JP2016039492A - Transfer function approximation device, program therefor and method therefor - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide a filter approximation device capable of approximating an input transfer function accurately.SOLUTION: A filter approximation device 1 includes transfer function decomposition means 10 for decomposing an input transfer function into an unstable transfer function and a stable transfer function, transfer function division means 20 for dividing the denominator polynomial of an unstable transfer function by a numerator polynomial, determination means 30 for making the transfer function division means 20 execute division, until all coefficients of a remainder polynomial are reduced, matched filter calculation means 40 for calculating a matched filter of a quotient polynomial, and transfer function approximation means 50 for approximating the input transfer function by the product of a matched filter and stable transfer function.SELECTED DRAWING: Figure 1

Description

  The present invention relates to a transfer function approximating device that approximates an input transfer function so that unstable poles are not included, a program thereof, and a method thereof.

  The stability of a transfer function of a discrete time system is determined by the poles included in the transfer function. The transfer function is stable when all poles are contained within the unit circle on the complex plane. At this time, the response of the transfer function converges toward zero. On the other hand, the transfer function becomes unstable when even one pole exists on or outside the unit circle. In the example of FIG. 5, the poles P1 to P3 are included in the unit circle, but the pole P4 exists on the unit circle and the pole P5 exists outside the unit circle. When the transfer function is unstable in this way, the response oscillates.

An unstable pole is a pole that exists on or outside the unit circle on the complex plane.
An unstable transfer function is a transfer function that includes an unstable pole and whose response oscillates.
The stable transfer function is a transfer function that does not include an unstable pole and converges a response.

  The stability of the transfer function is important when constructing a control system. For example, a sound field control system is intended to compensate for characteristics of a certain sound field and cancel noise and reverberation, and is often designed as an inverse system of the transfer function of the sound field. Usually, an inverse system is composed of an unstable transfer function having an unstable pole because the sound field to be controlled has an unstable zero. Since such a control system cannot be implemented, it is approximated by a stable transfer function.

For example, Non-Patent Document 1 proposes a technique in which an unstable transfer function is divided into a function having an all-pass characteristic and a minimum phase function, and an all-pass function including an unstable pole is approximated by a delay. Further, Non-Patent Document 2 proposes a technique in which unstable poles are bundled from unstable transfer functions, and the unstable poles are approximated by stable transfer functions by division called a division method and truncation of the remainder.
The function having the all-pass characteristic is a transfer function including an unstable zero point and / or a pole, and a pole and a zero point having a reciprocal complex conjugate thereof.

Matsui et al., Binaural Reproduction of 22.2 Multichannel Sound over Loudspeakers, presented at AES 129th Conv., 2010. Ro et al., Robust model following control system of non-minimum phase discrete-time system using approximate inverse system, Electrical theory C, 121, 11, 2001.

  However, the technique described in Non-Patent Document 1 accurately approximates the amplitude characteristic of the transfer function, but the phase characteristic becomes a linear phase approximation, so that the accuracy of the approximated unstable transfer function is lowered. In the technique described in Non-Patent Document 2, the truncation process is repeated until the remainder polynomial is stabilized, so that the error in the amplitude characteristic and the phase characteristic increases.

  In view of the above problems, it is an object of the present invention to provide a transfer function approximating apparatus, a program thereof, and a method thereof that can accurately approximate an input transfer function.

  In order to solve the above-described problems, a transfer function approximating device according to the present invention is configured to include a transfer function decomposing unit, a transfer function dividing unit, a determining unit, a matched filter calculating unit, and a transfer function approximating unit. .

  According to such a configuration, the transfer function approximating device uses the transfer function decomposing means to convert the input transfer function into an unstable transfer function including an unstable transfer function including an unstable pole and an inverse of the complex conjugate of the unstable pole, and an unstable transfer function. Decompose into stable transfer functions other than the stable transfer function. In this way, the transfer function approximating apparatus bundles together the unstable pole and the zero point having the inverse relation of the complex conjugate as an unstable transfer function. In addition, the unstabilized unstable transfer function has an all-pass characteristic.

The transfer function approximation device calculates the quotient polynomial and the remainder polynomial for the reciprocal of the unstable transfer function by dividing the denominator polynomial of the unstable transfer function decomposed by the transfer function decomposing means by the numerator polynomial by the transfer function dividing means. To do.
Here, dividing the denominator polynomial of the unstable transfer function by the numerator polynomial is equivalent to taking the inverse of the unstable transfer function and dividing.

  The transfer function approximating device causes the transfer function dividing unit to execute division until all the coefficients of the remainder polynomial become small so that the determination unit performs approximation using only the quotient polynomial without handling the remainder polynomial. As a result, the reciprocal of the unstable transfer function can be approximated only by the quotient polynomial, ignoring the remainder polynomial.

The transfer function approximation device calculates a matched filter of the quotient polynomial calculated by the transfer function division unit by the matched filter calculation unit. This matched filter is also called a matched filter. Then, the transfer function approximating device approximates the input transfer function by the product of the matched filter calculated by the matched filter calculating unit and the stable transfer function by the transfer function approximating unit.
Here, since the unstable transfer function has all-pass characteristics, the inverse of the unstable transfer function also has all-pass characteristics. Since the inverse of the unstable transfer function is approximated by a quotient polynomial, the matched filter of the quotient polynomial also has an all-pass characteristic.

  The present invention can also be realized by a transfer function approximation program that causes a computer to operate as each means of a transfer function approximation device. The present invention can also be realized by a transfer function approximating method in which each means of the transfer function approximating apparatus sequentially executes processing.

  According to the present invention, since the unstabilized unstable transfer function has all-pass characteristics, the reciprocal of the unstable transfer function also has all-pass characteristics, and the matched filter also has all-pass characteristics. Therefore, according to the present invention, the amplitude characteristic of the unstable transfer function can be approximated with high accuracy. Further, according to the present invention, the phase is once inverted when the reciprocal of the unstable transfer function is taken, but the phase is inverted again by calculating the matched filter, so that the phase characteristic of the unstable transfer function is restored. Can be approximated accurately. Thus, according to the present invention, the input transfer function can be approximated with high accuracy.

It is a block diagram which shows the structure of the filter approximation apparatus which concerns on 1st Embodiment of this invention. It is a flowchart which shows operation | movement of the filter approximation apparatus of FIG. In 2nd Embodiment of this invention, it is a block diagram which shows the structure of a trans-oral reproduction | regeneration system. It is a block diagram which shows the structure of the filter approximation apparatus which concerns on 2nd Embodiment of this invention. It is explanatory drawing of a prior art.

  Hereinafter, embodiments of the present invention will be described in detail with reference to the drawings as appropriate. In each embodiment, means having the same function are denoted by the same reference numerals and description thereof is omitted.

(First embodiment)
[Configuration of filter approximation device]
With reference to FIG. 1, the structure of the filter approximation apparatus (transfer function approximation apparatus) 1 which concerns on 1st Embodiment of this invention is demonstrated.
The filter approximation device 1 approximates the input transfer function H including an unstable pole so that the unstable pole is not included. As shown in FIG. 1, the filter approximation device 1 includes a transfer function decomposition unit 10, a transfer function division unit 20, a determination unit 30, a matched filter calculation unit 40, and a transfer function approximation unit 50.

The transfer function decomposing means 10 receives an input transfer function H and decomposes the input transfer function H into an unstable transfer function and a stable transfer function.
In this embodiment, the transfer function decomposing means 10 receives a filter having a coefficient of the input transfer function H from the outside (for example, a transoral reproduction system). The input transfer function H is an unstable transfer function that is represented by a ratio of rational functions and includes an unstable pole.

As in the following equation (1), an input transfer function H is unstable transfer function H that contains the zero point in the relationship between the inverse of the complex conjugate of the unstable poles and unstable poles ^- and unstable transfer function H ^- except Is defined as the product of the stable transfer function H ⁺ . This unstable transfer function H ⁻ has an all-pass characteristic. In addition, the root of the denominator polynomial of the transfer function is the pole, and the root of the numerator polynomial is the zero point. From this, the unstable transfer function H ⁻ has an unstable one of the roots of the denominator polynomial of the input transfer function H as a pole, and has a zero that is in the relationship of the inverse of the complex conjugate.

For example, consider the case where the input transfer function H = 1 / (0.5-2.25q ⁻¹ + q ⁻² ). In this case, the transfer function decomposing means 10 factors the denominator polynomial of the input transfer function H into “(0.25−q ⁻¹ ) (2−q ⁻¹ )”. At this time, the roots as the poles of the input transfer function H are “4” and “0.5”, respectively. Therefore, the transfer function decomposing means 10 uses the equation (1) to convert the input transfer function H into an unstable transfer function H ⁻ = (q ⁻¹ -4) / (0.25−q ⁻¹ ) and stable. The transfer function H ⁺ = 1 / ((q ⁻¹ -4) (2-q ⁻¹ )). Thereafter, the transfer function decomposing means 10 outputs the decomposed unstable transfer function H ⁻ and stable transfer function H ⁺ to the transfer function dividing means 20.

The transfer function dividing means 20 calculates a quotient polynomial and a remainder polynomial for the reciprocal of the unstable transfer function H ⁻ by dividing the denominator polynomial of the unstable transfer function H ⁻ input from the transfer function decomposing means 10 by the numerator polynomial. (First division).

Here, as shown in the following equation (2), the numerator polynomial of the unstable transfer function H ⁻ is set as H ⁻ _A and the denominator polynomial is set as H ⁻ _B. In this case, the transfer function dividing means 20, as in the following equation (3), the numerator polynomial H ^- _A by the denominator polynomial H ^- Divides _B, and quotient polynomial quotient and remainder of the time C and the remainder polynomial D And Thereafter, the transfer function dividing unit 20 outputs the stable transfer function H ⁺ input from the transfer function decomposing unit 10 and the calculated quotient polynomial C and remainder polynomial D to the determining unit 30.

Further, the transfer function dividing means 20, every time the division execution command to be described later is inputted from the judging unit 30, the numerator polynomial H of the remainder polynomial D ^- performing division by _A (the second and subsequent division). In other words, the transfer function dividing means 20, until all coefficients of the remainder polynomial D decreases, the remainder polynomial D numerator polynomial H ^- repeating dividing by _A.

The determination unit 30 causes the transfer function division unit 20 to perform division until all coefficients of the remainder polynomial D become small in order to perform approximation using only the quotient polynomial C without handling the remainder polynomial D.
In the present embodiment, the determination unit 30 determines whether or not all the coefficients of the remainder polynomial D are small by threshold determination for all the coefficients of the remainder polynomial D. For example, the determination unit 30 determines whether or not the threshold T is preset and all the coefficients of the remainder polynomial D are less than the threshold T.

When all the coefficients of the remainder polynomial D are not less than the threshold value T, the determination unit 30 instructs the transfer function division unit 20 to execute division (division execution command). At this time, the determination unit 30 calculates (counts) the number of division execution command outputs as the number of division iterations N.
When the transfer function dividing means 20 repeats the division, the approximation accuracy is improved, but the number of division iterations N is increased, so that a delay due to approximation (modeling delay) increases.

When all the coefficients of the remainder polynomial D are less than the threshold value T, the determination unit 30 matches the stable transfer function H ⁺ input from the transfer function division unit 20 with the calculated quotient polynomial C and the number of division iterations N. Output to 40.

<Specific examples of division and threshold determination>
Here, a specific example of dividing the denominator polynomial H ⁻ _B = 3q ⁻¹ + 4q ⁻² + 5q ⁻³ by the numerator polynomial H ⁻ _A = q ⁻¹ will be given, and the division by the transfer function division unit 20 and the determination unit 30 The threshold determination will be described.

In the first division, the transfer function dividing means 20 divides the denominator polynomial H ⁻ _B = 3q ⁻¹ + 4q ⁻² + 5q ⁻³ by “q ⁻¹ ”, so that the quotient is “3” and the remainder is “4q ⁻² ”. + 5q ⁻³ ”. In this case, the determination unit 30 performs threshold determination for each of the coefficients “4, 5” of the remainder polynomial D. Here, since the coefficients “4, 5” of the remainder polynomial D are not less than the threshold value T, the transfer function dividing unit 20 performs the second division.

In the second division, the transfer function dividing means 20 divides the remainder polynomial D = 4q ⁻² + 5q ⁻³ generated by the first division by “q ⁻¹ ”, so that the quotient is “3 + 4q ⁻¹ ” and the remainder is “5q ⁻³ ”. In this case, the determination unit 30 performs threshold determination on the coefficient “5” of the remainder polynomial D.
Until all the coefficients of the remainder polynomial D are less than the threshold value T, the division by the transfer function dividing unit 20 and the threshold value determination by the determining unit 30 are repeated.

Hereinafter, the description of the configuration of the filter approximation device will be continued.
Matched filter calculating means 40 is for calculating the matched filter C _M of the input from the judgment unit 30 quotient polynomial C.

Now consider a matched filter of the quotient polynomial C. The frequency transfer function of the matched filter is a complex conjugate of the frequency transfer function of the target signal. That is, this is equivalent to the time axis inversion of the target signal. Therefore, the quotient polynomial C can be described by the following formula (4). In this case, the matched filter C _M is expressed by the following formula (5). The quotient polynomial C and matched filter C _M have the same amplitude characteristic, phase characteristic proceeds in reverse with a delay.
Note that c ₀ ,..., C _N−1 represent coefficients of the quotient polynomial C, and q represents a shift operator.

That is, the matched filter calculating means 40, using equation (5) to calculate the matched filter C _M of the quotient polynomial C obtained in repeated division. Then, the matched filter calculating means 40 outputted a stable transfer function H ⁺ input from deciding section 30, the calculated and matched filter C _M to the transfer function approximation unit 50.

Transfer function approximation unit 50 is an approximation of the input transfer function H in the product of the matched filter C _M inputted from the matched filter calculating means 40 and stable transfer function H ^+.
Hereinafter, the transfer function approximated by the product of the matched filter and the stable transfer function is referred to as an “approximate transfer function”.

Here, since the transfer function dividing means 20 repeats the division, the remainder polynomial D becomes so small that it can be ignored. Therefore, the unstable transfer function H ⁻ can be approximated by a quotient polynomial C from Equation (3). As a result, unstable transfer function H ^- uses a matched filter C _M, can be approximated as the following equation (6). Further, according to the equations (1) and (6), the transfer function approximating means 50 can approximate the input transfer function H with the approximate transfer function C _M H ⁺ using the following equation (7). Thereafter, the transfer function approximating means 50 outputs the approximate transfer function C _M H ⁺ to the outside (for example, a transoral reproduction system).

[Operation of filter approximation device]
The operation of the filter approximation device 1 will be described with reference to FIG. 2 (see FIG. 1 as appropriate).
The filter approximating apparatus 1 decomposes the input transfer function H into the unstable transfer function H ⁻ and the stable transfer function H ⁺ by using the transfer function decomposition means 10 using the equation (1) (step S1).

Filter approximation device 1, the transfer function division means 20, using equation (3), unstable transfer function H ^- the denominator polynomial H ^- the _B numerator polynomial H ^- dividing by _A (Step S2).
The filter approximation device 1 determines whether or not all the coefficients of the remainder polynomial D are less than the threshold value T by the determination unit 30 (step S3).

If all the coefficients of the remainder polynomial D is not less than the threshold value T (No in step S3), and the filter approximation device 1 returns to the process in step S2, the transfer function division means 20, the remainder polynomial D numerator polynomial H ^- still _A Will divide.

When all the coefficients of the remainder polynomial D are less than the threshold value T (Yes in step S3), the filter approximation device 1 calculates the matched filter C _M of the quotient polynomial C by using the matched filter calculation means 40 using the equation (5). (Step S4).
Filter approximation device 1, the transfer function approximation unit 50, using equation (7) approximates the input transfer function H by matched filter C _M (step S5).

[Action / Effect]
In the filter approximating device 1 according to the first embodiment of the present invention, since the unstabilized unstable transfer function H ⁻ has an all-pass characteristic, the reciprocal of the unstable transfer function H ⁻ also has an all-pass characteristic. The matched filter also has an all-pass characteristic. Therefore, the filter approximation device 1 is unstable transfer function H ^- amplitude characteristics can be accurately approximated in. Further, the filter approximating device 1 temporarily inverts the phase when taking the reciprocal of the unstable transfer function H ⁻ , but calculating the matched filter causes the phase to invert and return to the original state although there is a delay. stable transfer function H ^- phase characteristic can be accurately approximated. As described above, the filter approximating apparatus 1 can accurately approximate the amplitude characteristic and the phase characteristic of the unstable transfer function H ⁻ and stabilize the input transfer function H, although with a certain delay.

(Second Embodiment)
A filter approximation device 1 according to a second embodiment of the present invention will be described with reference to FIGS. In the second embodiment, the filter approximation device 1 is used in the transoral reproduction system 100.

[Outline of transoral playback]
With reference to FIG. 3, the outline of the trans-oral reproduction system 100 will be described.
The head-related transfer function (HRTF) from the position corresponding to the center of the head in the free sound field to the binaural tympanic membrane position or the ear canal entrance through the position of the external sound source. Defined as acoustic transfer function. The HRTF includes many feature quantities related to sound image localization perception such as interaural time difference, level difference, and spectral cue on frequency characteristics. For this reason, the sound image can be localized in an arbitrary direction by causing the HRTF to act on the sound source signal, in other words, by convolving the head impulse response, which is an HRTF region representation, with the sound source signal. The signal generated in this way is called a binaural signal. A reproduction method for reproducing binaural signals using headphones is called a binaural reproduction method.

  When a binaural signal is reproduced by a speaker, leakage (crosstalk) to the opposite ear also occurs in addition to signal propagation from the speaker to the same ear. Therefore, it is necessary to perform compensation processing for suppressing the crosstalk and transmitting only a desired signal to each ear. This process is called crosstalk cancellation. A three-dimensional sound reproduction method using a speaker realized by this processing is referred to as transoral reproduction.

As shown in FIG. 3, in the transoral reproduction system 100, the number of control points is m (1 ≦ i ≦ m) and the number of secondary sound sources is n (1 ≦ j ≦ n). G _ij (q) represents an acoustic transfer function from the j-th secondary sound source to the i-th control point, that is, HRTF. X _i (q) represents a desired transfer function at each control point. H _ij (q) serves as a controller 120 for crosstalk cancellation at each control point.
In FIG. 3, X _i (q) and G _ij (q) are denoted by reference numerals 110 and 140, respectively. Reference numeral 130 represents a speaker, reference numeral 150 represents a head position, and reference numeral 160 represents a sound source.

  Input / output signals of the trans-oral reproduction system 100 are expressed by the following expressions (8) to (12). Here, u (k) represents an input signal of the transoral reproduction system 100. Further, y (k) represents an output signal of the transoral reproduction system 100. T represents transposition.

  Since the output signal y (k) is a signal obtained by applying the transfer function X (q) to the input signal u (k), it can be expressed by the following formula (13). As described above, when the shift operator q is used, the convolution operation in the time domain can be described in the form of a matrix product. For this reason, the controller 120 can be designed by an algebraic inverse matrix operation like the following formula (14).

However, since Expression (14) becomes unstable, it cannot be mounted as it is. Therefore, after designing the unstable controller 120 once, the filter approximating device 1 ties up the unstable transfer function H ⁻ from the transfer functions H (q) constituting each controller 120. Then, the filter approximating device 1 approximates the unstable transfer function H ⁻ as a delayed inverse system, whereby a stable controller 120 can be realized.

<Usage example of filter approximation device>
With reference to FIG. 4, the usage example of the filter approximation apparatus 1 of FIG. 1 is demonstrated (refer FIG. 3 suitably).
In FIG. 4, only one controller 120 and one speaker 130 are shown for easy understanding of the drawing.

As shown in FIG. 4, the filter approximation device 1 approximates the transfer function H (q) used in the transoral reproduction system 100.
Specifically, the filter approximating apparatus 1 inputs the transfer function H (q) constituting each controller 120 as an input transfer function, and performs the same processing as in the first embodiment. Then, the filter approximation device 1 outputs an approximate transfer function of the transfer function H (q) to each controller 120. Thereafter, each controller 120 performs transoral reproduction using the approximate transfer function input from the filter approximation device 1.

[Action / Effect]
The filter approximation device 1 according to the second embodiment of the present invention can accurately approximate the transfer function H (q) for the same reason as in the first embodiment. Therefore, if the filter approximation device 1 is used, the controller 120 can be easily mounted.

  In each of the above-described embodiments, the filter approximation device 1 has been described as independent hardware, but the present invention is not limited to this. For example, the filter approximating apparatus 1 can also be realized by a filter approximating program that causes hardware resources such as a CPU, a memory, and a hard disk included in a computer to operate cooperatively as the above-described units. This program may be distributed through a communication line, or may be distributed by writing in a recording medium such as a CD-ROM or a flash memory.

  The present invention can be used in general control systems using the reverse system. For example, the present invention can be used in an acoustic control system such as a sound field reproduction technique and a noise canceling process.

1 Filter approximation device (transfer function approximation device)
DESCRIPTION OF SYMBOLS 10 Transfer function decomposition means 20 Transfer function division means 30 Determination means 40 Matched filter calculation means 50 Transfer function approximation means

Claims (5)

A transfer function approximating device for approximating an input transfer function including an unstable pole represented by a ratio of rational functions so as not to include the unstable pole,
A transfer function that decomposes the input transfer function into an unstable transfer function including a zero that is in the relationship of the inverse of the unstable pole and the complex conjugate of the unstable pole, and a stable transfer function other than the unstable transfer function Decomposition means;
A transfer function dividing means for calculating a quotient polynomial and a remainder polynomial for the reciprocal of the unstable transfer function by dividing the denominator polynomial of the unstable transfer function decomposed by the transfer function decomposing means by a numerator polynomial;
In order to perform approximation only with the quotient polynomial without dealing with the remainder polynomial, determination means for causing the transfer function division means to perform division until all the coefficients of the remainder polynomial are reduced,
Matched filter calculating means for calculating a matched filter of a quotient polynomial calculated by the transfer function dividing means;
Transfer function approximating means for approximating the input transfer function by the product of the matched filter calculated by the matched filter calculating means and the stable transfer function;
A transfer function approximating device comprising:   The transfer function approximating apparatus according to claim 1, wherein the determination unit determines whether or not all coefficients of the remainder polynomial are small by threshold determination for all coefficients of the remainder polynomial.   The transfer function approximating device according to claim 1, wherein the transfer function decomposing means decomposes a transfer function for crosstalk cancellation as the input transfer function.   A transfer function approximating program for causing a computer to function as the transfer function approximating device according to any one of claims 1 to 3. A transfer function approximation method for approximating an input transfer function including an unstable pole represented by a ratio of rational functions so as not to include the unstable pole,
A transfer function that decomposes the input transfer function into an unstable transfer function including a zero that is in the relationship of the inverse of the unstable pole and the complex conjugate of the unstable pole, and a stable transfer function other than the unstable transfer function A decomposition step;
A transfer function division step of calculating a quotient polynomial and a remainder polynomial for the reciprocal of the unstable transfer function by dividing the denominator polynomial of the unstable transfer function decomposed in the transfer function decomposition step by a numerator polynomial;
In order to determine whether or not all the coefficients of the remainder polynomial are small by threshold determination for all the coefficients of the remainder polynomial, and to perform approximation only with the quotient polynomial without treating the remainder polynomial, all the coefficients of the remainder polynomial A determination step for returning to the transfer function division step until
A matched filter calculating step for calculating a matched filter of the quotient polynomial calculated in the transfer function dividing step;
A transfer function approximating step for approximating the input transfer function by a product of the matched filter calculated in the matched filter calculating step and the stable transfer function;
The transfer function approximation method characterized by performing sequentially.

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