JP3033871B2 - Cross spectrum operation device, transfer function operation device, and coherence function operation device - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a cross-spectrum calculator for obtaining a cross spectrum between time-series signals, a transfer function calculator for obtaining a transfer function, and a coherence function calculator for obtaining a coherence function.

[0002]

2. Description of the Related Art Conventionally, there has been known a method of obtaining a cross spectrum between time-series signals or using the cross spectrum to obtain a transfer function and a coherence function in order to examine characteristics of a signal transmission system. Since the operation for obtaining the cross spectrum is included in the operation for obtaining the transfer function and the coherence function, the transfer function and the coherence function will be described below.

In the signal transmission system 1 shown in FIG. 2, the impulse response of the system is h (t), the input signal is x (t), the output signal is y (t), and the disturbance noise mixed in the output signal is u.
(T), and x (t) and u (t) are stationary white noises and are uncorrelated with each other. Observable signals are x (t) and y
(T). At this time, the output signal y (t) can be expressed by the following equation.

Y (t) = h (t) * x (t) + u (t) (1) where * represents a convolution operation. Here, Expression (1) is discretized using τ as a sampling interval, and discrete Fourier transform of N points (hereinafter, referred to as DFT) is performed, and then Expression (2) is obtained. Y (k) = H (k) X (k) + U (k) (2) The input signal x (t) and the output signal y of the target transmission system
As shown in FIG. 3, (t) is cut out at time windows for every N points, and a series of numbers is assigned to each time window. The input signal is the i-th block,
The output signal is called an m-th block (i and m are arbitrary integers).

At this time, both sides of the equation (2) are multiplied by the complex conjugate X ^* (k) of X (k), and the results obtained between the corresponding input / output blocks are averaged many times. When the cross spectrum between them is obtained, the following equation is obtained. E [X ^* (k) Y (k)] = H (k) E [X ^* (k) X (k)] + E [X ^* (k) U (k)] (3) where E [ ] Represents a multiple average. The second term on the right side of the equation (3) is a term due to disturbance noise, and has no correlation between the input signal and the disturbance noise. Therefore, the term asymptotically approaches 0 as the number of times of averaging increases.

E [X ^* (k) Y (k)] = H (k) E [X ^* (k) X (k)] (4) Accordingly, the transfer function H (k) at the discrete frequency is It can be estimated by the formula.

[0007]

(Equation 1)

Similarly, the coherence function γ ² (k) of the signal transmission system 1 is given by the following equation.

[0009]

(Equation 2)

[0010]

In order to obtain a cross spectrum, a transfer function and a coherence function obtained by an operation using the cross spectrum, each of the input signal and the output signal must be cut out by a certain finite length time window. .
Normally, the start time and the time window length T (T = Nτ, τ is a sampling interval and N is the number of sampling points) of the time window from which the input signal is cut out and the time window from which the output signal is cut out are the same. In this case, when the Q of the target transmission system is large or the like, an estimation error (resolution bias error) due to the fact that the length of the used time window is not sufficient compared to the length of the impulse response of the transmission system.
Occurs. This resolution bias error is caused by the fact that the response to the signal within the input time window leaks out of the output time window, and the signal component correlated with the input signal is not completely reflected in the output time window. It is. Even if the time window length is made sufficiently long, the response to the signal near the rear end in the input time window always leaks outside the output time window, so that a resolution bias error remains. In.

Accordingly, the present invention provides a cross spectrum calculator, a transfer function calculator, and a coherence function calculator capable of obtaining a cross spectrum, a transfer function, and a coherence function having a smaller resolution deviation error as compared with the prior art. Aim.

[0012]

Means for Solving the Problems FIGS. 1 (a), (b),
(C) is a block diagram showing a configuration of a cross spectrum calculation device, a transfer function calculation device, and a coherence function calculation device of the present invention, respectively. The cross spectrum calculation device (a) of the present invention includes a first calculation means 12 and a second calculation means 14. The first time-series signal and the second time-series signal are input to the first calculation unit 12. The first time series signal and the second time series signal may be signals that are not related to each other at all for obtaining the cross spectrum, but usually, the input signal and the output signal of a predetermined signal transmission system are To be elected. First arithmetic unit 1
As shown in FIG. 3, the first time-series signal and the second time-series signal input to 2 are conceptually divided into a plurality of blocks at time intervals corresponding to each other. , The first time-series signal and the second time-series signal divided in this way are mutually m blocks (m = 0, 1, 2,..., M; M is a predetermined positive A total of (M + 1) cross spectra (herein referred to as delayed block cross spectra) of blocks shifted by an integer are calculated. A total of M + 1 delayed block cross spectra obtained by the first calculating means 12 are input to the second calculating means 14. The second calculating means 14 calculates the first time-series signal and the second time-series signal based on the input M + 1 cross spectra.
A cross spectrum with the time series signal is calculated and output from the second calculating means.

Further, the transfer function calculating device of the present invention (FIG. 1)
(B)) The third arithmetic means 22 and the fourth arithmetic means 24
It is composed of The first arithmetic unit 22 receives a first time-series signal input to a predetermined signal transmission system and a second time-series signal output from the signal transmission system. The first time series signal and the second time series signal are, as in the case of the above-described cross spectrum calculation device of the present invention,
As shown in FIG. 3, the blocks are conceptually divided into a plurality of blocks at each time interval corresponding to each other. , M blocks (m = 0, 1, 2,..., M; M is a predetermined positive integer)
A total of M + 1 transfer functions of blocks displaced by just one are calculated. Here, this transfer function is referred to as a delayed block transfer function. A total of (M + 1) delayed block transfer functions obtained by the third calculating means 22 are input to the fourth calculating means 24. The fourth calculation means 24 calculates and outputs the transfer function of the signal transfer system based on the input N + 1 delayed block transfer functions.

The transfer function calculating device of the present invention calculates a transfer function using the cross spectrum obtained by the above-mentioned cross spectrum calculating device of the present invention. This is also one embodiment of the cross spectrum calculation device of the present invention. Further, the coherence function calculation device of the present invention (FIG. 1 (c)) includes a fifth calculation means 32 and a sixth calculation means 34. The first time-series signal input to a predetermined signal transmission system and the second time-series signal output from the signal transmission system are input to the fifth arithmetic unit 32. These first time-series signal and second time-series signal are conceptually, as shown in FIG. 3, as in the case of the above-described cross-spectrum calculation device of the present invention and the transfer function calculation device of the present invention. Each of the first time series signal and the second time series signal thus divided is divided into a plurality of blocks at time intervals corresponding to each other.
There are a total of M + 1 coherence functions (herein referred to as a delayed block coherence function) of blocks shifted from each other by m blocks (m = 0, 1, 2,..., M; M is a predetermined positive integer). Is calculated. A total of M + 1 delayed block coherence functions obtained by the fifth calculating means 32 are input to the sixth calculating means 34. In the sixth calculating means 34, the input M +
A coherence function of the signal transmission system is obtained based on one delayed block coherence function, and is output from the sixth arithmetic unit.

The coherence function calculation device of the present invention also calculates a coherence function using the cross spectrum obtained by the cross spectrum calculation device of the present invention, similarly to the transfer function calculation device of the present invention. Therefore, the coherence function calculation device of the present invention is also one aspect of the cross spectrum calculation device of the present invention.

Here, each of the first to sixth arithmetic means may be constituted by hardware so as to obtain desired functions, but the present invention is not limited thereto. Alternatively, the functions may be realized by inputting data to a computer or the like and realizing each function by software. Further, in the present invention, as an intermediate output, a delayed block cross spectrum, a delayed block transfer function,
Alternatively, it is not always necessary to obtain a delayed block coherence function.
, The third computing means 22 and the fourth computing means 24, or the fifth computing means 32 and the sixth computing means 34 may be integrally formed.

[0017]

According to the present invention, a time delay is given to the time window on the output side as compared with the time window on the input side, and each signal is cut out. First, a cross spectrum, a transfer function, and a coherence function therebetween are obtained, and then, A method of obtaining the overall cross spectrum, transfer function, and coherence function using the above method.

As a result, a resolution bias error is suppressed,
As will be compared in the examples described later, a cross spectrum, a transfer function, and a coherence function with higher precision than the conventional method are obtained.

[0019]

EXAMPLES Theoretical considerations and comparison of the present invention and the conventional method will be described below. However, as described above, since the cross spectrum is included in the transfer function and the coherence function, the consideration and comparison of extracting only the cross spectrum overlap with the consideration and comparison of the transfer function and the like. Therefore, consideration and comparison of only the cross spectrum are omitted here. 1. 1. Transfer Function Estimation Method According to the Present Invention 1.1 Estimation of Transfer Function for Each Block Here, first, attention is paid only to the 0th block shown in FIG. The response and the response leaking from the block are evaluated. For this reason, among the response signals of the transmission system based on the input signal x ₀ (n) of the 0th block, the output signal obtained by adding the disturbance noise to the components in the mth block (m is an integer of 0 or more) is represented by y _m ( n). x ₀ (n), y _m
The signals obtained by adding (M-1) × N zero-point signals after (n) are expressed as x ′ ₀ (n) and y ′ _m (n), respectively.
The following equations hold respectively.

[0020]

(Equation 3)

MN point DF of x ' ₀ (n) and y' _m (n)
Let T be X ′ ₀ (k) and Y ′ _m (k), respectively, and estimate the blocked delay transfer function between them as follows.

[0022]

(Equation 4)

Here, G ' _m (k) is the average value of the cross spectrum of X' ₀ (k) and Y ' _m (k). H ′ _m (k) obtained in this manner is the ratio of the power of the input signal to the power of the output component correlated therewith. If the input signal is stationary and the transmission system is linear, the input signal Is determined only by the difference between the block positions of the input signal and the output signal. The denominator of the equation (9) is expressed as follows.

[0024]

(Equation 5)

Here, it is assumed that z ₀ = exp (−j2π / MN). When the variance of the input signal x ₀ (n) is set to σ _x ² , the following _holds : E [x ₀ (n) x ₀ (q)] = σ _x ² δ (n−q) In the above equation, since the contribution of the power σ _x ² (k) = σ _x ² / N of the k-th frequency component is considered,

[0026]

(Equation 6)

Therefore,

[0028]

(Equation 7)

## EQU2 ## 1.2 Calculation of Overall DFT by DFT for Each Block As shown in FIG. 4, an arbitrary signal r (t) is cut out by M blocks continuously at every N points, and 0 to each block. Number M-1. Signal r _m in the m-th block (n) is _{r m (0), ...,} r m of (N-1) N
Has samples. As in the equations (7) and (8), r
_m the (M-1) signal added to the signal of the N zeros after (n) represents the r _'m (n). MN for r ′ _m (n)
When the spectrum obtained by performing the point DFT is represented by R ′ _m (k), the following equation is established from the definition of DFT.

[0030]

(Equation 8)

Next, let us consider the M blocks as one block. In that case, the signal r (n) in the block has MN samples r (0),..., R (MN−
1). If a spectrum obtained by applying the MN point DFT to r (n) is represented as R (k), the following equation is established.

[0032]

(Equation 9)

The expression in {} of the expression (13) is rewritten as follows.

[0034]

(Equation 10)

[0035] Here, n = 0,1, ..., was used in the N-1 r (n + mN ) = r m (n) becomes the relationship holds. (13)
4) By substituting the equation,

[0036]

[Equation 11]

By substituting equation (12) into equation (15),

[0038]

(Equation 12)

## EQU1 ## 1.3 Estimation of Overall Transfer Function Using Transfer Function for Each Block As shown in FIG. 5, the input signal x ₀ (n) in the 0th block
And the output signals y ₀ (n),
.., Y _M-1 (n). x ' ₀ (n) and y' ₀
The delayed block transfer functions of (n),..., X ′ ₀ (n) and y ′ _M−1 (n) are respectively H ′ ₀ from equation (9).
(K),..., H ′ _M−1 (k). H ′ ₀
(K), ..., H ' M-1 blocks signal h of the impulse response of each of the _{(k)' 0 (n)} , ..., h 'M-1 (n)
H ′ _m (n) as the MN point DFT of
Consider the signal h _B (n) synthesized up to the (M-1) block (see FIG. 6).

When a spectrum obtained by applying the MN point DFT to h _B (n) is represented as H _B (k), the following equation is established from the equation (16).

[0041]

(Equation 13)

That is, the estimated value H _B of the transfer function of the transfer system
(K) is each delayed block transfer function H ′ _m (k)
Is multiplied by each coefficient exp (−j2πkm / M), and is added to each other. As is apparent from the equation (17), the estimated value G _B (k) of the cross spectrum between the input and output signals is calculated assuming that each delayed block cross spectrum is G ′ _m (k).

[0043]

[Equation 14]

Is as follows. 2. As in the conventional normal transfer function estimation method (H _A (k)) 7, the input signal x of the 0~ (M-1) block
_{0 (n), ..., x} M-1 (n) and the 0~ (M-1) output signal of the block y ₀ (n), ..., focusing on y _M-1 (n).

The signals obtained by assuming the M blocks as one block and combining them are x _M0 (n) and y _M0 (n), respectively.
And Let the MN point DFT of x _M0 (n) and y _M0 (n) be X _M0 (k) and Y _M0 (k), respectively, and let the transfer function between them be H _A (k). H _A (k) is represented by the following equation.

[0046]

(Equation 15)

From equation (16), X _M0 (k) and Y _M0
(K) is represented by the following equation.

[0048]

(Equation 16)

From equation (19), E [X ^* _m0 (k) X
_M0 (k)] is calculated as follows.

[0050]

[Equation 17]

Here, E [X ′ _i ^* (k) X ′ _j (k)] = N ² σ ² _x (k) δ
(Ij). Also, E [X ^* _M0 (k) Y
_M0 (k)] is calculated as follows.

[0052]

(Equation 18)

Here, E [X ′ _i ^* (k) Y ′ _j (k)] = 0 when i> j due to causality, and i < j because the input / output signal is stationary.
E [X ′ _i ^* (k) Y ′ _j (k)] = E [X ′ ₀ ^*
(K) _Y'ji (k)] (= _G'ji (k)). Therefore, the following equation holds.

[0054]

[Equation 19]

By substituting equations (20) and (21) into equation (18), the following equation is obtained.

[0056]

(Equation 20)

From equation (11), H ′ _m (k) = G ′ _m (k) / N ² σ ² _X (k), so equation (22) is as follows.

[0058]

(Equation 21)

H _A (k) is a transfer function obtained by a conventionally used normal estimation method without giving a time delay to a block from which an input signal and an output signal are cut out. 3. Definition of true transfer function The unit impulse response h (n) is assumed by the following equation.

[0060]

(Equation 22)

H (n) given by the equation (24) is cut out by the block of the MN point, and the MN point DFT is performed.

[0062]

(Equation 23)

[0063] In (25), and given the extreme that the _{M → ∞ (p i z 0} k) the following equation because it is ^MN → 0.

[0064]

(Equation 24)

H (k) given by equation (26) represents a true transfer function. 4. H _B (k), calculated (m> 1) N-point length of the cross spectrum G _'0m (k) of the theoretical solution 4.1 X' ₀ and (k) Z 'and _{0 m} (k) of H _A (k) Of the response signals of the transmission system due to the unit impulse δ (n−q) at the q position in the 0 th block of the
The spectrum of the signal obtained by adding (M-1) N zeros to the response signal included in the block is represented by _F'0m (k; q), (m
> 1), this results in h (n) of h (n) in equation (24).
+ MN-q) and can be defined as the following equation by Fourier transform of the components at N points.

[0066]

(Equation 25)

By substituting equation (24),

[0068]

(Equation 26)

By using F ′ _0m (k; q),
The response signal z _{0 m} of the m blocks due to input x ₀ (n) of block 0 (m> 1) (n) , (M-1) signal subjected to N zeros additional z _{'0 m} (n ) Spectrum Z ′ _0m
(K) is given by the following equation.

[0070]

[Equation 27]

By substituting equation (28) into equation (29),

[0072]

[Equation 28]

Here, assuming that the cross spectrum between X ′ ₀ (k) and Z ′ _0m (k) is G ′ _0m (k), G ′ _0m (k)
Is given by the following equation.

[0074]

(Equation 29)

Here, the variance of the input signal x ₀ (n) is represented by σ ² _x
Placing ^{_{a, E [x 0 * (n}} ) x 0 (q)] = σ 2 x δ (n-
q) holds. Further, in the above equation, since the contribution of the power σ ² _x (k) = σ ² _x / N of the k-th frequency component is considered,

[0076]

[Equation 30]

4.2 Calculation of Cross Spectrum G _'00 (k) of X' ₀ (k) and Z _'00 (k) If m = 0 in equation (29), the input signal x of the 0th block ₀ (n)
, The response signal z ₀₀ (n) of the 0th block due to
-1) The spectrum Z _'00 (k) of the signal z' ₀₀ (n) to which N zeros have been added is represented by the following equation.

[0078]

(Equation 31)

However, F ′ ₀₀ (k; q) is different from F ′ _0m (k; q) defined by the equation (27) when m = 0 is set in the window having the length of N points. Since the Fourier transform is related to the impulse response h (n−q) (q < n <N) where n = q, it is defined by the following equation.

[0080]

(Equation 32)

Further, the following equation can be obtained by substituting the equation (24) and rearranging and modifying the equation in the same manner as the equation (28).

[0082]

[Equation 33]

By substituting equation (34) for equation (32), the following equation is obtained.

[0084]

(Equation 34)

Here, assuming that the cross spectrum of X ′ ₀ (k) and Z ′ ₀₀ (k) is G ′ ₀₀ (k), G ′ ₀₀ (k)
₀₀ (k) is given by the following equation.

[0086]

(Equation 35)

4.3 G ′ _m (k) and G ′ _0m (k),
Relationship with G ′ ₀₀ (k) Of the response of the transmission system due to the input signal x _i (n) in the _i- th block, the component in the m-th block is represented by z
It is represented by _im (n). Since the response from the input signal in the previous block is all added to the output signal y _m (n) of the m-th block, the disturbance noise in the m-th block is represented by u _m.
As (n), y _m (n) is given by the following equation.

[0088]

[Equation 36]

Therefore,

[0090]

(37)

Expression (38) is expressed in the frequency domain.

[0092]

(38)

Here, y ′ _m (n),
The spectra obtained by performing the MN-point DFT on z ′ _im (n) and u ′ _m (n) are Y ′ _m (k),
Z ′ _im (k) and U ′ _m (k). Therefore,
X _'0 (k) and Y' mean value G of the cross spectrum of _{m (k) 'm (k} ) is given by the following equation.

[0094]

[Equation 39]

[0095] Here, z _im (n) is out of the response of the transfer system by x _i (n), is a component in a block of the m blocks, the input signal x _i (n) and the disturbance noise u _m (n ) Are stationary and uncorrelated white noises, so the following equation holds.

[0096]

(Equation 40)

By substituting these two equations into equation (40), G ' _m (k) is expressed by the following equation. G ′ _m (k) = E [X ′ ₀ ^* (k) Z ′ _0m (k)] (41) G ′ _0m (k) in equation (31) and G ′ ₀₀ (k) in equation (36)
Is used, G ′ _m (k) is expressed as follows.

[0098]

[Equation 41]

4.4 Transfer Function H _B by this Estimation Method
(K) theoretical solution

[0100]

(Equation 42)

Therefore, using equation (42), equation (17)
The equation can be expressed as the following equation.

[0102]

[Equation 43]

By substituting equations (31) and (36),

[0104]

[Equation 44]

Here, [] is calculated as follows.

[0106]

[Equation 45]

Therefore, H _B (k) is rearranged as follows.

[0108]

[Equation 46]

4.5 Transfer Function H _A by Normal Estimation Method
When H _A (k) is calculated in the same manner as the theoretical solution H _B (k) of (k),

[0110]

[Equation 47]

5. Calculation result Assumed by equation (24)

[0112]

[Equation 48]

In the equation, P = 1 and C _i = 1, and ∠p _i
= P / 4 (constant), | p _i |
H _B (k) and H _A (k) when changed to 9, 0.5
Are divided into an amplitude and a phase, respectively, as shown in FIGS.
1, FIG. 12 to FIG. 15, and FIG. 16 to FIG.
(However, N = 16 and M = 3.) As a result, a normal estimation method is used for both the amplitude and the phase.
It can be seen that this estimation method H _B (k) has a smaller error from the true transfer function than H _A (k). 6. Coherence Function According to Present Invention 6.1 Introduction of Delayed Block Coherence Function Now, as shown in FIG. 3, an input signal of a target transmission system is cut out at a time window for every N points as shown in FIG. Give a series of numbers. Each of the input signals is called an i-th block, and the output signal is called an m-th block (i and m are arbitrary integers). First, focusing on only the 0th block as an input, the response to the input within the focused time window on the output side and the response leaking from the time window are evaluated. For this, the O-th
Among the response signals of the transmission system based on the input signal x ₀ (n) of the block, an output signal obtained by adding disturbance noise to components in the m-th block (m is an integer of 0 or more) is defined as y _m (n). x ₀
Signals obtained by adding (M−1) × N zero-point signals after (n) and y _m (n) are x ′ ₀ (n) and y ′ _m , respectively.
When expressed as (n), the equations (7) and (8) hold.

The N-point DFTs of x ₀ (n) and y _m (n) are represented as X ₀ (k) and Y _m (k), respectively, and x ′ ₀
The MN point DFTs of (n) and y ′ _m (n) are respectively X ′ ₀
(K) and Y ′ _m (k). D for each of these blocks
If FT is used, the ordinary coherence function γ ₀ ² (k) can be described as follows.

[0115]

[Equation 49]

Next, in obtaining the coherence function,
From the input side, the position of the time window on the output side to be analyzed is
Assuming that the score N of the FT is one block, m blocks (m>
Considering that the calculation is delayed by 0), it is defined by the following equation. (Delayed block coherence function)

[0117]

[Equation 50]

6.2 Calculation of Overall Coherence Function (Multiple Coherence Function) by Individual Delayed Block Coherence Function k of Power Spectrum of Input Signal, Output Signal and Disturbance Noise
Σ _x ² (k), σ _y ² (k), σ
_When expressed as _u ² (k), the following equations hold based on equation (10).

E [X ′ ₀ ^* (k) X ′ ₀ (k)] = N ² σ _x ² (k) E [Y ′ ₀ ^* (k) Y ′ ₀ (k)] = N ² σ _y ² (K) E [U ′ ₀ ^* (k) U ′ ₀ (k)] = N ² σ _u ² (k) (47) The output signal in the 0th block includes the time window before the 0th block. The response component of the transmission system by the input signal is included. Therefore, the power spectrum of the output of the 0th block is expressed by the following equation using H ′ _m (k) defined by the equation (9) and the spectrum U ′ ₀ (k) of the disturbance entering the output of the 0th block. Can be represented.

[0120]

(Equation 51)

As shown in FIG. 20, the input signal x (t)
Are the-(M-1) to 0-th blocks, and the output signal y (t)
Focuses on the 0th block. The signals x ' ₀ (n), x' _-1 (n), ..., x 'for each block of x (t)
_{- (M-1) (n} ) are each _{H '0 (k), H} ' 1
(K),..., H ′ _M−1 (k), and responds to the signal y ′ ₀ (n) in the zeroth block of y (t). this is,
It can be considered as a multi-input linear system as shown in FIG. The multiple relevance function of this system (multiple
The coherence function is defined by the following equation.

[0122]

(Equation 52)

By the way, when the equations (11) and (47) are substituted into the delayed block coherence function (46) considered above,

[0124]

(Equation 53)

By substituting equation (50) into equation (49), the following equation is obtained.

[0126]

(Equation 54)

When M is considered as ∞

[0128]

[Equation 55]

From equations (47) and (48),

[0130]

[Equation 56]

Therefore, the equation (51) becomes the following equation.

[0132]

[Equation 57]

When the expressions are summarized and summarized,

[0134]

[Equation 58]

7. Theoretical Solution of Coherence Function From equation (39), the average value of the power spectrum of the output signal y ′ _m (n) is given by the following equation.

[0136]

[Equation 59]

Here, from the assumption of the stationarity of the input signal and the disturbance noise,

[0138]

[Equation 60]

The following relationship was used. The delayed block coherence function γ ′ _m ² (k) obtained from the input signal of the 0-th block and the output signal of the m-th block is “the power spectrum of the observation signal y ′ _m (n) of the m-th block”, This is the ratio of the power spectrum of the response signal _z'0m (n) of the m-th block caused by the input signal x'0 (n) of the _zeroth block, and is expressed by the following equation.

[0140]

[Equation 61]

Next, when P = 1 in the equation (24) (h
(N) = a c _i p _i ^n), to derive an analytical solution of (53) multiple coherence function ^{_{γ 'm · M 2 (=}} γ' given by formula _B ² (k)). From equations (28) and (29), Z ′
The power spectrum of _0m (k), (m > 1) is represented by the following equation.

[0142]

(Equation 62)

From equations (32) and (34), Z ′
Power spectrum of ₀₀ (k) is expressed by the following equation.

[0144]

[Equation 63]

Here,

[0146]

[Equation 64]

The following equation is calculated from the equation (57).

[0148]

[Equation 65]

Here,

[0150]

[Equation 66]

When | p _i | <1, | p _i | ^{2N (M−1)} →
Since 0 (M → ∞),

[0152]

[Equation 67]

Then,

[0154]

[Equation 68]

The equations (53), (56), (58),
From equations (59) and (60), the multiple coherence function γ
_B ² (k) is given by the following equation.

[0156]

[Equation 69]

[0157] ordinary coherence function γ _A ² (k) when the length of the time window and MNτ is, γ ₀ ² (k) at N
Is replaced by MN.

[0158]

[Equation 70]

8. Calculation result Assumed by equation (24)

[0160]

[Equation 71]

Where P = 1, c _i = 1, ∠p _i = π
/ 4, and | p _i | = 0.95, and σ _u ² (k) / σ
γ when _x ² (k) is changed to 0, 1, 10, 100
The values of _B ² (k) and γ _A ² (k) are shown in FIGS.
Is shown in the graph. (However, N = 16 and M = 3.) As a result, the value of the multiple coherence is larger than that of the normal coherence over the entire frequency band. In particular, when there is no disturbance noise of σ _u ² (k) / σ _x ² (k) = 0, and under ideal conditions where the transfer system is linear,
As shown in FIG. 22, the value drops near the resonance frequency in the ordinary coherence, but it is improved well in the multiple coherence.

[0162]

As described in detail above, the present invention provides
A time delay (including a case of zero time delay) is given to the output side block with respect to the input side block. First, a cross spectrum, a transfer function, and a coherence function are obtained between the blocks to which the time delay is given. Since this method adopts a method of calculating the entire cross spectrum, transfer function, and coherence function using the method, a high-accuracy cross spectrum, transfer function, and coherence function having less resolution deviation error than the conventional method can be obtained.

[Brief description of the drawings]

FIG. 1 is a block diagram showing a configuration of a cross spectrum calculation device, a transfer function calculation device, and a coherence function calculation device of the present invention.

FIG. 2 is a diagram showing a target signal transmission system.

FIG. 3 is a schematic diagram showing division of an input output signal.

FIG. 4 is a schematic diagram showing an N-point time window and an NM-point time window.

FIG. 5 is a schematic diagram illustrating a transfer function estimation method in consideration of a time delay.

FIG. 6 is a schematic diagram showing a combined impulse response.

FIG. 7 is a schematic diagram illustrating a normal transfer function estimation method.

FIG. 8 is a graph of the amplitude of the transfer function when | p _i | = 0.95.

FIG. 9 is a graph showing the difference between the true value and the amplitude of the transfer function when | p _i | = 0.95.

FIG. 10 is a graph of the phase of the transfer function when | p _i | = 0.95.

FIG. 11 is a graph showing the difference between the true value and the phase of the transfer function when | p _i | = 0.95.

FIG. 12 is a graph of the amplitude of the transfer function when | p _i | = 0.90.

FIG. 13 is a graph showing the difference between the true value and the amplitude of the transfer function when | p _i | = 0.90.

FIG. 14 is a graph of the phase of the transfer function when | p _i | = 0.90.

FIG. 15 is a graph showing the difference between the true value and the phase of the transfer function when | p _i | = 0.90.

FIG. 16 is a graph of the amplitude of the transfer function when | p _i | = 0.50.

FIG. 17 is a graph showing the difference between the true value and the amplitude of the transfer function when | p _i | = 0.50.

FIG. 18 is a graph of the phase of the transfer function when | p _i | = 0.50.

FIG. 19 is a graph of the difference between the true value and the phase of the transfer function when | p _i | = 0.50.

FIG. 20 is a schematic diagram showing an input / output block of interest.

FIG. 21 is a schematic diagram showing a multi-input linear system.

FIG. 22 is a graph of a coherence function when σ _u ² (k) / σ _x ² (k) = 0.

FIG. 23 is a graph of a coherence function when σ _u ² (k) / σ _x ² (k) = 1.

FIG. 24 is a graph of a coherence function when σ _u ² (k) / σ _x ² (k) = 10.

FIG. 25 is a graph of a coherence function when σ _u ² (k) / σ _x ² (k) = 100.

[Explanation of symbols]

 12 first computing means 14 second computing means 22 third computing means 24 fourth computing means 32 fifth computing means 34 sixth computing means

────────────────────────────────────────────────── ─── Continued on the front page (72) Inventor Masaki Imai 3-2-9 Endo Building, 3-2-9 Yawata, Aoba-ku, Sendai, Miyagi Prefecture (72) Inventor Hiroshi Kanai 4-181-1-301 Nakayama, Aoba-ku, Aoba-ku, Sendai, Miyagi Prefecture. (72) Inventor Kenken Nakabachi 4-3-1-18 Kaigamori, Aoba-ku, Sendai, Miyagi Prefecture (56) References JP-A-4-29068 (JP, A) (58) Fields investigated (Int. Cl. ⁷ , DB G01R 27/28 G06F 17/10

Claims (3)

(57) [Claims] 1. A first time series signal and a second time series signal, each of which is divided into a plurality of blocks at each time interval corresponding to each other, are m blocks (m = 0, 1,
.., M; M is a predetermined positive integer). First arithmetic means for calculating a total of M + 1 cross spectra of blocks shifted from each other by the first block, based on the M + 1 cross spectra.
A second calculating means for calculating a cross spectrum between the time series signal and the second time series signal. 2. A first time series signal input to a predetermined signal transmission system and a second time series output from the signal transmission system, each of which is divided into a plurality of blocks at each time interval corresponding to each other. For signals, m blocks (m =
0, 1, 2,..., M; M is a predetermined positive integer).
And a fourth calculating means for obtaining a transfer function of the signal transfer system based on the M + 1 transfer functions. 3. A first time series signal input to a predetermined signal transmission system and a second time series output from the signal transmission system, each of which is divided into a plurality of blocks at each time interval corresponding to each other. For signals, m blocks (m =
0, 1, 2,..., M; M is a predetermined positive integer). Fifth calculating means for calculating a total of M + 1 coherence functions between blocks, and the signal based on the M + 1 coherence functions. 6. A coherence function calculation device comprising: a sixth calculation means for obtaining a coherence function of a transfer system.