JP2005249967A - Method and apparatus for frequency analysis - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a method and apparatus for frequency analysis that can analyze unsteady time-series signals. <P>SOLUTION: A response function calculation part 3 divides time-series signals s<SB>1</SB>for each L (L: a positive integer) sections and uses time series signals s<SB>l</SB>which are successive, section by section, to calculate a response function ak up to (K)th order (K: a positive integer). Then a complex frequency calculation part 5 uses the response frequency a<SB>k</SB>to calculate K complex frequencies F<SB>k</SB>=f<SB>k</SB>+ib<SB>k</SB>satisfying 1-Σ<SB>k</SB>a<SB>k</SB>z<SP>-k</SP>=Π<SB>k</SB>(1-e<SP>2πiFkΔT</SP>z<SP>-1</SP>). <P>COPYRIGHT: (C)2005,JPO&NCIPI

Description

  The present invention relates to a frequency analysis method and apparatus capable of analyzing a time waveform of a non-stationary signal.

  For analysis of signal waveforms, it is common to use a power spectrum obtained by Fourier transform. However, since the Fourier transform is intended to characterize a time series using a real frequency, it is assumed that the time series is periodic. For this reason, there is a problem that it is unsuitable for analysis of transient phenomena and analysis of non-stationary signals such as randomly generated pulse signals and signals with attenuation / divergence.

  Transient phenomena can be analyzed by using wavelet analysis, but there is a problem that the physical meaning of information obtained by the analysis is unclear. In addition, wavelet analysis has a problem that it requires a large amount of calculation and is difficult to perform real-time analysis. Furthermore, there is a problem that it is difficult to achieve both time and frequency resolution, such that the frequency resolution decreases when the time resolution is increased.

On the other hand, formant analysis is used to analyze human speech. However, the technology based on formant analysis is based on the premise that the signal is stationary in terms of intervals. It was not used for the analysis of signals that it did not have. For example, Patent Document 1 is known as an application of formant analysis to other than human speech. However, in this document as well, signal continuity is assumed.
Japanese Patent No. 3453130

  As described above, the conventional technique has a problem that an unsteady signal cannot be effectively analyzed.

  The present invention has been made in view of the above, and an object of the present invention is to provide a frequency analysis method and apparatus capable of analyzing an unsteady time series signal.

The frequency analysis method according to the first aspect of the present invention includes a step of reading the time series signal from a storage unit that stores the time series signal, a step of dividing the time series signal into sections of finite length, and the time series signal. And calculating a complex frequency F _k when the time series signal is represented as a set of waveforms represented by an equation including a term of e ^2πiFkt for each section.

In the present invention, the time series signal is divided into sections of finite length, and for each section, the time series signal is characterized as one or more complex frequencies F characterized as a collection of waveforms represented by an equation including e ^2πiFkt. By calculating _k = f _k + ib _k , an imaginary part b _k that cannot be obtained by Fourier transform is obtained, and the time waveform of the non-stationary signal can be analyzed.

Frequency analysis method according to a second aspect of the present invention includes the steps of calculating the response function a _k by using the time-series signal, this with 1-Σ _k a _k z ^-k to factor the response function a _k Complex frequency F _k satisfying the relationship of 1−Σ _k a _k z ^−k = Π _k (1−e ^2πiFkΔT z ⁻¹ ) using Π _k (1−e ^2πiFkΔT z ⁻¹ ) obtained by factoring. And a step of calculating.

In the present invention, the response function a _k is calculated using the time series signal S _l, and the complex frequency satisfying the relationship of 1−Σ _k a _k z ^−k = Π _k (1−e ^2πFkΔT z ⁻¹ ). By calculating F _k , one or more complex frequencies characterizing the time series signal are calculated.

Frequency analysis method according to a third aspect of the present invention, the time-series signal and s _l, the polynomial _{y (t) = Σ m c} m (t-t O) when using ^{m Σ} _l ^‖s _l -y (t) A step of calculating the undetermined coefficient _cm so that ‖ is minimized, and 未_t ‖D · y using this undetermined coefficient _cm and the operator D = Π _k (d / dt- _ak ) comprising the steps of: (t) ‖dt calculates the unknown coefficients a _k that minimizes the steps of calculating a complex frequency F _k satisfies the relationship a _k = 2πiF _k using the calculated a _k, to have a Features.

In the present invention, Σ _l ‖s _l -y (t) ‖ using the time series signal s _l and the polynomial y (t) = Σ _{m cm} (t−t _O ) ^m is minimized. After calculating the undetermined coefficient c _m and calculating the undetermined coefficient a _k that minimizes ∫ _t ‖D · y (t) ‖dt using the operator D = Π _k (d / dt- _ak ) By calculating the complex frequency F _k that satisfies the relationship a _k = 2πiF _k using the calculated coefficient a _k , one or more complex frequencies characterizing the time series signal are calculated.

Frequency analysis method according to the fourth invention, the time-series signal and s _l, sigma _l when 'a _l s' time series signals s was _l = _{Σ k} a _k s _lk approximating this ‖s a step of calculating a prediction coefficient a _k that minimizes _l −s ′ _l １, 1−Σ _k a _k z ^{−k using the} prediction coefficient a _k as a coefficient, and Π _k ( having calculating a ^1-e 2πiFkΔT z ^-1) and 1-Σ _k a _k z complex frequency F _k satisfies the relationship _{^{-k = Π k (1-e}} 2πiFkΔT z -1) with the It is characterized by.

In the present invention, the prediction coefficient a _k is calculated using the time series signal S _l, and the complex frequency satisfying the relationship of 1−Σ _k a _k z ^−k = Π _k (1−e ^2πFkΔT z ⁻¹ ). By calculating F _k , one or more complex frequencies characterizing the time series signal are calculated.

Frequency analysis method according to the fifth present invention includes the steps of calculating the Z transform Σ _k s _k z ^-k when the time-series signal is a _{s l, Σ k s k z} -k The normalized c _k was calculated sigma _k d _k / a coefficients a _k and d _k by using the 1 + Σ _k c _k z ^-k which a coefficient ^{_{(1-a k z -1)}} , Σ k d k / ( 1-a _k z ^-1 ) = Π _k (1-e ^{2πiF "kΔT} z ^-1 ) / Π _k (1-e ^{2π iF'kΔT} z ^-1 ) K 'complex frequencies F' _k = calculating f ′ _k + ib ′ _k and K ″ complex frequencies F ″ _k = f ″ _k + ib ″ _k .

In the present invention, the time series signal s _l was calculated by Z-transform Σ _k s _k z ^-k, this Σ _k s _k z ^-k the the normalized c _k a coefficient 1 + sigma _k Σ _k d _k / (1−a _k z ⁻¹ ) with a _k and d _k as coefficients is calculated using c _k z ^−k , and Σ _k d _k / (1−a _k z ⁻¹ ) = K ′ complex frequencies F ′ _k = f ′ _k + ib ′ _k and K ”satisfying the relationship of Π _k (1-e ^{2πiF" kΔT} z ^-1 ) / Π _k (1-e ^{2π iF'k ΔT} z ^-1 ) One or more complex frequencies characterizing the time-series signal are calculated by calculating the complex frequencies F " _k = f" _k + ib " _k .

  A frequency analysis method according to a sixth aspect of the present invention includes a step of calculating a spectrum envelope using the complex frequency.

  In the present invention, by calculating a spectrum envelope using a complex frequency, a characteristic frequency can be found as a frequency located at the peak of the spectrum envelope.

According to a seventh aspect of the present invention, there is provided a frequency analysis device using a storage unit that stores a time series signal, a division unit that reads the time series signal from the storage unit and divides the time series signal into sections of a finite length, and the time series signal. And a complex frequency calculating means for calculating a complex frequency F _k when a time series signal is expressed as a collection of waveforms represented by an equation including e ^2πiFkt for each section.

A frequency analysis device according to an eighth aspect of the present invention includes response function calculation means for calculating a response function a _k using the time series signal, and the complex frequency calculation means uses the response function a _k as a coefficient. 1-Σ _k a _k z ^-k and ^{_{1-Σ k a k z -k}} = Π k (1-e 2πiFkΔT using a _k [pi obtained by factorization ^(1-e 2πiFkΔT z ^-1) This The complex frequency F _k satisfying the relationship of z ⁻¹ ) is calculated.

Frequency analyzer according to the present invention of the ninth, the time-series signal and s _l, the polynomial _{y (t) = Σ m c} m (t-t O) Σ l ‖s l -y when using ^m (t) Calculate the undetermined coefficient _cm so that ‖ is minimized, and use this undetermined coefficient _cm and the operator D = Π _k (d / dt- _ak ) to calculate ∫ _t ‖D · y (t ) ‖Dt comprising a differential coefficient calculating means for calculating the unknown coefficients a _k, which is minimized, the complex frequency calculation means, a complex frequency F _k satisfies the relationship a _k = 2πiF _k using the calculated a _k It is characterized by calculating.

Frequency analyzer according to the present invention of the tenth, the time-series signal and s _l, sigma _l when 'a _l s' time series signals s was _l = _{Σ k} a _k s _lk approximating this ‖s a prediction coefficient calculation unit that calculates a prediction coefficient a _k that minimizes _l −s ′ _l 、, and the complex frequency calculation unit includes 1−Σ _k a _k z ^{−k using the} prediction coefficient a _k as a coefficient; Complex frequency satisfying the relationship of 1−Σ _k a _k z ^−k = Π _k (1−e ^2πiFkΔT z ⁻¹ ) using Π _k (1−e ^2πiFkΔT z ⁻¹ ) obtained by factoring this. _Fk is calculated.

Frequency analyzer according to the present invention of the 11th has a Z conversion calculation means for calculating a Z transform Σ _k s _k z ^-k when the time-series signal is a s _l, the complex frequency calculation means, sigma Σ _k d _k / (1−a _k z ⁻¹⁾ with a _k and d _k as coefficients using 1 + Σ _k c _k z ^−k with _{k k} as a coefficient normalized _k s _k z ^−k ) And Σ _k d _k / (1−a _k z ⁻¹ ) = Π _k (1-e ^{2πiF ”kΔT} z ⁻¹ ) / Π _k (1-e ^2πiF′kΔT z ⁻¹ ) K ′ complex frequencies F ′ _k = f ′ _k + ib ′ _k and K ″ complex frequencies F ″ _k = f ” _k + ib” _k are calculated.

  A frequency analysis apparatus according to a twelfth aspect of the present invention includes a spectrum envelope calculation unit that calculates a spectrum envelope using the complex frequency.

  The frequency analysis method and apparatus according to the present invention can analyze a non-stationary time series signal.

  Hereinafter, embodiments of the present invention will be described with reference to the drawings.

[First Embodiment]
FIG. 1 is a functional block diagram showing the configuration of the frequency analysis apparatus according to the present embodiment. The frequency analysis apparatus shown in FIG. 1 includes a measurement unit 1, an analog / digital (A / D) conversion unit 2, a response function calculation unit 3, a response function optimization unit 4, a complex frequency calculation unit 5, and a display unit. 6 and a storage unit 11.

  This frequency analysis apparatus may be configured by dedicated hardware, or may be configured by a computer, and processing in each unit may be executed by a program installed in the computer. The storage unit 11 is configured by a device capable of storing data such as a magnetic storage device, an optical disc device, and a memory.

The measurement unit 1 measures an unsteady analog signal s (t) and outputs it to the A / D conversion unit 2. In the A / D converter 2, for example, the sampling frequency is set to 40 kHz or more for frequency analysis up to 20 kHz, and the sampling frequency is set to 1 GHz or more for frequency analysis up to 500 MHz. set the frequency at least twice the value, the analog signal s (t) to obtain a time series signal s _l digitized, and temporarily stored in the memory unit 11 of the time-series signal s _l. Here, the sampling interval determined from the sampling frequency is ΔT. A / D conversion section 2 reads a series signal s _l when the storage unit 11, and outputs the response function calculation unit 3.

Response function calculating section 3, time series signals s _l L (L is a positive integer) is divided into every individual sections, each section, the following simultaneous equations using the series signal s _l when successive a By solving for _k , the response function a _k is calculated up to Kth order (K is a positive integer) and stored in the memory.

s ₁ = Σ _k a _k s _lk
s _l-1 = Σ _k a _k s _l-1-k
...
s _lK = Σ _k a _k s _lKk (1)
Here, the range of k is 1 to K. K is the number of complex frequencies to be calculated, and is a value of 2 or more. L is set to a value not less than twice K as required. When the value of L is larger than K, the response function optimizing unit 4 calculates the optimum response function a _k using the least square method or the like.

Next, consider the following equation using the response function a _k as a coefficient.

1-Σ _k a _k z ^-k (2)
Here, the range of k is 1 to K. Considering factoring equation (2), the following equation holds.

1-Σ _k a _k z ^-k
= Π _k (1-e ^2πiFkΔT z ^-1 ) (3)
The complex frequency calculation unit 5 reads the response function a _k obtained in advance from the memory, and calculates K complex frequencies F _k = f _k + ib _k satisfying the expression (3) using the response function a _k. And output to the display unit 6.

As shown in FIG. 2, the display unit 6 displays f _k and b _k on a screen showing a graph with the horizontal axis representing time and the vertical axis representing magnitude.

Waveform of the time-series signal s _l can be expressed in the form of _{Σ k Σ t'k c t'k h (} t-t 'k) e 2πiFk (t-t'k). Here, t ′ _k is the generation time of each waveform, and may be multiple times for each F _k . c _t′k is a function that takes the size of each waveform, and h (t) takes 0 in the range of t <0 and 1 in the range of t ≧ 0. The same applies to the following embodiments.

Therefore, according to this embodiment, time-series signals s _l is divided into sections of finite length, one characterized as a collection of waveforms represented a time series signal with an expression that contains e ^2PaiiFkt in each section By calculating the above complex frequency F _k = f _k + ib _k , an imaginary part b _k that cannot be obtained by Fourier transform is obtained, and the time waveform of the unsteady signal can be analyzed.

According to the present embodiment, the response function calculation unit 3 calculates the response function a _k using the time series signal _sl , and the complex frequency calculation unit 5 calculates 1−Σ _k a _k z ^−k = Π _k (1 -e ^2πiFkΔT z ^-1 ) to calculate the complex frequency F _k , one or more complex frequencies characterizing the time series signal are calculated, and the time waveform of the non-stationary signal can be analyzed. it can.

[Second Embodiment]
FIG. 3 is a functional block diagram showing the configuration of the frequency analysis apparatus according to the present embodiment. The frequency analysis apparatus shown in the figure has a configuration including a differential coefficient calculation unit 7 instead of the response function calculation unit 3 and the response function optimization unit 4 shown in FIG. In addition, the same code | symbol is attached | subjected about the thing same as FIG. 1, or an approximate thing, and the overlapping description is abbreviate | omitted here.

The time series signal s _l, given that modeled with a polynomial of the following equation for the c _m with undetermined coefficients.

y (t) = Σ _m c _m (t−t _O ) ^m (4)
Here, the range of m is 0 to M (M is a positive integer). Differential coefficient calculating unit 7, by using a series signal s _l when read from the storage unit 11, for example by the least squares method to calculate the unknown coefficients c _m the following equation is minimized and stored in the memory.

Σ _l ‖s _l -y (t _l ) ‖ (5)
Here, the range of l is 0 to L (L is a positive integer). Next, consider an operator D of the following expression represented by a linear differential equation with a _k as an undetermined coefficient.

D = Π _k (d / dt-a _k ) (6)
For the undetermined coefficient a _k included in the operator D, the differential coefficient calculating unit 7 uses the undetermined coefficient _cm read from the memory to calculate the undetermined coefficient a _k that minimizes the following equation using, for example, the least square method. Store in memory.

∫ _t ‖ D · y (t) ‖ dt… (7)
Here, the range of t is t _a to t _b . t _l is the time that gives the l-th value of the time series signals s _l is t _a and t _b are the time t _O giving the leading value of the time series signals s _l used to calculate the coefficients a _k by using the time t _L to give a final value, given by the range satisfying the inequality _{t O ≦ t a ≦ t b} ≦ t L.

The complex frequency calculation unit 5 reads the coefficient a _k from the memory, and calculates a complex frequency F _k = f _k + ib _k that satisfies the following relational expression using the coefficient a _k .

a _k = 2πiF _k (8)
The display unit 6 displays the K complex frequencies F _k = f _k + ib _k thus obtained for each section. Here, as shown in FIG. 4, the complex frequency F _k is displayed on a screen showing a graph with the real part of the complex frequency on the horizontal axis and the imaginary part on the vertical axis. Of course, as shown in FIG. 2, the time axis may be displayed on the horizontal axis and the size on the vertical axis.

Therefore, according to this embodiment, the differential coefficient calculation unit 7, the time-series signal s _l and polynomial _{y (t) = Σ m c} m (t-t O) were used and ^{m Σ} _l ^‖s _l - The undetermined coefficient c _m is calculated so that y (t) ‖ is minimized, and ∫ _t ‖D · y (t) ‖dt using the operator D = Π _k (d / dt- _ak ) is minimized. after calculating the unknown coefficients a _k to be, by the complex frequency calculator 5, by calculating the complex frequency F _k satisfying a _k = 2πiF _k the relationship with the coefficients a _k, characterizing the time series signal 1 More than one complex frequency is calculated, and the time waveform of the non-stationary signal can be analyzed.

[Third Embodiment]
FIG. 5 is a functional block diagram showing the configuration of the frequency analysis apparatus according to the present embodiment. The frequency analysis apparatus of FIG. 9 includes a prediction coefficient calculation unit 8 instead of the response function calculation unit 3 and the response function optimization unit 4 of FIG. 1 and a spectrum envelope calculation unit 9 at the output stage of the complex frequency calculation unit 5. It is the structure provided with. In addition, the same code | symbol is attached | subjected about the thing same as FIG. 1, or an approximate thing, and the overlapping description is abbreviate | omitted here.

The prediction coefficient calculation unit 8 calculates a prediction coefficient _ak by a linear prediction method (Linear Predictive Coding, LPC). Specifically, the following processing is performed.

First, consider the following time series signal s' _l approximating the series signal s _l when read from the storage unit 11 by the A / D converter 2.

s ′ _l = Σ _k a _k s _lk (9)
Here, the range of k is 1 to K. The prediction coefficient calculation unit 8 uses the time series signal s ′ _l to calculate a prediction coefficient a _k that minimizes the following equation and stores it in the memory.

Σ _l ‖s _l -s' _l ‖… (10)
Here, the range of l is l _O -L + 1~l _O. Next, consider the following equation using the prediction coefficient a _k .

1-Σ _k a _k z ^-k (11)
Here, the range of k is 1 to K. Considering factoring equation (11), the following equation holds.

1-Σ _k a _k z ^-k
= Π _k (1-e ^2πiFkΔT z ^-1 ) (12)
The complex frequency calculation unit 5 reads the prediction coefficient a _k from the memory, calculates K complex frequencies F _k = f _k + ib _k satisfying the equation (12) using the prediction coefficient a _k , and a spectrum envelope calculation unit Output to 9.

  The spectrum envelope calculation unit 9 calculates the spectrum envelope G (f) by the following equation.

G (f) = 1 / | 1-e ^{2πi (Fk-f) ΔT} | ² (13)
As shown in FIG. 6, the display unit 6 displays a spectrum envelope G (f) on a screen showing a graph with the horizontal axis representing the frequency f.

Therefore, according to this embodiment, the prediction coefficient calculation unit 8 calculates the prediction coefficient a _k using the time series signal _sl , and the complex frequency calculation unit 5 calculates 1−Σ _k a _k z ^−k = Π _k. By calculating the complex frequency F _k satisfying the relationship (1-e ^2πiFkΔT z ⁻¹ ), one or more complex frequencies characterizing the time series signal are calculated, and the time waveform of the non-stationary signal is analyzed. be able to.

According to the present embodiment, the spectral envelope calculation unit 9 calculates the spectral envelope G (f) using K complex frequencies F _k = f _k + ib _k , so that a characteristic frequency is obtained. f can be found as the frequency located at the peak of the spectral envelope.

  The display unit 6 may perform display as described in the above embodiments. Moreover, in each said embodiment, you may make it display a spectrum envelope on the display part 6. FIG.

[Fourth Embodiment]
FIG. 7 is a functional block diagram showing the configuration of the frequency analysis apparatus according to the present embodiment. The frequency analysis apparatus shown in the figure has a configuration including a Z-transform calculation unit 10 instead of the response function calculation unit 3 and the response function optimization unit 4 shown in FIG. In addition, the same code | symbol is attached | subjected about the thing same as FIG. 1, or an approximate thing, and the overlapping description is abbreviate | omitted here.

In Z transform calculating unit 10, time-series signals s _l from the storage unit 11 K + 1 pieces of consecutive read by the A / D converter 2 for Z transform by the following equation.

Σ _k sk _k ^-k (14)
Here, the range of k is 0-K. The Z conversion calculation unit 10 normalizes this and gives the following expression using _ck as a coefficient.

1 + Σ _k c _k z ^-k (15)
Here, the range of k is 1 to K. K is the number of complex frequencies to be obtained, and is a value of 2 or more.

The complex frequency calculation unit 5 calculates the following equation using a _k and d _k as coefficients using equation (15).

Σ _k d _k / (1-a _k z ⁻¹ ) (16)
Here, the range of k is 1 to K ′ (K ′ is a positive integer). The calculation of the equation (16) is performed according to the correspondence relationship of the following equation.

Σ _k d _k / (1-a _k z ^-1 )
= Σ _k d _k (1 + Σ _m a _k ^m z ^-m ) (17)
= Σ _k d _k + Σ _m (Σ _k d _k a _k ^m ) z ^−m (18)
_{= 1 + Σ k c k z} -k + O (z -K-1) ... (19)
Here, the range of k in Formula (17) and Formula (18) is 1 to K ′, and the range of m is 1 to ∞. The range of k in the formula (19) is 1 to K. O (z ^−K−1 ) is a residual term when calculating the complex frequency.

  The complex frequency calculation unit 5 transforms the equation (16) to obtain the following equation.

Σ _k d _k / (1-a _k z ^-1 )
= Σ _k d _k Π _{k ′ ≠ k} (1-a _{k ′} z ⁻¹ ) / Π _k (1-a _k z ⁻¹ ) (20)
= Π _k (1-a ′ _k z ^-1 ) / Π _k (1-a _k z ^-1 ) (21)
^{_{= Π k (1-e 2πiF}} "kΔT z -1) / Π k (1-e 2πiF'kΔT z -1) ... (22)
Here, the range of k and k ′ in Expression (20) is 1 to K. The range of k in the numerators of the formulas (21) and (22) is 1 to K ″, and the range of k in the denominator is 1 to K ′.

Subsequently, the complex frequency calculation unit 5 satisfies K ′ complex frequencies F ′ _k = f ′ _k + ib ′ _k and K ″ complex frequencies F ″ _k = f ” _k + ib satisfying the relationship of Expression (22). "Calculate _k .

In the display unit 6, K ′ complex frequencies F ′ _k on a screen showing a graph with the horizontal axis representing the real part, the vertical axis representing the imaginary part, and the horizontal axis representing time and the vertical axis representing the imaginary part, Display K "number of complex frequencies F" _k .

Therefore, according to the present embodiment, the Z conversion calculation unit 10 _performs Z conversion on the time series signal ^sl to calculate Σ _k s _k z ^-k , and the complex frequency calculation unit 5 calculates Σ _k s _k z ^−. Σ _k d _k / (1−a _k z ⁻¹ ) with a _k and d _k as coefficients is calculated by using 1 + Σ _k c _k z ^−k with _{k k} as a coefficient, normalized ^k , K ′ pieces satisfying the relationship of Σ _k d _k / (1−a _k z ⁻¹ ) = ( _k (1−e ^{2πiF ”kΔT} z ⁻¹ ) / Π _k (1−e ^{2π iF′kΔT} z ⁻¹ ) By calculating the complex frequencies F ′ _k = f ′ _k + ib ′ _k and K ”complex frequencies F” _k = f ” _k + ib” _k , one or more complex frequencies characterizing the time-series signal are calculated. The time waveform of the non-stationary signal can be analyzed.

[Example]
Hereinafter, a result when a complex frequency is calculated using an actual time series signal will be described.

Figure 9 is a diagram showing a waveform of a time-series signal generated waveforms represented by ^y (t) = e -λt sin2πωt randomly. Here, the frequency ω is slowly raised and lowered. The waveform decay rate λ is given as a constant.

  FIG. 10 is a diagram showing a waveform obtained by Fourier transforming the time series signal of FIG. As shown in FIG. 10, the characteristic frequency ω included in the non-stationary signal cannot be found by the Fourier analysis.

  FIG. 11 shows a spectrum envelope calculated using the complex frequency calculated for the time series signal shown in FIG. 9 using the prediction coefficient as described in the third embodiment. FIG. As shown in FIG. 11, in this embodiment, the characteristic frequency ω included in the non-stationary signal can be found as a frequency located at the peak p of the spectral envelope.

  FIG. 12 is a graph in which the time change of the peak p in FIG. 11 is plotted. The line shown as the frequency axis in FIG. 11 corresponds to the frequency axis in FIG. Thus, in this embodiment, it is possible to find a temporal change in the characteristic frequency ω included in the non-stationary signal.

  The formant analysis is recognized as being used for analysis of a sound signal in a stationary audible frequency band for each section. It became clear that it can be applied to analysis of radio waves and electrical signals. It can also be applied to those single pulse signals.

  The technique in each of the above embodiments can be applied not only to the analysis of time series signals on the time axis, but also to the analysis of spatial time series signals indicating patterns and patterns.

It is a functional block diagram which shows the structure of the frequency analyzer in 1st Embodiment. It is a figure which shows a mode that the complex frequency calculated in 1st Embodiment was displayed. It is a functional block diagram which shows the structure of the frequency analyzer in 2nd Embodiment. It is a figure which shows a mode that the complex frequency calculated in 2nd Embodiment was displayed. It is a functional block diagram which shows the structure of the frequency analyzer in 3rd Embodiment. It is a figure which shows a mode that the complex frequency calculated in 3rd Embodiment was displayed. It is a functional block diagram which shows the structure of the frequency analyzer in 4th Embodiment. It is a figure which shows a mode that the complex frequency calculated in 4th Embodiment was displayed. y a ^(t) = e -λt sin2πωt format waveform that is a diagram showing a waveform illustrating a time series signal which is generated at random. It is a figure which shows the waveform which carried out the Fourier-transform of the time series signal of FIG. It is a figure which shows the spectrum envelope calculated using the complex frequency for the time series signal of FIG. 9, using the prediction coefficient. It is a figure which shows the graph which plotted the time change of the peak p in FIG.

Explanation of symbols

DESCRIPTION OF SYMBOLS 1 ... Measurement part 2 ... Conversion part 3 ... Response function calculation part 4 ... Response function optimization part 5 ... Complex frequency calculation part 6 ... Display part 7 ... Differential coefficient calculation part 8 ... Prediction coefficient calculation part 9 ... Spectrum envelope calculation part 10 ... Z conversion calculation unit 11 ... Storage unit

Claims (12)

Reading the time series signal from the storage means storing the time series signal;
Dividing the time series signal into sections of finite length;
Calculating a complex frequency F _k when the time series signal is represented as a set of waveforms represented by an equation including e ^2πiFkt for each section using the time series signal;
A frequency analysis method characterized by comprising: Calculating a response function a _k using the time series signal;
The response function a _k 1-sigma and coefficient _k a _k z ^-k and which _k [pi obtained by factorization ^(1-e 2πiFkΔT z ^-1) and 1-Σ _k a _k z with ^- calculating a complex frequency F _k satisfying a relationship of ^k = Π _k (1-e ^2πiFkΔT z ⁻¹ );
The frequency analysis method according to claim 1, further comprising: An undetermined coefficient so that Σ _l ‖s _l -y (t) ‖ is minimized when the time series signal is s _l and a polynomial y (t) = Σ _{m cm} (t−t ₀ ) ^m is used. calculating c _m ;
Calculating an undetermined coefficient a _k that minimizes ∫ _t ‖D · y (t) ‖dt using the undetermined coefficient _cm and an operator D = Π _k (d / dt- _ak );
Calculating a complex frequency F _k satisfies the relationship a _k = 2πiF _k using the calculated a _k,
The frequency analysis method according to claim 1, further comprising: Prediction coefficient that minimizes Σ _l ‖s _l -s' _lと し when the time series signal is s _l and the time series signal s ′ _l approximating this is s ′ _l = Σ _k a _k s _lk calculating a _k ;
The prediction coefficients a _k and a coefficient 1-Σ _k a _k z ^-k and which _k [pi obtained by factorization ^(1-e 2πFkΔT z ^-1) and was used 1-Σ _k a _k z ^- calculating a complex frequency F _k satisfying a relationship of ^k = Π _k (1-e ^2πFkΔT z ⁻¹ );
The frequency analysis method according to claim 1, further comprising: Calculating a Z transform Σ _k s _k z ^-k when the time-series signal is a s _{l,
^{Σ k s k z -k 1 +}} Σ a normalized c _k a coefficient _k c _k z and coefficients a _k and d _k with ^{_{-k Σ k d k / (1}} -a k z - ¹ ) is calculated, and Σ _k d _k / (1−a _k z ⁻¹ ) = ｅ _k (1-e ^{2πiF ”kΔT} z ⁻¹ ) / Π _k (1-e ^2πiF′kΔT z ⁻¹ ) Calculating K ′ complex frequencies F ′ _k = f ′ _k + ib ′ _k and K ″ complex frequencies F ″ _k = f ” _k + ib ″ _k satisfying
The frequency analysis method according to claim 1, further comprising:   The frequency analysis method according to claim 1, further comprising a step of calculating a spectrum envelope using the complex frequency. Storage means for storing time-series signals;
Dividing means for reading out the time-series signal from the storage means and dividing it into sections of finite length;
Complex frequency calculation means for calculating a complex frequency F _k when the time series signal is represented as a collection of waveforms represented by an equation including e ^2πiFkt for each section using the time series signal;
A frequency analysis apparatus comprising: Response function calculating means for calculating a response function a _k using the time series signal,
The complex frequency calculation means uses 1-Σ _k a _k z ^-k having the response function a _k as a coefficient and Π _k (1-e ^2πiFkΔT z ⁻¹ ) obtained by factoring the response function a _{k. ^{-Σ k a k z -k = Π}} k (1-e 2πiFkΔT z -1) frequency analyzer according to claim 7, wherein the calculating the complex frequency F _k satisfies the relationship. An undetermined coefficient so that Σ _l ‖s _l -y (t) ‖ is minimized when the time series signal is s ₁ and a polynomial y (t) = Σ _{m cm} (t−t _O ) ^m is used. c _m is calculated, and the undetermined coefficient a _k that minimizes ∫ _t ‖D · y (t) ‖dt using the undetermined coefficient _cm and the operator D = Π _k (d / dt- _ak ) is calculated. A differential coefficient calculating means for calculating,
The complex frequency calculating means, the frequency analyzer according to claim 7, wherein the calculating the complex frequency F _k satisfies the relationship a _k = 2πiF _k using the calculated a _k. Prediction coefficient that minimizes Σ _l ‖s _l −s ′ _lと し when the time series signal is s _l and the time series signal s ′ _l approximating this is s ′ _l = Σ _k a _k s _lk a prediction coefficient calculating means for calculating a _k ;
The complex frequency calculating means 1 uses 1-Σ _k a _k z ^-k having the prediction coefficient a _k as a coefficient and Π _k (1-e ^2πiFkΔT z ⁻¹ ) obtained by factoring this 1 ^{_{-Σ k a k z -k = Π}} k (1-e 2πiFkΔT z -1) frequency analyzer according to claim 7, wherein the calculating the complex frequency F _k satisfies the relationship. Comprises a Z conversion calculation means for calculating a Z transform Σ _k s _k z ^-k when the time-series signal is a s _l,
The complex frequency calculating ^{_{means, Σ k s k z -k 1}} + Σ a normalized c _k a coefficient _k c _k z ^-k and coefficients a _k and d _k by using a sigma _k d _k / (1-a _k z ^-1 ) is calculated, and Σ _k d _k / (1-a _k z ^-1 ) = Π _k (1-e ^{2πiF "kΔT} z ^-1 ) / Π _k (1-e ^{2πiF '} _k ′ complex frequencies F ′ _k = f ′ _k + ib ′ _k and K ″ complex frequencies F ″ _k = f ” _k + ib” _k satisfying the relationship of ^kΔT z ⁻¹ ) The frequency analysis apparatus according to claim 7. The frequency analysis apparatus according to claim 7, further comprising: a spectrum envelope calculation unit that calculates a spectrum envelope using the complex frequency.

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