              
                Y
                i
              
              
            
            2
          
        
      
    
    ,
  


estimate the speech posterior probability η1,i(α0,i-τ,θi-τ) and the non-speech posterior probability η0,i(α0,i-τ,θi-τ) for the current frame i, as given below, by using the complex spectrum Yi of the observed signal and the first variance σy,i,12 of the observed signal in the current frame i and the speech prior probability α1,i-τ, the non-speech prior probability α0,i-τ, and the variance σv,i-τ2 of the noise signal estimated in the past frame where s=0 or s=1




  
    
      
    
    ⁢
    
      
        σ
        
          x
          ,
          
            i
            -
            τ
          
        
        2
      
      =
      
        
          σ
          
            y
            ,
            i
            ,
            1
          
          2
        
        -
        
          σ
          
            v
            ,
            
              i
              -
              τ
            
          
          2
        
      
    
  




  
    
      
    
    ⁢
    
      
        p
        ⁡
        
          (
          
            
              
                Y
                i
              
              |
              
                H
                0
              
            
            ;
            
              θ
              
                i
                -
                τ
              
            
          
          )
        
      
      =
      
        
          1
          
            πσ
            
              v
              ,
              
                i
                -
                τ
              
            
            2
          
        
        ⁢
        
          e
          
            
              
                
                
                  Y
                  i
                
                
              
              2
            
            
              σ
              
                v
                ,
                
                  i
                  -
                  τ
                
              
              2
            
          
        
      
    
  




  
    
      
    
    ⁢
    
      
        p
        ⁡
        
          (
          
            
              
                Y
                i
              
              |
              
                H
                1
              
            
            ;
            
              θ
              
                i
                -
                τ
              
            
          
          )
        
      
      =
      
        
          1
          
            π
            ⁡
            
              (
              
                
                  σ
                  
                    v
                    ,
                    
                      i
                      -
                      τ
                    
                  
                  2
                
                +
                
                  σ
                  
                    x
                    ,
                    
                      i
                      -
                      τ
                    
                  
                  2
                
              
              )
            
          
        
        ⁢
        
          e
          
            
              
                
                
                  Y
                  i
                
                
              
              2
            
            
              
                σ
                
                  v
                  ,
                  
                    i
                    -
                    τ
                  
                
                2
              
              +
              
                σ
                
                  x
                  ,
                  
                    i
                    -
                    τ
                  
                
                2
              
            
          
        
      
    
  




  
    
      
        η
        
          s
          ,
          i
        
      
      ⁡
      
        (
        
          
            α
            
              0
              ,
              
                i
                -
                τ
              
            
          
          ,
          
            θ
            
              i
              -
              τ
            
          
        
        )
      
    
    =
    
      
        
          α
          
            s
            ,
            
              i
              -
              τ
            
          
        
        ⁢
        
          p
          ⁡
          
            (
            
              
                
                  Y
                  i
                
                |
                
                  H
                  s
                
              
              ;
              
                θ
                
                  i
                  -
                  τ
                
              
            
            )
          
        
      
      
        
          
            α
            
              0
              ,
              
                i
                -
                τ
              
            
          
          ⁢
          
            p
            ⁡
            
              (
              
                
                  
                    Y
                    i
                  
                  |
                  
                    H
                    0
                  
                
                ;
                
                  θ
                  
                    i
                    -
                    τ
                  
                
              
              )
            
          
        
        +
        
          
            (
            
              1
              -
              
                α
                
                  0
                  ,
                  
                    i
                    -
                    τ
                  
                
              
            
            )
          
          ⁢
          
            p
            ⁡
            
              (
              
                
                  
                    Y
                    i
                  
                  |
                  
                    H
                    1
                  
                
                ;
                
                  θ
                  
                    i
                    -
                    τ
                  
                
              
              )
            
          
        
      
    
  


estimate the speech prior probability α1,i and the non-speech prior probability α0,i, as given below, by using the speech posterior probability η1,i(α0,i-τ,θi-τ) and the non-speech posterior probability η0,i(α0,i-τ,θi-τ) estimated in the current frame i




  
    
      c
      
        s
        ,
        i
      
    
    =
    
      
        λ
        ⁢
        
          
        
        ⁢
        
          c
          
            s
            ,
            
              i
              -
              τ
            
          
        
      
      +
      
        
          η
          
            s
            ,
            i
          
        
        ⁡
        
          (
          
            
              α
              
                0
                ,
                
                  i
                  -
                  τ
                
              
            
            ,
            
              θ
              
                i
                -
                τ
              
            
          
          )
        
      
    
  




  
    
      c
      i
    
    =
    
      
        c
        
          0
          ,
          i
        
      
      +
      
        c
        
          1
          ,
          i
        
      
    
  




  
    
      
        α
        
          s
          ,
          i
        
      
      =
      
        
          c
          
            s
            ,
            i
          
        
        
          c
          i
        
      
    
    ,
  


estimate the variance σv,i2 of the noise signal in the current frame i, as given below, by using the complex spectrum Yi of the observed signal, the non-speech posterior probability η0,1(α0,i-τ,θi-τ) estimated in the current frame i, and the variance σv,i-τ2 of the noise signal estimated in the past frame i−τ




  
    
      β
      
        0
        ,
        i
      
    
    =
    
      
        
          η
          
            0
            ,
            i
          
        
        ⁡
        
          (
          
            
              α
              
                0
                ,
                
                  i
                  -
                  τ
                
              
            
            ,
            
              θ
              
                i
                -
                τ
              
            
          
          )
        
      
      
        c
        
          0
          ,
          i
        
      
    
  




  
    
      
        σ
        
          v
          ,
          i
        
        2
      
      =
      
        
          
            (
            
              1
              -
              
                β
                
                  0
                  ,
                  i
                
              
            
            )
          
          ⁢
          
            σ
            
              v
              ,
              
                i
                -
                τ
              
            
            2
          
        
        +
        
          
            β
            
              0
              ,
              i
            
          
          ⁢
          
            
              
              
                Y
                i
              
              
            
            2
          
        
      
    
    ,
    and
  


estimate the second variance σy,i,22 of the observed signal in the current frame i, as given below, by using the complex spectrum Yi of the observed signal in the current frame i, the speech posterior probability η1,i(α0,i-τ,θi-τ) estimated in the current frame i, and the second variance σy,i-τ,22 of the observed signal estimated in the past frame i−τ




  
    
      β
      
        1
        ,
        i
      
    
    =
    
      
        
          n
          
            1
            ,
            i
          
        
        ⁡
        
          (
          
            
              α
              
                0
                ,
                
                  i
                  -
                  τ
                
              
            
            ,
            
              θ
              
                i
                -
                τ
              
            
          
          )
        
      
      
        c
        
          1
          ,
          i
        
      
    
  




  
    
      σ
      
        y
        ,
        i
        ,
        2
      
      2
    
    =
    
      
        
          (
          
            1
            -
            
              β
              
                1
                ,
                i
              
            
          
          )
        
        ⁢
        
          σ
          
            y
            ,
            
              i
              -
              τ
            
            ,
            2
          
          2
        
      
      +
      
        
          β
          
            1
            ,
            i
          
        
        ⁢
        
          
            
            
              Y
              i
            
            
          
          2
        
        ⁢
        
          c
          .

8. A noise estimation method comprising:
a step, by circuitry of a noise estimation apparatus, of receiving, as an input, complex spectra of inputted observed waveform signals, which are acoustic signals that include clean speech mixed with a noise signal, up to a current frame;
obtaining a variance of the noise signal, where the noise signal follows a complex Gaussian distribution, such that a value of weighted addition of sums becomes large, wherein:
each of the sums is obtained by adding a first product and a second product; the first product in each frame is a product of a log likelihood of a model of an observed signal expressed by a Gaussian distribution in a speech segment and a speech posterior probability; and the second product in each frame is a product of a log likelihood of a model of an observed signal expressed by a Gaussian distribution in a non-speech segment and a non-speech posterior probability; and
the method includes estimating, by the circuitry, a variance σv,i2 of the noise signal in the current frame i by weighted addition of a complex spectrum Yi of an observed signal in the current frame i and a variance σv,i-τ2 of the noise signal estimated in a past frame where τ is an integer greater than 1, on the basis of a non-speech posterior probability estimated in the current frame, and
outputting the variance σv,i2 of the noise signal for cancellation of the noise signal from the acoustic signals, wherein the cancellation of the noise signal includes subtracting a power spectrum of the noise signal, which is estimated based on the outputted variance σv,i2 from a power spectrum of the observed waveform signals.

9. The noise estimation method according to claim 8, wherein in the step, the observed waveform signals include an observed signal in the current frame, and the variance of the noise signal, a speech prior probability, a non-speech prior probability and a variance of a desired signal such that the value of the weighted addition of the sums becomes large are obtained.

10. The noise estimation method according to claim 8, wherein a greater weight in the weighted addition is assigned to a frame closer to the current frame.

11. The noise estimation method according to claim 9, wherein a greater weight in the weighted addition is assigned to a frame closer to the current frame.

12. The noise estimation method according to one of claims 8-10 and 11, further comprising:
a first observed signal variance estimation step of estimating a first variance σy,i,12 of the observed signal in the current frame i by weighted addition of the complex spectrum Yi of the observed signal in the current frame i and a second variance σy,i-τ,22 of the observed signal estimated in the past frame i−τ, on the basis of the speech posterior probability estimated in the past frame i−τ;
a posterior probability estimation step of estimating a speech posterior probability η1,i(α0,i-τ,θi-τ) and a non-speech posterior probability η0,i(α0,i-τ,θi-τ) for the current frame i by using the complex spectrum Yi of the observed signal and the first variance σy,i,12 of the observed signal in the current frame and a speech prior probability α1,i,τ and a non-speech prior probability α0,i-τ estimated in the past frame i−τ, assuming that the complex spectrum Yi of the observed signal in the non-speech segment follows a Gaussian distribution determined by the variance σv,i-τ2 of the noise signal and assuming that the complex spectrum Yi of the observed signal in the speech segment follows a Gaussian distribution determined by the variance σv,i-τ2 of the noise signal and the first variance σy,i,12 of the observed signal, and
a prior probability estimation step of estimating values obtained by weighted addition of speech posterior probabilities and weighted addition of non-speech posterior probabilities estimated up to the current frame i as a speech prior probability α1,i and a non-speech prior probability α0,i, respectively; and
a second observed signal variance estimation step of estimating a second variance σy,i,22 of the observed signal in the current frame i by weighted addition of the complex spectrum Yi of the observed signal in the current frame i and the second variance σy,i-τ,22 of the observed signal estimated in the past frame i−τ, on the basis of the speech posterior probability estimated in the current frame i.

13. The noise estimation method according to one of claims 8-10 and 11, further comprising:
a posterior probability estimation step of estimating a speech posterior probability η1,i(α0,i-τ,θi-τ) and a non-speech posterior probability η0,i(α0,i-τ,θi-τ) for the current frame i by using the complex spectrum Yi of the observed signal in the current frame i and a variance σy,i-τ2 of the observed signal, a speech prior probability α1,i-τ, and a non-speech prior probability α0,i-τ estimated in the past frame i−τ, assuming that the complex spectrum Yi of the observed signal in the non-speech segment follows a Gaussian distribution determined by the variance σy,i-τ2 of the noise signal and assuming that the complex spectrum Yi of the observed signal in the speech segment follows a Gaussian distribution determined by the variance σv,i-τ2 of the noise signal and a variance σy,i2 of the observed signal;
a prior probability estimation step of estimating values obtained by weighted addition of speech posterior probabilities and weighted addition of non-speech posterior probabilities estimated up to the current frame i as a speech prior probability α1,i and a non-speech prior probability α0,i, respectively; and
an observed signal variance estimation step of estimating the variance σy,i2 of the observed signal in the current frame i by weighted addition of the complex spectrum Yi of the observed signal in the current frame i and the variance σy,i-τ2 of the observed signal estimated in the past frame i−τ, on the basis of the speech posterior probability estimated in the current frame i.

14. A non-transitory computer-readable recording medium having recorded thereon a noise estimation program which when executed by a noise estimation apparatus, causes the noise estimation apparatus to perform a method comprising:
a step, by circuitry of a noise estimation apparatus, of receiving, as an input, complex spectra of inputted observed waveform signals, which are acoustic signals that include clean speech mixed with a noise signal, up to a current frame;
obtaining a variance of the noise signal, where the noise signal follows a complex Gaussian distribution, such that a value of weighted addition of sums becomes large, wherein:
each of the sums is obtained by adding a first product and a second product, the first product in each frame is a product of a log likelihood of a model of an observed signal expressed by a Gaussian distribution in a speech segment and a speech posterior probability; and the second product in each frame is a product of a log likelihood of a model of an observed signal expressed by a Gaussian distribution in a non-speech segment and a non-speech posterior probability; and
the method includes estimating, by the circuitry, a variance σv,i2 of the noise signal in the current frame i by weighted addition of a complex spectrum Yi of an observed signal in the current frame i and a variance σv,i-τ2 of the noise signal estimated in a past frame where τ is an integer greater than 1, on the basis of a non-speech posterior probability estimated in the current frame, and
outputting the variance σv,i2 of the noise signal for cancellation of the noise signal from the acoustic signals, wherein the cancellation of the noise signal includes subtracting a power spectrum of the noise signal, which is estimated based on the outputted variance σv,i2, from a power spectrum of the observed waveform signals.