JP2011175114A - Signal processing method and device - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To perform stable and fast independent component analysis in signal processing using independent component analysis. <P>SOLUTION: In a signal processing method and device for separating an observation signal, in which a plurality of signals are mixed, into separation signals by using independent component analysis, an auxiliary function is designed and an auxiliary variable and a separation matrix W are alternately updated and repeated by using the auxiliary function, and thereby, the separation matrix is calculated by monotonously decreasing an objective function J(W) in a step of obtaining the separation matrix for minimizing the objective function J(W) of independent component analysis. <P>COPYRIGHT: (C)2011,JPO&INPIT

Description

  The present invention relates to a signal processing method and apparatus, and more particularly, to a mixed signal processing method and apparatus using independent component analysis.

  Independent component analysis is a method for estimating an original signal from only a mixed signal based on the assumption of signal source independence. At present, it is widely used for acoustic signals such as speech and music, biological signals such as EEG, images, and communications. This is a signal processing technique used in the area. There are many documents on independent component analysis, and non-patent documents 1 and 2 can be cited as examples. For patent applications related to independent component analysis, Patent Documents 1 and 2 can be cited as examples.

In the signal processing using independent component analysis, the separation matrix estimation means for obtaining a separation matrix for separating the mixed signal by learning by applying independent component analysis to the observation signal composed of the mixed signal, and the observation signal Means for separating the mixed signal to obtain a separated signal by dividing the obtained separation matrix.
Independent component analysis is a nonlinear optimization problem that uses the elements of the separation matrix as parameters, and two iterative methods, FastICA and the natural gradient method, have been used to solve the nonlinear optimization problem to obtain the separation matrix. Are known.

  FastICA is a technique for obtaining a separation matrix at high speed by limiting the separation matrix to an orthogonal matrix with data whitening as preprocessing, but the observation signal data length is short or the observation signal contains outliers. In some cases, there is a problem that the accuracy of the obtained solution is lowered.

The natural gradient method estimates a separation matrix not only as an orthogonal matrix but as a general matrix. If an appropriate step size is used, convergence to a highly accurate solution can be expected, but step size tuning is required. Specifically, if the step size is too small, the convergence is slow, and if the step size is too large, there is a problem that the solution may not be obtained because it diverges during the iterative calculation.
JP2003-84793A JP2007-178590 JP2008-304718 JP2009-63761 Aapo Hyvarinen, Juha Karhunenand Erkki Oja, Independent component analysis, John Wiley & Sons, New York, NY, USA, 2001. A. Hyvarinen, “Fast and RobustFixed-Point Algorithms for Independent Component Analysis,” IEEE Trans. NeuralNetworks, vol. 10, no. 3, pp. 626.634, 1999. H. Sawada, R. Mukai, S. Araki, and S. Makino, “Polar Coordinate Based Nonlinear Function for Frequency-domainBlind Source Separation,” IEICE Trans. Fundamentals, vol. E86-A, no. 3, pp.590- 596, Mar. 2003. Hirokazu Kameoka, Junki Ono, Shigeki Hiyama, “Derivation of Parameter Optimization Algorithm for Sine Wave Superposition Model,” IEICE Technical Report, Vol. 106, EA2006-97, pp.49-54, Dec, 2006 . Hirokazu Kameoka, Junki Ono, Shigeki Hiyama, “Derivation of parameter optimization algorithm for sine wave superposition model,” IEICE Technical Report, Vol. 106, EA2006-97, pp.49-54, Dec, 2006 .

  In the signal processing using independent component analysis, the present invention does not limit the solution for obtaining a separation matrix to an orthogonal matrix, and does not require parameter tuning and ensures stable convergence. The purpose is to do.

  The technical means adopted by the present invention in order to solve such a problem is a signal processing method for separating an observation signal, in which a plurality of signals are mixed, into a separated signal using independent component analysis. In the step of obtaining a separation matrix for minimizing the function J (W), an auxiliary function is designed, and the auxiliary function and the separation matrix W are alternately updated by using the auxiliary function, thereby making the objective function J (W) monotonic. The separation matrix is obtained by decreasing the number.

とし、
目的関数Ｊ（Ｗ）を最小化する分離行列を求めるステップにおいて、
補助関数

を用いて補助変数と分離行列の更新を交互に繰り返すことで、目的関数Ｊ（Ｗ）を単調減少させて、分離行列を求めることを特徴とする信号処理方法、である。 More specifically, the present invention is a signal processing method for separating an observation signal in which a plurality of signals are mixed into a separated signal using independent component analysis,
Independent component analysis objective function

age,
In the step of obtaining a separation matrix that minimizes the objective function J (W),
Auxiliary function

The signal processing method is characterized by obtaining the separation matrix by monotonically decreasing the objective function J (W) by alternately repeating the updating of the auxiliary variable and the separation matrix.

である。
（ｂ）目的関数の第１項において、

は分離信号のコントラスト関数であり、コントラスト関数G(z)は、

を満たす。
ここで、G_R(|z|)=G(z)であり、ｚは実変数あるいは複素変数である。
また、G_R(r)は、連続で微分可能な実変数rの関数であり、G_R´(r)/rが連続
であり、r＞0 で単調減少である。
（ｃ）補助変数は

である。
（ｄ）補助関数の第１項のＶ_ｋは

である。
ここで、

である。
（ｅ）補助関数のＲは分離行列Ｗに依存しない定数、である。 (A) The separation matrix is

It is.
(B) In the first term of the objective function,

Is the contrast function of the separated signal, and the contrast function G (z) is

Meet.
Here, G _R (| z |) = G (z), and z is a real variable or a complex variable.
G _R (r) is a continuously differentiable function of the real variable r, G _R ′ (r) / r is continuous, and monotonically decreases when r> 0.
(C) Auxiliary variables are

It is.
(D) V _k of the first term of the auxiliary function is

It is.
here,

It is.
(E) R of the auxiliary function is a constant that does not depend on the separation matrix W.

  In this specification, the objective function and the auxiliary function are defined by mathematical expressions. However, mathematical expressions (including modifications, improvements, etc.) that are considered equivalent to those skilled in the art to the mathematical expressions described in this specification are claimed. Those skilled in the art will understand that they are within the scope of the invention described in the section.

に基づいて行われる。
１つの態様では、ｗ_ｋの更新は、分離行列Ｗの各行ベクトルw_ｋに対して、１つずつw_ｋを更新する。
１つの態様では、ｗ_ｋの更新は、分離行列Ｗの各行ベクトルw_ｋに対して、２つの行ベクトルのペアを更新する。 In one aspect, the update of w _k is

Based on.
In one embodiment, the update of w _k, to the row vector w _k of the separating matrix W, and updates the one by 1 w _k.
In one aspect, updating w _k updates two row vector pairs for each row vector w _k of the separation matrix W.

The signal processing method according to the present invention is executed by a computer as a hardware configuration, and the signal processing apparatus according to the present invention includes a computer (specifically, an input device, an output device, a CPU, a storage device (ROM, RAM). Etc.) and a bus for connecting them.
The present invention can also be provided as a computer program that causes a computer to execute signal processing using independent component analysis.
In one aspect, the computer program causes the computer to execute according to the update rules set forth in Table 1.
In one aspect, the computer program causes the computer to execute according to the update rules set forth in Table 2.

By adopting independent component analysis using an auxiliary function method, the present invention can estimate a separation matrix as a general matrix as well as an orthogonal matrix as in the gradient method. To provide a stable and high-speed independent component analysis in which parameter tuning is unnecessary and convergence is guaranteed.
By performing signal processing using stable and high-speed independent component analysis, a separated signal can be acquired from the mixed signal at high speed and with high accuracy.

It is a figure which compares the convergence speed of the method which concerns on this invention, and the conventional method about a stationary signal. The left figure shows the number of signal sources = 2, and the right figure shows the number of signal sources = 6. It is a figure which compares the convergence speed of the method which concerns on this invention, and the conventional method about an unsteady signal. The left figure shows the number of signal sources = 2, and the right figure shows the number of signal sources = 6. It is a figure which compares the convergence speed of the method which concerns on this invention, and the conventional method about the signal which has a big outlier. The left figure shows the number of signal sources = 2, and the right figure shows the number of signal sources = 6.

In this embodiment, by applying the optimization framework called the auxiliary function method to independent component analysis, the solution is not limited to an orthogonal matrix for a dominant Gaussian signal source, and convergence is achieved without parameter tuning. A new learning rule based on the quadratic auxiliary function of independent component analysis is derived.

Hereinafter, the signal processing embodiment according to the present invention will be described focusing on the independent component analysis using the auxiliary function method, but the optimization calculation in the independent component analysis using the auxiliary function method is not a pure mathematical method. These are used as components of signal processing of observation signals in which a plurality of signals are mixed. A typical example of signal processing to which the present invention is applied is blind sound source separation, but signal processing using independent component analysis is widely applicable to blind signal source separation in general, and other than acoustic signals. Also, for example, it can be used for separation of biological signals such as brain waves and signals in image processing.

ｘは、観測信号（観測により得られる多次元確率変数ベクトル）であり、x=(x₁・・・x_K）^tで表す。
sは、信号源（例えば、音源）信号であり、s=(s₁・・・s_K）^tで表さす。
Ａは、混合行列である。
尚、本明細書全般において、変数は複素数値を取るものとして扱うが、エルミート共役^hを転置^tで置き換えることで変数を実数値として扱うことができることは当業者に理解される。 [A] Objective function of independent component analysis An observation model (mixing process) of independent component analysis is shown below.

x is an observation signal (a multidimensional random variable vector obtained by observation), and is represented by x = (x ₁ ... x _K ) ^t .
s is a signal source (for example, a sound source) signal, and is represented by s = (s ₁ ... s _K ) ^t .
A is a mixing matrix.
Note that, throughout this specification, variables are treated as complex values, but it will be understood by those skilled in the art that variables can be treated as real values by replacing Hermite conjugate ^h with transpose ^t .

として、

のように信号を変換し、各成分（分離信号）y=(y₁・・・y_K）^tが、信号源信号sに一致するような分離行列Ｗを見つけることによって行われる（スケールとパーミュテーション問題を除いて）。 The separation process in the independent component analysis is the separation matrix W

As

Is performed by finding a separation matrix W such that each component (separation signal) y = (y ₁ ... Y _K ) ^t matches the source signal s (scale and par) Except for the mutation problem).

ここで、G(y)はコントラスト関数（信号の独立性を測る関数）と呼ばれる非２次関数であり、信号源の確率密度関数p(y)が既知である場合には、G(y)=-logp(y)と定めることにより、Jの最小化は対数尤度最大化と等価となる。Ｊ(Ｗ)の非線形性により、一般に

を解析的に解くことはできず、反復解法が必要となる。従来の反復解法としては既述のように、FastICAや自然勾配法が用いられていた。 As an index of independence, non-Gaussianity, mutual information, likelihood, and the like have been used, but in any case, it is known that it results in an objective function minimization problem of the following form (non- Patent Document 1).

Here, G (y) is a non-quadratic function called a contrast function (a function for measuring signal independence). If the probability density function p (y) of the signal source is known, G (y) By defining = -logp (y), J minimization is equivalent to log likelihood maximization. Generally due to the nonlinearity of J (W)

Cannot be solved analytically, and an iterative solution is required. As described above, FastICA and natural gradient method have been used as conventional iterative solutions.

を満たすパラメータθ=θ^＊を見つけることである。
簡単な解法は、

を解くことであるが、多くの場合、目的関数J(θ)は非線形であり、反復解法が必要となる。 [B] Application of auxiliary function method to independent component analysis [B-1] Principle of auxiliary function method The optimization problem of the objective function J (θ) is expressed by the following equation:

Is to find a parameter θ = θ ^* that satisfies
The simple solution is

In many cases, the objective function J (θ) is nonlinear, and an iterative solution is required.

を満たすような補助関数

を設計する。 In the auxiliary function method, for nonlinear objective function J (θ),

Auxiliary functions that satisfy

To design.

のように交互に反復的に適用する。尚、iは繰り返し回数である。
補助関数法では、補助関数をパラメータθと補助変数θ（バー）に対して交互に反復的に適用することにより目的関数J(θ)を単調減少させ、目的関数J(θ)を極小とするパラメータθを得る最適化手法である。 The auxiliary function method uses the auxiliary function Q instead of directly minimizing the objective function J (θ).

Apply alternately and repeatedly as follows. Note that i is the number of repetitions.
In the auxiliary function method, the objective function J (θ) is monotonously decreased by alternately and repeatedly applying the auxiliary function to the parameter θ and the auxiliary variable θ (bar), and the objective function J (θ) is minimized. This is an optimization method for obtaining the parameter θ.

よって、仮に式（５）を解析的に解くことができない場合であっても、式（８）、（９）が解析的に解けるような式（７）を満たす補助関数を設計できる場合には、補助関数法は、効果的な反復解法が得られる。しかしながら、補助関数の設計の可否は、目的関数の具体的な形状に依存するため、補助関数法の原理が既知であったとしても、独立成分分析に適用可能な補助関数を簡単に設計できるものではない。
補助関数法の原理については、特許文献３、４、非特許文献４、５に記載されており、必要に応じてこれらの文献を参照することができる。 The principle of the auxiliary function method is based on the fact that the objective function J (θ) is not increased in the update, as shown in the following proof.

Therefore, even if the equation (5) cannot be solved analytically, an auxiliary function satisfying the equation (7) that can solve the equations (8) and (9) analytically can be designed. The auxiliary function method provides an effective iterative solution. However, whether or not an auxiliary function can be designed depends on the specific shape of the objective function, so even if the principle of the auxiliary function method is known, an auxiliary function applicable to independent component analysis can be easily designed. is not.
The principle of the auxiliary function method is described in Patent Documents 3 and 4 and Non-Patent Documents 4 and 5. These documents can be referred to as necessary.

[B-2] Quadratic auxiliary function of contrast function In general, the quadratic function is a promising candidate of an auxiliary function because it can be easily minimized and has been conventionally used as an auxiliary function. A function is not applied, and a secondary auxiliary function applicable to independent component analysis is not known.
The inventors tried to design an auxiliary function of a quadratic form of w _k for the first term (nonlinear term) of Equation (1), and obtained the following theorem.

が任意のrについて成り立ち、等号はr=±r₀のときにのみ成り立つ。
定理１の証明は、本発明を実施する上では必ずしも必要としないので、定理１の証明は省略する。 [Theorem 1]
For an even function G _R (r) of a real variable r that is continuously differentiable, if G´ _R (r) / r is continuous and r> 0 and is monotonically decreasing,

Holds for any r, and the equal sign holds only when r = ± r ₀ .
Since the proof of Theorem 1 is not always necessary for carrying out the present invention, the proof of Theorem 1 is omitted.

したがって、優ガウス性の信号源に対してよく用いられる下記コントラスト関数は、定理１で必要としている条件を満たす。

The following contrast function (see Non-Patent Document 2) satisfies the conditions required by Theorem 1.

Therefore, the following contrast function often used for a dominant Gaussian signal source satisfies the condition required by Theorem 1.

に対して、G(r)が偶関数で、原点以外で微分可能、G_R(r)とG_R´(r)/r が r≧０で狭義単調減少、であるとき、p(r)は優ガウス的である、という（A. Benveniste, M.
Metivier, and P. Priouret, Adaptive algorithms and stochastic approximations,
Springer-Verlag, 1990.）。
定理１は、信号源の確率密度分布

が優ガウス性であって、G´_R(r)/rが原点で連続であるいかなるコントラスト関数(G_R(r)＝-logp(r))にも適用可能である。 The condition of G _R (r) required for Theorem 1 is closely related to Gaussianity. The definition of the Gaussian distribution is as follows.
Probability density distribution of real variable r

On the other hand, when G (r) is an even function and can be differentiated at other than the origin, G _R (r) and G _R ′ (r) / r are strictly monotonically decreasing when r ≧ 0, and p (r) Are gaussian (A. Benveniste, M.
Metivier, and P. Priouret, Adaptive algorithms and stochastic approximations,
Springer-Verlag, 1990.).
Theorem 1 is the probability density distribution of the signal source

Can be applied to any contrast function (G _R (r) = − logp (r)) where G ′ _R (r) / r is continuous at the origin.

のように定義する。
ここで、G_R(r)は、連続で微分可能な実変数rの関数であり、G´_R(r)/rが連続であり、r＞0で単調減少である。
定理１とＳ_Ｇの定義から定理１´が導かれる。
［定理１´］
全てのG(z)∈Ｓ_Ｇにおいて、G_R(|z|)=G(z)であり、

が全てのｚについて成り立ち、等号は|z|=|z₀|のときにのみ成り立つ。
例えば、非特許文献３に記載されている極座標表示に基づくコントラスト関数もＳ_Ｇに含まれる。 In order to extend Theorem 1 to the case of complex variables, a set of contrast functions of real and complex variables is uniformly defined.
Here, a set S _G of real functions of real variables / complex variables z is expressed as

Define as follows.
Here, G _R (r) is a continuously differentiable function of the real variable r, G ′ _R (r) / r is continuous, and monotonically decreases when r> 0.
Theorem 1 'is derived from the definition of Theorem 1 and _{S G.}
[Theorem 1 ']
G _R (| z |) = G (z) for all G (z) ∈S _G ,

Holds for all z, and the equal sign holds only when | z | = | z ₀ |.
For example, the contrast function based on polar coordinates described in Non-Patent Document 3 is also included in the S _G.

を設計する。
ただし、Ｒは分離行列Ｗに依存しない定数、また

に対し、

である。このとき

が任意のＷについて成り立ち、等号は

のときにのみ成り立つ。ただし、φ_kは任意の位相を表す。
式（１８）は、式（１５）に

を代入し、K=1からk=Kまで足し合わせて、期待値を取ることで得られる。 The quadratic auxiliary functions applied to the independent component analysis are as follows.
[Theorem 2]
For all G (z) ∈S _G , G _R (| z |) = G (z),

To design.
Where R is a constant independent of the separation matrix W,

Whereas

It is. At this time

Holds for any W, and the equal sign is

Only holds when However, φ _k represents an arbitrary phase.
Equation (18) is transformed into Equation (15)

Can be obtained by substituting and adding from K = 1 to k = K and taking the expected value.

に関して、変数

を交互に更新しながら最小化することにより、目的関数Ｊ（Ｗ）を単調減少させることにより得られる。
定理２より、補助変数

についてのＱの最小化は単に

とすればよいので、ここでは、分離行列ＷについてのＱの最小化に着目して説明する。 [C] Derivation of update rule [C-1] Differentiation of auxiliary function Based on the principle of the auxiliary function method, the update rule is an auxiliary function.

With respect to variables

Is obtained by monotonically decreasing the objective function J (W) by minimizing while alternately updating.
From Theorem 2, auxiliary variables

The minimization of Q for is simply

Therefore, here, the description will be given focusing on the minimization of Q with respect to the separation matrix W.

から、

が得られる。
ここで、log|detＷ|の微分は、

とすると、

となり、したがって、

となる。
これを

に変形できる。
複素共役の要素の表現は、

であり、δ_lkはクロネッカーのデルタである。式（１９）の両辺に左からw_l ^hを掛けて、式（２５）をこれらに適用すると、Ｗの各行ベクトルw_kのＫ^２個の連立方程式

を得る。
全てのw_kを同時に更新するためには、式（２６）を解く必要があるが、現時点では閉形式の解はＫ＝２の場合しか求まっていない。
本実施形態では、Ｋ個のｗ_ｋを一度に更新する代わりに、１つずつ更新していく第１学習則を用いた解法と、２つずつのペアについて更新していく第２学習則を用いた解法を提案する。 In order to minimize the auxiliary function Q with respect to the separation matrix W, the equation (16) is differentiated by w _k or w _k ^* ( ^* is a complex conjugate), and these are set to 0.

From

Is obtained.
Where log | detW |

Then,

And therefore

It becomes.
this

Can be transformed into
The representation of the complex conjugate element is

Δ _lk is the Kronecker delta. By multiplying both sides of equation (19) by w _l ^h from the left and applying equation (25) to these, K ^two simultaneous equations for each row vector w _k of W

Get.
In order to update all w _k at the same time, Equation (26) needs to be solved, but at the present time, a closed-form solution is obtained only when K = 2.
In this embodiment, instead of updating K w _k at a time, a solution using a first learning rule that updates one by one and a second learning rule that updates two pairs each The solution used is proposed.

から、

となる。
式（２８）から、w_kは

の全てに対して直交する。このようなベクトルは、任意ベクトルａから、

とし、

のようにｃ_ｋを決定することで得られる。ベクトル形式では

と書くことができ、ここで

である。

から、

が得られ、

が得られる。 [C-2] First Learning Rule The first learning rule is a learning rule that sequentially updates each of w _k . w When _l is fixed,

From

It becomes.
From equation (28), w _k is

Is orthogonal to all of. Such a vector can be derived from an arbitrary vector a

age,

It is obtained by determining _ck as follows. In vector format

Where you can write

It is.

From

Is obtained,

Is obtained.

となる。
補助変数

はＶ_ｋにのみ含まれているため、補助変数

の更新は更新されたｗ_ｋを用いてＶ_ｋを計算することと等価である。 Note that w _k in the previous update is a. Combining normalization to satisfy equation (27), the update rule for w _k is

It becomes.
Auxiliary variable

Is included only in V _k , so the auxiliary variable

Is equivalent to calculating V _k using the updated w _k .

表１に、ＭＡＴＬＡＢでの実行を示す。

In summary, the update rules are as follows.

Table 1 shows the execution with MATLAB.

となる。
式（４２）、（４３）は、Ｖ_１ｗ_１、Ｖ_２ｗ_１の両方がｗ_２に対して直交していることを示している。ｗ_２に対して直交する方向は二次元空間において一意なので、Ｖ_１ｗ_１、Ｖ_２ｗ_１は

のように平行する。ここで、γは定数である。
同様に、Ｖ_１ｗ_２、Ｖ_２ｗ_２は平行である。したがって、式（４２）、（４３）を満たすｗ_１、ｗ_２は、一般化固有値問題の解として得ることができ、Ｖ_１、Ｖ_２が特異でない場合には、これらはＶ_２ ^−１Ｖ_１の固有ベクトルとして得られる。
式（４１）、（４４）を正規化して、

を得る。 [C-3] Second Learning Rule The second learning rule is based on the closed form when K = 2 in Equation (26). If K = 2, then equation (26) becomes

It becomes.
Expressions (42) and (43) indicate that both V ₁ w ₁ and V ₂ w ₁ are orthogonal to w ₂ . Since the direction orthogonal to w ₂ is unique in the two-dimensional space, V ₁ w ₁ and V ₂ w ₁ are

Parallel to each other. Here, γ is a constant.
Similarly, V ₁ w ₂ and V ₂ w ₂ are parallel. Therefore, w ₁ and w ₂ satisfying the equations (42) and (43) can be obtained as a solution of the generalized eigenvalue problem, and when V ₁ and V ₂ are not singular, they are V ₂ ⁻¹ V Obtained as _one eigenvector.
Normalizing equations (41) and (44),

Get.

ここで、ｍ＜ｎである。これは、乗法的更新

と等価である。ここで、Ｉ_δはｍｍ，ｍｎ，ｎｍ，ｎｎ要素を除く単位行列であり、それぞれ、ａ，ｂ，ｃ，ｄである。 In order to apply the closed form solution to the general case (K> 2), consider the following pair update.

Here, m <n. This is a multiplicative update

Is equivalent to Here, I _δ is a unit matrix excluding mm, mn, nm, and nn elements, and is a, b, c, and d, respectively.

新しい変数ｕ

を導入すると、

が得られる。 In this update, the auxiliary function can be described as follows:

New variable u

Introduced

Is obtained.

The auxiliary function format in the pair update in the equations (48) and (49) is the same as in the case of K = 2. Thus, the optimal a, b, c, d that minimizes equation (53) can be obtained from h _m , h _n , which is the generalized eigenvalue problem: U _m h _m = γU _n h _n Two solutions, followed by normalization as shown in equations (46), (47).
Therefore, the following update rule is iteratively executed for all m, n pairs (m <n). Table 2 shows the execution with MATLAB.

Step 1: Perform the following calculation.

Step 2: Find two solutions h _m and h _n for the generalized eigenvalue problem of the 2 × 2 matrix.

Step 3: Perform normalization.

Step 4: Update w _m and w _n .

各信号源において、信号源数をＫ＝２，６とした条件で、瞬時混合により１０００サンプルの観測信号を生成し、これを白色化した後に各手法を適用した。混合行列は、各要素を平均０、分散１の複素ガウス分布に従う乱数により生成した。
いずれの手法においてもコントラスト関数は式（１２´）を用いた。反復的に推定される分離行列の計算時間と、それによって得られる分離信号のＳＮ比を１００試行で平均し、それぞれ縦軸、横軸としてグラフ化したものを図１乃至３に示す。なお、ＳＮ比はProjection Backによる定数倍推定に正解パーミュテーションを与えることで求めている。

計算は2.66GHzのCPUをもつノートPC上のMatlabで行った。 [D] Comparison Experiment of Separation Performance Experiments were conducted to compare the convergence speed and stability of the first and second solutions with the conventional natural gradient method (step size 0.05, 0.1, 0.2) and FastICA. Intended to simulate (1) stationary signal, (2) non-stationary signal, and (3) signal with large outliers, the phase is uniformly distributed, and the amplitude independently follows the probability density function below An artificial complex-valued signal source was used.

In each signal source, 1000 samples of observation signals were generated by instantaneous mixing under the condition that the number of signal sources was K = 2, 6, and each method was applied after whitening this. The mixing matrix was generated by random numbers according to a complex Gaussian distribution with an average of 0 and a variance of 1 for each element.
In any of the methods, the contrast function is the expression (12 ′). FIGS. 1 to 3 show the calculation time of the separation matrix estimated repeatedly and the S / N ratio of the separation signal obtained thereby averaged over 100 trials, which are plotted as a vertical axis and a horizontal axis, respectively. The S / N ratio is obtained by giving a correct permutation to the constant multiple estimation by Projection Back.

The calculation was performed with Matlab on a notebook PC with a 2.66 GHz CPU.

In the gradient method, the larger the step size, the faster the convergence, but the solution tends to diverge and the average separation performance decreases (see the result for signal type (3)).
FastICA converges very quickly on a stationary signal, but the separation performance may decrease for nonstationary signals (because independent components in finite-length observations are not necessarily uncorrelated).
In contrast to the features of these conventional methods, Solution 1 and Solution 2 do not diverge regardless of the signal, and the average separation performance at the time of convergence is high.
The convergence speeds of Solution 1 and Solution 2 are faster when Solution 2 is faster when the number of signal sources is 2 and when Solution 1 is faster when the number of signal sources is 6, but both are faster than the gradient method and are similar to FastICA (conditions The characteristics of the proposed method can be confirmed.

The signal processing according to the present invention can be used in a wide range of areas such as acoustic signals such as voice and music (typically blind sound source separation), biological signals such as EEG, images (for example, feature extraction), and communication.

Claims (8)

A signal processing method for separating an observation signal in which a plurality of signals are mixed into a separated signal using independent component analysis,
Independent component analysis objective function

age,
In the step of obtaining a separation matrix that minimizes the objective function J (W),
Auxiliary function

A signal processing method characterized by obtaining an isolation matrix by monotonically decreasing the objective function J (W) by alternately repeating the updating of the auxiliary variable and the isolation matrix using.
here,
(A) The separation matrix is

(B) In the first term of the objective function,

Is the contrast function of the separated signal, and the contrast function G (z) is

Meet,
Where G _R (| z |) = G (z) and z is a real or complex variable,
(C) Auxiliary variables are

(D) V _k of the first term of the auxiliary function is

here,

(E) R of the auxiliary function is a constant independent of the separation matrix W,
It is. The update of w _k is

The signal processing method according to claim 1, wherein the signal processing method is performed based on: The update rule for w _k is as follows:

The signal processing method according to claim 2 performed according to: The update rule for updating w _k is
Performing the following calculations:

Finding two solutions h _m , h _n of a generalized eigenvalue problem of a 2 × 2 matrix based on the following equations:

The following normalization steps;

Updating w _m and w _n as follows:

The signal processing method according to claim 2, comprising:   The computer program for making a computer perform each procedure of any one of Claims 1-4. The computer program for making a computer perform each procedure of Claim 3 according to the update rule of Table 1. FIG.

The computer program for making a computer perform each procedure of Claim 4 according to the update rule of Table 2. FIG.

A signal processing device that separates an observation signal in which a plurality of signals are mixed into a separated signal using independent component analysis,
Independent component analysis objective function

age,
A separation matrix estimating means for obtaining a separation matrix that minimizes the objective function J (W);
The separation matrix estimation means includes:
Auxiliary function

A signal processing apparatus characterized by obtaining an isolation matrix by monotonically decreasing the objective function J (W) by alternately repeating the updating of the auxiliary variable and the isolation matrix using.
here,
(A) The separation matrix is

(B) In the first term of the objective function,

Is the contrast function of the separated signal, and the contrast function G (z) is

Meet,
Where G _R (| z |) = G (z) and z is a real or complex variable,
(C) Auxiliary variables are

(D) V _k of the first term of the auxiliary function is

here,

(E) R of the auxiliary function is a constant independent of the separation matrix W,
It is.

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