JP2009163320A - Data interpolation method, data interpolation device, data interpolation program, spectrum interpolation method, spectrum interpolation device, and spectrum interpolation program - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To highly precisely calculate the fine coefficients of an arbitrary point other than sample points in a data interpolation method for calculating the fine coefficients of an arbitrary point between adjacent sample points from a sampled data group. <P>SOLUTION: A sampled data group obtained by sampling a J-dimensional periodic analog signal is acquired, and the fine coefficients of data f<SB>p</SB>(t<SB>1</SB>, through t<SB>J</SB>) of an arbitrary point in a J-dimensional space are calculated according to a formula (6.1) based on the obtained sampled data group. <P>COPYRIGHT: (C)2009,JPO&INPIT

Description

  The present invention relates to a data interpolation method and a data interpolation device for obtaining a derivative of an arbitrary point in the middle between adjacent sample points from a sampled data group obtained by sampling a J-dimensional periodic analog signal, And a data interpolation program that is executed in an information processing apparatus such as a computer and operates the information processing apparatus as a data interpolation apparatus, and a J-dimensional discrete Fourier variable applied to the J-dimensional sampled data group. Spectral interpolation method and spectral interpolation apparatus for obtaining a derivative of a spectrum of an arbitrary frequency between adjacent discrete points not appearing in the discrete spectrum from the J-dimensional discrete spectrum, and information processing such as a computer A spectrum interpolation program that is executed in the apparatus and operates the information processing apparatus as a spectrum interpolation apparatus. On the lamb.

  Conventionally, data interpolation operations for obtaining data at points other than the sampling points from the sampling data group have been performed in various fields.

  One of the data interpolation methods conventionally employed is a zero point addition method. In this zero point addition method, zero points (sample points having data values of zero) are added at equal intervals as many times as you want to interpolate between adjacent sample points in the sampled data sequence. This is a technique of increasing the number of times and passing through a digital low-pass filter. According to this zero point addition method, as long as the filtering process for preventing aliasing noise is properly performed in the first sampling, the data of the sample point added as the zero point can be accurately reproduced.

  However, in this zero point addition method, it is necessary to add zero points at regular intervals between the original sample points, and although fixed point data determined by the number of additional points can be reproduced accurately, data at an arbitrary time can be obtained. It is impossible to reproduce.

  In addition, as another method of data interpolation methods that have been conventionally employed, there is a method that uses the Sanya & Shannon sampling theorem. Although details will be omitted, if this sampling theorem is adopted, mathematically, the data at an arbitrary time t can be obtained as long as the filtering process for preventing aliasing noise is properly performed in the first sampling. Can be played.

  However, this sampling theorem requires an infinite number of sample points. However, in reality, there is a problem that it is necessary to perform an operation using data of a finite number of sample points, and only a result including a certain large error can be obtained.

  On the other hand, the present applicant has proposed a data interpolation method for obtaining data of an arbitrary point with high accuracy (mathematically strictly) based on a limited number of sampled data in Patent Document 1. .

  However, the data interpolation method proposed in Patent Document 1 can be applied only to one-dimensional data, and cannot be applied to multi-dimensional data such as two-dimensional image data.

  For example, spectrum analysis of time-varying waveforms has been frequently used in various fields. For example, FFT calculation methods are used for this spectrum analysis. In this FFT, a time waveform is regarded as a periodic function. Sampling at a predetermined time interval, obtaining N discrete sampling point values for one period, and calculating based on the sampling point values, N spectral frequency spectrum components are obtained. Desired.

  However, in the above FFT method, since it is possible to obtain spectral components of discrete frequencies, the spectral values of frequencies between discrete points are not known, and there is a limit to performing detailed spectral analysis. is there. Although it is conceivable to increase the number of sampling points (N mentioned above), in that case, the amount of calculation of the FFT becomes enormous and unrealistic, and even if the number of sampling points is increased, it is still discrete. The spectrum value of an arbitrary frequency cannot be obtained.

  On the other hand, the present applicant has proposed in Japanese Patent Application No. 2006-150057 a spectral interpolation method capable of interpolating a spectral value of an arbitrary frequency from a discrete spectrum with extremely high accuracy.

  However, this spectral interpolation method is applicable only to one-dimensional spectra, and is applicable to, for example, a two-dimensional spectrum such as a two-dimensional spectrum obtained by performing a two-dimensional FFT operation on two-dimensional image data. I can't do it.

  Furthermore, the present applicant has proposed a data interpolation method and a spectrum interpolation method applicable to a plurality of dimensions in Japanese Patent Application No. 2007-149971.

  However, even if the data of an arbitrary point or the spectrum of an arbitrary point is obtained by, for example, the data interpolation method or spectral interpolation method proposed in Japanese Patent Application No. 2007-149971, what about the derivative at that point? The problem is whether to find it strictly. Normally, the approximate derivative is obtained by obtaining the data of a plurality of surrounding points including the point and obtaining the difference, but this method only obtains the approximate value of the derivative. It can only be done.

Non-patent documents 1 to 5 are listed for the following explanation.
JP 2002-278948 A Koichi Makino, Kotoshi Yokota, Kohei Yamamoto, Toshiaki Okada, Koichi Yoshihisa, “Measurement of Directionality of Automobile Noise Using Cross Correlation Method,” Proc. 827-828, March 2007. Hitoshi Namba, Meiji Muneyasu, “A Method for Tracking Moving Objects Using POC,” IEICE Technical Report, EA 2007-15, pp. 49-54, May 2007. Yasuhiro Koshiyama, Kiyoshi Nishikawa, Hitoshi Kiya, “A method for estimating the sampling rate conversion ratio of an audio signal based on the phase-only correlation method,” IEICE Tech. 55-60, May 2007. L. E. Franks (translated by Hirose Hirose, Seigo Kato, Hiroshi Yasuda), Signal Theory, Sangyo Tosho, 1974, Tokyo. (L. E. Franks, “Signal Theory,” Prentice-Hall Inc. Anglewood Cliffs, 1969, N.J.) A. Papoulis, “The Fourier integrals and its applications,” (McGraw-Hill, Inc., New York, 1962 (Reissued 1987)), p. 134.

  In view of the above circumstances, the present invention increases the derivative of an arbitrary point other than a sampling point based on a sampling data group consisting of a limited number of sampling data of J dimensions (J is a positive integer of 1 or more). Data interpolation method and data interpolation apparatus that can be obtained with high accuracy, and the information processing apparatus that is executed in the information processing apparatus such as a computer can accurately calculate the derivative of any point other than the sample point. A data interpolation program that operates as a data interpolation device that can be obtained, and interpolates the derivative of a spectrum of an arbitrary frequency from a finite number of discrete spectra in the J dimension (J is a positive integer of 1 or more) with extremely high accuracy. Spectral interpolation method and spectrum interpolation apparatus, and information processing apparatus that can be executed in an information processing apparatus such as a computer to make the information processing apparatus of any frequency And to provide a spectrum interpolation program for operating the derivative of torque as a spectral interpolation apparatus capable of interpolating a very high accuracy.

The data interpolation method of the present invention that achieves the above object is as follows.
Obtaining a sampling data group obtained by sampling a periodic analog signal of J dimension (J is a positive integer of 1 or more);
Based on the acquired sampling data group, the derivative of the data f _p (t ₁ ,..., T _J ) at an arbitrary point in the J-dimensional space is expressed by the equation

  However,

Is an integer that allows zero,
(T ₁ ,..., T _J ) are arbitrary points in the J-dimensional space (t ₁ , ₀ ,..., T _J , ₀ ) are arbitrary reference points (Δt ₁ ,..., Δt _J in the J-dimensional space. ) Is the sample interval (N ₁ ,..., N _J ) is the number of samples f _{p, n1,..., NJ over} one period (T ₁ ,..., T _J )
f _{p, n1,..., nJ} = f _p (t _1,0 + n ₁ · Δt ₁ ,..., t _{J, 0} + n _J · Δt _J )
And ψ ₁ (t),..., Ψ _J (t) are interpolation functions,

(J = 1, ..., J)
Is,
It is obtained according to the following.

  According to the data interpolation method of the present invention, the derivation of the above-described equation will be described later, but the derivative of an arbitrary point can be obtained with extremely high accuracy without any error at least in principle according to the equation.

In addition, the data interpolation device of the present invention that achieves the above-described object provides:
An acquisition unit that acquires a sampling data group obtained by sampling a periodic analog signal of J dimension (J is a positive integer of 1 or more);
Based on the acquired sampling data group, the derivative of the data f _p (t ₁ ,..., T _J ) at an arbitrary point in the J-dimensional space is expressed by the equation

  However,

Is an integer that allows zero,
(T ₁ ,..., T _J ) are arbitrary points in the J-dimensional space (t ₁ , ₀ ,..., T _J , ₀ ) are arbitrary reference points (Δt ₁ ,..., Δt _J in the J-dimensional space. ) Is the sample interval (N ₁ ,..., N _J ) is the number of samples f _{p, n1,..., NJ over} one period (T ₁ ,..., T _J )
f _{p, n1,..., nJ} = f _p (t _1,0 + n ₁ · Δt ₁ ,..., t _{J, 0} + n _J · Δt _J )
And ψ ₁ (t),..., Ψ _J (t) are interpolation functions,

(J = 1, ..., J)
Is,
And an arithmetic unit to be obtained according to the above.

Furthermore, the data interpolation program of the present invention that achieves the above object is
Executed in an information processing apparatus that executes a program, and acquires a sampled data group obtained by sampling a periodic analog signal of J dimension (J is a positive integer of 1 or more) from the information processing apparatus. An acquisition unit;
Based on the acquired sampling data group, the derivative of the data f _p (t ₁ ,..., T _J ) at an arbitrary point in the J-dimensional space is expressed by the equation

  However,

Is an integer that allows zero,
(T ₁ ,..., T _J ) are arbitrary points in the J-dimensional space (t ₁ , ₀ ,..., T _J , ₀ ) are arbitrary reference points (Δt ₁ ,..., Δt _J in the J-dimensional space. ) Is the sample interval (N ₁ ,..., N _J ) is the number of samples f _{p, n1,..., NJ over} one period (T ₁ ,..., T _J )
f _{p, n1,..., nJ} = f _p (t _1,0 + n ₁ · Δt ₁ ,..., t _{J, 0} + n _J · Δt _J )
And ψ ₁ (t),..., Ψ _J (t) are interpolation functions,

(J = 1, ..., J)
Is,
And operating as a data interpolating device provided with a calculation unit obtained according to the above.

Moreover, the spectrum interpolation method of the present invention that achieves the above-mentioned object is as follows.
Obtain a J-dimensional discrete spectrum obtained by sampling a J-dimensional (J is a positive integer of 1 or more) periodic analog signal and applying a J-dimensional discrete Fourier transform;
Based on the acquired J-dimensional discrete spectrum, the spectral value of an arbitrary point in the J-dimensional frequency space

The derivative of

  However,

Is an integer that accepts zero (ω ₁ ,..., Ω _J ) is an arbitrary point in the J-dimensional frequency space (ω _1,0 ,..., Ω _{J, 0} ) is an arbitrary reference point in the J-dimensional frequency space ( Δω _1, ..., Δω _J) is the discrete spectrum interval _{(N 1, ..., N J} ) is one period (omega _S1, ..., the number of samples over a _{ω SJ) F p, k1,} ..., kJ is discrete The value of the spectrum,
F _{p, k1,..., KJ} = F _p (ω _1,0 + k ₁ · Δω ₁ ,..., Ω _{J, 0} + k _J · Δω _J )
And Ψ ₁ (ω),..., Ψ _J (ω) are interpolation functions,

(J = 1, ..., J)
Is,
It is obtained according to the following.

  According to the spectral interpolation method of the present invention, the derivation of the above formula will be described later, but the derivative of the spectrum of an arbitrary point in the J-dimensional frequency space can be determined in accordance with the formula at least in principle without error. It can be obtained with high accuracy.

In addition, the spectrum interpolation device of the present invention that achieves the above-described object is
An acquisition unit for acquiring a J-dimensional discrete spectrum obtained by sampling a J-dimensional (J is a positive integer of 1 or more) periodic analog signal and performing J-dimensional discrete Fourier transform;
Based on the acquired J-dimensional discrete spectrum, the spectral value of an arbitrary point in the J-dimensional frequency space

The derivative of

  However,

Is an integer that accepts zero (ω ₁ ,..., Ω _J ) is an arbitrary point in the J-dimensional frequency space (ω _1,0 ,..., Ω _{J, 0} ) is an arbitrary reference point in the J-dimensional frequency space ( Δω _1, ..., Δω _J) the discrete spectral interval _{(N 1, ..., N J} ) is one period (omega _S1, ..., sample size across _{ω SJ) F p, k1,} ..., kJ is discrete The value of the spectrum,
F _{p, k1,..., KJ} = F _p (ω _1,0 + k ₁ · Δω ₁ ,..., Ω _{J, 0} + k _J · Δω _J )
And Ψ ₁ (ω),..., Ψ _J (ω) are interpolation functions,

(J = 1, ..., J)
Is,
And an arithmetic unit to be obtained according to the above.

Furthermore, the spectrum interpolation program of the present invention for achieving the above object is
The information processing apparatus is executed in an information processing apparatus that executes a program.
An acquisition unit for acquiring a J-dimensional discrete spectrum obtained by sampling a J-dimensional (J is a positive integer of 1 or more) periodic analog signal and performing J-dimensional discrete Fourier transform;
Based on the acquired J-dimensional discrete spectrum, the spectral value of an arbitrary point in the J-dimensional frequency space

The derivative of

  However,

Is an integer that accepts zero (ω ₁ ,..., Ω _J ) is an arbitrary point in the J-dimensional frequency space (ω _1,0 ,..., Ω _{J, 0} ) is an arbitrary reference point in the J-dimensional frequency space ( Δω _1, ..., Δω _J) is the discrete spectrum interval _{(N 1, ..., N J} ) is one period (omega _S1, ..., the number of samples over a _{ω SJ) F p, k1,} ..., kJ is discrete The value of the spectrum,
F _{p, k1,..., KJ} = F _p (ω _1,0 + k ₁ · Δω ₁ ,..., Ω _{J, 0} + k _J · Δω _J )
And Ψ ₁ (ω),..., Ψ _J (ω) are interpolation functions,

(J = 1, ..., J)
Is,
And operating as a spectrum interpolation device provided with a calculation unit obtained according to the above.

  According to the present invention described above, the derivative of an arbitrary point in the J-dimensional space can be obtained with extremely high accuracy from a finite number of discrete data, and the arbitrary coefficient in the J-dimensional frequency space can be obtained from the finite number of discrete spectra. The derivative of the spectrum at the point can be obtained with extremely high accuracy.

In the following, first, the theoretical background of the present invention will be described.
1. 1. Introduction Here, the sampling theorem for a periodic signal proved for a one-dimensional signal is expanded as a sampling theorem for a multidimensional periodic function, and a periodic signal interpolation formula using sample points is derived as a byproduct. Furthermore, from this result, a formula for calculating a derivative for a multidimensional periodic function is derived.
2.1 Summary of sampling theorems for periodic signals in the dimensional time domain (see Patent Document 1)
Now the cycle

Period analog signal

And That is,

And

Since is a periodic signal, Fourier series expansion is possible, and is expressed as follows.

In particular, if the signal is band limited,

Suppose that At this time,

The following interpolation formula holds. However, the interpolation function is

Given in.
(註 2.1) The conditions of (2.3) are from (2.2)

The spectral component in the range is zero or the spectral component is not zero.

It is easily shown that it means that it is limited to a certain section. However,

Is the sampling time interval,

It was.
3. Summary of sampling theorems for periodic signals in the one-dimensional frequency range (see Appendix A)
Now the cycle

The periodic spectral function of

And That is,

And

Since is a periodic signal, Fourier series expansion is possible, and is expressed as follows.

Now the signal is localized in the time domain,

Suppose that At this time,

The following interpolation formula holds. However,

Or

It is.
(註 3.1)

From (3.2), the condition is that in the time domain,

It can be assumed that the time domain component of a certain range is zero, or the time domain component is not zero,

It means that it is limited within the section. However,

Is the sampling angular frequency interval,

It was.
4). Sampling theorem and interpolation formula for multidimensional periodic functions (1)
Now the cycle

Period analog signal

And That is,

Is assumed to hold. Assuming that this two-dimensional signal is band-limited, using the results in Section 2,

Get. The above is further generalized as follows.

  Now the cycle

Period analog signal

And That is,

Is assumed to hold. Assuming that this J-dimensional signal is band-limited, using the results in Section 2,

Get.
(Note 4.1) The bandwidth limitation condition is an extension to multi-dimension in the case of one dimension.

The spectral component in the range is zero or the spectral component is not zero.

It is easily shown that it means that it is limited to a certain section. However,

Is the sampling time interval,

It was.
5. Sampling theorem and interpolation formula for multidimensional periodic functions (2)
Now the cycle

Period analog signal

And That is,

Is assumed to hold. Assuming that this two-dimensional signal is localized in the time domain, using the results in Section 3,

Get. The above is further generalized as follows.

  Now the cycle

Period analog signal

And That is,

Is assumed to hold. Assuming that this J-dimensional signal is localized in the time domain, using the results in Section 3,

Get.
(Note 5.1) The condition of time domain localization is as an extension to multi-dimension in the case of one dimension.

It can be assumed that the time domain component of a certain range is zero, or the time domain component is not zero,

It means that it is limited within the section. However,

Is the sampling angular frequency interval,

It was.
6). Sampling theorems and interpolation formulas for derivatives of multidimensional periodic functions (Part 1)
In the above-described interpolation formula, the interpolation function is a differentiable function. Therefore, the interpolation formula for the derivative can be obtained by differentiating both sides of the interpolation formula. First, by differentiating (4.4),

Get. However,

Is an integer that allows zero. In particular, the first derivative for one variable is given.

In order to find the limit value at, the denominator and numerator are differentiated twice,

If we take the limit of, we get

In order to find the limit value at, the denominator and numerator are differentiated twice,

If we take the limit of, we get

  In addition, higher-order derivatives can be calculated in principle,

Since it is difficult to obtain the limit value at, it is possible to obtain a sample value corresponding to the first derivative and apply an interpolation formula for the first derivative to it.

  That is, the higher-order derivative is obtained similarly,

The extreme operation at is even more complicated. Therefore, it may be one method to repeat the operation of obtaining the derivative of the previous rank from only the sample point and performing the first derivative based on the derivative. Or

Is a power of 2, using FFT or the like to obtain the derivative value of the corresponding rank only at the sample points, and performing simple interpolation based on it is also one method. Let's go.
7). Sampling theorems and interpolation formulas for derivatives of multidimensional periodic functions (Part 2)
In the above-described interpolation formula, the interpolation function is a differentiable function. Therefore, the interpolation formula for the derivative can be obtained by differentiating both sides of the interpolation formula. First, by differentiating (5.4),

Get. However,

Is an integer that allows zero.

  In the following, in particular, a first derivative with respect to one variable is given. The first derivative is

Or

Given in.

  Again,

Find the limit value at. At that time, the function form of the second term of (7.4) is the same as that considered in (6.8), and therefore the second term of (7.4) is

Note that At this time, from (7.4)

Get.

  As in the case of the data interpolation method described above, the higher-order derivative is also obtained in the same way,

The extreme operation at is even more complicated. Therefore, it may be one method to repeat the operation of obtaining the derivative of the previous rank from only the sample point and performing the first derivative based on the derivative. Or

Is a power of 2, using FFT or the like to obtain the derivative value of the corresponding rank only at the sample points, and performing simple interpolation based on it is also one method. Let's go.
(Application examples of sampling theorem and interpolation formula for derivatives of J-dimensional periodic functions)
8). Introduction Here are some examples of the application of the sampling theorem and interpolation formula for the derivatives of multidimensional periodic functions derived in the attachment.
9. Direct application example As the simplest application example, for example, when a sampling point for a displacement signal is given, the value of velocity and acceleration is estimated in a form including interpolation in addition to the sampling point. Can be mentioned.

  FIG. 1 is a graph showing velocity and acceleration values determined from a sample point with respect to a displacement signal by simulation.

  A circle indicates a displacement signal at discrete points 0, 1,... 5, 6, 7,... 10 on the horizontal axis, and a graph a indicates a speed change curve obtained from displacement signals at five discrete points 0 to 5. The graph b represents the acceleration change curve obtained from the displacement signals at five discrete points 6-10.

Graphs a and b accurately represent not only the sample point of the displacement signal but also the velocity and acceleration at any point other than the sample point.
10. High-accuracy estimation method of delay time using differential interpolation method for peak position search of cross-correlation function In the estimation of delay time using peak position of cross-correlation function, when using digital signal processing, sampling time interval Limits the accuracy (Non-Patent Documents 1 to 3). One way to overcome this limitation is to use an interpolation formula. In this method, the peak position can be continuously searched, so that the accuracy increases dramatically. However, when the search point is close to the peak, the search accuracy inevitably decreases. At this stage, if the differential interpolation method is used, that is, if the differential value becomes zero at the peak position, and the search is switched to the zero search of the derivative, the search accuracy is further improved. This method can be applied not only to a one-dimensional problem but also to a peak position search of a multidimensional cross-correlation function.
11. Instantaneous frequency estimation As shown in Appendix B, the signal

The instantaneous frequency of

Given in. However,

Is the Hilbert transform. When this is calculated by digital signal processing, the value at the sample point is known, but if you want the value at a point other than the sample point, that is, the interpolated value, use the interpolation method or the differential interpolation method. It should be used. In particular,

Can be obtained directly from (11.1) using interpolation and differential interpolation, or the instantaneous frequency of only (11.1) at the sampling point using FFT. It is also conceivable to perform interpolation by considering this as sampled data.
12 Estimating group delay time As shown in Appendix C, the frequency response is

The group delay time of the system given by

Given in. Again, when an interpolation value other than the sample points is desired, an interpolation formula or a differential interpolation formula is useful. Group delay time has a wide range of applications, such as when discussing distortion in the transmission system.
(Appendix A Sampling Theorem and Interpolation Formula for Periodic Functions in the Frequency Domain)
Now the cycle

The periodic spectral function of

And That is,

And

Since is a periodic signal, Fourier series expansion is possible, and is expressed as follows.

On the other hand, discrete Fourier transform and inverse transform are

It is expressed. However,

Is the sampling angular frequency interval,

Was abbreviated. At this time, from (A.2) and (A.5),

I can say. In particular, the signal is localized in the time domain,

Suppose that
(Note 1)

From (A.2), the condition

This means that the time domain component in a certain range can be regarded as zero.
At this time, from (A.8)

It becomes. Put (A.9) and (A.10) in (A.2),

Get. Therefore,

(A.11) becomes

It becomes.

  Given by (A.12)

Is readily shown to have the following properties:
1) Periodicity

2) Scoring (

,and,

Value at

3)

Symmetry around

Put it. here,

It is. First, for the real part,

Than,

It can be seen that there is symmetry around. Next, for the imaginary part,

Than,

It can be seen that there is distortion symmetry around.

  In particular,

If is even,

For the value at

I can say.
(Appendix B Instantaneous Frequency (See Non-Patent Document 4))
Signal now

Hilbert transform, inverse transform,

Defined in Where the sign

Represents the Cauchy principal value integral. From these, the signal

The analytic signal of

Given in. Sine wave now

Consider this as a complex envelope function

Narrowband signal amplitude modulated with

think of. At this time, the signal

The instantaneous angular frequency of

Given in.
(Appendix C: Group delay time (see Non-Patent Document 5))
Now the frequency response of the system is

It shall be given by This system is

It is assumed that the passbands are concentrated in a narrow band around At this time, the impulse response of the system is

However,

It was. That is, the envelope delay time is defined as the group delay time.

  A specific calculation method of the group delay time is given as follows. By differentiating (C.1) by angular frequency,

As such, the group delay time is given.

  Hereinafter, embodiments of the present invention will be described.

  FIG. 2 is an external perspective view of the computer. This computer operates as a data interpolation device and a spectrum interpolation device according to an embodiment of the present invention by executing a program to be described later.

  The computer 100 shown in FIG. 2 has a main body device 110, an image display device 120 that displays an image on a display screen 121 according to an instruction from the main body device 110, and a main body device 110 according to key operations. In addition, a keyboard 130 for inputting various information and a mouse 140 for inputting an instruction corresponding to, for example, an icon or the like displayed at that position by designating an arbitrary position on the display screen 121 are provided. The main body device 110 has an FD loading port 111 for loading a flexible disk (hereinafter abbreviated as FD) and a CD-ROM loading port 112 for loading a CD-ROM.

  FIG. 3 is a hardware configuration diagram of the computer 100 whose appearance is shown in FIG.

  As shown in FIG. 3, the main body device 110 includes a CPU 113 that executes various programs, a main memory 114 that reads programs stored in the hard disk device 115 and develops them for execution by the CPU 113, and various programs. The hard disk device 115 in which data and data are stored, the FD drive 116 that accesses the FD 400, the CD-ROM 401 are loaded, the CD-ROM drive 117 that accesses the loaded CD-ROM 401, and a signal is input from an external sensor or the like 2 and an output interface 119 for outputting calculation results to an external device. These various elements, and the image display device 120, keyboard 130, and mouse 140 shown in FIG. Connected to each other through .

  The CD-ROM 401 stores a program for operating the computer 100 as an embodiment of the data interpolation device or spectrum interpolation device of the present invention. The CD-ROM 401 is loaded into the CD-ROM drive 117, and the program stored in the CD-ROM 401 is uploaded to the computer and stored in the hard disk device 115. When this program is activated and executed, the computer 100 operates as an embodiment of the data interpolation device or spectrum interpolation device of the present invention.

  In the above, the CD-ROM 401 is exemplified as the storage medium for storing the program. However, the storage medium for storing the program is not limited to the CD-ROM, and other optical disks, MOs, FDs, and magnetic tapes. Or a storage medium such as

  FIG. 4 is a conceptual diagram showing a CD-ROM in which a data interpolation program is stored.

  A CD-ROM 401 shown in FIG. 4 stores a data interpolation program 410 including an acquisition unit 411 and a calculation unit 412.

  The computer 100 shown in FIGS. 2 and 3 receives a J-dimensional sampling data group from the outside, and the acquisition unit 411 acquires the sampling data group that has been captured by the computer 100 and stored in, for example, the hard disk device 115. Is a program component that plays a role of acquiring the sampled data group by reading out from the hard disk device. The acquisition unit 411 includes a program element that causes the computer 100 to include an A / D converter, and causes the computer 100 to input an analog signal and perform A / D conversion to generate a sampling data group. The acquisition unit may be included.

The arithmetic unit 412 based on the acquired sample data set, data f _p of an arbitrary point in the J-dimensional space _{(t 1, ..., t J} ) a derivative of formula

  However,

Is an integer that allows zero,
(T ₁ ,..., T _J ) are arbitrary points in the J-dimensional space (t ₁ , ₀ ,..., T _J , ₀ ) are arbitrary reference points (Δt ₁ ,..., Δt _J in the J-dimensional space. ) Is the sample interval (N ₁ ,..., N _J ) is the number of samples f _{p, n1,..., NJ over} one period (T ₁ ,..., T _J )
f _{p, n1,..., nJ} = f _p (t _1,0 + n ₁ · Δt ₁ ,..., t _{J, 0} + n _J · Δt _J )
And ψ ₁ (t),..., Ψ _J (t) are interpolation functions,

(J = 1, ..., J)
Is,
It is a program part to be obtained according to

  FIG. 5 is a functional configuration diagram of the data interpolation apparatus as an embodiment of the present invention.

  The data interpolation apparatus 420 shown in FIG. 5 includes an acquisition unit 421 and a calculation unit 422. The acquisition unit 421 and the calculation unit 422 are functions realized by executing the acquisition unit 411 and the calculation unit 412 that are program components of the data interpolation program 410 illustrated in FIG. Therefore, the acquisition unit 421 and the calculation unit 422 of FIG. 5 are configured by the acquisition unit 411 and the calculation unit 412 of FIG.

  When the data interpolation apparatus of the present invention is used, the data of the sample point with respect to the displacement signal as described above can be acquired, and the speed and acceleration at an arbitrary time point can be strictly determined based on the data.

  As another application example, waveform data of discrete sample points can be acquired by using the data interpolation device of the present invention, and an instantaneous frequency at an arbitrary time point can be strictly determined.

  FIG. 6 is a flowchart of a data interpolation method as one embodiment of the present invention.

  The data interpolation method 430 shown in FIG. 6 includes an acquisition step a1 and a calculation step a2. The acquisition step a1 and the calculation step a2 are executed in the computer 100 by executing the acquisition unit 411 and the calculation unit 412 which are program parts of the data interpolation program 410 shown in FIG. Step.

  FIG. 7 is a conceptual diagram showing a CD-ROM storing a spectrum interpolation program as one embodiment of the present invention.

  A CD-ROM 401 shown in FIG. 7 stores a spectrum interpolation program 510 including an acquisition unit 511 and a calculation unit 512.

  The computer 100 shown in FIGS. 2 and 3 is also equipped with a function of an FFT analyzer that takes a J-dimensional signal from the outside and performs an FFT operation to obtain a discrete spectrum. It is a program component that plays a role of acquiring a discrete spectrum obtained by reading out a discrete spectrum obtained by a dimensional FFT operation, for example, stored in the hard disk device 115 from the hard disk device 115. The acquisition unit 511 may be an acquisition unit including a program element that causes the computer 100 to perform an FFT operation.

  In addition, the calculation unit 512 calculates a spectrum value at an arbitrary point in the J-dimensional frequency space based on the acquired discrete spectrum.

The derivative of

  However,

Is an integer that accepts zero (ω ₁ ,..., Ω _J ) is an arbitrary point in the J-dimensional frequency space (ω _1,0 ,..., Ω _{J, 0} ) is an arbitrary reference point in the J-dimensional frequency space ( Δω _1, ..., Δω _J) the discrete spectral interval _{(N 1, ..., N J} ) is one period (omega _S1, ..., sample size across _{ω SJ) F p, k1,} ..., kJ is discrete The value of the spectrum,
F _{p, k1,..., KJ} = F _p (ω _1,0 + k ₁ · Δω ₁ ,..., Ω _{J, 0} + k _J · Δω _J )
And Ψ ₁ (ω),..., Ψ _J (ω) are interpolation functions,

(J = 1, ..., J)
Is,
It is a program part to be obtained according to

  FIG. 8 is a functional configuration diagram of the spectrum interpolation device as one embodiment of the present invention.

  The spectrum interpolation device 520 shown in FIG. 8 includes an acquisition unit 521 and a calculation unit 522. The acquisition unit 521 and the calculation unit 522 are functions realized by executing the acquisition unit 511 and the calculation unit 512, which are the program components of the spectrum interpolation program 510 illustrated in FIG. Therefore, the acquisition unit 521 and the calculation unit 522 of FIG. 8 are configured by the acquisition unit 511 and the calculation unit 512 of FIG.

  Using the spectrum interpolation device of the present invention, as described above, frequency data of discrete sampling points on the frequency axis is acquired, and the group delay time is strictly determined based on the frequency data of those sampling points. can do.

  FIG. 9 is a flowchart of a spectrum interpolation method as one embodiment of the present invention.

  The spectrum interpolation method 530 shown in FIG. 9 includes an acquisition step b1 and a calculation step b2. The acquisition step b1 and the calculation step b2 are executed in the computer 100 by executing the acquisition unit 511 and the calculation unit 512, which are program components of the data interpolation program 510 shown in FIG. Step.

It is the graph which showed the value of the speed and acceleration which were determined from the sample point with respect to the displacement signal by simulation. It is an external appearance perspective view of a computer. FIG. 3 is a hardware configuration diagram of a computer whose appearance is shown in FIG. 2. It is a conceptual diagram which shows CD-ROM in which the data interpolation program was memorize | stored. It is a functional block diagram of the data interpolation apparatus as one Embodiment of this invention. It is a flowchart of the data interpolation method as one Embodiment of this invention. It is a conceptual diagram which shows CD-ROM with which the spectrum interpolation program as one Embodiment of this invention was memorize | stored. It is a functional block diagram of the spectrum interpolation apparatus as one Embodiment of this invention. It is a flowchart of the spectrum interpolation method as one embodiment of the present invention.

Explanation of symbols

100 Computer 110 Main Device 120 Image Display Device 130 Keyboard 140 Mouse 401 CD-ROM
410 Data Interpolation Program 411, 421, 511, 521 Acquisition Unit 412, 422, 512, 522 Calculation Unit 420 Data Interpolation Device 510 Spectrum Interpolation Program 520 Spectrum Interpolation Device

Claims (6)

Obtaining a sampling data group obtained by sampling a periodic analog signal of J dimension (J is a positive integer of 1 or more);
Based on the acquired sampling data group, the derivative of the data f _p (t ₁ ,..., T _J ) at an arbitrary point in the J-dimensional space is expressed by the equation

However,

Is an integer that allows zero,
(T ₁ ,..., T _J ) are arbitrary points in the J-dimensional space (t ₁ , ₀ ,..., T _J , ₀ ) are arbitrary reference points (Δt ₁ ,..., Δt _J in the J-dimensional space. ) Is the sample interval (N ₁ ,..., N _J ) is the number of samples f _{p, n1,..., NJ over} one period (T ₁ ,..., T _J )
f _{p, n1,..., nJ} = f _p (t _1,0 + n ₁ · Δt ₁ ,..., t _{J, 0} + n _J · Δt _J )
And ψ ₁ (t),..., Ψ _J (t) are interpolation functions,

(J = 1, ..., J)
Is,
A data interpolation method characterized by being obtained according to: An acquisition unit that acquires a sampling data group obtained by sampling a periodic analog signal of J dimension (J is a positive integer of 1 or more);
Based on the acquired sampling data group, the derivative of the data f _p (t ₁ ,..., T _J ) at an arbitrary point in the J-dimensional space is expressed by the equation

However,

Is an integer that allows zero,
(T ₁ ,..., T _J ) are arbitrary points in the J-dimensional space (t ₁ , ₀ ,..., T _J , ₀ ) are arbitrary reference points (Δt ₁ ,..., Δt _J in the J-dimensional space. ) Is the sample interval (N ₁ ,..., N _J ) is the number of samples f _{p, n1,..., NJ over} one period (T ₁ ,..., T _J )
f _{p, n1,..., nJ} = f _p (t _1,0 + n ₁ · Δt ₁ ,..., t _{J, 0} + n _J · Δt _J )
And ψ ₁ (t),..., Ψ _J (t) are interpolation functions,

(J = 1, ..., J)
Is,
And a data interpolating device characterized by comprising: Executed in an information processing apparatus that executes a program, and acquires a sampled data group obtained by sampling a periodic analog signal of J dimension (J is a positive integer of 1 or more) from the information processing apparatus. An acquisition unit;
Based on the acquired sampling data group, the derivative of the data f _p (t ₁ ,..., T _J ) at an arbitrary point in the J-dimensional space is expressed by the equation

However,

Is an integer that allows zero,
(T ₁ ,..., T _J ) are arbitrary points in the J-dimensional space (t ₁ , ₀ ,..., T _J , ₀ ) are arbitrary reference points (Δt ₁ ,..., Δt _J in the J-dimensional space. ) Is the sample interval (N ₁ ,..., N _J ) is the number of samples f _{p, n1,..., NJ over} one period (T ₁ ,..., T _J )
f _{p, n1,..., nJ} = f _p (t _1,0 + n ₁ · Δt ₁ ,..., t _{J, 0} + n _J · Δt _J )
And ψ ₁ (t),..., Ψ _J (t) are interpolation functions,

(J = 1, ..., J)
Is,
A data interpolation program which is operated as a data interpolation apparatus including a calculation unit obtained according to the above. Obtain a J-dimensional discrete spectrum obtained by sampling a J-dimensional (J is a positive integer of 1 or more) periodic analog signal and applying a J-dimensional discrete Fourier transform;
Based on the acquired J-dimensional discrete spectrum, the spectral value of an arbitrary point in the J-dimensional frequency space

The derivative of

However,

Is an integer that accepts zero (ω ₁ ,..., Ω _J ) is an arbitrary point in the J-dimensional frequency space (ω _1,0 ,..., Ω _{J, 0} ) is an arbitrary reference point in the J-dimensional frequency space ( Δω _1, ..., Δω _J) the discrete spectral interval _{(N 1, ..., N J} ) is one period (omega _S1, ..., sample size across _{ω SJ) F p, k1,} ..., kJ is discrete The value of the spectrum,
F _{p, k1,..., KJ} = F _p (ω _1,0 + k ₁ · Δω ₁ ,..., Ω _{J, 0} + k _J · Δω _J )
And Ψ ₁ (ω),..., Ψ _J (ω) are interpolation functions,

(J = 1, ..., J)
Is,
A spectral interpolation method characterized in that it is obtained according to: An acquisition unit for acquiring a J-dimensional discrete spectrum obtained by sampling a J-dimensional (J is a positive integer of 1 or more) periodic analog signal and performing J-dimensional discrete Fourier transform;
Based on the acquired J-dimensional discrete spectrum, the spectral value of an arbitrary point in the J-dimensional frequency space

The derivative of

However,

Is an integer that accepts zero (ω ₁ ,..., Ω _J ) is an arbitrary point in the J-dimensional frequency space (ω _1,0 ,..., Ω _{J, 0} ) is an arbitrary reference point in the J-dimensional frequency space ( Δω _1, ..., Δω _J) the discrete spectral interval _{(N 1, ..., N J} ) is one period (omega _S1, ..., sample size across _{ω SJ) F p, k1,} ..., kJ is discrete The value of the spectrum,
F _{p, k1,..., KJ} = F _p (ω _1,0 + k ₁ · Δω ₁ ,..., Ω _{J, 0} + k _J · Δω _J )
And Ψ ₁ (ω),..., Ψ _J (ω) are interpolation functions,

(J = 1, ..., J)
Is,
A spectrum interpolation device comprising: an arithmetic unit that is calculated according to The information processing apparatus is executed in an information processing apparatus that executes a program.
An acquisition unit for acquiring a J-dimensional discrete spectrum obtained by sampling a J-dimensional (J is a positive integer of 1 or more) periodic analog signal and performing J-dimensional discrete Fourier transform;
Based on the acquired J-dimensional discrete spectrum, the spectral value of an arbitrary point in the J-dimensional frequency space

The derivative of

However,

Is an integer that accepts zero (ω ₁ ,..., Ω _J ) is an arbitrary point in the J-dimensional frequency space (ω _1,0 ,..., Ω _{J, 0} ) is an arbitrary reference point in the J-dimensional frequency space ( Δω _1, ..., Δω _J) the discrete spectral interval _{(N 1, ..., N J} ) is one period (omega _S1, ..., sample size across _{ω SJ) F p, k1,} ..., kJ is discrete The value of the spectrum,
F _{p, k1,..., KJ} = F _p (ω _1,0 + k ₁ · Δω ₁ ,..., Ω _{J, 0} + k _J · Δω _J )
And Ψ ₁ (ω),..., Ψ _J (ω) are interpolation functions,

(J = 1, ..., J)
Is,
A spectrum interpolation program which is operated as a spectrum interpolation device including a calculation unit obtained according to the above.

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