JP5318042B2 - Signal analysis apparatus, signal analysis method, and signal analysis program - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a signal analysis apparatus for extracting dynamic characteristics of a time-series signal. <P>SOLUTION: When a signal analysis apparatus extracts dynamic characteristics of a time-series signal by estimating an input signal parameter, a filter property parameter, and a residual signal parameter from an observation signal while the observation signal is indicated by a sum of an output signal in a signal generation system and a residual signal obtained by convolution between an input signal and an impulse response signal indicating a filter property, the signal analysis apparatus comprises: a parameter initial value generation unit for generating initial values of the parameters; a signal separation unit for separating the observation signal; a model parameter updating unit for updating a model parameter to maximize a target function with regard to the model parameter; a parameter convergence determination unit for making the signal separation unit and the model parameter updating unit perform processes again until the model parameter satisfies a predetermined criterion; and a parameter output unit for outputting the model parameter if it is judged that the model parameter satisfies a predetermined criterion. <P>COPYRIGHT: (C)2012,JPO&amp;INPIT

Description

  The present invention relates to a signal analysis apparatus, a signal analysis method, and a signal analysis program that extract dynamic characteristic features of time series signals.

  The prior art will be described using a basic frequency sequence extracted from a singing voice acoustic signal as an example. In the basic frequency series of this singing voice, the pitch target value series that the singer wants to sing and various dynamic fluctuation components (overshoot, vibrato, etc.) based on singing power, singing style, personality, and emotion are overlaid in a complex manner. Has been. Singing voice is one of the important elements that characterize many genres of music, and various studies focusing on the fundamental frequency series of singing voice are currently being actively conducted.

  If feature extraction of the pitch target value sequence included in the fundamental frequency sequence can be performed, application to hamming search and automatic music transcription is expected. In particular, in the Hamming search for searching for music from a singing voice, it is necessary to correctly estimate a pitch target value series intended by the singer from the basic frequency series of sung singing voices, and to collate with the melody of the music database.

  On the other hand, it is known that dynamic fluctuation components of the fundamental frequency series such as overshoot and vibrato are components that affect singing voice perception and personality perception. Therefore, it can be a useful measure for description of singing style, similar singing voice search using the singing style, and automatic evaluation of singing ability. It is also an indispensable component for more expressive and diverse singing voice synthesis. Therefore, in the conventional research, a model for controlling a dynamic fluctuation component of the fundamental frequency of a singing voice using a linear quadratic system has been proposed (for example, see Non-Patent Documents 1, 2, and 3).

における減衰率ζを調整することによって、指数減衰（ζ＞１）、減衰振動（０＜ζ＜１、オーバーシュートに対応する）、臨界制動（ζ＝１）、定常振動（ζ＝０、ビブラートに対応する）からなる様々な振動現象を表現する。 In these studies, the Fujisaki model, which expresses the fundamental frequency pattern of Japanese speech, was referenced. The Fujisaki model uses the impulse response and step response of the critical braking secondary system, and the phrase component that slowly falls from the beginning of the Japanese phrase toward the end of the phrase, and the accent that rises and falls sharply corresponding to the phrase The fundamental frequency sequence is described by expressing the components and superimposing them. However, the control of the sudden rise and fall of the fundamental frequency accompanying the melody of the singing voice and the periodic vibration such as vibrato cannot be expressed by the critical braking system. Therefore, in the fundamental frequency control model of singing voice, the transfer function of the secondary system

By adjusting the damping rate ζ at, exponential damping (ζ> 1), damping vibration (0 <ζ <1, corresponding to overshoot), critical braking (ζ = 1), steady vibration (ζ = 0, vibrato) To represent various vibration phenomena.

  In Non-Patent Document 3, an expressive singing voice synthesized sound is realized by using a fundamental frequency sequence obtained by convolving the impulse response of Expression (1) with a stepped signal representing a musical scale. However, in these conventional techniques, control parameters (attenuation rate ζ and natural frequency Ω) are determined manually or based on rules. On the other hand, in Non-Patent Document 4, a stochastic frame in which the input stepwise signal and the control parameter of the linear secondary system are both unknown and are estimated simultaneously from only the observed fundamental frequency sequence. Work was proposed. This is a learning algorithm that iteratively estimates the maximum likelihood model parameter using a hidden Markov model (HMM) that expresses a pitch target value series and a parametric expression of Equation (1) based on difference approximation.

N. Minematsu, B. Matsuoka, and K. Hirose, “Prosodic Modeling of Nagauta Singing and Its Evaluation,” in Proc. SpeechProsody 2004, pp. 487-490, Mar. 2004 T. Saitou, M. Unoki, and M. Akagi, “Development of an F0 control Model Based on F0 Dynamic Characteristics for Singing-Voice Synthesis,” Speech Communication, vol.46, pp. 405-417, 2005 T. Saitou, M. Goto, M. Unoki, and M. Akagi, “Speech-To-Singing Synthesis: Converting Speaking Voices to Singing Voices by Controlling Acoustic Features Unique to Singing Voices,” in Proc. WASPAA 2007, pp. 215 -218, Oct. 2007 Y. Ohishi, H. Kameoka, K. Kashino, and K. Takeda, “Parameter Estimation Method of F0 Control Model for Singing Voices,” in Proc INTERSPEECH 2008, pp. 139-142, Sep. 2008

  However, in the prior art of Non-Patent Document 4 described above, the model parameter estimation performance was poor. That is, there is a problem that an error between the observed fundamental frequency sequence and the fundamental frequency sequence re-synthesized according to the parameter increases. This is because various dynamic characteristics such as pitch rise and vibrato are mixed in the fundamental frequency sequence, and if a second-order parameter is directly estimated by the maximum likelihood method, it is pulled to a specific dynamic characteristic. This is because an overfitting problem occurs. On the other hand, a method of dividing the fundamental frequency sequence into frames and estimating model parameters for each frame has also been proposed, but the range of influence of the secondary system that generates each dynamic characteristic is unclear on the sequence, so There is a problem that model parameters cannot be estimated appropriately.

  The present invention has been made in view of such circumstances, and when extracting the dynamic characteristic features of the time series signal, the input stepwise signal and the control parameter of the linear secondary system are both unknown. An object of the present invention is to provide a signal analysis apparatus, a signal analysis method, and a signal analysis program capable of accurately estimating model parameters from only the observed fundamental frequency series.

In the present invention, an output signal [y] and a residual signal [ε] of a signal generation system obtained by convolving an observation signal [o] with an input signal [f] and an impulse response signal [h] representing filter characteristics. And the input signal parameter [u] constituting the model representing the input signal and the filter characteristic parameter [{a, b} or {a ₀ constituting the model representing the filter characteristic from the observed signal. , A ₁ , a ₂ } or {w ₁ , w ₂ ,..., W _I }] and a residual signal parameter [β] constituting the model representing the residual signal, thereby estimating the time series signal. A signal analysis apparatus for extracting a dynamic characteristic feature, comprising: a parameter initial value generation unit that generates initial values of the input signal parameter, the filter characteristic parameter, and the residual signal parameter from the observed signal; A set of a filter characteristic parameter, the input signal parameter, and the residual signal parameter is a model parameter [Θ], and the observation signal is configured by the input signal parameter and the filter characteristic parameter using the model parameter. A set of a signal separation unit for separating the output signal of the signal generation system and the residual signal configured by the residual signal parameter, the observation signal, the model parameter, the output signal, and the residual signal When the Q function [Equation 27] obtained by adding the prior probability of the model parameter to the conditional expected value of the log likelihood function is given as an objective function, the objective function is maximized with respect to the model parameter. A model parameter updating unit for updating the model parameter so as to be It is determined whether or not a predetermined standard is satisfied, and when it is determined that the predetermined standard is not satisfied, the processing by the signal separation unit and the model parameter update unit is performed again until the predetermined standard is satisfied. A parameter convergence determination unit; and a parameter output unit configured to output the model parameter when the parameter convergence determination unit determines that the model parameter satisfies a predetermined criterion.

  In the present invention, the input signal is a step signal [Equation 17], the output signal is stochastically modeled as following a multidimensional Gaussian distribution [Equation 21], and the residual signal is expressed as Gaussian white noise. It is characterized by being probabilistically modeled [Equation 22].

According to the present invention, the filter characteristic of the signal generation system is expressed by a filter [Expression 5] derived by a difference method, and the filter characteristic parameter is a parameter [a = 1 / Ω ² inversely proportional to the square of the natural frequency. , Equation 5] and a parameter [b = 2ζ / Ω, Equation 5] proportional to the attenuation factor and inversely proportional to the natural frequency.

  In the present invention, the filter characteristic of the signal generation system is represented by a filter [Equation 10] configured based on an autoregressive process, and the filter characteristic parameter is an autoregressive parameter [Equation 9]. To do.

In the present invention, the filter characteristic of the signal generation system is represented by a filter [Equation 15] configured by a weighted linear sum of a plurality of second-order filters, and the filter characteristic parameter is a weight of each second-order filter. [{W ₁ , w ₂ ,..., W _I }].

  According to the present invention, the model parameter update unit sets the auxiliary function of the objective function composed of auxiliary variables to a parameter [a] that is inversely proportional to the square of the natural frequency and the auxiliary function that is proportional to the attenuation rate and the eigenfunction. By solving simultaneous equations [Equations 36 and 37] consisting of equations obtained by differentiating each of the parameters [b] inversely proportional to the frequency with respect to the filter properties parameters, the values of the filter properties parameters [{a, b}] are obtained. A filter characteristic parameter updating unit for updating, an input signal parameter updating unit for updating an input signal parameter by solving an equation [Equation 38] obtained by differentiating the auxiliary function with the input signal parameter, and the auxiliary function By solving the equation [Equation 38] obtained by differentiating the signal with the residual signal parameter. Characterized in that it is composed of a residual signal parameter update section for updating.

In the present invention, the model parameter update unit is configured to obtain an equation obtained by differentiating the objective function by each autoregressive parameter [{a ₀ , a ₁ , a ₂ }] included in the filter characteristic parameter [formula 40 , Equation 41, Equation 42] are solved for the filter characteristic parameter, and the filter characteristic parameter updating unit for updating the value of the filter characteristic parameter and the objective function are obtained by differentiating with the input signal parameter. By solving the equation [Equation 43], the input signal parameter updating unit for updating the input signal parameter and the equation [Equation 43] obtained by differentiating the objective function with the residual signal parameter are obtained. And a residual signal parameter updating unit for updating the difference signal parameter.

In the present invention, the model parameter updating unit converts the auxiliary function of the objective function composed of auxiliary variables into weights [{w ₁ , w ₂ ,..., W _{I of the} second-order filters that are the filter characteristic parameters]. }], A nonlinear simultaneous equation [Equation 49 or 57] consisting of equations obtained by differentiating each of them by solving for the filter characteristic parameter, the filter characteristic parameter updating unit for updating the value of the filter characteristic parameter, and the auxiliary function Is obtained by differentiating the input signal parameter by solving the equation [Equation 50 or 58] obtained by differentiating the input signal parameter, and the auxiliary function is differentiated by the residual signal parameter. A residual signal that updates the residual signal parameters by solving the equation [Eq. 50 or Eq. 58] Characterized in that it is composed of a parameter update unit.

  According to the present invention, the signal separation unit includes an expected value [Equation (28)] of complete data composed of the output signal and the residual signal when the observation signal and the model parameter are given, The observation signal is separated into an output signal and a residual signal using autocorrelation [Equation (29)] of complete data.

  The present invention represents an observation signal as a sum of an output signal of a signal generation system obtained by convolution of an input signal and an impulse response signal representing a filter characteristic and a residual signal, and the input signal is expressed from the observation signal. The dynamic characteristics of the time-series signal by estimating the input signal parameters constituting the model to be represented, the filter characteristic parameters constituting the model representing the filter characteristics, and the residual signal parameters constituting the model representing the residual signal A signal analysis method in a signal analysis apparatus for extracting features, the parameter initial value generating step for generating initial values of the input signal parameter, the filter characteristic parameter, and the residual signal parameter from the observed signal, and the filter characteristic A set of a parameter, the input signal parameter, and the residual signal parameter is a model parameter. The observation signal is separated into the output signal of the signal generation system configured by the input signal parameter and the filter characteristic parameter and the residual signal configured by the residual signal parameter using the model parameter. A priori probability of the model parameter to a conditional expectation value of a log-likelihood function when given a signal separation step, the observed signal, the model parameter, and a set of the output signal and the residual signal. A model parameter update step of updating the model parameter so as to maximize the objective function with respect to the model parameter, using the function obtained by addition as an objective function, and whether or not the model parameter satisfies a predetermined criterion If it is determined that the predetermined standard is not satisfied, the predetermined standard is satisfied. Until it is determined that the model parameter satisfies a predetermined criterion by the parameter convergence determination step that causes the signal separation step and the model parameter update step to be performed again, and the parameter convergence determination step, the model parameter And a parameter output step for outputting.

  The present invention represents an observation signal as a sum of an output signal of a signal generation system obtained by convolution of an input signal and an impulse response signal representing a filter characteristic and a residual signal, and the input signal is expressed from the observation signal. The dynamic characteristics of the time-series signal by estimating the input signal parameters constituting the model to be represented, the filter characteristic parameters constituting the model representing the filter characteristics, and the residual signal parameters constituting the model representing the residual signal A signal analysis program for causing a computer on a signal analysis device for extracting features to perform signal analysis, wherein parameter initials for generating initial values of the input signal parameter, the filter characteristic parameter, and the residual signal parameter from the observed signal A value generation step, the filter characteristic parameter, the input signal parameter, and the residual signal parameter. A set with a meter is used as a model parameter, and the observation signal is configured by the output signal of the signal generation system configured by the input signal parameter and the filter characteristic parameter and the residual signal parameter by using the model parameter. A conditional expectation value of a log-likelihood function when given a signal separation step for separating the residual signal, the observed signal, the model parameter, and the set of the output signal and the residual signal. A function obtained by adding the prior probabilities of the model parameters to the objective function, a model parameter updating step for updating the model parameter so as to maximize the objective function with respect to the model parameter; and Judgment is made whether or not the standard is satisfied, and the predetermined standard is not satisfied When it is determined that the model parameter satisfies the predetermined criterion, the parameter separation determining step for performing again the processing by the signal separation step and the model parameter updating step until the predetermined criterion is satisfied, and the model parameter satisfies the predetermined criterion by the parameter convergence determining step. If it is determined, the computer is caused to perform a parameter output step of outputting the model parameter.

  According to the present invention, it is possible to accurately estimate a model parameter from only an observed fundamental frequency sequence when both the input stepped signal and the control parameter of the linear secondary system are unknown. It is done.

It is explanatory drawing which shows the concept of a linear secondary system model. It is explanatory drawing which shows the concept of a signal generation system. It is a block diagram which shows the structure of the signal analysis apparatus with respect to the fundamental frequency series of the singing voice in the 1st Embodiment of this invention. It is explanatory drawing which shows the signal decomposition model with respect to the fundamental frequency series of a singing voice. It is a block diagram which shows the structure of the signal analysis apparatus with respect to the mel frequency cepstrum coefficient (MFCC) series of the audio | voice signal in the 5th Embodiment of this invention. It is a figure which shows the comparison result of signal (micro | micron | mu) produced | generated by the model parameter in 4th Embodiment, and the observation fundamental frequency series o. It is a figure which shows the average value for every singer of (zeta) and (omega) estimated by 4th Embodiment. It is a figure which shows frequency distribution for every song person of u estimated by 4th Embodiment.

  Hereinafter, a signal analyzing apparatus according to an embodiment of the present invention will be described with reference to the drawings. First, in the signal analysis apparatus of the present invention, the principle of accurately estimating model parameters from only the observed fundamental frequency sequence will be described under the assumption that both the input stepwise signal and the control parameter of the linear quadratic system are unknown. To do.

  FIG. 1 is an explanatory diagram showing the concept of a linear quadratic system model. As shown in FIG. 1, the observation signal o (t) is divided into a plurality of sections, and the observation signal in each section is converted into a step signal (input signal) f (t) and a filter characteristic (an impulse response of a linear secondary system). ) H (t) and residual signal (residue signal) ε (t).

によって計算される系の出力信号を表す。残差信号ε（ｔ）は観測信号と出力信号の残差信号を表す。ただし、観測信号は離散的な信号系列であるため、これらの要素に分解するために、式（１）の２次系の伝達関数を離散時間表現する４つの手法のいずれかを用いる。 Here, the relationship between the input signal f (t) and the observation signal o (t) will be described with reference to FIG. FIG. 2 is an explanatory diagram showing the concept of the signal generation system. The input signal f (t) means a signal composed of a target value. The linear secondary system filter h (t) expresses dynamic characteristics such as temporal rise of an observation signal and overshoot according to the secondary system. The output signal y (t) of the linear second-order filter is a convolution of f (t) and h (t)

Represents the output signal of the system calculated by The residual signal ε (t) represents the residual signal between the observation signal and the output signal. However, since the observation signal is a discrete signal sequence, any one of four methods for expressing the transfer function of the second-order system of Expression (1) in discrete time is used to decompose it into these elements.

となり、ｆ（ｔ）とｙ（ｔ）の関係を表す２階微分方程式が導かれる。ここでｆ（ｔ）とｙ（ｔ）をサンプリング周期Δで離散化して、

と表し（Ｎは信号の長さを表す）、差分法を利用すると、

のように各時刻の微分値を近似できる。 Here, four methods for expressing the transfer function of the linear quadratic system in discrete time will be described. First, the discrete time expression method (referred to as Method 1) by the difference method will be described. When formula (1) is inversely transformed by Laplace,

Thus, a second-order differential equation representing the relationship between f (t) and y (t) is derived. Here, f (t) and y (t) are discretized with a sampling period Δ,

(N represents the length of the signal), and using the difference method,

The differential value at each time can be approximated as follows.

と書ける。ここで、（ａＡ＋ｂＢ＋Ｃ）が２次系の逆フィルタのインパルス応答に相当し、Ａ、Ｂ、ＣはそれぞれＮ×Ｎの行列であり、

と書ける。 Using this approximation, equation (3) becomes

Can be written. Here, (aA + bB + C) corresponds to the impulse response of the inverse filter of the second-order system, and A, B, and C are each an N × N matrix,

Can be written.

  Of course, the difference method can use a plurality of approximation methods (center difference, forward difference, backward difference, use of Sinc function, etc.) in addition to equation (4). This corresponds to changing the configuration. In Method 1, a and b in Expression (5) are filter characteristic parameters.

と近似すると、ｓ領域とｚ領域の関係は、

となる。式（１）の逆フィルタの伝達関数はｚ領域で、

と書ける。 Next, a discrete time expression method (referred to as method 2) based on the autoregressive model will be described. Consider the mapping from the s region to the z region in equation (1).

And the relationship between the s region and the z region is

It becomes. The transfer function of the inverse filter of equation (1) is in the z region,

Can be written.

とすると、式（８）は、

と書ける。Ｕ_０＝Ｉ_Ｎ（Ｉ_ＮはＮ×Ｎの単位行列）であり、Ｕ_１、Ｕ_２はそれぞれ

からなるＮ×Ｎの行列とする。手法２では、式（１０）のａ_０，ａ_１，ａ_２がフィルタ特性パラメータとなる。 This is isomorphic to the second-order autoregressive model,

Then, equation (8) becomes

Can be written. U ₀ = I _N (I _N is an N × N unit matrix), and U ₁ and U ₂ are respectively

An N × N matrix consisting of In Method 2, a ₀ , a ₁ , and a ₂ in Expression (10) are filter characteristic parameters.

Next, a discrete time expression method (method 3 and method 4) based on a linear sum of a plurality of vibration bases will be described. The impulse response obtained by the Laplace inverse transform of the transfer function of Equation (1) is classified as follows according to the value of ζ.

で書ける。ただし、ｈ（ｔ）は式（１２）のように複数の場合からなるので、行列Φを以下のように構成する。

When these impulse responses are discretized based on the sampling period Δ, the input / output relationship of the system can be described in a format such as y = Φf. For example, when ζ = 1, Φ is a lower triangular matrix

You can write in However, since h (t) consists of a plurality of cases as shown in Equation (12), the matrix Φ is configured as follows.

Here, ζ and Ω are manually determined in advance, and impulse responses {Φ ⁽¹⁾ , Φ ⁽²⁾ ,..., Φ ^(I) } representing I vibration phenomena are calculated. Then, Φ ⁻¹ is approximated by a linear weighted sum of these inverse matrices ^{ｉ (i)} : = (Φ ^(t) ) ⁻¹ (representing the impulse response of the inverse filter. These are hereinafter referred to as vibration bases). To do. In Method 3, w: = {w ₁ , w ₂ ,..., W _I } in Expression (14) is a filter characteristic parameter, and these parameters are estimated in the framework of a regression problem.

On the other hand, in the method 4, it is assumed that the filter characteristic parameter w: = {w ₁ , w ₂ ,..., W _I } is sparse. This means that Φ ⁻¹ is represented by only a few vibration bases. As will be described later, this can be easily realized by assuming a prior probability of w.

と表現される。 The impulse response Φ of the system may be expressed by a linear weighted sum of Φ ⁽ⁱ⁾ , but the reason why the impulse response Φ ⁻¹ of the inverse filter is expressed by the linear weighted sum of Υ ⁽ⁱ⁾ will be described later. This is to eliminate the complexity of deriving the parameter learning algorithm. Therefore, the input / output relationship of the system is

It is expressed.

と表現される。 From the four methods described above (formula (5), formula (10), and formula (15)), the transfer function of formula (1) is all converted into the form of Ψy = f. Ψ is respectively

It is expressed.

  Next, statistical modeling of a linear quadratic system based on a Gaussian process will be described. The input / output relationship of the system expressed by Equation (5), Equation (10), and Equation (15) can be expressed as a Gaussian process (literature: CE Rasmussen and CKI Williams, Gaussian Processes for Machine Learning. MIT Press, Cambridge, Mass, USA). , 2006.) for statistical modeling.

を用意する。ここで、スカラー値ｕは相対音高を表すパラメータ、１_ＮはＮ個の１の値が並ぶベクトルとする。このベクトルｕを平均とする多次元ガウス分布から生成される確率変数として、入力信号ｆを確率的にモデル化する。

Here, modeling of the input step signal will be described. The input signal f is assumed to be a step signal. Therefore, a vector that always has the same value

Prepare. Here, the scalar value u is a parameter representing relative pitch, and _N is a vector in which N 1 values are arranged. The input signal f is stochastically modeled as a random variable generated from a multidimensional Gaussian distribution with the vector u as an average.

である。よって、ｙが従う確率分布は、

と書ける。 Here, α is a super parameter representing dispersion, and is manually set in advance. Therefore, the only input signal parameter mentioned in the claims is u. Since the output y is a linear combination (y = Ψ ⁻¹ f) of the variable set f according to the Gaussian distribution, y itself also follows the Gaussian distribution. Its mean and covariance are

It is. Therefore, the probability distribution that y follows is

Can be written.

  It should be noted that this model is an example of a Gaussian process. The important point of the Gaussian process is that the simultaneous distribution of N elements of y is completely described by statistics up to second order such as mean and covariance. In normal Gaussian processes, the mean is often zero, and the covariance is the linear sum of kernel matrices (Multiple Kernel Learning; literature: FR Bach, GRG Lanckriet, and MI Jordan, “Multiple kernel learning, conic In general, it is composed of duality, and the smo algorithm, "in Proc. ICML 2004, pp. 6-13, July 2004). With this technique, rather than assuming that a plurality of observed values (in this case, elements of y) follow independent and identical distributions, the correlation between the observed values can be considered. In recent years, signal modeling using Gaussian processes using Multiple Kernel Learning has attracted attention in the field of machine learning.

On the other hand, in the above-described model, the matrix Ψ ⁻¹ is included in the mean and covariance terms of Equation (21). Then, as in Expression (16), Ψ is expressed by a linear sum of several bases. Therefore, here we refer to a signal model based on a special Gaussian process, which is different from Multiple Kernel Learning.

を導入し、観測信号

は、系の出力信号ｙに残差信号εが加わった信号

と仮定する。 Next, the likelihood function and the prior probability will be described. Residual signal subject to Gaussian white noise

Introduced the observation signal

Is a signal obtained by adding a residual signal ε to the output signal y of the system

Assume that

となる。ここで、μ：＝Ψ^−１ｕ，Σ：＝αΨ^−１（Ψ^−１）^Ｔ＋βＩ_Ｎとする。 Here, β is a super parameter representing the variance of the residual signal, which corresponds to the residual signal parameter in the claims. Assuming that y and ε are independent from each other, from the definition of the Gaussian process, the log likelihood function of the model parameter θ = {Ψ, u, β} when the observation signal o is given is

It becomes. ^{Here, μ: = Ψ -1 u,} Σ: = αΨ -1 (Ψ -1) and ^T + beta I _N.

とし、Ｐ（ａ）、Ｐ（ｂ）、Ｐ（ａ_０）、Ｐ（ａ_１）、Ｐ（ａ_２）はそれぞれ一様分布を仮定する。手法３では、同様にＰ（ｗ_１），Ｐ（ｗ_２），・・・，Ｐ（ｗ_Ｉ）はそれぞれ一様分布を仮定する。一方、手法４では、パラメータｗにスパースな制約をもたせるために、一般化正規分布

に従うものとする。ただし、ｐ、λは一般化正規分布の形状を規定する定数であり、０＜ｐ＜２のときｐ（ｗ）は優ガウス的となり、スパースネスを測るための尺度となる。 The prior probability P (Θ) of Θ assumes the independence P (Θ) = P (ψ) P (u) P (β) of each element, and u and β are assumed to follow a uniform distribution. P (Ψ) assumes the independence of the filter characteristic parameters of the methods 1 to 4 described above,

And P (a), P (b), P (a ₀ ), P (a ₁ ), and P (a ₂ ) are assumed to have a uniform distribution. In method 3, similarly, P (w ₁ ), P (w ₂ ),..., P (w _I ) are assumed to have a uniform distribution. On the other hand, in Method 4, a generalized normal distribution is used in order to place a sparse constraint on the parameter w.

Shall be followed. However, p and λ are constants that define the shape of the generalized normal distribution. When 0 <p <2, p (w) becomes dominant Gaussian and is a scale for measuring sparseness.

  Next, a parameter learning algorithm based on the EM method will be described. Given a fundamental frequency sequence o, we want to determine an estimate of the parameter Θ that maximizes the posterior probability P (Θ | o) ∝P (o | Θ) P (Θ). However, it is difficult to analytically determine the optimal solution for the posterior (MAP) estimate of Θ. The reason is that the observation signal o is composed of the sum of the output signal y and the residual signal ε, and the likelihood function is non-linear with respect to the filter characteristic parameter constituting Ψ. Here, in order to deal with each problem, the following two measures are applied.

  The first is to distribute o to y and ε in E-step of EM method. The second is an auxiliary function method (reference: H. Kameoka, N. Ono, K. Kashino, and S. Sagayama, “Complex NMF: A New Sparse Representation for Acoustic Signals,” in Proc. ICASSP 2009 , pp. 3437-3440, April 2009) to design the auxiliary function of the Q function.

と表現する。ｘと現在のパラメータΘ´が与えられたときの、対数尤度関数の条件付き期待値を計算し、さらにｌｏｇＰ（Θ）を加算すると、以下のようなＱ関数を得る。

Next, the definition of complete data will be described. The first step in applying the EM method to this MAP estimation problem is to define complete data. Here, y and ε are regarded as complete data x, and the EM method is applied. The relationship between incomplete data and complete data

It expresses. When the conditional expected value of the log likelihood function when x and the current parameter Θ ′ are given is calculated and logP (Θ) is added, the following Q function is obtained.

である。ここで、ｔｒ（・）は行列のトレースを表し、条件付ガウス分布の性質から、

と書ける。ここで、式（２８）は、不完全データｏ（観測信号）と現在のパラメータが与えられた時の、完全データｘ（出力信号と残差信号で構成される）の期待値であり、式（２９）は、不完全データｏ（観測信号）と現在のパラメータが与えられた時の、完全データｘ（出力信号と残差信号で構成される）の自己相関である。 Note that the symbol with c above = indicates that the left side and the right side are equal except for the constant term.

It is. Where tr (•) represents the trace of the matrix, and from the nature of the conditional Gaussian distribution,

Can be written. Here, Expression (28) is an expected value of complete data x (consisting of an output signal and a residual signal) when incomplete data o (observation signal) and the current parameter are given. (29) is an autocorrelation of complete data x (consisting of an output signal and a residual signal) when incomplete data o (observation signal) and the current parameter are given.

のように区分表現する。￣ｘ_ｙと￣ｘ_εは（￣はそれぞれｘの頭に付く、以下の説明においても同様）はどちらも長さＮのベクトルであり、Ｒ_ｙとＲ_εはどちらもＮ×Ｎの正方行列を表す。 In the E-step of the EM method, the model parameter updated immediately before is substituted into Θ ′, and E [x | o; Θ ′] and E [xx ^T | o; Θ ′] are calculated. For later calculations, let E [x | o; Θ ′] and E [xx ^T | o; Θ ′] correspond to y and ε.

It is divided and expressed as ￣ _xy and ￣ x _ε (￣ is prefixed with x, respectively in the following description) are both vectors of length N, and R _y and R _ε are both N × N square matrices. Represents.

と改めて定義される。ここで、Ａ_ｎ，ｎとは行列Ａのｎ行ｎ列目の要素を表す。パラメータ集合Θに関する最大化問題を解く更新式を前述の補助関数法により導く。式（３１）で与えられる目的関数の補助関数を以下の不等式を用いて導く。

ここで、等号成立は、

の場合である。補助変数γ_１＝｛γ_ａ，１，γ_ｂ，１，γ_ａ，２，γ_ｂ，２，・・・，γ_ａ，Ｎ，γ_ｂ，Ｎ｝を導入すると、補助関数は、

とすれば、ｆ_１（ａ，ｂ，ｕ，β）≧ｆ_１ ^＋（ａ，ｂ，ｕ，β，γ_１）が成り立ち、式（３３）が成立するとき、

となるため、ｆ_１ ^＋（ａ，ｂ，ｕ，β，γ_１）は補助関数の定義を満たす。 Next, an M-step update formula for maximizing the Q function for each model parameter of the above-described methods 1 to 4 will be described. First, the M-step update formula of Method 1 will be described. Since Ψ = aA + bB + C, taking the relevant term from Equation (27), the objective function to be maximized is

Is redefined. Here, A _{n, n} represents an element in the nth row and the nth column of the matrix A. An update formula for solving the maximization problem with respect to the parameter set Θ is derived by the auxiliary function method described above. The auxiliary function of the objective function given by Equation (31) is derived using the following inequality.

Here, the establishment of the equal sign is

This is the case. When the auxiliary variable γ ₁ = {γ _{a, 1} , γ _{b, 1} , γ _{a, 2} , γ _{b, 2} ,..., Γ _{a, N} , γ _{b, N} } is introduced, the auxiliary function is

Then, f ₁ (a, b, u, β) ≧ f ₁ ⁺ (a, b, u, β, γ ₁ ) holds, and when equation (33) holds,

Therefore, f ₁ ⁺ (a, b, u, β, γ ₁ ) satisfies the definition of the auxiliary function.

を得る。式（５）から分かるように、ａとｂは共に正の値となる制約の下で、Coordinate descent法を適用して、式（３６）と式（３７）からなる連立方程式を解き、ａとｂを求める。 Therefore, if f ₁ ⁺ (a, b, u, β, γ ₁ ) is differentiated with respect to each of a and b and set to 0,

Get. As can be seen from Equation (5), under the constraint that both a and b are positive values, the Coordinate descent method is applied to solve the simultaneous equations of Equation (36) and Equation (37), and a and b is obtained.

を得る。これらの式を用いて、ｕとβを更新する。 On the other hand, if f ₁ ⁺ (a, b, u, β, γ ₁ ) is differentiated with respect to u and β, and is set to 0,

Get. Using these equations, u and β are updated.

The parameter learning algorithm based on the EM method related to the above method 1 is summarized as follows.
Initialization: An initial value is given to the parameter Θ = {a, b, u, β}. E-step: E [x | o; Θ ′], E [xx ^T | o; Θ ′], γ ₁ = {γ _{a, 1} , γ _{b, 1} , γ _{a, 2} , γ _{b, 2} ,. .., γ _{a, N} , γ _{b, N} } update. M-step: Update of model parameter Θ = {a, b, u, β} from equations (36), (37), and (38). Convergence determination: If the value of the auxiliary function of Expression (34) has not converged, return to E-step as Θ ′ = Θ.

と改めて定義される。ｆ_２（ａ_０，ａ_１，ａ_２，ｕ，β）をａ_０，ａ_１，ａ_２に関してそれぞれ微分して０とおくと，

を得る。式（９）に示すように、ａ_０とａ_２は正の値、ａ_１は負の値となる制約の下で、Coordinate descent法を適用して、式（４０）、（４１）、（４２）からなる連立方程式を解き、ａ_０，ａ_１，ａ_２を求める。 Next, the M-step update formula of Method 2 will be described. Since Ψ = a ₀ U ₀ + a ₁ U ₁ + a ₂ U ₂ , when the relevant terms are taken from equation (27), the objective function to be maximized is

Is redefined. If f ₂ (a ₀ , a ₁ , a ₂ , u, β) is differentiated with respect to a ₀ , a ₁ , a ₂ and set to 0,

Get. As shown in Expression (9), the Coordinate descent method is applied under the constraint that a ₀ and a ₂ are positive values and a ₁ is a negative value, and Expressions (40), (41), ( 42) are solved to obtain a ₀ , a ₁ , and a ₂ .

を得る。これらの式を用いて、ｕとβを更新する。 On the other hand, if f ₂ (a ₀ , a ₁ , a ₂ , u, β) is differentiated with respect to u and β, and set to 0,

Get. Using these equations, u and β are updated.

The parameter learning algorithm based on the EM method related to the above method 2 is summarized as follows. Initialization: An initial value is given to the parameter Θ = {a ₀ , a ₁ , a ₂ , u, β}. E-step: Update of E [x | o; Θ ′], E [xx ^T | o; Θ ′].
M-step: Update of the model parameter Θ = {a ₀ , a ₁ , a ₂ , u, β} from the equations (40), (41), (42), (43). Convergence determination: If the value of the objective function in Expression (39) has not converged, return to E-step as Θ ′ = Θ.

と改めて定義される。ここで、Υ_ｎ，ｎ ^（ｉ）はΥ^（ｉ）のｎ行ｎ列目の対角要素を表す。この最大化問題を解く更新式を補助関数法により導く。式（４４）で与えられる目的関数の補助関数を以下の不等式を用いて導く。

Next, the M-step update formula of Method 3 will be described. Since Ψ = w ₁ Υ ⁽¹⁾ + w ₂ Υ ⁽²⁾ +... + W _I Υ ^(I) , when the relevant term is taken from equation (27), the objective function to be maximized is

Is redefined. Here, Υ _{n, n} ⁽ⁱ⁾ represents the diagonal element of the n-th row and the n-th column of Υ ⁽ⁱ⁾ . An update formula that solves this maximization problem is derived by the auxiliary function method. The auxiliary function of the objective function given by Expression (44) is derived using the following inequality.

と定義する。このとき、

が成り立ち、等号成立は、

のときであるため、式（４６）は補助関数の定義を満たす。 Here, the auxiliary variable γ ₃ = {γ _1,1 ,..., Γ _{I, N} } is defined, and the auxiliary function is defined as

It is defined as At this time,

The establishment of the equality sign

(46) satisfies the definition of the auxiliary function.

を得る。Coordinate descent法を利用して、ｉ´＝１，２，・・・，Ｉに関する非線形連立方程式を解くと、ｗを求めることができる。一方、ｆ_３ ^＋（ｗ，ｕ，β，γ_３）をｕ、βそれぞれに関して微分して０とおくと、

を得る。これらの式を用いて、ｕとβを更新する。 Differentiating equation (46) with respect to w _{i ′} and _setting it to 0,

Get. By solving the nonlinear simultaneous equations related to i ′ = 1, 2,..., Using the coordinate descent method, w can be obtained. On the other hand, if f ₃ ⁺ (w, u, β, γ ₃ ) is differentiated with respect to u and β, and set to 0,

Get. Using these equations, u and β are updated.

The parameter learning algorithm based on the EM method related to the above method 3 is summarized as follows.
Initialization: An initial value is given to the parameter Θ = {w, u, β}. E-step: Update of E [x | o; Θ ′], E [xx ^T | o; Θ ′], γ ₃ . M-step: Update of model parameter Θ = {w, u, β} from equations (49) and (50). Convergence determination: If the value of the auxiliary function in the equation (46) has not converged, return to E-step as Θ ′ = Θ.

と改めて定義される。ここで、Υ_ｎ，ｎ ^（ｉ）はΥ^（ｉ）のｎ行ｎ列目の対角要素を表す。この最大化問題を解く更新式を補助関数法により導く。式（５１）で与えられる目的関数の補助関数を以下の２つの不等式を用いて導く。

Next, the M-step update formula of Method 4 will be described. Since Ψ = w ₁ Υ ⁽¹⁾ + w ₂ Υ ⁽²⁾ +... + W _I Υ ^(I) , when the relevant term is taken from equation (27), the objective function to be maximized is

Is redefined. Here, Υ _{n, n} ⁽ⁱ⁾ represents the diagonal element of the n-th row and the n-th column of Υ ⁽ⁱ⁾ . An update formula that solves this maximization problem is derived by the auxiliary function method. The auxiliary function of the objective function given by equation (51) is derived using the following two inequalities.

と定義する。このとき、

が成り立ち、等号成立は、

のときであるため、式（５４）は補助関数の定義を満たす。
式（５４）をｗ_ｉ´に関して微分して０とおくと、

を得る。Coordinate descent法を利用して、ｉ´＝１，２，・・・，Ｉに関する非線形連立方程式を解くと、ｗを求めることができる。 Here, the auxiliary variable _{¯w: = {¯w 1, ¯w} 2, ···, ¯w}, γ 4 = {γ 1,1, ···, γ I, N} define the auxiliary Function

It is defined as At this time,

The establishment of the equality sign

(54) satisfies the definition of the auxiliary function.
Differentiating equation (54) with respect to w _{i ′} and _setting it to 0,

Get. By solving the nonlinear simultaneous equations related to i ′ = 1, 2,..., Using the coordinate descent method, w can be obtained.

を得る。これらの式を用いて、ｕとβを更新する。 On the other hand, when f ₄ ⁺ (w, u, β, γ ₄ ) is differentiated with respect to u and β, and set to 0,

Get. Using these equations, u and β are updated.

The parameter learning algorithm based on the EM method related to the above method 4 is summarized as follows.
Initialization: An initial value is given to the parameter Θ = {w, u, β}. E-step: E [x | o; Θ'], E [xx T | o; Θ'], ¯w, γ 4 of the update. M-step: Update of the model parameter Θ = {w, u, β} from the equations (57) and (58). Convergence determination: If the value of the auxiliary function of Expression (54) has not converged, return to E-step as Θ ′ = Θ.

<First Embodiment>
Next, the configuration of the signal analyzing apparatus according to the first embodiment of the present invention will be described with reference to FIG. FIG. 3 is a block diagram showing the configuration of the embodiment. As shown in FIG. 3, the signal analysis device is configured by a computer device, and includes a fundamental frequency extraction unit 1, a segment division unit 2, a parameter initial value generation unit 3, a signal separation unit 4, a filter characteristic parameter update unit 5, an input signal. A parameter update unit 6, a residual signal parameter update unit 7, a parameter convergence determination unit 8, and a parameter output unit 9 are provided. The filter characteristic parameter update unit 5, the input signal parameter update unit 6, and the residual signal parameter update unit 7 constitute a model parameter update unit 10. The first embodiment is configured to perform signal analysis using the method 1 described above.

  The fundamental frequency extraction unit 1 extracts an observed fundamental frequency time series from the input singing voice acoustic signal. This processing can be realized by a well-known technique. For example, literature: A de Cheveign´e and H. Kawahara, “YIN, a fundamental frequency estimator for speech and music,” Journal of the Acoustical Society of America, vol.111, no .4, pp. 1917-1930, 2002, the fundamental frequency estimation method YIN is used to estimate the fundamental frequency from the singing voice signal every 5 ms.

The segment dividing unit 2 divides the estimated fundamental frequency sequence into several segments. As shown in FIG. 4, each segment has a start point and an end point at the moment when it rises from one pitch to another. As a method of dividing into segments, manual work, k-mean method, Viterbi algorithm or the like is used. Divided segment o = [o ₁ , o ₂ ,..., O _N ] ^T (N represents the length of the fundamental frequency sequence in the segment and varies from segment to segment)
Model parameters are estimated every time. As preprocessing, normalization is performed by subtracting the fundamental frequency value o ₁ at the beginning of the segment from all the fundamental frequency values of the segment.

The parameter initial value generation unit 3 determines an initial value of the model parameter Θ ₁ = {a, b, u, β}. As for a and b, when ζ = 1.0 and Ω = 0.1, a = 100 and b = 20 calculated from the equation (5) are set as initial values. The initial value of u is the median value of the elements of the observation signal o. The initial value of β is β = 100. These are all determined experimentally. Α is fixed to α = 2.

The signal separation unit 4 separates the observation signal into the output signal and the residual signal using the output signal derived based on the definition of the Gaussian process and the expected value of the residual signal based on the EM algorithm as a signal separation reference. Here, equations (28) and (29) are calculated using the current model parameter Θ ₁ ′ = {a, b, u, β}, and E [x | o based on equation (30). ; Θ ₁ ′] and E [xx ^T | o; Θ ₁ ′] are divided into ￣x _y , ￣x _ε , R _y , and R _ε . Also, the auxiliary variable γ ₁ is calculated based on the equation (33).

を計算する。 The filter characteristic parameter updating unit 5 updates the values of a and b that are filter characteristic parameters. As can be seen from Equation (5), under the constraint that both a and b are positive values, the Coordinate descent method is applied to solve the simultaneous equations of Equation (36) and Equation (37), and a and b is obtained. First, as initial values, a = 0 and b = 0 are set. And, considering equation (36) as an equation for a,

Calculate

を計算する。式（５９）、（６０）による更新を交互に繰り返し、ａとｂの値がそれぞれ変化しなくなるまで更新を続ける。 Next, considering equation (37) as an equation for b,

Calculate Updates according to equations (59) and (60) are alternately repeated until the values of a and b no longer change.

  The input signal parameter update unit 6 calculates Ψ of Expression (16) using a and b updated by the filter characteristic parameter update unit 5, and based on Expression (38), the value of the input signal parameter u Update.

  The residual signal parameter updating unit 7 updates the value of the residual signal parameter β based on Expression (38).

The parameter convergence determination unit 8 includes ￣x _y , R _y , R _ε calculated by the signal separation unit 4 and the model parameter Θ ₁ = {a, b, u, β} updated by the model parameter update unit 10, respectively. Is used to calculate the value of the objective function of equation (31). If an error between the value of the objective function of the equation (31) calculated using the model parameter before the update and the value of the objective function of the equation (31) calculated using the model parameter after the update is less than a predetermined threshold value If it is, it is determined that it has converged. If it has converged, the parameter output unit 9 outputs the model parameter Θ ₁ = {a, b, u, β} and then proceeds to the process of the segment dividing unit 2 in order to proceed to model parameter estimation in the next segment. To do. On the other hand, when it does not converge, it returns to the processing of the signal separation unit 4 as Θ ₁ ′ = Θ ₁ .

  In addition to the method using the objective function, as a method for determining whether or not the convergence has occurred, the values of the model parameters may be compared before and after the update, or a predetermined number of iterations has been reached. The determination may be made based on whether or not.

<Second Embodiment>
Next, the configuration of the signal analyzing apparatus according to the second embodiment of the present invention will be described. The second embodiment is configured to perform signal analysis using the method 2 described above. The configuration of the signal analyzing apparatus in the second embodiment is the same as that shown in FIG. 3, and the processing operations of the fundamental frequency extracting unit 1 and the segment dividing unit 2 are the same as those in the first embodiment. The second embodiment differs in processing operations of other configurations.

The parameter initial value generation unit 3 determines an initial value of the model parameter Θ ₂ = {a ₀ , a ₁ , a ₂ , u, β}. a ₀ , a ₁ , and a ₂ are calculated from Equation (9) when ζ = 1.0 and Ω = 0.1, and a ₀ = 121, a ₁ = −220, and a ₂ = 100. Use the initial value. The initial value of u is the median value of the elements of the observation signal o. The initial value of β is β = 100. These are all determined experimentally. Α is fixed to α = 2.

The signal separation unit 4 separates the observation signal into the output signal and the residual signal using the output signal derived based on the definition of the Gaussian process and the expected value of the residual signal based on the EM algorithm as a signal separation reference. Equations (28) and (29) are calculated using the current model parameter Θ ₂ ′ = {a ₀ , a ₁ , a ₂ , u, β}, and E [x | O; Θ ₂ ′] and E [xx ^T | o; Θ ₂ ′] are divided into ￣ _xy , ￣ _{x ε} , R _y , and R _ε .

を計算する。 The filter characteristic parameter update unit 5 updates the values of a ₀ , a ₁ , and a ₂ that are filter characteristic parameters. As can be seen from equation (9), the coordinate descent method is applied under the constraint that a ₀ and a ₂ are positive values and a ₁ is a negative value, and equations (40), (41), ( 42) are solved to obtain a ₀ , a ₁ , and a ₂ . First, as initial values, a ₀ = 0, a1 = 0, and a ₂ = 0 are set. And considering equation (40) as an equation for a ₀ ,

Calculate

を計算する。 Next, considering equation (41) as an equation for a ₁ ,

Calculate

を計算する。そして、式（６１）、（６２）、（６３）による更新を順番に繰り返し、ａ_０，ａ_１，ａ_２の値がそれぞれ変化しなくなるまで更新を続ける。 Next, considering equation (42) as an equation for a ₂ ,

Calculate Then, the updating by the formulas (61), (62), and (63) is repeated in order, and the updating is continued until the values of a ₀ , a ₁ , and a ₂ no longer change.

The input signal parameter updating unit 6 calculates Ψ of Expression (16) using a ₀ , a ₁ , and a ₂ updated by the filter characteristic parameter updating unit 5, and inputs based on Expression (43). Update the value of the signal parameter u.

  The residual signal parameter update unit 7 updates the value of the residual signal parameter β based on the equation (43).

The parameter convergence determination unit 8 includes ￣ _{x y} , R _y , R _ε calculated by the signal separation unit 4 and the model parameter Θ ₂ = {a ₀ , a ₁ , a ₂ , updated by the model parameter update unit 10. u, β} is used to calculate the value of the objective function of Equation (39). If the value has converged, the parameter output unit 9 outputs the model parameter Θ ₂ = {a ₀ , a ₁ , a ₂ , u, β}, and shifts to model parameter estimation in the next segment. The process proceeds to the processing operation of the segment division unit 2. On the other hand, when it does not converge, it shifts to the processing operation of the signal separation unit 4 as Θ ₂ ′ = Θ ₂ .

<Third Embodiment>
Next, the configuration of the signal analyzing apparatus according to the third embodiment of the present invention will be described. The third embodiment is configured to perform signal analysis using the method 3 described above. The configuration of the signal analysis apparatus in the third embodiment is the same as that shown in FIG. 3, and the processing operations of the fundamental frequency extraction unit 1 and the segment division unit 2 are the same as those in the first embodiment. The third embodiment differs in processing operations of other configurations.

The parameter initial value generation unit 3 determines an initial value of the model parameter Θ ₃ = {w, u, β}. First, in order to create {Υ ⁽¹⁾ , Υ ⁽²⁾ ,..., Υ ^(I) }, ζ is from 0 to 2 in increments of 0.02, and Ω is 0.05 to 0. .3 is changed in increments of 0.005. As a result, I = 3100. The initial values of w = {w ₁ , w ₂ ,... w _I } are all set to 1 / I. The initial value of u is the median value of the elements of the observation signal o. The initial value of β is β = 100. These are all determined experimentally. Α is fixed to α = 2.

The signal separation unit 4 separates the observation signal into the output signal and the residual signal using the output signal derived based on the definition of the Gaussian process and the expected value of the residual signal based on the EM algorithm as a signal separation reference. Equations (28) and (29) are calculated using the current model parameter Θ ₃ ′ = {w, u, β}, and E [x | o; Θ ₃ ′] is calculated based on equation (30). And E [xx ^T | o; Θ ₃ ′] are divided into ￣ _xy , ￣ _{x ε} , R _y , and R _ε . Further, the auxiliary variable γ ₃ is also calculated based on the equation (48).

と変形する。ｉ´＝１，２，・・・，Ｉに関して、式（６４）による更新を順番に繰り返し、ｗ＝｛ｗ_１，ｗ_２，・・・ｗ_Ｉ｝の値がそれぞれ変化しなくなるまで更新を続ける。 The filter characteristic parameter updating unit 5 updates the value of w that is a filter characteristic parameter. Using the coordinate descent method, w can be obtained by solving the nonlinear simultaneous equations related to i ′ = 1, 2,... First, {w ₁ , w ₂ ,... W _I } are all set to 0 as initial values. And considering equation (49) as an equation for w _{1 ′} ,

And deformed. i'= 1, 2, · · ·, with respect to I, repeatedly updated according to formula (64) in _turn, the update to _{w = {w 1, w 2} , ··· w I} is the value of does not change, respectively to continue.

  The input signal parameter updating unit 6 calculates Ψ of Expression (16) using w updated by the filter characteristic parameter updating unit 5, and updates the value of the input signal parameter u based on Expression (50). To do.

  The residual signal parameter updating unit 7 updates the value of the residual signal parameter β based on Expression (50).

The parameter convergence determination unit 8 uses ￣x _y , R _y , R _ε calculated by the signal separation unit 4 and the model parameter Θ ₃ = {w, u, β} updated by the model parameter update unit 10. Thus, the value of the objective function of Expression (44) is calculated. If the value has converged, the parameter output unit 9 outputs the model parameter Θ ₃ = {w, u, β} and proceeds to model parameter estimation in the next segment. Migrate to On the other hand, when it does not converge, it shifts to the processing operation of the signal separation unit 4 as Θ ₃ ′ = Θ ₃ .

<Fourth Embodiment>
Next, the configuration of the signal analyzing apparatus according to the fourth embodiment of the present invention will be described. The fourth embodiment is configured to perform signal analysis using the method 4 described above. The configuration of the signal analyzing apparatus in the fourth embodiment is the same as that shown in FIG. 3, and the processing operations of the fundamental frequency extracting unit 1 and the segment dividing unit 2 are the same as those in the first embodiment. The fourth embodiment differs in processing operations of other configurations.

The parameter initial value generation unit 3 is the same as that of the third embodiment except that the model parameter is Θ ₄ = {w, u, β}.

The signal separation unit 4 separates the observation signal into the output signal and the residual signal using the output signal derived based on the definition of the Gaussian process and the expected value of the residual signal based on the EM algorithm as a signal separation reference. Using the current model parameter Θ ₄ ′ = {w, u, β}, equations (28) and (29) are calculated, and E [x | o; Θ ₄ ′] is calculated based on equation (30). And E [xx ^T | o; Θ ₄ ′] are divided into ￣ _xy , ￣ _{x ε} , R _y , and R _ε . Also, the auxiliary variable γ ₄ is calculated based on the equation (56).

と変形する。ｉ´＝１，２，・・・，Ｉに関して、式（６５）による更新を順番に繰り返し、ｗ＝｛ｗ_１，ｗ_２，・・・ｗ_Ｉ｝の値がそれぞれ変化しなくなるまで更新を続ける。 The filter characteristic parameter updating unit 5 updates the value of w that is a filter characteristic parameter. Using the coordinate descent method, w can be obtained by solving the nonlinear simultaneous equations related to i ′ = 1, 2,... First, {w ₁ , w ₂ ,... W _I } are all set to 0 as initial values. Then, considering equation (57) as an equation for w _{1 ′} ,

And deformed. i'= 1, 2, · · ·, with respect to I, repeatedly updated according to formula (65) in _turn, the update to _{w = {w 1, w 2} , ··· w I} is the value of does not change, respectively to continue.

  The input signal parameter updating unit 6 calculates Ψ of Expression (16) using w updated by the filter characteristic parameter updating unit 5, and updates the value of the input signal parameter u based on Expression (58). To do.

  The residual signal parameter updating unit 7 updates the value of the residual signal parameter β based on Expression (58).

The parameter convergence determination unit 8 uses ￣ _xy , R _y , R _ε calculated by the signal separation unit and the model parameter Θ ₄ = {w, u, β} updated by the model parameter update unit 10. The value of the objective function of equation (51) is calculated. If the value has converged, the parameter output unit 9 outputs the model parameter Θ ₄ = {w, u, β} and proceeds to model parameter estimation in the next segment. Migrate to On the other hand, when it does not converge, it shifts to the processing operation of the signal separation unit 4 as Θ ₄ ′ = Θ ₄ .

<Fifth Embodiment>
Next, the configuration of a signal analyzing apparatus according to the fifth embodiment of the present invention will be described with reference to FIG. The fourth embodiment is configured to perform signal analysis of normal voice signals (including speech and singing voices) using the methods 1 to 4 described above. The configuration of the signal analyzing apparatus in the fifth embodiment differs from the first to fourth embodiments in that a mel cepstrum coefficient extracting unit 11 is provided instead of the fundamental frequency extracting unit 1 shown in FIG. In the signal analysis device according to the fifth embodiment, a time series of mel frequency cepstrum coefficients (MFCC) extracted from an audio signal is convoluted with a step-like input signal in the same manner as a basic frequency sequence of singing voices. It is assumed that the input signal representing the phoneme string and the filter characteristic are separated from the observed MFCC signal.

  The mel cepstrum coefficient extraction unit 11 performs frequency analysis on the audio signal and extracts a time series of mel cepstrum coefficients (usually a vector of about 12 dimensions). The signal analysis apparatus according to the fifth embodiment performs an analysis process for each time-series signal of each element of the MFCC vector.

  Segment dividing unit 2, parameter initial value generating unit 3, signal separating unit 4, filter characteristic parameter updating unit 5, input signal parameter updating unit 6, residual signal parameter updating unit 7, parameter convergence determination unit 8 and parameter output unit 9 The processing operation is based on any one of the first to fourth embodiments described above.

<Experimental result>
Next, in order to show the effects and operations of the present invention, the results of experiments using the signal analysis apparatus according to the embodiment of the present invention will be described below. Here, the conventional method of the first embodiment, the second embodiment, the fourth embodiment and the non-patent document 4 is implemented, and the input step signal (pitch target value sequence) and filter of the fundamental frequency sequence are implemented. The decomposition performance into an impulse response signal (singing dynamic variation component) representing the characteristic is evaluated.

  In the first evaluation experiment, it is confirmed whether the present invention can solve the local minimization problem. First, 100 parameters u are randomly determined, and 100 input step signals are artificially created based on Expression (18). Here, N = 300 and α = 2. Similarly, 100 ζ and Ω are respectively determined at random, and 100 impulse response signals are artificially created based on Expression (12). Based on these artificial signals and equation (22), 100 observation signals are artificially created. Here, β = 100. In this experiment, model parameters are estimated for each of these observation signals.

As an evaluation scale, for each observed signal,
(1) A square error between the estimated model parameter u and u determined at random for creating the observation signal. (2) An impulse response signal of the system constituted by the estimated model parameter, Root Mean Square Error (RMSE) of impulse response signal based on ζ and Ω determined randomly for creation
Calculate In both cases, if the error is small, it can be said that the resolution performance of the observation signal into the input step signal and the impulse response signal is high (a local minimization problem can be avoided). Table 1 shows the average values of the respective errors, and the signal analysis apparatus according to the fourth embodiment has the smallest error.

In the second evaluation experiment, the convergence performance of the parameter learning algorithm is evaluated using an actual fundamental frequency sequence extracted from the singing voice acoustic signal. As a singing voice database, a database consisting of singing voices of classical vocalists, pop singers, and amateurs who have not been trained in music (one male and one female each) was used. The singer sang the song without accompaniment. The songs are “Kirakira Hoshi” and “Song of Joy” (the song part of Beethoven's 9th 4th movement was written by Toichiro Iwasa). Singing with Japanese lyrics (two patterns of lyrics) and singing with Hamming. The fundamental frequency is estimated every 5 ms using the YIN described above. The frequency o _Hz expressed in Hz is converted into a logarithmic scale frequency o _cent expressed as _cent as follows.

  By this conversion, a semitone corresponds to 100 cent. Next, as shown in FIG. 4, the estimated fundamental frequency sequence is manually divided into segments. As a result, the total number of segments was 1323. Estimate model parameters for each segment. As an evaluation measure, for each segment, the root mean square error (RMSE) between the observed signal o and the signal μ (see equation (23)) recombined with the estimated model parameters was calculated. The average value of the RMSE is shown in the right part of Table 1. Also in this evaluation experiment, the signal analysis apparatus according to the fourth embodiment has the smallest error.

  FIG. 6 shows the observed signal and the signal μ recombined with the model parameters of the fourth embodiment. Looking at the observed signal, it can be seen that each segment has a complex overlap of dynamic characteristics related to the rise of the pitch and dynamic characteristics such as vibrato that vibrate when the pitch stabilizes. Since the present invention has means for separating each using the EM method, the parameter estimation performance is improved as compared with the conventional method. In particular, the signal analysis apparatus according to the fourth embodiment expresses the filter characteristics of the second-order system with linear sums of various vibration bases, and further has sparse restrictions on the weight parameters. Compared with, the problem of overfitting to the observed signal is solved and the error is minimized.

FIG. 7 shows the average values for each singer of ζ and Ω estimated by the signal analyzing apparatus according to the fourth embodiment. Paying attention to the largest value w _i of the element of w estimated in each segment, the average value over all segments of ζ and Ω corresponding thereto was calculated for each singer. A small value of ζ means that the vibration phenomenon is a damped vibration such as an overshoot. On the other hand, a small value of Ω means that the pitch rise time is long. Therefore, since amateur singers have poor singing skills, the values of ζ and Ω are both smaller than those of other singers. FIG. 8 shows the frequency distribution of each u singer estimated by the signal analyzing apparatus according to the fourth embodiment. For vocalists and pop singers, a distribution peak is observed at a position that is an integral multiple of a semitone (100 cent). On the other hand, for amateur singers, the peak is unclear. This also means that it is difficult for amateur singers to sing with the correct scale (accurate semitones) because of their poor singing skills.

  Thus, the observed signal is composed of the sum of the output signal of the system obtained by convolution of the input step signal and the impulse response signal representing the filter characteristics, and the residual signal. Is a process of generating a signal for each segment divided by the time when the sound rises (the pitch changes) as the start point and the end point. In addition, since various dynamic characteristics exist in the observed signal, the signal generation process is modeled from the viewpoint of a Gaussian process (Bayesian approach), and the process of separating the observed signal into an output signal and a residual signal is applied. . Then, the input signal parameter and the filter characteristic parameter are estimated based on the separated output signal. On the other hand, a residual signal parameter is estimated based on the separated residual signal. By repeating this estimation flow many times, the model parameter can be estimated with accuracy.

  As described above, assuming that the input signal is a step signal and assuming that the filter characteristic of the system follows a linear quadratic system, the input signal and filter characteristic of the signal generation system can be obtained from the observed time series signal. (Impulse response signal), because the residual signal is estimated, from the fundamental frequency (F0) sequence of the singing voice, the pitch target value sequence that the singer wants to sing, and singing dynamic fluctuation components such as vibrato and overshoot Feature extraction can be performed.

  3 and 5 are recorded on a computer-readable recording medium, and the program recorded on the recording medium is read into a computer system and executed. Signal analysis processing may be performed. Here, the “computer system” includes an OS and hardware such as peripheral devices. The “computer system” includes a WWW system having a homepage providing environment (or display environment). The “computer-readable recording medium” refers to a storage device such as a flexible medium, a magneto-optical disk, a portable medium such as a ROM and a CD-ROM, and a hard disk incorporated in a computer system. Further, the “computer-readable recording medium” refers to a volatile memory (RAM) in a computer system that becomes a server or a client when a program is transmitted via a network such as the Internet or a communication line such as a telephone line. In addition, those holding programs for a certain period of time are also included.

  The program may be transmitted from a computer system storing the program in a storage device or the like to another computer system via a transmission medium or by a transmission wave in the transmission medium. Here, the “transmission medium” for transmitting the program refers to a medium having a function of transmitting information, such as a network (communication network) such as the Internet or a communication line (communication line) such as a telephone line. The program may be for realizing a part of the functions described above. Furthermore, what can implement | achieve the function mentioned above in combination with the program already recorded on the computer system, what is called a difference file (difference program) may be sufficient.

  By extracting the dynamic characteristic features of time series signals, it can be applied to applications where signal analysis is indispensable.

  DESCRIPTION OF SYMBOLS 1 ... Fundamental frequency extraction part, 2 ... Segment division part, 3 ... Parameter initial value generation part, 4 ... Signal separation part, 5 ... Filter characteristic parameter update part, 6 ... Input Signal parameter update unit, 7 ... residual signal parameter update unit, 8 ... parameter convergence determination unit, 9 ... parameter output unit, 10 ... model parameter update unit, 11 ... mel cepstrum coefficient extraction Part

Claims (11)

The observed signal is represented by the sum of the output signal and residual signal of the signal generation system obtained by convolution of the input signal and the impulse response signal representing the filter characteristics, and a model representing the input signal is constructed from the observed signal A dynamic characteristic feature of a time-series signal is extracted by estimating an input signal parameter to be processed, a filter characteristic parameter constituting a model representing the filter characteristic, and a residual signal parameter constituting a model representing the residual signal A signal analyzer,
A parameter initial value generating unit that generates initial values of the input signal parameter, the filter characteristic parameter, and the residual signal parameter from the observed signal;
A set of the filter characteristic parameter, the input signal parameter, and the residual signal parameter is used as a model parameter, and the observation signal is defined by the input signal parameter and the filter characteristic parameter using the model parameter. A signal separation unit for separating the output signal of the generation system and the residual signal configured by the residual signal parameters;
A function obtained by adding the prior probability of the model parameter to the conditional expected value of the log likelihood function when the observed signal, the model parameter, and the set of the output signal and the residual signal are given A model parameter updating unit that updates the model parameter so as to maximize the objective function with respect to the model parameter,
It is determined whether or not the model parameter satisfies a predetermined criterion, and when it is determined that the predetermined parameter is not satisfied, processing by the signal separation unit and the model parameter update unit is performed until the predetermined criterion is satisfied. A parameter convergence determination unit to be performed again;
A parameter output unit that outputs the model parameter when the parameter convergence determination unit determines that the model parameter satisfies a predetermined criterion;
A signal analyzing apparatus comprising: The input signal is a step signal;
The output signal is stochastically modeled as following a multidimensional Gaussian distribution;
The signal analysis apparatus according to claim 1, wherein the residual signal is probabilistically modeled as Gaussian white noise. The filter characteristic of the signal generation system is represented by a filter derived by a difference method,
The signal analysis apparatus according to claim 2, wherein the filter characteristic parameters are a parameter that is inversely proportional to the square of the natural frequency and a parameter that is proportional to the attenuation factor and inversely proportional to the natural frequency.   The signal analysis apparatus according to claim 2, wherein the filter characteristic of the signal generation system is represented by a filter configured based on an autoregressive process, and the filter characteristic parameter is an autoregressive parameter.   The filter characteristic of the signal generation system is represented by a filter constituted by a weighted linear sum of a plurality of secondary system filters, and the filter characteristic parameter is a weight of each secondary system filter. The signal analysis apparatus according to claim 2. The model parameter update unit
Equations obtained by differentiating the auxiliary function of the objective function composed of auxiliary variables with a parameter inversely proportional to the square of the natural frequency and a parameter proportional to the attenuation rate and inversely proportional to the natural frequency. A filter characteristic parameter updating unit that updates the value of the filter characteristic parameter by solving the simultaneous equation consisting of
An input signal parameter updating unit for updating the input signal parameter by solving an equation obtained by differentiating the auxiliary function with the input signal parameter;
4. The signal analysis according to claim 3, further comprising a residual signal parameter for updating the residual signal parameter by solving an equation obtained by differentiating the auxiliary function with the residual signal parameter. apparatus. The model parameter update unit
A filter characteristic parameter updating unit that updates the value of the filter characteristic parameter by solving for the filter characteristic parameter a simultaneous equation consisting of equations obtained by differentiating the objective function with each autoregressive parameter included in the filter characteristic parameter. When,
An input signal parameter update unit for updating the input signal parameter by solving an equation obtained by differentiating the objective function with the input signal parameter;
5. The signal analysis according to claim 4, further comprising a residual signal parameter for updating the residual signal parameter by solving an equation obtained by differentiating the objective function with the residual signal parameter. apparatus. The model parameter update unit
By solving a nonlinear simultaneous equation consisting of equations obtained by differentiating the auxiliary function of the objective function composed of auxiliary variables by the weights of the respective second-order filters, which are the filter characteristic parameters, with respect to the filter characteristic parameters, A filter characteristic parameter update unit for updating the value of the filter characteristic parameter;
An input signal parameter updating unit for updating the input signal parameter by solving an equation obtained by differentiating the auxiliary function with the input signal parameter;
6. The signal analysis according to claim 5, further comprising: a residual signal parameter for updating the residual signal parameter by solving an equation obtained by differentiating the auxiliary function with the residual signal parameter. apparatus.   The signal separation unit uses the expected value of complete data composed of the output signal and the residual signal when the observation signal and the model parameter are given, and the autocorrelation of the complete data, 9. The signal analyzing apparatus according to claim 2, wherein the observation signal is separated into an output signal and a residual signal. The observed signal is represented by the sum of the output signal and residual signal of the signal generation system obtained by convolution of the input signal and the impulse response signal representing the filter characteristics, and a model representing the input signal is constructed from the observed signal A dynamic characteristic feature of a time-series signal is extracted by estimating an input signal parameter to be processed, a filter characteristic parameter constituting a model representing the filter characteristic, and a residual signal parameter constituting a model representing the residual signal A signal analysis method in a signal analyzer,
A parameter initial value generating step of generating initial values of the input signal parameter, the filter characteristic parameter, and the residual signal parameter from the observed signal;
A set of the filter characteristic parameter, the input signal parameter, and the residual signal parameter is used as a model parameter, and the observation signal is defined by the input signal parameter and the filter characteristic parameter using the model parameter. A signal separation step of separating the output signal of the generation system and the residual signal constituted by the residual signal parameters;
A function obtained by adding the prior probability of the model parameter to the conditional expected value of the log likelihood function when the observed signal, the model parameter, and the set of the output signal and the residual signal are given A model parameter update step for updating the model parameter so as to maximize the objective function with respect to the model parameter,
It is determined whether or not the model parameter satisfies a predetermined criterion, and when it is determined that the model parameter does not satisfy the predetermined criterion, processing by the signal separation step and the model parameter update step is performed until the predetermined criterion is satisfied. A parameter convergence determination step to be performed again;
A parameter output step for outputting the model parameter when the model parameter is determined to satisfy a predetermined criterion by the parameter convergence determination step;
A signal analysis method characterized by comprising: The observed signal is represented by the sum of the output signal and residual signal of the signal generation system obtained by convolution of the input signal and the impulse response signal representing the filter characteristics, and a model representing the input signal is constructed from the observed signal A dynamic characteristic feature of a time-series signal is extracted by estimating an input signal parameter to be processed, a filter characteristic parameter constituting a model representing the filter characteristic, and a residual signal parameter constituting a model representing the residual signal A signal analysis program for causing a computer on a signal analyzer to perform signal analysis,
A parameter initial value generating step of generating initial values of the input signal parameter, the filter characteristic parameter, and the residual signal parameter from the observed signal;
A set of the filter characteristic parameter, the input signal parameter, and the residual signal parameter is used as a model parameter, and the observation signal is defined by the input signal parameter and the filter characteristic parameter using the model parameter. A signal separation step of separating the output signal of the generation system and the residual signal constituted by the residual signal parameters;
A function obtained by adding the prior probability of the model parameter to the conditional expected value of the log likelihood function when the observed signal, the model parameter, and the set of the output signal and the residual signal are given A model parameter update step for updating the model parameter so as to maximize the objective function with respect to the model parameter,
It is determined whether or not the model parameter satisfies a predetermined criterion, and when it is determined that the model parameter does not satisfy the predetermined criterion, processing by the signal separation step and the model parameter update step is performed until the predetermined criterion is satisfied. A parameter convergence determination step to be performed again;
A parameter output step for outputting the model parameter when the model parameter is determined to satisfy a predetermined criterion by the parameter convergence determination step;
A signal analysis program for causing the computer to execute

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