JP2001051979A - Method and device for processing data - Google Patents

Abstract

PROBLEM TO BE SOLVED: To make a method and a device suitable for computer mounting. SOLUTION: A class (m), to which an object signal belongs, can be specified by generating plural discrete data streams based on plural functions classified by the number of times of permitting differentiation, performing correlative operation with the plural discrete data streams parallel to input data and finding a special point included in the input data on the basis of result of that correlative operation. Therefore, a discontinuous signal or signal having the special point can be efficiently analyzed, the source signal can be highly accurately approximated while using the signal, which can be described by the simple function, and various kinds of processing to the signal can be realized with a little operation quantity.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION The present invention relates to data describing various time-series data such as continuous, discontinuous, smooth (differentiable), and a combination of these signals in the natural world by approximating by function expansion. The present invention relates to a processing method and a data processing device.

[0002]

2. Description of the Related Art In the natural world, there are various kinds of time-series data such as continuous, discontinuous, smooth (differentiable), and signals composed of combinations thereof. In engineering fields such as voice, image, and communication, it is extremely important to accurately approximate and describe an observed signal by function expansion in order to accurately perform various information processing, such as noise removal and information compression. .

As a function system used for function expansion, a Fourier series is generally used. Since the expansion coefficient of the Fourier series has the physical meaning of frequency, it has been applied to many engineering fields for human audiovisual recognition. However, since the Fourier signal space is based on an exponential function that can be differentiated infinitely, discontinuous signals such as pulses and square waves and non-smooth signals such as triangular waves are efficiently handled by Fourier series. It is difficult. In addition, it has been pointed out that a structure localized on the time axis of a signal cannot be specified in the frequency domain.

[0004] As an orthogonal transformation that gives an optimal function expansion,
Although the KL (Karhunen-Loeve) transformation is known, a problem similar to the Fourier series expansion occurs for this. KL
Transformation is based on the input pattern (input signal) observed in the M-dimensional Euclidean space E ^M

[0005]

(Equation 1)

[0006] This is a transformation that projects the most efficiently to an arbitrary K-dimensional space. That is, the functional J (｛φ _k ｝ ^K _{k = 1} )

[0007]

(Equation 2)

Is defined by an orthonormal function that minimizes the functional J. Here, in the above functional J (｛φ _k ｝ ^K _{k = 1} ), “K” described in the upper right of “の” and “k = described in the lower right of“｝ ”
“1” should originally be described in the same column as shown in (Equation 2), but such description cannot be made in the specification, and therefore, the position is described as being shifted as described above. Similarly, in this specification, for example, the letter “A”
When the character "1" is described in the upper right of the character and the character "2" is to be described in the lower right of the character "A", the characters "1" and "2" cannot be expressed in the same row. "A ¹ _2"
And so on.

[0009] It is known that KL basis sets are often similar to Fourier basis sets for speech signals and image data generally observed in the natural world. In recent years,
In order to overcome the above-mentioned problem of Fourier basis sets, new function systems such as wavelets and splines have been attracting attention.

[0010]

The wavelet series is a hierarchical signal expansion in which a basis derived from one function (mother wavelet) by scaling / translation is applied. Since the resolution changes as the mother wavelet is enlarged or reduced, the position of the singular point can be extracted. However, since the expansion result greatly changes depending on the mother wavelet, it is necessary to appropriately select the mother wavelet according to the property of the target signal. At this stage, no unified method has been established for selecting an appropriate mother wavelet.

Although a spline function system composed of piecewise polynomials was originally derived for the purpose of generating a curve passing through given data points, ie, an interpolation curve, a spline function of an order suitable for the purpose is obtained. By using an expanded basis, signals that are difficult to handle with Fourier series can be described efficiently, and thus have come to be widely used in the field of function approximation. Above all, a series of functions called B-splines can be generated only by convolution of a square wave, so that they are easy to implement, and low-order B-splines can be approximated with a small amount of computation because of their small support. It also has the advantage. However, much research is still needed on how to determine the order of the spline to use.

In view of the usefulness and problems in the functional system as described above, the present invention can capture the structure of a signal localized on the time axis, and can generate a discontinuous signal or a signal having a singular point. An object of the present invention is to provide a data processing method and a data processing apparatus suitable for computer implementation, focusing on a fluency function system having a characteristic that it can be described efficiently.

[0013]

According to a first aspect of the present invention, there is provided a data processing method for generating a plurality of discrete data strings generated based on a plurality of functions classified according to the number of differentiable times. Generating, performing a correlation operation between the plurality of discrete data strings in parallel with the input data, and then determining a singular point included in the input data based on the result of the correlation operation. Features.

According to a second aspect of the present invention, in the data processing method according to the first aspect, the input data surrounded by the adjacent singular points can be differentiated based on a result of the correlation operation. It is characterized in that the signal is classified based on the number of times.

According to a third aspect of the present invention, there is provided a data processing method for converting input data into output data included in a polynomial signal space of a class (m-1) -order differentiable class m. The output data is obtained by multiplying each of the input data by a value of a predetermined conversion function corresponding to the class m and adding each multiplication result.

According to a fourth aspect of the present invention, there is provided the data processing method according to the third aspect, wherein the conversion function interpolates between discrete data using a signal included in the polynomial signal space of the class m. It is characterized in that the interpolation function used for calculating the interpolation value has biorthogonality.

According to a fifth aspect of the present invention, there is provided an input section for reading an instantaneous value of an input analog signal at a predetermined sampling interval, converting the instantaneous value into digital data, and outputting the digital data.
A data conversion unit that multiplies the plurality of digital data sequentially output from the input unit by a value of a predetermined conversion function and adds each multiplication result, and converts the analog signal into a polynomial signal of class m. It is characterized in that it is converted into digital data contained in a space.

According to a sixth aspect of the present invention, in the data processing apparatus of the fifth aspect, the conversion function is (m-
1) When interpolating between digital data output from the input unit using a signal included in a polynomial signal space of class m that can be differentiated, an interpolation function used to calculate an interpolation value And has biorthogonality.

According to a seventh aspect of the present invention, there is provided a data processing method comprising:
A data processing method for determining a class of a polynomial signal space including input data when considering a polynomial signal space of a class m that can be differentiated, and performing a predetermined thinning process on the discrete input data. A second step of performing an interpolation operation on the data after the thinning-out processing, using an interpolation function corresponding to each of a plurality of classes having different values of m; A third step of calculating an error between the interpolation data obtained in the step and the data before the decimation processing, and a class corresponding to the interpolation data having the smallest error calculated in the third step. And a fourth step of determining the class of the polynomial signal space including the input data.

The data processing method according to claim 8 is the data processing method according to claim 7, wherein a position at which the value of the class m determined in the fourth step changes is extracted as a singular point of the input data. It has 5 steps. According to a ninth aspect of the present invention, in the data processing device,
1) A data processing device for determining a class of a polynomial signal space including input data when considering a polynomial signal space of a class m that can be differentiated, and a predetermined thinning-out of discrete input data A thinning processing unit for performing processing;
An interpolation operation unit that performs an interpolation operation on the data after the decimation process is performed using an interpolation function corresponding to each of a plurality of classes having different values of m, and each obtained by the interpolation operation unit. An error calculation unit for calculating an error between the interpolation data for each class and the data before the thinning processing for each class; and a class determination for determining a class corresponding to the interpolation data having the smallest error calculated by the error calculation unit And a unit.

According to a tenth aspect of the present invention, there is provided a data processing apparatus according to the ninth aspect.
And a singularity extracting unit for extracting a position at which the value of the class m determined by the class determining unit changes as a singularity of the input data.

[0022]

DETAILED DESCRIPTION OF THE INVENTION The polynomial signal space ^m S considered by the fluency theory has its continuous differentiability m (= 1, 2,
3, ...) it is classified by, fluency digital / analog corresponding class m (D / A) function ^m [psi _k
(T) and fluency analog / digital (A / D)
It is spanned by the piecewise polynomial called the function ^m _* ^ψ _k (t). Here, in the above-mentioned A / D function ^m _* ψ _k (t), and are then subscripts upper left "[psi", "m", "[psi"
The suffix “*” at the lower left of “” should be originally written in the same column, but such description cannot be made in the specification, so the position is shifted as described above. . Similarly, in this specification, for example, the letter “A”
When the character "1" is described in the upper left of the character and the character "2" is to be described in the lower left of the character "A", the characters "1" and "2" cannot be described in the same row. ¹ ₂ A "
And so on.

The polynomial signal space ^ms is of the order (m-2)
It is composed of a signal that is (m−1) th order differentiable. Therefore, a square wave is classified as m = 1, a triangular wave is classified as m = 2, and a signal belonging to the Fourier signal space is classified as m = ∞ (FIG. 1).
reference). Also, it will be included signal also in ^m S constituted by the spline function bases.

The above D / A function and A / D function are:
Biorthogonality not orthogonal to self but orthogonal to one another

[0025]

(Equation 3)

Satisfies here,

[0027]

(Equation 4)

Where (u, v) is the inner product

[0029]

(Equation 5)

Represents the following. In _{^{addition, m ψ k (t),}} m * ψ
_k (t), and

[0031]

(Equation 6)

Is the following relation

[0033]

(Equation 7)

Is satisfied. This means that a signal that can be completely reconstructed by the D / A function of class m belongs to ^m S, and such a signal is discretized with the width h by the A / D function of class m, It matches the value of the signal sampled with the width h (see FIG. 2). However, the width h is assumed to be fixed. Therefore, if the target signal is ^m S
, The A / D function of class m works in the same way as the δ function, and the class m that minimizes the difference between the result of applying the A / D function and the original signal immediately gives the space to which the signal belongs. . As described above, the fluency theory provides a method of selecting an appropriate basis which is a problem in the spline function system and the wavelet series. Therefore, a discontinuous signal or a signal without smoothness for which Fourier series is difficult to apply can be efficiently described by applying an appropriate fluency function.

In general, analog signals

[0036]

(Equation 8)

With respect to

[0038]

(Equation 9)

X (t) is optimally approximated in the sense of the L ₂ norm

[0040]

(Equation 10)

Can be generated (see FIG. 3). A
The derivation of the expansion coefficient by the / D function can be said to be very useful in configuring a practical processing system in consideration of the fact that it is realized by an integration operation and is also strong against noise. In addition to the fact that approximation can be performed accurately, there is an advantage that the class of the target signal can be specified. The fluency function system of class m can describe a signal of class m or less that is differentiable. However, in the fluency function system, the convergence is faster for lower-order classes. By using the fluency function, the amount of calculation required for conversion is minimized. Therefore, it is possible to reduce the amount of calculation when mounting the program on a computer.

Although the fluency function system has excellent properties as described above, when applied to real data,
(1) There is a fluency function whose support (unit) is an infinite section. (2) When the support of the fluency function is discontinued and discretized in the time direction, biorthogonality is not guaranteed. (3) Continuous differentiable The problem is that it is difficult to specify the space to which the original signal belongs from discrete data whose property cannot be defined.

Next, problems in the fluency function system described above will be described. The support of the fluency function system is an infinite section, but when the processing system is implemented by a digital element such as a DSP (digital signal processor), it is necessary to terminate the support at a finite section. If support is simply discontinued, the A / D function, D
The bi-orthogonality of the / A function is not maintained, but no theoretical study has been made on the approximation error generated at that time. Therefore, first, the upper bound of the A / D function is derived, and based on this, the theoretical upper bound of the approximation error of the fluency function expansion is derived as a function of the truncation width of the support of the basis function. It became clear that the following fluency function requires a longer support width to maintain the approximation accuracy. Specifically, when sampling at 44.1 kHz for 60 minutes, in order to suppress the square error to -100 dB, 703.3 μs in class m = 2
c, 1135.0 μsec with m = 4, and class m =
7, it turned out that a support of 2020.3 μsec was required.

When the support of the fluency function is discontinued and discretized in the time direction, the shape of the biorthogonal function greatly differs from the theoretical value. Therefore, in the present invention,
Focusing on a compact D / A function, a discretized A / A function having a desired support width so that bi-orthogonality is guaranteed.
D function is newly defined. In the present invention, the newly defined discretized A / D function and D / A function are referred to as “down-sampling basis vector” and “up-sampling basis vector”, respectively, and these are collectively referred to as “fluency vector”. Shall be referred to. This newly defined fluency vector was applied to real data and compared with a conventional rate conversion method using an ideal low-pass filter. As a result, it has been clarified that the conventional method does not always perform an optimal approximation to all discrete data. In addition, it was confirmed that the class of the fluency vector performing the optimal approximation was not a single one, but changed with time. In addition, it has been found that approximation accuracy can be improved by selecting an appropriate one from a plurality of classes of fluency vectors according to the local property of the target signal.

In the fluency theory, a signal can be efficiently described by specifying a class to which a target signal belongs, and the amount of calculation required for conversion can be reduced. Thus, the signal x (t) the time which is assumed to belong to one of the polynomial signal space ^m S t _j = _j
h ′: x (t) discretized by (j = 0, 1, 2,...)
_{From j} ), the space to which x (t) belongs can be specified. The correlation between the downsampling basis vectors of each class in the input data all points, and as a result, (the smallest distance in L ₂ norm) shortest distance between the input data space belongs signal class m that gives a correlation value did. Furthermore, the method was applied to square waves, triangular waves, quadratic curves, etc., and its effectiveness was confirmed. As a result, it was confirmed that the present method was effective if the target signal was sampled to a certain degree.

In consideration of the above-mentioned problem of the fluency function system, the present invention defines a discrete fluency function that maintains biorthogonality based on the result of theoretically evaluating an error resulting from truncation of support, and optimizes the discrete fluency function for discrete data. Provide a way to determine the class. Hereinafter, the above (1)
Theoretical assessment of errors arising from discontinuation of support,
(2) Definition of discrete fluency function that maintains biorthogonality
(3) A method of determining an optimal class in discrete data will be described separately.

(1) Error resulting from discontinuation of support
In some fluency function systems, the support is defined in an infinite interval, so that in order to implement it on a computer, the support must be a finite interval.
However, when the support is simply discontinued, the biorthogonality of the A / D function and the D / A function is not maintained, and an approximation error occurs. Since the convergence speed of the low-order class fluency function system is fast, the error that occurs for the same support width (cutoff width) is smaller than that of an ideal low-pass filter that has been used in the past. No verification has been done. Therefore, the upper bound of the A / D function was derived, and from this and the upper bound of the D / A function, the upper bound of the approximation error caused by the truncation of the support was derived as a function of the support width. From this relational expression, it is possible to select the censoring width according to the permissible error, which is considered to be very useful in constructing an actual system.

Next, a polynomial signal space for L ₂ (R)
Positioning of ^m S, and signal and ^m S belonging to L ₂ (R)
The relationship between signals belonging to In fluency theory, signals are classified according to their continuous differentiability.

[0049]

[Equation 11]

B-spline basis of (m-1) order: (m = 1, 2,...)

[0051]

(Equation 12)

Let it be a polynomial signal space spanned by Here, the B spline function ^m φ _l (t) is defined as follows.

[0053]

(Equation 13)

The following three properties are given as properties of the B-spline basis shown in (Equation 12). • Support for φ _l (t) is local. That is,

[0055]

[Equation 14]

Φ _l (t) is shift invariant. That is,

[0057]

(Equation 15)

Φ _l (t) is symmetric with respect to lh.
That is,

[0059]

(Equation 16)

Here, if l = 0, the above (11)
ceremony

[0061]

[Equation 17]

Thus, φ ₀ (t) is an even function. m
When ≧ 2, ^m S becomes a (m−2) -order differentiable B-spline signal space. Further, ^m S becomes a Walsh signal space in the case of ^m = 1, m S in the case of m = ∞ is the Fourier signal space.

[0063] The s and analog signals belonging to the ^m S, time this

[0064]

(Equation 18)

Assuming that the sampling is made discrete, the sampling basis of ^m S is given by the following equation:

[0066]

[Equation 19]

Satisfy

[0068]

(Equation 20)

Is given by The expansion coefficient s
(T _k ) is a basis that is biorthogonal to the sampling basis

[0070]

(Equation 21)

Is given as follows.

[0072]

(Equation 22)

[0073] The expansion of the signal by the above-described (13) and (14) below is referred to as bi-orthogonal expansion in ^m S. These bases for ^m S are determined as follows:

[0074]

(Equation 23)

Here,

[0076]

(Equation 24)

Is as follows. The above-mentioned ^m _f B (f) is

[0078]

(Equation 25)

The Fourier transform of ^m _f G (f) ^-1 is

[0080]

(Equation 26)

This is a Fourier transform. Hereafter, ^m ψ _k (t)
Is the D / A function,

[0082]

[Equation 27]

[0083] The D / A base, ^m _* ψ _k a (t) A / D function,

[0084]

[Equation 28]

Is called an A / D basis. A / D function ^m _* { ₀ (t) and D / A function ^m } for classes m = 1 and m = 2
₀ (t) is shown in FIG. 4A is an A / D function of m = 1, FIG. 4B is a D / A function of m = 1, and FIG.
= 2, and FIG. 4D shows the D / A function of m = 2. Also, A / of m = 3 and m = ∞
FIG. 5 shows the D function and the D / A function. FIG. 5A shows m = 3.
A / D function of FIG. 5, FIG. 5B shows a D / A function of m = 3, and FIG.
(C) is an A / D function of m = ∞, and FIG. 5 (d) is a D / m function of m = ∞.
/ A functions are shown.

Next, the approximation error due to the suspension of the support will be described. The signal space L ₂ (R) is ^m S and its orthogonal space.

[0087]

(Equation 29)

Since it is composed of the direct sum of

[0089]

[Equation 30]

Is

[0091]

(Equation 31)

And

[0093]

(Equation 32)

The sum of

[0095]

[Equation 33]

S _o is a least square approximation of x to ^m S. Since the real signal is usually observed in a finite time, x

[0097]

(Equation 34)

Then, it is defined by a finite time [0, T]. Here, N is a positive integer. Hereinafter, approximation of x defined by the above equation (22) will be considered. An A / D function is applied to x, and the result is set to v _k .

[0099]

(Equation 35)

The equation (21) is substituted for x in the equation (23), and the following relation is obtained.

[0101]

[Equation 36]

By using

[0103]

(37)

Is derived. Equation (25) above gives:

[0105]

(38)

Least square approximation of

[0107]

[Equation 39]

Where

[0109]

(Equation 40)

This means that it can be represented by

[0111]

[Equation 41]

V _k : (k = 0, ± 1, ± 2...
..) Are all subspaces of the Hilbert space. Here, in the present specification, since a vector cannot be described in bold characters, by writing an arrow “→” to the right of a character representing a vector amount such as “vector v →” described above, Is a vector quantity.

In the Hilbert space, the inner product is

[0114]

(Equation 42)

The following norm is guaranteed.

[0116]

[Equation 43]

Since the equations (23) and (26) are expressed in the form of an intervalless integral and an infinite series, it is necessary to make these integral sections and series finite in order to implement them on a computer. Therefore, support for H> 0 is [-Hh, H
h] and the D / A function and A / D function

[0118]

[Equation 44]

[0119] Equations (30) and (3)
Substituting equation 1) into equations (23) and (26), t-kh =
Assuming that τ, the approximation by the D / A function and the A / D function in a finite section is as follows.

[0120]

[Equation 45]

Therefore, the least squares approximation s _{o of} x and the approximation when the support is truncated

[0122]

[Equation 46]

The relative error with respect to can be expressed as the following square norm.

[0124]

[Equation 47]

Next,

[0126]

[Equation 48]

The relative error between the least squares approximation and the approximation when the support is discontinued will be described. As a preparation, the upper bound of the D / A function and the A / D function will be described first. The upper bound of the D / A function is shown in Lemma 1 below. (Lemma 1): When m = 1,2

[0128]

[Equation 49]

[0129] 2. When m ≧ 3, ^m satisfying the following relationship
U and ^{m u (0 <m U <} ∞, 0 <m u <1) is present.

[0130]

[Equation 50]

If m = 1 or m = 2, the support for ^mサ ポ ー ト₀ is finite and need not be truncated. If m ≧ 3, |
^{m ψ} _k (t) | is | t | decreases exponentially with increasing. Next, as preparation for obtaining the upper bound of the A / D function, three lemmas are shown below.

(Lemma 2): For any integer l,
The following relationship holds:

[0133]

(Equation 51)

The above Lemma 2 is proved as follows. ^m _f B (f) can be transformed as follows.

[0135]

(Equation 52)

For any integer l,

[0137]

(Equation 53)

Is obtained. Here, q = 1 + p. (3
From equations 9) and (40),

[0139]

(Equation 54)

Is derived. This lemma is ^m _f B
(F) means a periodic function with a period of 1 / h. (Lemma 3):

[0141]

[Equation 55]

However,

[0143]

[Equation 56]

When given to have a [0144], A / D function ^m _* ^ψ ₀ (t) is a linear combination of B-spline function

[0145]

[Equation 57]

Can be represented by Lemma 3 mentioned above
Is proved as follows. Equation (13) is replaced by (14)
Substituting as ^m _* [psi ₀ equation, ^m _* ^ψ ₀ (t) can be expressed in the form of a linear combination of B splines.

[0147]

[Equation 58]

[0148] The _m d _p

[0149]

[Equation 59]

When [0150], _m d _p is as follows from the convolution theorem in the discrete Fourier transform.

[0151]

[Equation 60]

By applying Lemma 2 and Eq. (39) to Eq. (20), ^m _f G (f) becomes

[0153]

[Equation 61]

Becomes

[0155]

(Equation 62)

Is obtained. Furthermore, from equation (18),

[0157]

[Equation 63]

Is the inverse Fourier transform of 1 / ^2m _f B (f), so _m d _p is

[0159]

[Equation 64]

And

[0161]

[Equation 65]

[0162] Convolution of

[0163]

[Equation 66]

Is given by From equation (9),

[0165]

[Equation 67]

Since ^m φ ₀ (kh) = 0 at the time of (42), (42)
An expression is derived. Next, the upper bound of the A / D function is derived. (Lemma 4): When m = 1,

[0167]

[Equation 68]

[0168] 2. When m ≧ 2, ^m satisfying the following relationship
There are W and ^m w (0 < ^m W <∞, 0 < ^m w <1).

[0169]

[Equation 69]

The above Lemma 4 is proved as follows. When m = 1, from equations (13) and (14),

[0171]

[Equation 70]

Is obtained. Therefore, from equation (35),
Equation (49) is derived. When m ≧ 2, (42)
From the formula,

[0173]

[Equation 71]

Is obtained. _Am

[0175]

[Equation 72]

And the equation (9)

[0177]

[Equation 73]

By using the B spline function ^m φ
₀ (t) is

[0179]

[Equation 74]

In order to satisfy

[0181]

[Equation 75]

Is obtained. Also, (0 <^2mb <1, 0 <
^2mB <∞)^2mb, and ^2mFor B,
^2mβ_pIs

[0183]

[Equation 76]

Satisfies the following. Here, ² mB is normalized by h. In this case, | ^m d _p | of the upper bound,

[0185]

[Equation 77]

Is given by ^{where m} D and ^m d are

[0187]

[Equation 78]

Then, the upper bound is

[0189]

[Expression 79]

Is obtained. ^{Here, 0 <m d <1,0 <} m D
<∞. Since the B-spline function is symmetric according to equation (12), ^m _* ψ is obtained from equations (13) and (14).
It is derived that ₀ (t) is also symmetric.

[0191]

[Equation 80]

Therefore, to show that the equation (50) holds, it is sufficient to consider only t ≧ 0. (54) and (55) from the ^{_{equation, | m * ψ 0 (t}} )
The upper bound of | is given by

[0193]

[Equation 81]

[0194] Further, 0 <a relation ^{m d <1 <m d -1} , (62) formula,

[0195]

(Equation 82)

This can be modified as follows. ^m W and ^m w

[0197]

[Equation 83]

[0198] and is ^{_{defined, | m * ψ 0 (t}} ) | of the upper bound,

[0199]

[Equation 84]

Is given by Therefore,

[0201]

[Equation 85]

Equation (50) holds true. From Lemma 4, the case of ^m = 1, m _* ψ ₀ support need not abort because it is finite. If m ≧ 2, | ^m _* ψ
_k (t) | are | t | continue to decrease exponentially with the _{^{increase, | m ψ k (t)}} | Since the upper bound of also decreases exponentially, if you take the support width greater It is expected that the error caused by the truncation will also decrease rapidly.

Constant for each class of 2 ≧ m ≧ 10
^{m U, m u, m W} , the result of obtaining ^m w shown in FIG. From this result, it can be confirmed that for m ≧ 3, the convergence speed of the lower order class is faster than that of the higher order class. Next, using the above-mentioned lemma, the upper bound of the relative approximation error caused by the cutoff of the support is derived.

(Theorem 1): When m ≧ 2,

[0205]

[Equation 86]

Is established. Theorem 1 is proved as follows. From equations (22), (23) and (32),

[0207]

[Equation 87]

Equation (67) is the Cauchy-Schwartz inequality

[0209]

[Equation 88]

This is derived by applying the formula to the equation (67). Therefore,

[0211]

[Equation 89]

The following is obtained. Finally, assuming that T = (N-1) h and substituting this into Expression (72), Expression (65) is derived. This allows

[0213]

[Equation 90]

It has been found that the upper bound can be expressed as a function of the truncation width H. Next, using Theorem 1,

[0215]

[Equation 91]

The upper bound is derived. (Theorem 2): When m ≧ 2,

[0219]

(Equation 92)

The upper bound E (H) is given by the following equation.

[0219]

[Equation 93]

Theorem 2 is proved as follows.
From equations (26) and (33),

[0221]

[Equation 94]

Can be expressed as follows.

[0223]

[Equation 95]

Here, the Helder inequality

[0225]

[Equation 96]

By applying, the expression (75) becomes as follows.

[0227]

(97)

Further, inequalities in the integral of Helder

[0229]

[Equation 98]

By applying to the third term of equation (78),

[0231]

[Equation 99]

The following is obtained. On the other hand, the expression (80)

[0233]

[Equation 100]

Can be modified as follows.

[0235]

[Equation 101]

According to Lemma 1,

[0237]

[Equation 102]

The upper bound is

[0239]

[Equation 103]

Is given by Further, from equations (22) and (23) and _{Lemma 4} , the upper bound of || v → || ² _l2 is obtained as follows.

[0241]

[Equation 104]

By applying equations (82), (85) and (87) to equation (80) and using Theorem 1,

[0243]

[Equation 105]

The error caused by the discontinuation of the support is obtained by the equation (73). From Theorem 2, the necessary support width h · H can be obtained by Expression (89) when the allowable error is E (H). FIG. 7 shows that T = 360
0 sec, and the sampling interval is h = 44.1 kHz = 22.7 μsec corresponding to CD audio data.
(2)
0 log ₁₀ E (H)). From FIG.
To reduce the truncation error to -100 dB, m = 2 and H
H ≧ 50 when m ≧ 4 and H ≧ 89 when m = 7.
Therefore, the support width of the A / D function and the D / A function is the minimum. When m = 2, h · H = 22.7 × 31 = 703.3 μ
sec, h = H = 22.7 × 50 = 11135.0 when m = 4
μsec, when m = 7, h · H = 22.7 × 89 = 2020.3
μsec. As the class becomes higher, the convergence speed of the A / D function and the D / A function becomes slower. Therefore, in order to maintain the approximation accuracy, it is necessary to increase the support width of the higher order class.

(2) Discrete fluency relation maintaining biorthogonality
Definition of Numbers In order to construct a real system, it is necessary to support fluency functions in finite intervals. Therefore, the approximation error generated when the function is set to a finite interval by simply terminating the support has been examined as described above. Next, instead of simply censoring the function, the A / D function and D /
A / D functions are finite so that biorthogonality is established between A functions, and newly discretized A / D functions and D / A functions are defined.

A / D even when the support is a finite section
A / D function and D / A function newly discretized are defined so that biorthogonality is established between the function and the D / A function. For this purpose, a previously derived D / A function is used. Hereinafter, this function will be referred to as a C type, and the function system defined in the above description will be referred to as a B type to distinguish them from each other. An overview of the C type fluency function system will be given below, and the reason for using the C type will also be described. Hereinafter, unless otherwise noted, the fluency function system means the C type.

The D / A function ^m _c ψ _k (t) of the class m which has been already derived is a basis extending the signal space ^m S, and is expressed by the linear combination of the (m−1) -order spline function as follows: Is defined in the form.

[0248]

[Equation 106]

Since ^m _c ψ _l (t) is expressed by a linear combination of B-spline functions, similar to the B-type D / A function, the shift invariance shown in the following equation (91) and the following The symmetry with respect to lh shown in the equation (92) is established.

[0250]

[Equation 107]

When the shape is m = 1 and m = 2, the D / A functions of the C type and the B type are the same, and are respectively a square wave and a triangular wave. The only difference is that support for C-type D / A functions is compact for 3 <m <∞. b _m (j) ｛j = −m + 1,..., m−
1｝ is

[0252]

[Equation 108]

Then, it is derived as a coefficient sequence satisfying the following relationship.

[0254]

(Equation 109)

A _m (j) is

[0256]

[Equation 110]

When defined, this is a coefficient sequence placed so that equations (96) and (97) are equivalent. b ₂ , b
_3, with respect to b _4, respectively obtained as follows.

[0258]

(Equation 111)

The higher the class ^m _c ψ ₀ (t), the greater the support.

[0260]

[Equation 112]

Is a sinc function with infinite interval support,
That is,

[0262]

[Equation 113]

Is obtained. This point also matches the B type. h
FIGS. 8A and 8B show C-type D / A functions for m = 3 and m = 4 when = 1.
Next, the reason for using the C-type fluency function system in the data processing method of the present invention will be described. Since the memory resources of the computer are finite, it is necessary to select a fluency function system to be implemented. The data processing method of the present invention pays particular attention to a low-order class fluency function as a function system capable of efficiently describing a signal.

A differentiable signal belonging to the space ^m S can be described by a fluency function of class m or higher, but the difference in smoothness decreases as the order increases. Therefore, in expressing the smoothness, it is sufficiently possible to use a fluency function of about m = 3 and m = 4, and this cannot be differentiated by using the fluency functions of m = 1 and m = 2 together. It is considered that signals, sharp signals, and smooth signals can be widely expressed. On the other hand, as for the C-type D / A function, the lower the class, the smaller the support width thereof, so that it is possible to simultaneously reduce the amount of computation and memory required for approximation, which is extremely excellent in implementation. Has the property of For the above reasons, the present invention uses a C-type fluency function system.

As an application example utilizing the characteristics of the C type fluency D / A function of being compact, unequal interpolation can be considered. When interpolating a discrete signal, the data points are usually convolved with some sampling function (a sinc function is often used in the past). Since the sampling function has the property of intersecting the time axis at intervals of the discretization width, interpolation between data points can be performed by convolving the data points with the sampling function. However, in the case of signals discretized at unequal intervals, smooth interpolation cannot be performed simply by extending the point where the sampling function intersects the time axis with the sample points and convolving it with the data points. . This is because the time information of the data point is not reflected. That is, it is necessary to consider that a data point that is temporally distant from the position to be interpolated has a smaller effect than a data point that is located closer. Therefore, for data points that have been discretized at non-uniform intervals, the values when they are rearranged at even intervals are predicted, and these values are referred to as pseudo data points (hereinafter referred to as “pseudo sample points”). It is conceivable to perform the interpolation as the above. At this time, since the number of pseudo sample points to be obtained for a sampling function having a large support width increases, a C-type fluency D / A function having compact support is suitable.

X → t _j = jh ′: (j = 0, 1,...)
.., D-1 order discrete period signal discretized by d-1)

[0267]

[Equation 114]

[0268] If this is the case, this is converted into n discrete D / A bases.

[0269]

[Equation 115]

Let us consider approximation. That is, r =
when the d / n, consider ^m [psi ₀ sampling at intervals of h = h '· r. Therefore, (Equation 115)
Is a d-order D / A function discretely generated at intervals of h ′

[0271]

[Equation 116]

Is obtained. Also, since x → is a periodic signal, the D / A base shown in (Equation 115) is represented by r
It becomes a column vector of degree d circulated by k.

[0273]

[Formula 117]

Here, Ω is

[0275]

[Equation 118]

Is as follows. A matrix in which the D / A base shown in (Equation 115) forms a column vector

[0277]

[Equation 119]

[0278]

[0279]

[Equation 120]

Approximating x → is equivalent to the following error function

[0281]

[Equation 121]

Expansion coefficients for minimizing

[0283]

[Equation 122]

[0284] This is the problem of finding FIG. 9 shows the matrix expressed by (Equation 120) and its function. In FIG. 9, a vector s → is a least-squares approximation of x → by a matrix expressed by (Equation 120). As shown in FIG. 9, the matrix expressed by (Equation 120) expands the expansion coefficient sequence of order n by r (= d / n) times, and is called an up-sampling basis matrix of class m. 120), the discretized D / A bases constituting the columns in the matrix shown in (120)
/ A basis) is referred to as an up-sampling basis vector. Also, the ratio r at which the D / A base upsamples as shown in (Equation 115) is referred to as an approximate rate. On the other hand, a down-sampled basis matrix of approximate rate r

[0285]

[Equation 123]

Is to generate an expansion base that minimizes the above equation (103). Using the down-sampling basis matrix shown in (Equation 123), the error function represented by Expression (103) is rewritten,

[0287]

[Equation 124]

And this is

[0289]

[Equation 125]

The following relationship is satisfied. Here, || · || _F means Frobenius norm. If x → is an arbitrarily selected normalized vector, the optimal down-sampling basis matrix will minimize the following equation:

[0291]

[Equation 126]

The down-sampling basis matrix shown in (Equation 123) is a pseudo-inverse of the matrix shown in (Equation 120).

[0293]

[Equation 127]

Can be easily obtained by solving The first line

[0295]

[Equation 128]

Then, all other row vectors are

[0297]

[Equation 129]

Can be represented in a cyclic form.

[0299]

[Equation 130]

Therefore, the pseudo inverse matrix expressed by (Equation 127) is

[0301]

[Equation 131]

The following is obtained. FIG. 10 shows the down-sampling basis matrix and its function. FIG. 10 shows how the down-sampling basis matrix shown in (Expression 131) generates an expansion coefficient of order n from a signal of order d. FIG. 11 shows the characteristics of the down-sampled base vectors of m = 2 and m = 3 obtained as a result of actually obtaining the pseudo inverse matrix. FIG.
FIG. 11A shows the case where m = 2, and FIG. 11B shows the case where m = 3. The characteristics shown in FIG. 11 are obtained when h = 1. FIG. 11 shows a state in which the shape of the down-sampled basis vector changes little by little depending on the order d of the target signal and the number n of basis vectors. If n and d are made sufficiently large, the characteristics of the down-sampled basis vectors converge.

Next, the above-described down-sampling basis vector and up-sampling basis vector (hereinafter, referred to as
These are referred to as a "fluency vector" in a pair.)
An approximation result when a fluency vector is applied to real data will be described.

Since it is inefficient to derive the fluency vector according to the size of the data to be approximated, a signal of order d (≦ d ′) is approximated at a rate r to the order d ′ of the data to be approximated. Assume that: For example, d = 51
Assuming that 2, r = 4, the up-sampling basis matrix is composed of up-sampling basis vectors of d / r = 128, and the corresponding pseudo-inverse matrix is obtained to obtain the down-sampling basis matrix.

The down-sampling basis matrix and the up-sampling basis matrix obtained in this way may be applied to the target signal for each d-order block as it is, and the approximation may be made. The most symmetric downsampling basis vector from the downsampling basis matrix to prevent

[0306]

(Equation 132)

[0307] One is selected, and expansion coefficients are obtained while shifting this by r (see FIG. 12). The expansion coefficients thus obtained are expanded by embedding r-1 zeros between the respective elements and performing convolution with the up-sampling base vector while shifting by r. As pre-processing, zero average was taken so that the DC component of the symmetric signal became zero.

Next, two CDs are set based on the above-described method.
The result of approximation of audio data (44.1 kHz, 16 bits, monaural, 441,000 samples (for 10 seconds)) will be described. The target songs are as follows. Song 1. Donald Fagen. Trans-island Skyway. JSBach. Cello Sonata No. 3 in G major. Misc
Performed by ha Maisky and Martha Argerich.

The above-mentioned song 1 was selected as a signal containing a large amount of high frequency components, and song 2 was selected as a representative signal containing a large number of smooth components. The parameters of the fluency vector used for the approximation are as follows. Class: m = 1, 2, 3, ∞ Order: d = 512 (only when m = ∞, d = 512 and 24000) Approximation rate: r = 2, 4 Calculation accuracy: double precision

As described above, the data processing method of the present invention pays particular attention to low-order classes in the fluency function system. Therefore, in order to verify the usefulness of the low-order class, as a representative of the high-order class, approximation was made for a fluency vector of class m = ∞, which is applied as an ideal low-pass filter in the conventional rate conversion method, and compared. For the purpose of comparison with the conventional method, the usual downsampling / upsampling operation was applied to the class m = ∞. The method is as follows. First, an ideal low-pass filter having infinite interval support (fluency A / D function and D / A function of class m = ∞) is finite by applying a Gaussian window.
Next, after applying the finite low-pass filter to the target signal, 1 / r
Thin out. Finally, up-sampling is performed r times by convolving the thinned result and the finite low-pass filter (see FIG. 13).

FIG. 14 shows the result of approximation of music 1 and music 2. FIG. 14 shows the SN ratio for the target signal in each class and each approximate rate. The SN ratio was calculated based on the following equation.

[0312]

[Equation 133]

As shown in FIG. 14, the conventional method (m = ∞) does not always give a good approximation. Since the fluency function of m = ∞ has infinite interval support, especially d = 512
It can be confirmed that the order of the fluency function needs to be on the order of at least about 10 ⁴ in order to obtain an approximation accuracy equivalent to that of the low-order class at the time of (1). As an overall tendency, the approximation accuracy in the music 1 does not show much difference from the fluency vector of each class, but in the music 2, except for m = ∞, the SN ratio is higher in the higher class. This is considered to be because the music 1 contains many high frequency components. However, considering that the signal properties change with time, it cannot be concluded that a fluency vector of class m = 3 gives an optimal approximation in all sections.

Next, a description will be given of the result of dividing music 2 having a clear class difference into small sections and performing similar approximation. FIG. 15 shows that song 2 is composed of 151-350
0 sample (section 1) and 20351 to 20550
This shows the result of approximating r = 2 for the second 200 samples (section 2) and all sections. Also, FIG. 16 shows that r = 2 for the 200 samples from the 151st to 350th (section 1), the 200 samples from the 20351st to 20550th (section 2), and all sections.
4 shows the result of approximation. As shown in FIGS. 15 and 16, in both of the approximation rates r = 2 and r = 4, the class m = 1 gives a good approximation in the section 1, whereas the class m = 3 in the section 2 Gives a good approximation. Therefore, it can be understood that the fluency vector that optimally approximates the target signal changes with time.

FIG. 17 shows that the approximate rate r = 2 and the class m =
It is a figure which shows the detail of the approximation result with respect to area | region 1 by the fluency function of 1, 2, 3. FIG. 17B shows the original waveform,
17 (c), 17 (d), and 17 (c) show the square errors obtained by approximating the original waveform (b) with fluency vectors of classes m = 1, 2, and 3 for each data point.
(E). The class giving the smallest error is shown in FIG. M = 1 for red, m = for green
2, and blue represent m = 3, respectively. Note that, since colors cannot be expressed in the specification, in FIG. 17 and FIGS. 18, 27, 28, and 29 described below, red is hatched by upward diagonal lines, and green is represented by diagonally downward diagonal lines. Hatching and blue are represented by hatching with horizontal lines.

FIG. 18 shows that the approximate rate r = 2 and the class m =
It is a figure which shows the detailed approximation result with respect to the area 2 by the fluency function of 1, 2, and 3. FIG. 18B shows the original waveform,
18 (c), 18 (d), and 18 (e) show square errors obtained by approximating the original waveform with fluency vectors of classes m = 1, 2, and 3 for each data point. The class giving the smallest error is shown in FIG.
(A). Red represents m = 1, green represents m = 2, and blue represents m = 3.

As shown in FIG. 17 and FIG.
It can be seen that the class of the optimal fluency vector changes with time in both the section 2 and the section 2. In section 1, the rate at which the optimal fluency vector is class m = 1 is high, and in section 2, the rate at which the optimal fluency vector is class m = 3 is high. In FIG. 15 described above, class m = 1 gives a good approximation in section 1, whereas class m = 3 gives a good approximation in section 2. From these results, it can be understood that the use of the optimum fluency vector in accordance with the change in the property of the signal can improve the approximation accuracy as compared with the conventional method in which the approximation is performed with m = ∞ single class.
FIG. 19 shows the SN ratio when the one with the smallest error is selected (optimum approximation) from the approximate values actually obtained by the fluency vectors of each class. For comparison, the result of approximation for each single class is also described.

As described above, the biorthogonality is maintained.
As a result of defining a discretized D / A function and a discretized A / D function and applying the newly defined fluency vector, which is a finite and discretized fluency function, to real data, an optimal approximation is obtained. It became clear that the given fluency vector (optimum fluency vector) changed. This indicates that the fluency vector has the ability to capture changes in the local properties of the signal. By selecting the fluency vector that gives the best approximation according to the change in the signal, m = 1 unity This means that the approximation accuracy is improved compared to the conventional method of performing approximation by class.

(3) Determination of optimal class in discrete data
Constant Method Next, a method for identifying a signal space belongs discrete data and the relationship between the applied limitations and singularities be described. As described above, there are various advantages in specifying the signal space to which the target signal belongs. First, the polynomial signal space ^m S
For the signal x (t) belonging to, there is an advantage that it is guaranteed that the signal can be efficiently described by using the fluency function of the class m. Secondly, the lower the order of the fluency function system (C type) is, the smaller the class m is for the signal x (t).
The use of the fluency function has the advantage of minimizing the amount of computation required for conversion. Therefore,
By specifying the class to which the signal belongs, it is possible to reduce the amount of calculation in computer implementation. Thirdly, there is an advantage that Gibbs phenomena such as overshoot and undershoot can be mitigated by specifying an indifferentiable point.

[0320] polynomial signal space ^m times the assumed signal x (t) as belonging to any one of S t _j = jh ':
X (t _j ) discretized by (j = 0, 1, 2,...)
If it is possible to specify the space to which x (t) belongs,
The effect provided by the fluency function system as described above can also be expected for discrete data.

Next, as a preparation for explaining a method of determining an optimal class in discrete data, a “class determination scale” which is a scale for specifying a space to which a signal belongs in a continuous system is defined. Equation (6) described above

[0322]

[Equation 134]

And equation (7)

[0324]

[Equation 135]

Thus, the A / D function of m in a continuous system is
Coefficient value v (t) obtained by discretizing analog signal x (t) with width h
_k ) is generated. At this time, h → 0 (only in this case,
If “→” described to the right of “h” does not represent the meaning of a vector), the above equation converges on the signal s (t),
Without using the D / A function projection to ^m S of x (t) is obtained. Therefore,

[0326]

[Equation 136]

[0327]

[0328]

137

[0329] By calculating the above, the space class to which x (t) belongs can be specified. Hereinafter, the expression (108) is referred to as a “class determination scale”. Next, based on the class determined measure described above, polynomial signal space ^m any time the assumed signal x (t) as belonging to one t _j = jh of S ':
X → = x (t _j ) discretized by (j = 0, 1, 2,...)
, A method of specifying the space to which x (t) belongs will be described. Here, a point where the first-order differential coefficient does not exist in the original signal x (t) is defined as a singular point, and a point closest to the singular point of the original signal is defined as a singular point of the discrete signal.

If the class determination measure in the continuous system is valid for the discrete system, it is possible to specify the class to which the discrete signal belongs. Therefore, the discrete class decision measure for the d-th discrete signal is

[0331]

138

The definition is redefined. here,

[0333]

139

A down-sampling basis vector for sampling a d-order discrete signal discretized by the width h 'at h. At this time, the approximation rate is r = h / h ', and the number of basis vectors is n = d / r. In order to assert the validity of the class determination measure, for a signal of interest belonging to ^m S,

[0335]

[Equation 140]

It suffices that m be obtained as the lowest-sampled down-sampled base vector class that gives Even if this is not the case, practicality can be tolerated unless a downsampling basis vector of class m or less is selected. This is because the nature of Fluency functions or classes m can describe signals belonging to ^m S.

As the target signal,

[0338]

[Equation 141]

As examples of the signals belonging to the above, a zero-order function (DC wave), a linear function, a quadratic function, and a sine wave of which the differentiability is known are selected. ¹ with respect to the S and ² S, since it is allowed to have a singular point, prepared a signal having no signal and singularity with singularities for 0-order function and a linear function, consistent For the quadratic function and sine wave, the shape was adjusted to this. These are summarized in FIG. The pattern 1 shown in FIG.
Pattern 2 is a pattern in the case where the next function does not have a singular point. Pattern 2 is a pattern in the case where the zero-order function and the linear function have the singular point. 21 shows the signal of pattern 1 shown in FIG. 20, and FIG. 22 shows the signal of pattern 2 shown in FIG.

Next, the effectiveness of the class determination scale will be described. 20. For each of the functions shown in FIG. 20, generate 1024 discrete data for each function, apply classes m = 1 to m = 4 to each discrete data, and change the approximate rate r. To change h. At this time,
Apply a down-sampled basis vector to all points so that the results do not vary. That is,

[0341]

[Equation 142]

[0342]

[0343]

[Equation 143]

The validity of the class determination scale is verified. Here, Ω is as shown in the above equation (101). Note that only the central 512 points were used for evaluation so that discontinuous points at the boundary did not affect. The transition of the class determination scale D (h) obtained under the above conditions with respect to h is shown in FIGS. FIG. 23A shows the transition of the class determination scale in the zero-order function (zero-order function not including a singular point) shown in the above-described pattern 1, and FIG.
3 (b) shows the transition of the class determination measure in the linear function (linear function not including a singular point) shown in the above-described pattern 1. FIG. 24A shows the transition of the class determination scale in the quadratic function shown in Pattern 1, and FIG. 24B shows the transition of the class determination scale in the sine wave shown in Pattern 1. ing. FIG. 25
(A) shows the transition of the class determination scale in the zero-order function (zero-order function including the singular point) shown in the above-described pattern 2, and FIG.
The transition of the class determination scale in a quadratic function (a linear function including a singular point) is shown. FIG. 26A shows the transition of the class determination scale in the quadratic function shown in the pattern 2, and FIG. 26B shows the transition of the class determination scale in the sine wave shown in the pattern 2. ing.

Considering that an A / D function of class m or more can describe a signal of class m, a large difference should appear between the class determination measures given by the down-sampled basis vectors of class m-1 and class m. . FIG.
As shown in (a), no difference appears in the class determination scale between the classes for the zero-order function, but since the value is small compared to the order of the class determination scale in the other target signals, The fluency vector of the class also accurately approximates the target signal. Therefore, the zero-order function (DC wave) is identified to belong to ¹ S.

Also, as shown in FIG. 23 (b), for the linear function, the smaller the h, the larger the difference in the class determination scale becomes. 2 is selected. Thus, the primary function can be identified to belong to the ² S. Similarly, FIGS. 24 (a) and 26 (a)
As shown in, it is always selected class m ≧ 3 as downsampling basis vector to minimize the class determining scale even in a quadratic function, which from the space belongs quadratic function can be specified as ³ S. Further, as shown in FIGS. 24 (b) and 26 (b), although the difference between the classes is not so clear as to the quadratic function with respect to the sine wave, the classes m = 3 and m = 4 determine the class determination scale most. Since it is small, it can be specified that the sine wave is a smooth signal. Therefore, it is considered that a class determination method based on a class determination measure is effective for a signal that does not include a singular point.

On the other hand, as shown in FIGS. 25 (a) and 25 (b), when a singular point is included,
The order of the next function is larger than that of the class determination scale in the case where no singular point is included, and there is no difference. Therefore, when the target signal includes a singular point, the class determination scale cannot be said to be effective, and the effectiveness of the class determination scale in the discrete system is confirmed only for a signal that does not include the singular point.

Next, a method of specifying a class of a discrete signal based on the effectiveness of the above-described class determination measure will be described. First, in order to more accurately capture the target signal, an approximate rate r = 2 that minimizes h is used. Then, a down-sampling basis vector is applied to all the data points so that the value of the class determination scale does not vary for each point, and the class determination scale in the above equation (110) is obtained. This is performed by downsampling basis vectors in each class, to compare the class determination measure, the class of downsampling basis vectors to minimize the class determination scales and classes in t _k.

The results obtained by applying this method are shown in FIGS.
Shown in FIG. 27 shows a result for the zero-order function including the singular point shown in FIG. FIG. 27A shows a zero-order function as a target signal, and FIG.
7A shows a result of specifying a class in the target signal shown in FIG. FIG. 28 is a view similar to FIG.
It is a result for the linear function including the singular point shown in (b). FIG. 28A shows a linear function serving as a target signal, and FIG. 28B shows a result of specifying a class in the target signal shown in FIG. 28A. Also, FIG.
This is a result for a function that changes from a zero-order function (DC component) to a linear function (linear component). FIG. 29A shows a function that changes from a zero-order function as a target signal to a linear function, and FIG. 29B shows a result of specifying a class in the target signal shown in FIG. I have.

As shown in FIGS. 27 to 29, it is specified that the DC component belongs to ¹ S and the straight line belongs to ² S in places other than the singular point, but the boundary where the singular point exists is
^It is determined to belong to 3S. It is considered that the coefficient value and the class determination scale fluctuate around the singular point. In this case, the range of the influence of the singular point on the peripheral points increases as h increases, but by making h sufficiently small, the range of influence also converges. The class determination measure does not give a valid value for the singular point or its surrounding points, but if h is sufficiently small, the singular point can be extracted as a part where the class changes.

Next, a case where the above-described class determination method is applied to actual data will be described. Since it is inefficient to derive the fluency vector according to the size of the target data, the order d ′ of the approximate target data is
Assuming that a signal of (≦ d ′) is approximated at a rate r = 2, a down-sampling basis matrix is obtained. Then, one down-sampling basis vector having the highest symmetry is selected from the down-sampling basis matrix to prevent block noise from occurring, and the down-sampling basis vector is operated while being shifted by one. This is shown in FIG. To get the expansion coefficient,
As shown in FIG. 30 (a), while shifting the down-sampled basis vector by r according to the approximate rate, x →
Find the inner product with When a class determination measure is obtained, x is shifted by one as shown in FIG.
Apply the downsampling vector to all points of →.

Next, two CDs are set based on the above-described method.
The result of approximation of audio data (44.1 kHz, 16 bits, monaural, 441,000 samples (for 10 seconds)) will be described. The tune used as the approximation target signal is described in “(2) Definition of discrete fluency function maintaining biorthogonality”.
And the following are used.

Music 1. Donald Fagen. Trans-island Skyw
ay. Song 2. JSBach. Cello Sonata No. 3 in G major. Misc
Performed by ha Maisky and Martha Argerich. Song 1 described above was selected as a signal containing a large amount of high frequencies, and song 2 was selected as a signal containing a large amount of smooth components. The parameters of the fluency vector used for the approximation are as follows. The downsampled basis vectors used were generated by the method described above. The parameters are as shown below. Class: m = 1, 2, 3 Order: d = 512 Approximate rate: r = 2.

Under the above conditions, the approximation result generated by the fluency vector of the specified class using the method described above was selected. This is referred to as Method 1, the method for simply selecting the optimal approximation result is referred to as Method 2, and the results of comparing the two are shown in the first column of FIGS. Also, the point where the class of the target signal specified by the present method has changed is regarded as a singular point, and this point is removed from each of the results obtained by approximating the method 1, the method 2, and the single class, and the SN ratio is compared. The results obtained are shown in the second columns of FIGS. Pattern a, where the point where the class has changed is not removed
The case of removal is defined as pattern b. Each of FIGS. 31 to 33 is obtained by evaluating the target signal for each section.
31 shows the results for the 151st to 350th samples (section 1), FIG. 32 shows the results for the 20351st to 20551th samples (section 2), and FIG. 33 shows the results for all points.

FIG. 34 shows the number of points removed in the pattern b of each method. Section 1 contains many high frequency components,
The percentage of the removed points is also as high as 77.5%. Since the sampling width h of the down-sampled basis vector cannot be smaller than the discretization width of the target signal, the discretization width of the target signal needs to be sufficiently small in order to accurately specify the class. It is considered that the initial sampling width (sampling width of x (t _k )) is too large to specify the class of the section having many high frequency components. For this reason, it cannot be said that a singular point is always present in a portion where the specified class changes, and it is considered that the SN ratio in the present method in pattern b is significantly deteriorated as compared with the optimal approximation (method 2). Can be

On the other hand, in section 2, while the SN ratio of method 1 and pattern a is lower than that approximated by a single fluency vector of class m = 3, the SN ratio of method 1 for pattern b is lower. It is almost the same, and it is considered that many of the singularities have been removed (see FIG. 33). Fluency vectors of class m = 3 do not contain singularities
Since the signals belonging to ¹ S and ² S fully approximated,
It is reasonable that the removal of the singular point makes the approximation to the pattern b equal to the method 1 and the fluency vector of the class m = 3. Also,
This SN ratio is the SN obtained by the method 2 for the pattern b.
It is understandable that it approaches the ratio.

In consideration of the fact that the percentage of points removed in all sections is low, there is no small class variation as the tendency of all sections, and there are many sections in which the sampling width h of the down-sampling base vector is sufficient. b, the S / N ratio obtained by the method 1 and the method 2 is the class m
It is predicted that this is almost equivalent to the approximation by the fluency vector of = 3.

As described above, in the fluency theory, the signal can be efficiently described by specifying the class to which the target signal belongs, and the amount of calculation required for the conversion can be reduced. Therefore, considered that there is great significance to identify the class of the target signal, polynomial signal space ^m any one belonged to Assumes signal x (t) time t _j = jh of S ':
A method of specifying a space to which x (t) belongs from x (t _j ) discretized by (j = 0, 1, 2,...) Has been proposed.
As a preparation, we first defined a class decision criterion for discrete signals that enables class identification in continuous systems, and verified its effectiveness. As a result, the validity of the class determination measure in the discrete system was confirmed for signals that did not include singularities. It was also found that the fluency vectors of classes m = 3 and m = 4 had almost no difference in approximation accuracy. After confirming the effectiveness of the class decision measure in discrete systems, a class decision method based on this was proposed. As a result of applying this method to signals of known differentiability,
It was found that the boundary part where the class changed was a singular point candidate. This method was applied to actual data. From the results obtained by approximating the fluency vectors of the classes m = 1 to m = 3, the approximation result having the smallest error from the original waveform was selected, and the singular point candidates detected by the above-described method were removed. However, for a signal that does not contain many high-frequency components, an S / N ratio almost equivalent to the result of approximation with a class m = 3 single class was obtained. Class m or more flue ene sheet vector can be completely approximate the signals belonging to ^m S free of singularities. Therefore, the fact that an SN ratio equivalent to the approximation result of the highest-order class among the applied fluency vectors was obtained indicates that the extraction of the singular point was successful. On the other hand, when the change in the discretization width is remarkable, it has been found that it is difficult to specify the class because h is not sufficiently small, and it is difficult to extract the position of the singular point. As a conclusion, in a discrete system, the sampling width h cannot be smaller than the discretization width h ′ of the target signal. Therefore, if h ′ is made sufficiently small, the method of determining the class to which the discrete signal belongs by this method is effective. It can be said that there is.

Thus, the data processing method of the present invention
It is possible to capture the structure of the signal localized on the time axis,
It uses a fluency function system that has the feature of being able to efficiently describe discontinuous signals and signals with singularities, performs theoretical evaluation of errors resulting from support truncation, and maintains biorthogonality. A method is provided in which a discrete fluency function is newly defined and an optimal class in discrete data is determined using the newly defined discrete fluency function. Therefore, the signal can be described with higher approximation accuracy than the conventional method, and the amount of calculation required for conversion can be reduced.

Next, a specific example of a data processing device configured based on the above-described theory will be described. FIG.
FIG. 1 is a diagram illustrating a configuration of a data processing device according to an embodiment. The data processing device illustrated in FIG. 35 performs a process of converting an input analog signal into output data included in a polynomial signal space of a class (m−1) th-order differentiable class m. It is configured to include a conversion unit 110.

The input unit 100 receives an analog signal which continuously changes in response to a voice or the like, reads an instantaneous value of the analog signal at a predetermined sampling interval, and converts it into digital data. That is, the input unit 100 performs analog-digital (A / D) conversion using a δ function that has been conventionally performed.

The data conversion unit 110 multiplies a plurality of digital data sequentially output from the input unit 100 by a value of a predetermined conversion function, and adds each multiplication result. As the conversion function, the above-mentioned A / D function ^m _* ^ψ ₀ (t)
Is used. As described above, by using the data processing device shown in FIG. 35, a process of converting an input analog signal into digital data included in a class m polynomial signal space is performed by an integration operation (product-sum operation). Can be. Therefore, compared to the case where the instantaneous value of the analog signal is used as it is, it is less affected by noise, and it can be said that it is very useful in configuring a practical processing system.

FIG. 36 is a diagram showing a configuration of a data processing device of another embodiment. The data processing apparatus shown in FIG. 36 determines a class of a polynomial signal space including input data when considering a polynomial signal space of a class m (m−1) th-order differentiable, and includes A singular point to be extracted, and performs a thinning processing section 200, an interpolation calculation section 210, an error calculation section 220, and a class determination section 23.
0, and includes a singular point extraction unit 240.

The thinning processing section 200 performs a predetermined thinning process on discrete input data. Interpolator 210
Performs an interpolation operation on the data after the thinning process using interpolation functions corresponding to a plurality of classes having different values of the class m. As the interpolation function, the above-described D / A function ^m用 い₀ (t) is used.
The error calculator 220 calculates, for each class, an error between the interpolation data for each class obtained by the interpolation calculator and the data before the thinning processing. The class determination unit 230 determines a class corresponding to the interpolation data with the smallest error calculated by the error calculation unit 220. Singular point extraction unit 240
Extracts a position where the value of the class m determined by the class determination unit 230 changes as a singular point of the input data.

As described above, by using the data processing device shown in FIG. 36, it is possible to determine the class m to which the input data belongs. Further, by examining a portion where the class m changes, a singular point included in the input data can be easily extracted. In particular, by extracting a singular point included in the input data and determining the class m of the data between adjacent singular points, it is possible to efficiently analyze the contents of the data and perform an operation for approximating the data. The amount can be reduced.

[Brief description of the drawings]

FIG. 1 is a diagram illustrating an example of a signal belonging to a signal space ^ms .

[Figure 2] target signal is a diagram illustrating a flue ene sheet model if it belongs to ^m S.

A diagram [3] target signal will be described flu ene sheet model when not belonging to ^m S

FIG. 4 shows the A / D function and D / when m = 1 and m = 2.
FIG. 4 is a diagram illustrating characteristics of an A function.

FIG. 5 shows A / D functions and D / D when m = 3 and m = ∞.
FIG. 4 is a diagram illustrating characteristics of an A function.

FIG. 6 is a diagram illustrating attenuation parameters in an A / D function and a D / A function.

FIG. 7 is a diagram for explaining an upper bound (E (H)) of a truncation error with respect to a truncation width (H).

FIG. 8 is a diagram illustrating characteristics of a D / A function when h = 1.

FIG. 9 is a diagram illustrating an upsampling basis matrix.

FIG. 10 is a diagram illustrating a downsampling basis matrix.

FIG. 11 is a diagram illustrating characteristics of a down-sampled basis vector when h = 1.

FIG. 12 is a diagram illustrating downsampling of a target signal using a downsampling basis vector.

FIG. 13 is a diagram illustrating a normal rate conversion method.

FIG. 14 is a diagram showing an SN ratio of an approximation result for music 1 and music 2;

FIG. 15 is a diagram showing an SN ratio of an approximation result when the music 2 is divided into small sections with the approximation rate being 2;

FIG. 16 is a diagram showing an SN ratio of an approximation result when the music 2 is divided into small sections with an approximation rate of 4;

FIG. 17 is a diagram showing a square error between an original waveform and an approximation result in section 1 of music 2;

FIG. 18 is a diagram showing a square error between an original waveform and an approximation result in a section 2 of music 2;

FIG. 19 is a diagram showing an SN ratio of an approximate result and an SN ratio of an optimal approximate result in each class.

FIG. 20 is a diagram illustrating a target signal.

FIG. 21 is a view for explaining target signals belonging to pattern 1 shown in FIG. 20;

FIG. 22 is a view for explaining target signals belonging to pattern 2 shown in FIG. 20;

23 is a diagram showing a class determination measure of a zero-order function belonging to pattern 1 shown in FIG. 20 (a zero-order function not containing a singular point) and a linear function shown in pattern 1 (a linear function not containing a singular point); It is a figure showing a transition.

24 is a diagram illustrating transition of a class determination measure in a quadratic function and a sine wave belonging to the pattern 1 illustrated in FIG. 20;

FIG. 25 is a diagram illustrating transition of a class determination measure in a zero-order function (a zero-order function including a singular point) and a linear function (a linear function including a singular point) belonging to pattern 2 illustrated in FIG. 20;

FIG. 26 is a diagram showing transition of a class determination measure in a quadratic function and a sine wave belonging to the pattern 2 shown in FIG. 20;

FIG. 27 is a diagram illustrating a result of performing class identification on the zero-order function including the singular point illustrated in FIG. 22 based on a class determination scale.

FIG. 28 is a diagram illustrating a result of performing class identification on the linear function including the singular point illustrated in FIG. 22 based on a class determination measure.

FIG. 29 is a diagram illustrating a result of performing class identification on a function that changes from a zero-order function to a linear function based on a class determination measure.

FIG. 30 is a diagram illustrating an implementation policy of a class determination method.

FIG. 31 is a diagram illustrating an SN ratio when approximation based on a class determination method is performed on section 1;

FIG. 32 is a diagram illustrating an SN ratio when approximation based on a class determination method is performed on a section 2;

FIG. 33 is a diagram illustrating an SN ratio when approximation is performed on all sections based on a class determination method.

FIG. 34 is a diagram showing a ratio of the number of samples removed by the pattern b in FIGS. 31 to 33;

FIG. 35 is a configuration diagram of a data processing device of one embodiment.

FIG. 36 is a configuration diagram of a data processing device of another embodiment.

[Explanation of symbols]

 Reference Signs List 100 input unit 110 data conversion unit 200 thinning processing unit 210 interpolation calculation unit 220 error calculation unit 230 class determination unit 240 singular point extraction unit

Claims (10)

[Claims] 1. A method for generating a plurality of discrete data strings generated based on a plurality of functions classified according to the number of differentiable times, and performing parallel processing on input data with the plurality of discrete data strings. A data processing method for obtaining a singular point included in the input data based on a result of the correlation operation after performing the correlation operation. 2. The method according to claim 1, wherein the input data surrounded by the adjacent singular points is classified into signals based on the number of differentiable times based on a result of the correlation operation. Data processing method. 3. A data processing method for converting input data into output data included in a polynomial signal space of a class (m−1) th-order differentiable class m, wherein each of the plurality of discrete input data is , Multiplying a value of a predetermined conversion function corresponding to the class m, and adding the multiplication results to obtain the output data. 4. The interpolation function according to claim 3, wherein the conversion function is used to calculate an interpolated value when interpolating between discrete data using a signal included in a polynomial signal space of the class m. A data processing method characterized by having biorthogonality with respect to a use function. 5. An input unit for reading an instantaneous value of an input analog signal at a predetermined sampling interval, converting the instantaneous value into digital data and outputting the digital data, and for a plurality of the digital data sequentially output from the input unit. A data conversion unit that multiplies a value of a predetermined conversion function and adds each multiplication result, and converts the analog signal into digital data included in a polynomial signal space of class m. Processing equipment. 6. The conversion function according to claim 5, wherein the conversion function converts between digital data output from the input unit using a signal included in a (m−1) -order differentiable class m polynomial signal space. A data processing device, characterized in that it has biorthogonality with respect to an interpolation function used for calculating an interpolation value when performing interpolation. 7. A data processing method for determining a class of a polynomial signal space including input data when considering a polynomial signal space of (m−1) th-order differentiable class m, comprising: First to perform predetermined thinning processing on input data
And a second step of performing an interpolation operation on the data after the decimation process is performed using an interpolation function corresponding to each of a plurality of classes having different values of m; and A third step of calculating an error between the interpolated data obtained in the step and the data before the decimation process; and a class corresponding to the interpolated data having the smallest error calculated in the third step. Fourth decision as a class of polynomial signal space containing data
A data processing method, comprising the steps of: 8. The data according to claim 7, further comprising a fifth step of extracting, as a singular point of the input data, a position at which the value of the class m determined in the fourth step changes. Processing method. 9. A data processing device for determining a class of a polynomial signal space including input data when considering a polynomial signal space of (m-1) th-order differentiable class m, comprising: A thinning-out processing unit for performing a predetermined thinning-out process on input data; and an interpolation operation on the data after the thinning-out process, using an interpolation function corresponding to each of a plurality of classes having different values of m. And an error calculator for calculating, for each class, an error between the interpolated data for each class obtained by the interpolation calculator and the data before the thinning processing, and an error calculator for calculating the error. A class determination unit that determines a class corresponding to the interpolation data having the smallest error. 10. The data processing according to claim 9, further comprising a singularity extracting unit that extracts a position at which the value of the class m determined by the class determining unit changes as a singularity of the input data. apparatus.