JP2018141924A - Vocal tract spectrum estimation apparatus, method, and program - Google Patents

Abstract

A vocal tract spectrum can be accurately estimated from a speech signal. A time-frequency expansion unit 24 receives time-series data of an audio signal and outputs an observation spectrogram. The parameter estimator 36 represents, for each of the Gaussian distribution functions corresponding to each peak, an average represented by a set of K formant frequencies and a coupling coefficient, and a weight combined with a formant intensity for the set of K formant frequencies. The criterion expressing the magnitude of the error between the mixed Gaussian function model at each time and the observed spectrogram when the variance is expressed in terms of formant variance and coupling coefficient for a set of K formant frequencies is reduced. Then, a set of K formant frequencies and a coupling coefficient at each time, a formant intensity, a formant dispersion, and a coupling coefficient at each time are estimated. [Selection] Figure 5

Description

  The present invention relates to a vocal tract spectrum estimation apparatus, method, and program, and more particularly, to a vocal tract spectrum estimation apparatus, method, and program for estimating a vocal tract spectrum from a speech signal.

  According to the speech source filter theory (FIG. 1), the spectrum envelope corresponds to the resonance characteristics of the vocal tract (the vocal tract spectrum) and contains a wealth of information relating to voice quality and phonology. Since there are only a limited number of vowels and phonemes in normal speech, assuming that the vocal tract spectrum at each time can be approximated by a non-negative combination of a finite number of templates, the vocal tract spectrogram is represented by the product of two non-negative matrices. can do. Approximating a non-negative matrix with the product of two non-negative matrices is called non-negative matrix factorization (NMF), and the matrix product representation of vocal tract spectrograms using NMF is useful in some applications. It is. Two examples are given below.

  First of all, this expression allows the vocal tract spectrogram to be decomposed into a factor related to the voice quality of the speaker that does not depend on time and a factor related to the time-dependent speech content and speech style, so that only the former factor remains fixed. By reconstructing the spectrogram by converting, it is possible to change only the voice quality without changing the speech content (Non-Patent Documents 1 and 2). Such a technique is called voice conversion.

  The other is application to vocal tract spectrum estimation for speech analysis and synthesis. In general voice processing including voice synthesis and voice conversion, a technique for estimating a vocal tract spectrum from a voice signal is used in many scenes. If the speech signal for each short interval can be modeled as the output of a linear time-invariant system with a periodic delta function (pulse train) as input, the input and impulse response of this linear system correspond to the vocal cord source signal and vocal tract characteristics, respectively. To do. This assumption is equivalent to representing a speech spectrum in the frequency domain by the product of a vocal cord source spectrum and a vocal tract spectrum represented by a periodic delta function. Accordingly, the voice spectrum can be regarded as a sample of the vocal tract spectrum periodically (at a fundamental frequency interval). STRAIGHT, which is widely known as one of typical vocal tract spectrum estimation methods, is a method in which a speech signal is cut out with the width of the basic period, and the spectrum of the cut-out signal is used as an estimated value of the vocal tract spectrum. This corresponds to the fact that, in the frequency domain, a smooth interpolated peak of each harmonic component is regarded as a vocal tract spectrum. However, it is known that the vocal tract spectrum estimated value obtained by this method periodically changes with time depending on the offset of the cut-out frame even if stationary speech is the target. This is because each harmonic component interferes with each other, and an improved method has been proposed to eliminate this fluctuation component caused by interference between harmonic components. However, as described above, since the speech spectrum can be regarded as a sampling of the vocal tract spectrum at the fundamental frequency (F0) interval, the higher the speech F0, the fewer clues for estimating the vocal tract spectrum. This suggests that there is an essential limit to independent processing for each frame. On the other hand, since the types of vowels and phonemes are limited in normal utterances, a similar vocal tract spectrum appears at a plurality of different times. In principle, if multiple frames can be assumed to have a common vocal tract spectrum and if F0 is different in those frames, the actual observable vocal tract spectrum has more sample points than a single frame. Allows "super-resolution" estimation of the vocal tract spectrum. Based on this concept, a method has been proposed for estimating the vocal tract spectrum with high accuracy by using NMF to match only the fundamental frequency components and harmonic components in the speech spectrogram, and using the speech spectrum of multiple frames as a clue. (Non-patent Document 3).

R. Takashima, R. Aihara, T. Takiguchi, and Y. Ariki, “Exemplar-based voice conversion usin sparse representation in noisy environments,” IEICE Transactions on Information and Systems, vol. E96-A, no. 10, pp. 1946-1953, 2013. Z. Wu, T. Virtanen, T. Kinnunen, E.S.Chng, and H. Li, “Exemplar-based voice conversion using non-negative spectrogram deconvolution,” Proc. 8th ISCA Speech Synthesis Workshop, pp. 201-206, 2013. Nakamura, Kameoka, “Examination of vocal tract spectrum estimation method by missing data interpolation based on non-negative matrix factorization,” Oncology (Spring), 3-P-33, 393-396, 2016.

  As described above, the matrix product representation of the vocal tract spectrogram by NMF has the potential to provide a useful solution to various important issues in speech signal processing. However, since this representation can only represent the vocal tract spectrum by linear combination of templates, it is difficult to represent a continuous peak locus appearing in the vocal tract spectrogram as shown in FIG. 2 (FIG. 3). Several peaks appearing in this vocal tract spectrum are called formants, and each frequency corresponds to the resonance frequency of the vocal tract and is considered to be a quantity characterizing the type and quality of the vowel.

  The present invention has been made in view of the above circumstances, and an object thereof is to provide a vocal tract spectrum estimation apparatus, method, and program capable of accurately estimating a vocal tract spectrum from a speech signal.

  In order to achieve the above object, a vocal tract spectrum estimation device according to the present invention includes a time-frequency expansion unit that outputs time-series data of an audio signal and outputs an observation spectrogram representing a signal component at each time and each frequency. A mixed Gaussian function model in which each peak of the spectral envelope at each time is approximated by a Gaussian distribution function based on the observed spectrogram output by the time-frequency expansion unit, and each of the Gaussian distribution functions corresponding to each peak In contrast, the mean is represented by a set of K formant frequencies and coupling coefficients, the weight is represented by the formant intensity and the coupling coefficient for the set of K formant frequencies, and the variance is represented by the K formant frequencies. The mixed Ga at each time, expressed as formant dispersion and coupling coefficient for the set The set of K formant frequencies and the coupling coefficient at each time for each of the Gauss distribution functions, so as to reduce the criterion representing the magnitude of error between the ss function model and the observed spectrogram, and the K A parameter estimating unit that estimates formant intensity for a set of formant frequencies, and formant dispersion and a coupling coefficient at each time for a set of K formant frequencies.

  In the vocal tract spectrum estimation method according to the present invention, the time-frequency expansion unit receives time-series data of a speech signal, outputs an observation spectrogram representing a signal component at each time and each frequency, and the parameter estimation unit A mixed Gaussian function model that approximates each peak of the spectral envelope at each time with a Gaussian distribution function based on the observed spectrogram output by the time-frequency expansion unit, and for each Gaussian distribution function corresponding to each peak , The mean is represented by a set of K formant frequencies and coupling coefficients, the weight is represented by the formant intensity and the coupling coefficient for the set of K formant frequencies, and the variance is represented for the set of K formant frequencies. The mixed Gauss at each time, expressed as formant dispersion and coupling coefficient The set of K formant frequencies and the coupling coefficient at each time for each of the Gaussian distribution functions and the K number of coefficients for each of the Gaussian distribution functions so as to reduce the criterion representing the magnitude of error between the numerical model and the observed spectrogram. The formant intensity for a set of formant frequencies, the formant variance and the coupling coefficient at each time for a set of K formant frequencies are estimated.

  Moreover, the program of this invention is a program for functioning a computer as each part which comprises said vocal tract spectrum estimation apparatus.

  As described above, according to the vocal tract spectrum estimation apparatus, method, and program of the present invention, it is a mixed Gaussian function model that approximates each peak of the spectrum envelope at each time with a Gaussian distribution function, and corresponds to each peak. For each of the Gaussian distribution functions, the mean is represented by a set of K formant frequencies and a coupling coefficient, the weight is represented by the formant intensity and the coupling coefficient for the set of K formant frequencies, and the variance is The Gaussian distribution so as to reduce the criterion representing the magnitude of error between the mixed Gaussian function model at each time and the observed spectrogram when expressed by formant dispersion and coupling coefficient for a set of K formant frequencies. The set of K formant frequencies and each time for each of the functions Estimate the coupling coefficient, the formant intensity for the set of K formant frequencies, the formant variance for the set of K formant frequencies, and the coupling coefficient at each time to accurately estimate the vocal tract spectrum from the speech signal. can do.

It is a figure for demonstrating a source filter theory. It is a figure which shows the vocal tract spectrogram estimated by STRAIGHT analysis. It is a figure which shows the nonnegative matrix product which approximated the estimated vocal tract spectrogram. It is a figure which shows the expression of the spectrum envelope by GMM. It is a block diagram which shows the functional structure of the vocal tract spectrum estimation apparatus which concerns on embodiment of this invention. It is a flowchart figure which shows the parameter estimation processing routine in the vocal tract spectrum estimation apparatus which concerns on embodiment of this invention. It is a figure which shows the vocal tract spectrogram estimated by STRAIGHT analysis. It is a figure which shows the estimation result estimated with the method which concerns on this Embodiment with respect to the vocal tract spectrogram of FIG. It is a figure which shows the vocal tract spectrogram estimated by STRAIGHT analysis. It is a figure which shows the estimation result estimated with the method based on this Embodiment with respect to the vocal tract spectrogram of FIG.

  Hereinafter, embodiments of the present invention will be described in detail with reference to the drawings.

<Outline of the present embodiment>
In the embodiment of the present invention, instead of the conventional vocal tract spectrogram expression by linear combination of spectral templates, a formant frequency set and a formant frequency set template are considered, and the formant frequency set and formant frequency at each time by their linear combination are considered. A new vocal tract spectrogram representation with a set representation inside is proposed, and an analysis algorithm is proposed to estimate each template and their coupling coefficient at each time from the observed speech spectrogram. Specifically, we consider a mixed Gaussian function model (Gaussian Mixture Model; GMM) that approximates each peak of the spectral envelope with a Gaussian distribution function, and builds a model that expresses the peak frequency and weight of each Gaussian function as a linear combination of templates. We propose an optimization algorithm that estimates formant frequency sets and formant frequency set templates and their coupling coefficients to fit this spectral envelope model to the observed speech spectrum as much as possible.

<Vocal tract spectrogram model>
The type and quality of vowels is characterized by a formant frequency set. As can be seen from FIG. 2, the formant frequency during speech tends to continuously change over time. This is due to physical constraints of the vocal organs. Further, since the types of vowels used in normal utterances are limited, it can be assumed that the formant frequency set at each time can be expressed by a convex combination of a finite number of templates corresponding to each vowel. This is equivalent to expressing a matrix having each column vector as a formant frequency set at each time by two matrix products. Hereinafter, this is called formant frequency matrix product representation. In the following, a vocal tract spectrogram model incorporating the above formant frequency matrix product representation is constructed.

  A Gaussian Mixture Model (GMM) F (ω, t) that approximates each peak of the spectral envelope with a Gaussian distribution function

(Fig. 4). Where ω is frequency, t is time, I +1 is Gaussian function

Number of

Are the mean, variance, and weight of the i-th Gaussian function, respectively. Also,

Is

Shall be satisfied.

Is multiplied by 2.

This is because. In addition,

When

It is. Therefore, under the condition of Equation (3)

It is. Accordingly, β (t) represents the scale of the spectrum at time t 1.

The formant frequency of the kth vowel template

Formant strength

And Following the previous discussion,

The

It is expressed as a convex combination of K formant frequency sets and formant intensity set templates. However,

It is.

Similarly for formant dispersion

Using

It expresses. More than

Is an unknown parameter to be estimated. Note that any variable must be non-negative.

The formant frequency and formant intensity usually change continuously over time. there,

Smooth non-negative functions centered around time t _j (Gauss function, Hanning function, etc.) to ensure that becomes a smooth function

Using

The

Expressed as

Instead of

Can also be used as a parameter to be estimated. However,

And

<Parameter estimation algorithm by auxiliary function method>
<Formulation of optimization problem>
Observation spectrogram

age,

When

Unknown parameters so that is as close as possible

An optimization algorithm for obtaining is described. here

When

For example, I divergence

It can be measured with. However,

Is the set of unknown parameters. Also,

For the purpose of avoiding becoming too small

Consider the penalty function (logarithm negative of symmetric Dirichlet distribution). Since the symmetric Dirichlet distribution is maximized when all variables are equal, Equation (14) becomes

At the minimum. Below,

Consider the case where is expressed by equation (9) (

The delta function

If you replace

Is equivalent to a variable). From the above, the optimization problem is as follows.

  The stopping point of the above optimization problem can be searched by using a projection gradient method or the like, but here, an optimization algorithm in which convergence to the stopping point is guaranteed is derived based on the principle of the auxiliary function method.

<Auxiliary function method>

The

The objective function we want to minimize with respect to

A function that satisfies

Is called an auxiliary function, and α is called an auxiliary variable. If you can design such an auxiliary function,

When

By alternately repeating the objective function

The stop point can be obtained. This optimization method is called an auxiliary function method.

<Auxiliary function design>
First, negative logarithmic functions are more convex than Jensen's inequalities

I can say. However,

Is

Is a non-negative variable that satisfies

This holds true. Similarly,

I can say. However,

Is

Is a non-negative variable that satisfies

This holds true. Second, the positive logarithmic function is

I can say. However,

Is a non-negative variable and the equal sign is

This holds true. Since the quadratic function is a convex function, again from Jensen's inequality

I can say. However,

Is

Is a non-negative variable that satisfies

This holds true. Finally, since the reciprocal function is a convex function, Jensen's inequality is

I can say. However,

Is

Is a non-negative variable that satisfies

This holds true.

  In summary,

But the right side is

Satisfies the requirement as an auxiliary function. However,

Is an auxiliary variable

Is a set of Also,

Is a parameter

Also auxiliary variables

This is a summary of constant terms that do not depend on.

<Parameter update formula>
More auxiliary functions

Can be used to derive an update formula for each parameter.

<W _{i, k} update formula>

Than,

Update formula

Get.

_<a i,k Nokoshinshiki_>

Since it is necessary to satisfy the condition of Equation (7), Lagrangian

The partial derivative of

By putting

Than,

Update formula

Get.

<L _{j, k} update formula>

It is necessary to satisfy the condition of equation (11).

Minimize

Let's take a normalization method after obtaining.

The partial derivative of

By putting

Get. Therefore,

Minimize

Is

It becomes. In equation (38)

After updating

Normalize by

<C _{i, k} update formula>

Than,

Update formula

Get.

<Update formula of d _k (t)>

Than,

Update formula

Get.

<Update formula of β (t)>

Than,

Update formula

Get.

<Optimization algorithm>
The above optimization algorithm based on the auxiliary function method is as follows. ~ 3. It can be summarized as follows.

1.

Is initialized.

2. Update the auxiliary variable.

3. Update the parameters.

<Configuration of Vocal Tract Spectrum Estimation Device According to Embodiment of the Present Invention>
Next, the configuration of the vocal tract spectrum estimation apparatus according to the embodiment of the present invention will be described. As shown in FIG. 5, the vocal tract spectrum estimation apparatus 100 according to the embodiment of the present invention includes a CPU, a RAM, a ROM for storing a program and various data for executing a parameter estimation processing routine described later, It can comprise with the computer which includes. Functionally, the vocal tract spectrum estimation apparatus 100 includes an input unit 10, a calculation unit 20, and an output unit 90 as shown in FIG.

  The input unit 10 receives time-series data of an audio signal.

  The calculation unit 20 includes a time frequency expansion unit 24 and a parameter estimation unit 36.

  The time-frequency expansion unit 24 calculates an observation spectrogram Y that is an amplitude spectrogram or a power spectrogram representing a signal component of each frequency at each time based on the time-series data of the audio signal. In this embodiment, time frequency expansion such as short-time Fourier transform and wavelet transform is performed.

The parameter estimation unit 36, based on the observation spectrogram Y output by the time-frequency expansion unit 24, approximates each peak of the spectral envelope at each time with a Gaussian distribution function Gi (ω, t), and a mixed Gaussian function model F (ω T), for each of the Gaussian distribution functions G _i (ω, t) corresponding to each peak, the average μ _i (t) is a set of K formant frequencies w _{i, 1} ,. , w _{i, k} and coupling coefficient h _k (t), and weight α _i (t) is expressed as formant intensity a _{i, 1} ,..., a _{i, k} and coupling coefficient for a set of K formant frequencies. h _k (t) and the variance σ _i ² (t) is represented by the formant variances c _{i, 1} ,..., c _{i, k} and the coupling coefficient d _k (t) for a set of K formant frequencies. The mixed Gaussian function model F (ω, t) at each time and Observation spectrogram Y (omega, t) represents the magnitude of the error between, so as to reduce the criteria above (15), for each of the Gauss distribution function G _i (omega, t), K pieces of formant frequencies set _{w i, 1, ···, w} i, parameters for representing a _k and the time of the coupling coefficient hk (t) l _{j, k} and, formant intensity for a set of K formant frequencies a _{i, 1} ,..., A _{i, k} and the formant variances c _{i, 1} ,..., C _{i, k} for a set of K formant frequencies and the coupling coefficient h _k (t) at each time, and A weight β (t) is estimated.

  Specifically, the parameter estimation unit 36 includes an initial value setting unit 40, an auxiliary variable update unit 42, a parameter update unit 44, and a convergence determination unit 46.

The initial value setting unit 40 uses unknown parameters

An initial value is set for each of.

The auxiliary variable update unit 42 is an initial value or a parameter updated last time.

On the basis of the above equation (46) to equation (50)

Update.

The parameter update unit 44 includes the observation spectrogram Y output by the time frequency expansion unit 24 and the auxiliary variable updated by the auxiliary variable update unit 42.

And parameters that are initial values or updated last time

Based on the above, the parameters according to the above equations (51) to (57)

Update.

  The convergence determination unit 46 determines whether or not the convergence condition is satisfied, and repeats the update process in the auxiliary variable update unit 42 and the update process in the parameter update unit 44 until the convergence condition is satisfied, and is finally estimated. The output parameters are output from the output unit 90.

  As the convergence condition, for example, the fact that the number of repetitions has reached the upper limit number can be used. Alternatively, as the convergence condition, it can be used that the difference between the value of the criterion of the above formula (15) and the value of the previous criterion is equal to or less than a predetermined threshold value.

<Operation of Vocal Tract Spectrum Estimation Device According to Embodiment of the Present Invention>
Next, the operation of the vocal tract spectrum estimation apparatus 100 according to the embodiment of the present invention will be described. First, when time series data of a speech signal is received at the input unit 10, the vocal tract spectrum estimation apparatus 100 executes a parameter estimation processing routine shown in FIG.

  First, in step S100, an observation spectrogram Y is calculated based on the time series data of the audio signal received by the input unit 10.

In step S102, the unknown parameter

An initial value is set for each of.

In step S104, the parameter is an initial value or updated last time.

On the basis of the above equation (46) to equation (50)

Update.

Next, in step S106, the observation spectrogram Y obtained in step S100 and the auxiliary variable updated by the auxiliary variable update unit 42 are displayed.

And parameters that are initial values or updated last time

Based on the above, the parameters according to the above equations (51) to (57)

Update.

  Next, in step S108, it is determined whether a convergence condition is satisfied. If the convergence condition is satisfied, the process proceeds to step S110. If the convergence condition is not satisfied, the process proceeds to step S104, and the processes in steps S104 to S106 are repeated.

In step S110, the parameter finally updated in step S106 above.

Is output from the output unit 90, and the parameter estimation processing routine is terminated.

<Experimental example>
As shown in Figs.

In contrast, it was estimated by the method according to the embodiment of the present invention.

Are shown in FIGS. The number of templates was K = 10.

It was confirmed that the formant frequency trajectory was successfully estimated.

As described above, according to the vocal tract spectrum estimation apparatus according to the embodiment of the present invention, the mixed Gaussian function model F () in which each peak of the spectrum envelope at each time is approximated by the Gaussian distribution function Gi (ω, t). ω, t), for each Gaussian distribution function G _i (ω, t) corresponding to each peak, the average μ _i (t) is the set of K formant frequencies w _{i, 1} ,. .., W _{i, k} and coupling coefficient h _k (t), weight α _i (t) combined with formant strengths a _{i, 1} ,..., A _{i, k} for a set of K formant frequencies Expressed by the coefficient h _k (t), the variance σ _i ² (t) is expressed by the formant variances c _{i, 1} ,..., C _{i, k} and the coupling coefficient d _k (t) for a set of K formant frequencies. Mixed Gaussian function model F (ω, t) at each time and observation spectrolog Arm Y (ω, t) so as to reduce the criteria representative of the magnitude of the error between, Gauss distribution function G _i (ω, t) for each of the set w _{i, 1} of the K formant frequencies, - · ·, w _i, the parameters l _j to represent _k and the time of the coupling coefficient hk _(t), and _k, formant intensity a _{i, 1} for a set of K formant _frequencies, ···, a i, _k and, formant variance for a set of K formant frequencies _{c i, 1, ···, c} i, and the coupling coefficient of _k and the time h _k (t), the weight beta (t) and an estimate of the time By doing so, the vocal tract spectrum can be accurately estimated from the audio signal.

  Note that the present invention is not limited to the above-described embodiment, and various modifications and applications are possible without departing from the gist of the present invention.

  In addition, since the order of the parameters to be updated is arbitrary, the order of the above embodiments is not limited.

  Further, in the present specification, the embodiment has been described in which the program is installed in advance. However, the program can be provided by being stored in a computer-readable recording medium or provided via a network. It is also possible to do.

DESCRIPTION OF SYMBOLS 10 Input part 20 Operation part 24 Time frequency expansion part 36 Parameter estimation part 40 Initial value setting part 42 Auxiliary variable update part 44 Parameter update part 46 Convergence determination part 90 Output part 100 Vocal tract spectrum estimation apparatus

Claims (7)

A time-frequency expansion unit that outputs time-series data of an audio signal and outputs an observation spectrogram representing a signal component at each time and each frequency;
A mixed Gaussian function model in which each peak of the spectral envelope at each time is approximated by a Gaussian distribution function based on the observation spectrogram output by the time-frequency expansion unit, and each Gaussian distribution function corresponding to each peak On the other hand, the mean is represented by a set of K formant frequencies and a coupling coefficient, the weight is represented by the formant intensity and the coupling coefficient for the set of K formant frequencies, and the variance is the set of the K formant frequencies. For each of the Gaussian distribution functions, the criterion representing the magnitude of error between the mixed Gaussian function model at each time and the observed spectrogram when expressed in terms of formant dispersion and coupling coefficient for Set of K formant frequencies and coupling coefficient at each time A formant intensity for the set of the K formant frequency, a parameter estimation unit that estimates the coupling coefficient of the formant dispersion and the time for the set of K formant frequencies,
A vocal tract spectrum estimation apparatus including:   The vocal tract spectrum estimation apparatus according to claim 1, wherein the criterion further includes a penalty term relating to a magnitude of formant intensity for a set of K formant frequencies for each of the Gauss distribution functions. The parameter estimation unit includes:
For each of the Gaussian distribution functions, the set of K formant frequencies and the coupling coefficient at each time, and the set of K formant frequencies, so as to reduce the auxiliary function that is the upper bound function of the criterion. A parameter updater for updating the formant intensity, the formant dispersion for the set of K formant frequencies and the coupling coefficient at each time;
A convergence determination unit that repeats the update by the parameter update unit until a predetermined convergence condition is satisfied;
The vocal tract spectrum estimation apparatus according to claim 1, comprising: The time-frequency expansion unit receives the time-series data of the audio signal and outputs an observation spectrogram representing the signal components at each time and each frequency,
A parameter estimation unit is a mixed Gaussian function model in which each peak of a spectrum envelope at each time is approximated by a Gaussian distribution function based on the observation spectrogram output from the time-frequency expansion unit, and a Gaussian corresponding to each peak For each distribution function, the mean is represented by a set of K formant frequencies and coupling coefficients, the weight is represented by the formant intensity and the coupling coefficient for the set of K formant frequencies, and the variance is represented by the K pieces. Of the Gaussian distribution function so as to reduce the criterion representing the magnitude of the error between the mixed Gaussian function model at each time and the observed spectrogram when expressed in formant dispersion and coupling coefficient for a set of formant frequencies of A set of K formant frequencies for each Beauty and coupling coefficient at each time, and formant intensity for the set of the K formant frequencies, vocal tract spectrum estimation method for estimating the coupling coefficient of the formant dispersion and the time for the set of K formant frequencies.   5. The vocal tract spectrum estimation method according to claim 4, wherein the criterion further includes a penalty term relating to the magnitude of formant intensity with respect to a set of K formant frequencies for each of the Gauss distribution functions. By the parameter estimation unit estimating,
A parameter updating unit reduces the auxiliary function, which is an upper bound function of the criterion, for each of the Gaussian distribution functions, the set of K formant frequencies and the coupling coefficient at each time, and the K Update the formant intensity for the set of formant frequencies, the formant variance for the set of K formant frequencies, and the coupling coefficient at each time,
6. The vocal tract spectrum estimation method according to claim 4, wherein the convergence determination unit includes repeating the update by the parameter update unit until a predetermined convergence condition is satisfied.   The program for functioning a computer as each part of the vocal tract spectrum estimation apparatus of any one of Claims 1-3.