WO2022064602A1

WO2022064602A1 - Signal processing device, method, and program

Info

Publication number: WO2022064602A1
Application number: PCT/JP2020/036059
Authority: WO
Inventors: 林太郎池下; 智広中谷
Original assignee: 日本電信電話株式会社
Priority date: 2020-09-24
Filing date: 2020-09-24
Publication date: 2022-03-31
Also published as: JPWO2022064602A1

Abstract

This signal processing device comprises: an initial setting unit 1 that sets a separation matrix W to an initial value, the separation matrix W being W = [w₁, ···, w_K]∈C^K×K where K is a positive integer of 2 or more and the number of microphones and sound sources; an auxiliary function updating unit 2 that updates an auxiliary function g(W) by updating, on the basis of the separation matrix W and an observation signal, a positive-definite Hermitian matrix V₁, ···, V_K appearing in the auxiliary function g(W), with h denoting a complex conjugate transpose; a separation matrix updating unit 3 that updates the separation matrix W by solving an optimization problem which minimizes the updated auxiliary function g(W) with respect to the separation matrix W by the multiplicative update algorithm of the separation matrix W; and a control unit 4 that performs control such that processing of the auxiliary function updating unit and processing of the separation matrix updating unit are alternately repeatedly performed until a predetermined convergence condition is satisfied.

Description

Signal processing equipment, methods and programs

The present invention relates to a sound source separation technique for estimating an acoustic signal of an individual sound source from a mixed acoustic signal collected by a plurality of microphones.

The optimization algorithm (AuxIVA) based on the auxiliary function method, which was developed for independent vector analysis (IVA), which is a method for separating sound sources based on statistical independence between sound sources, is stable and stable. It has been confirmed by experiments that it operates at high speed (see, for example, Non-Patent Document 1). In AuxIVA, it is necessary to repeatedly solve the max-det problem, which is an optimization problem that minimizes the auxiliary function (g (W) in Eq. (1) of Non-Patent Document 1) with respect to the separation matrix W. As an optimization algorithm for solving this problem, an algorithm called an iterative projection method (IP1 of Non-Patent Document 1 and IP2 of Non-Patent Document 2) has been proposed so far.

However, with the conventional optimization algorithms IP1 and IP2 that solve the Max-det problem, the dimension of the parameters that can be updated per iteration is small, and convergence may be slow.

An object of the present invention is to provide a signal processing device, method and program in which the dimension of a parameter that can be updated per iteration is larger than before and the convergence is faster than before.

In the signal processing device according to one aspect of the present invention, K is a positive integer of 2 or more, the number of microphones and sound sources, and the separation matrix W is W = [w ₁ , ..., w _K ] ∈ C ^{K × K.} The initial setting part that sets the separation matrix W to the initial value, and h represents the complex conjugate translocation, and the positive constant value Elmeat matrix V ₁ , ..., V _K appearing in the auxiliary function g (W) is the separation matrix W and observation. Multiplying the separation matrix W with the auxiliary function updater that updates the auxiliary function g (W) by updating based on the signal and the optimization problem that minimizes the updated auxiliary function g (W) for the separation matrix W. A separation matrix update unit that updates the separation matrix W by solving with an update algorithm, and a control unit that controls the processing of the auxiliary function update unit and the separation matrix update unit to be alternately and repeatedly executed until a predetermined convergence condition is satisfied. And have.

The dimension of the parameter that can be updated per iteration is larger than before, and the convergence can be made faster than before.

FIG. 1 is a diagram showing an example of a functional configuration of a signal processing device. FIG. 2 is a diagram showing an example of a processing procedure of a signal processing method. FIG. 3 is a diagram showing an example of Algorithm 1. FIG. 4 is a diagram showing an example of the algorithm 2. FIG. 5 is a diagram showing an example of the algorithm 3. FIG. 6 is a diagram showing an example of a functional configuration of a computer.

Hereinafter, embodiments of the present invention will be described in detail. In the drawings, the components having the same function are given the same number, and duplicate description is omitted.

[Signal processing device and method]
As shown in FIG. 1, the signal processing device includes, for example, an initial setting unit 1, an auxiliary function update unit 2, a separation matrix update unit 3, and a control unit 4.

The signal processing method is, for example, realized by each component of the signal processing apparatus performing the processing of steps S1 to S4 described below and shown in FIG.

Hereinafter, each component of the signal processing device will be described. It is assumed that the following processing is performed for each frequency bin.

<Initial setting unit 1>
The initial setting unit 1 sets the separation matrix W to the initial value (step S1).

The initial value may be a predetermined value or a random number.

Here, the separation matrix W is W = [w ₁ ,…, w _K ] ∈ C ^{K × K.} K is a positive integer greater than or equal to 2 and is the number of microphones and sound sources. It is assumed that the number of microphones and the number of sound sources are the same. As k = 1, ..., K, w _k ∈ C ^K is a filter for separating the signals emitted from the sound source k.

<Auxiliary function update unit 2>
The separation matrix W is input to the auxiliary function update unit 2. In the first process of the auxiliary function update unit 2, the separation matrix W set by the initial setting unit 1 is input. In the second and subsequent processes of the auxiliary function update unit 2, the separation matrix W updated by the separation matrix update unit 3 described later is input. Further, an observation signal is input to the auxiliary function update unit 2.

The auxiliary function update unit 2 updates the auxiliary function g (W) by updating the definite matrix Hermitian matrices V ₁ , ..., V _K based on the input separation matrix W and the observed signal (step S2).

The updated auxiliary function g (W) is output to the separation matrix update unit 3.

The observation signal is a signal that is the target of sound source separation.

The definite-value Hermitian matrix V ₁ ,…, V _K ∈ S ₊₊ ^K is a matrix that appears in the auxiliary function g (W) defined by the following equation. S ₊₊ represents the set of the entire positive constant Hermitian matrix. The h on the right shoulder of the matrix or vector represents the complex conjugate transpose of the matrix or vector. In the following equation, the base of log is the natural logarithm e.

The definite matrix Hermitian matrix V ₁ , ..., V _K is updated based on the separation matrix W and the observation signal by, for example, the method described in Non-Patent Document 1.

<Separation matrix update unit 3>
The auxiliary function g (W) updated by the auxiliary function update unit 2 is input to the separation matrix update unit 3.

The separation matrix update unit 3 updates the separation matrix W by solving the optimization problem that minimizes the updated auxiliary function g (W) with respect to the separation matrix W by the multiplication update algorithm of the separation matrix W (step S3). ).

The updated separation matrix W is output to the control unit 4 and the auxiliary function update unit 2.

As the multiplication update algorithm of the separation matrix W, any of the algorithms 1 to 3 described below can be adopted.

<< Algorithm 1 >>
FIG. 3 shows the algorithm 1.

In step 2 of the algorithm 1, the separation matrix update unit 3 determines a subset F ⊆ C ^{K × K} of the separation matrix W.

In step 3 of the algorithm 1, the separation matrix updater 3 determines the matrix Q ∈ F that minimizes the value of g (WQ) based on the determined F.

In step 4 of the algorithm 1, the separation matrix update unit 3 calculates WQ from the determined matrix Q, and the calculation result is the updated separation matrix W.

The separation matrix update unit 3 repeats the processes of step 2 to step 4 of the algorithm 1. As described as "1 repeat" and "until convergence" in the algorithm 1, the separation matrix update unit 3 may repeat the processes of steps 2 to 4 of the algorithm 1 until a predetermined convergence condition is satisfied.

In this way, the separation matrix update unit 3 determines the matrix Q ∈ F that minimizes the value of g (WQ), calculates WQ from the determined matrix Q, and updates the calculation result of the separation matrix. Repeat the process of setting W.

In this way, by updating the separation matrix using the multiplication update algorithm, the dimension of the parameter that can be updated in one iteration can be increased. As a result, the convergence becomes faster than before.

<< Algorithm 2 >>
FIG. 4 shows the algorithm 2.

Algorithm 2 can be said to be a subordinate concept of Algorithm 1. Specifically, if it is considered that the processing of Step 3 to Step 4 of Algorithm 1 is realized by the processing of Step 3 to Step 7 of Algorithm 2 or the processing of Step 8 to Step 12 of Algorithm 2, Algorithm 2 Can be said to be a subordinate concept of Algorithm 1. Hereinafter, the parts different from Algorithm 1 will be mainly described.

In the algorithm 2, in step 2, the separation matrix update unit 3 alternately selects the set _FL and the set F _U as F. In other words, when the separation matrix update unit 3 selects the set FL in the previous process of step 2, the separation matrix update unit 3 _selects the set F _U in the current process of step 2. Further, when the separation matrix update unit 3 selects the set F _U in the previous process of step 2, the separation matrix update unit 3 selects the set _FL in the process of this step 2. In the process of the first step 2, the separation matrix update unit 3 may select either the set _FL or the set F _U.

Here, FL is a set _FL ∈ _C ^{K × K} of the entire matrix which is a lower triangular matrix and whose lower right component is 1. Further, FU is a set _FU ∈ _C ^{K × K} of the entire matrix which is an upper triangular matrix and whose upper left component is 1. For example, when K = 4, Q _L ⊆ F _L and Q _U ⊆ F _U are the following matrices. In the following Q _L and Q _U , * is an arbitrary number, and the value of the blank element of the matrix is 0.

When _FL is selected in the process of step 2, the separation matrix update unit 3 performs the processes of steps 3 to 7. That is, in this case, the separation matrix update unit 3 performs the processes of steps 4 to 7 for each of i = 1, ..., K-1.

In step 4, the separation matrix update unit 3 processes V _{i: K} ← W _{i: K} ^h V _i W _{i: K} ∈ S ₊₊ ^{K-i + 1} . Let A be any matrix or vector, and let A _{i: K} be a matrix or vector consisting of the i-th to K-th elements of A. For example, _{Wi: K} = [w _i ,…, w _K ] ∈ C ^{K × (K-i + 1)} .

In step 5, the separation matrix update unit 3 processes p _{i: K} ← V _{i: K} ^-1 e ₁ ^{(K-i + 1)} ∈ C ^{K-i + 1} . Here, let a and b be arbitrary positive integers, e _a ^(b) is a vector in which the number of elements is b, the ath element is 1, and the other elements are 0.

In step 6, the separation matrix update unit 3 performs the process of q _{i: K} ← p _{i: K} (p _{i: K} ^h V _{i: K} p _{i: K} ) ^-1/2 .

In step 7, the separation matrix update unit 3 processes w _i ← W _{i: K} q _{i: K} ∈ C ^K.

When _FU is selected in the process of step 2, the separation matrix update unit 3 performs the processes of steps 8 to 12. That is, in this case, the separation matrix update unit 3 performs the processes of steps 8 to 12 for each of i = K, ..., 2.

In step 9, the separation matrix update unit 3 processes V _{1: i} ← W _{1: i} ^h V _i W _{1: i} ∈ S ₊₊ ⁱ .

In step 10, the separation matrix update unit 3 processes p _{1: i} ← V _{1: i} ^-1 e _i ⁽ⁱ⁾ ∈ C ⁱ .

In step 11, the separation matrix update unit 3 processes q _{1: i} ← p _{1: i} (p _{1: i} ^h V _{1: i} p _{1: i} ) ^{-1 / 2} ∈ C ⁱ .

In step 12, the separation matrix update unit 3 processes w _i ← W _{1: i} q _{1: i} ∈ C ^K.

The separation matrix update unit 3 repeats the processes from step 2 to step 12 of the algorithm 2. As described as "1 repeat" and "until convergence" in the algorithm 2, the separation matrix update unit 3 may repeat the processes of step 2 to step 12 of the algorithm 2 until a predetermined convergence condition is satisfied.

With the conventional method IP1, only the K dimension could be updated at each iteration. On the other hand, the algorithm 2 allows the 2 + 3 + ... + K dimensions of the W parameter dimension K ² to be updated at each iteration. As a result, the convergence becomes faster than before.

The theoretical background of Algorithm 2 will be described below.

It is known that for any regular matrix Q, there is a permutation matrix P, and PQ can be LU decomposed. That is, it is known that L ∈ FL and U ∈ FU for which PQ = LU exist. Furthermore, for general Q, it is known that P can be taken as an identity matrix. From this, the max-det problem

I give up solving directly and decide to solve the following two optimization problems alternately.

In the following, we derive an algorithm that exactly solves Q ∈ argmin {g (WQ) | Q ∈ F}. First, consider the case of F = F _U. If we set the vector consisting of the non-zero components of the i-th column of Q ∈ F _U as q _{1: i} ∈ C _i , and write W _{1: i} = [w ₁ ,…, w _i ] ∈ C ^{K × i} . , G (WQ) is expressed as follows, except for the constant term.

However, q _ii is the (i, i) component of Q. The global optimal solution to the problem of minimizing this g (WQ) with respect to Q can be obtained as follows because g is separated for each q _{1: i} .

Next, consider the case of F = FL. Let q _{i: K} ∈ C ^{K-i + 1} be the vector consisting of the non-zero components of the i-th column of Q ∈ FL, and W _{i: K} = [w _i ,…, w _K ] ∈ C ^{K × ( Assuming K-i + 1)} , g (WQ) is expressed as follows, except for the constant term.

The global optimum solution of the problem that minimizes this g (WQ) with respect to Q can be obtained as follows in the same manner as before.

From this, the algorithm 2 is as shown in FIG.

<< Algorithm 3 >>
FIG. 5 shows the algorithm 3.

Algorithm 2 can be said to be a subordinate concept of Algorithm 1. Specifically, if it is considered that the processing of Step 3 to Step 4 of Algorithm 1 is realized by the processing of Step 3 to Step 10 of Algorithm 3 or the processing of Step 11 to Step 18 of Algorithm 3, Algorithm 3 Can be said to be a subordinate concept of Algorithm 1. Hereinafter, the parts different from Algorithm 1 will be mainly described.

In Algorithm 3, K is even.

In the algorithm 3, in step 2, the separation matrix update unit 3 alternately selects the set _FL2 and the set F _U2 as F. In other words, when the separation matrix update unit 3 selects the set F _L2 in the previous process of step 2, the separation matrix update unit 3 selects the set F _U 2 in the process of this step 2. Further, when the separation matrix update unit 3 selects the set F _U2 in the previous process of step 2, the separation matrix update unit 3 selects the set F _L 2 in the process of this step 2. The separation matrix update unit 3 may select either the set FL _{2 or the set F U} ₂ in the process of the first step 2.

Here, _FL 2 is a set of all matrices in which the lower right block is I ₂ and is a triangular matrix under the block, F _L 2 ∈ C ^{K × K.} Also, F _{U 2} is a set of all matrices F _{U 2} ∈ C ^{K × K} , which is a triangular matrix on the block and the upper left block is I ₂ . For example, when K = 6, Q _L2 ⊆ F _L2 and Q _U2 ⊆ F _{U 2} are the following matrices. It is assumed that the block is a 2 × 2 matrix. In the following Q _L2 and Q _{U 2} , * is an arbitrary number, and the value of the blank element of the matrix is 0.

When _FL2 is selected in the process of step 2, the separation matrix update unit 3 performs the process of step 3 to step 10. That is, in this case, the separation matrix update unit 3 performs the processes from step 4 to step 10 for each of i = 1, ..., K / 2-1.

In steps 4 to 6, the separation matrix updater 3 performs V _{j: K} ← W _{2i-1: K} ^h V _j W _{2i-1: K} ∈ S ₊₊ ^K- for each of j = 2i-1, 2i. The process of ^{2i + 2} and the process of setting the 2 × 2 submatrix in the upper left of V _{j: K} ^-1 to G _{j: K} are performed.

In step 7, the separation matrix updater 3 solves the following 2 × 2 generalized eigenvalues problem. That is, a _2i-1 and a _2i satisfying the following equations are obtained. Here, a _2i-1 and a _2i are 2 × 1 matrices.

In steps 8 to 10, the separation matrix update unit 3 performs the following processing for each of j = 2i-1, 2i. Here, where m is a positive integer of 1 or more, 0 _m is a matrix of m × 1 in which each element is 0.

When _FU is selected in the process of step 2, the separation matrix update unit 3 performs the processes of steps 11 to 18. That is, in this case, the separation matrix update unit 3 performs the processes from step 12 to step 18 for each of i = K / 2, ..., 2.

In steps 12 to 14, the separation matrix updater 3 performs the process of V _{1: j} ← W _{1: 2i} ^h V _j W _{1: 2i} ∈ S ₊₊ ²ⁱ and V for each of j = 2i-1, 2i. The process is performed so that the 2 × 2 main submatrix at the lower right of _{1: j} ^-1 is G _{1: j} .

In step 15, the separation matrix updater 3 solves the following 2 × 2 generalized eigenvalues problem. That is, a _2i-1 and a _2i satisfying the following equations are obtained.

In steps 16 to 18, the separation matrix update unit 3 performs the following processing for each of j = 2i-1, 2i.

The separation matrix update unit 3 repeats the processes from step 2 to step 18 of the algorithm 3. As described as "1 repeat" and "until convergence" in the algorithm 3, the separation matrix updater 3 may repeat the processes of step 2 to step 18 of the algorithm 3 until a predetermined convergence condition is satisfied.

With the conventional method IP1, only the K dimension could be updated at each iteration. On the other hand, the algorithm 2 can update the 8 + 16 + ... + 2K dimension of the W parameter dimension K ² at each iteration. As a result, the convergence becomes faster than before.

The theoretical background of Algorithm 3 will be described below.

First, consider the case of F = F _U2 . Let q _2i-1 , q _2i ∈ C ²ⁱ be the vector consisting of the non-zero components of the (2i-1) and 2i columns of Q ∈ F _U2 , and the (2i-1) and 2i columns of Q. Let us write the 2 × 2 diagonal block in Q _{2i-1: 2i} ∈ C ^{2 × 2} . Also, let W _{1: 2i} = [w ₁ ,…, w _2i ] ∈ C ^{K × 2i} . Then, g (WQ) is expressed as follows, except for the constant term.

The stagnation condition for Q ∈ F _{U 2} of g (WQ) can be written as follows for each q _2i-1 and q _2i ∈ C _2i (i = 2,…, K ² ).

Q that satisfies this retention condition and the global optimal solution Q ∈ argmin {g (WQ) | Q ∈ F _U2 } can be obtained through generalized eigenvalue decomposition as shown in Proposition 4 of Reference 1. The description is omitted here due to space limitations.

[Reference 1] R. Ikeshita, N. Ito, T. Nakatani, and H. Sawada, "Independent Low-rank Matrix Analysis with Decorrelation Learning," Proc. IEEE Workshop Applications of Signal Processing to Audio and Acoustics (WASPAA), pp. 283 --287, Oct. 2019.
Next, consider the case of F = F _L2 . Let q _2i-1 , q _2i ∈ C ^{K-2i + 2} be the vector consisting of the non-zero components of the (2i-1) and 2i columns of Q ∈ F _L2 , and W _{(2i-1) :.} If we write _K = [w _2i-1 ,…, w _K ] ∈ C ^{K × (K-2i + 2)} , g (WQ) is expressed as follows, except for the constant term.

The global optimal solution of the problem of minimizing this g (WQ) with respect to Q can be obtained by Proposition 4 of Reference 1 as before.

From this, the algorithm 3 is as shown in FIG.

<Control unit 4>
The separation matrix W updated by the separation matrix update unit 3 is input to the control unit 4.

The control unit 4 controls so that the processes of the auxiliary function update unit 2 and the separation matrix update unit 3 are alternately and repeatedly executed until a predetermined convergence condition is satisfied (step S4).

An example of a predetermined convergence condition is until a predetermined number of repetitions is reached or the update amount of each parameter becomes equal to or less than a predetermined threshold value.

When a predetermined convergence condition is satisfied, the control unit 4 outputs a separation matrix W satisfying the predetermined convergence condition. For example, the separation matrix W is output to the separation unit 5 described later.

[Modification example]
Although the embodiments of the present invention have been described above, the specific configuration is not limited to these embodiments, and even if the design is appropriately changed without departing from the spirit of the present invention, the specific configuration is not limited to these embodiments. Needless to say, it is included in the present invention.

For example, the signal processing device may further include a separation unit 5.

The observation signal X and the separation matrix W output by the control unit 4 are input to the separation unit 5.

The separation unit 5 obtains K separation signals from the observation signal X by using the separation matrix W obtained by the signal processing device, in other words, the separation matrix W output by the control unit 4.

For example, by calculating W ^h X, the signal of each sound source k (k = 1, ..., K) can be obtained.

The various processes described in the embodiments are not only executed in chronological order according to the order described, but may also be executed in parallel or individually as required by the processing capacity of the device that executes the processes.

For example, data may be exchanged directly between the constituent units of the signal processing device, or may be performed via a storage unit (not shown).

[Program, recording medium]
The processing of each part of each device described above may be realized by a computer, and in this case, the processing content of the function that each device should have is described by a program. Then, by loading this program into the storage unit 1020 of the computer shown in FIG. 6 and operating it in the arithmetic processing unit 1010, the input unit 1030, the output unit 1040, etc., various processing functions in each of the above devices are realized on the computer. Will be done.

The program that describes this processing content can be recorded on a computer-readable recording medium. The computer-readable recording medium is, for example, a non-temporary recording medium, specifically, a magnetic recording device, an optical disk, or the like.

In addition, the distribution of this program is carried out, for example, by selling, transferring, renting, etc. a portable recording medium such as a DVD or CD-ROM in which the program is recorded. Further, the program may be stored in the storage device of the server computer, and the program may be distributed by transferring the program from the server computer to another computer via the network.

A computer that executes such a program, for example, first transfers a program recorded on a portable recording medium or a program transferred from a server computer to an auxiliary recording unit 1050, which is its own non-temporary storage device. Store. Then, at the time of executing the process, the computer reads the program stored in the auxiliary recording unit 1050, which is its own non-temporary storage device, into the storage unit 1020, and executes the process according to the read program. Further, as another execution form of this program, a computer may read the program directly from the portable recording medium into the storage unit 1020 and execute the processing according to the program, and further, the program from the server computer to this computer may be executed. Each time the computer is transferred, the processing according to the received program may be executed sequentially. In addition, the above processing is executed by a so-called ASP (Application Service Provider) type service that realizes the processing function only by the execution instruction and result acquisition without transferring the program from the server computer to this computer. May be. The program in this embodiment includes information used for processing by a computer and equivalent to the program (data that is not a direct command to the computer but has a property that regulates the processing of the computer, etc.).

Further, in this form, the present device is configured by executing a predetermined program on a computer, but at least a part of these processing contents may be realized by hardware.

Needless to say, other changes can be made as appropriate without departing from the spirit of the present invention.

Claims

K is a positive integer of 2 or more, the number of microphones and sound sources, the separation matrix W is W = [w 1 , ..., w K ] ∈ C K × K , and the separation matrix W is set as the initial value. Initial setting part and
h represents the complex conjugate transpose, and the auxiliary function g (W) is updated by updating the definite matrix Hermitian V 1 , ..., V K appearing in the auxiliary function g (W) based on the separation matrix W and the observed signal. Auxiliary function update part and

The separation matrix update unit that updates the separation matrix W by solving the optimization problem that minimizes the updated auxiliary function g (W) for the separation matrix W by the multiplication update algorithm of the separation matrix W.
A control unit that controls the processing of the auxiliary function update unit and the separation matrix update unit to be alternately and repeatedly executed until a predetermined convergence condition is satisfied.
Signal processing equipment including.
The signal processing device according to claim 1.
F is a subset of the separation matrix W, and the separation matrix updater determines the matrix Q ∈ F that minimizes the value of g (WQ), calculates WQ from the determined matrix Q, and the calculation result. Is repeatedly processed as the separation matrix W after updating.
Signal processing device.
The signal processing device according to claim 2.
F is a set of all matrices in which the lower right component is 1 in the lower triangular matrix FL ∈ C K × K or a set of all matrices in which the upper left component is 1 in the upper triangular matrix F U ∈ C K × K , and the separation matrix updater alternately selects the set FL and the set F U as the F.
Signal processing device.
The signal processing device according to claim 2.
K is an even number, the block is a 2x2 matrix,
The F is a set of all matrices in which the lower right block is I 2 in the lower block triangular matrix FL 2 ∈ C K × K or the entire matrix in which the upper left block is I 2 in the upper triangular matrix. The set F U2 ∈ C K × K , and the separation matrix updater alternately selects the set FL2 and the set F U2 as the F.
Signal processing device.
The signal processing device according to any one of claims 1 to 4.
Using the separation matrix W obtained by the signal processing device, a separation unit for obtaining K separation signals from the observation signal X is further included.
Signal processing device.
In the initial setting part, K is a positive integer of 2 or more, the number of microphones and sound sources, the separation matrix W is W = [w 1 , ..., w K ] ∈ C K × K , and the separation matrix W. And the initial setting step to set to the initial value,
The auxiliary function updater represents the complex conjugate translocation, and the auxiliary function g is updated by updating the positive-definite Hermitian matrix V 1 , ..., V K appearing in the auxiliary function g (W) based on the separation matrix W and the observed signal. Auxiliary function update step to update (W) and

The separation matrix update unit updates the separation matrix W by solving the optimization problem that minimizes the updated auxiliary function g (W) for the separation matrix W by the multiplication update algorithm of the separation matrix W. Steps and
A control step for controlling the control unit to alternately and repeatedly execute the processes of the auxiliary function update unit and the separation matrix update unit until a predetermined convergence condition is satisfied.
Signal processing methods including.
A program for operating a computer as each part of the signal processing device according to any one of claims 1 to 5.