Uncertainty in Natural Observation Theory of Normal Type and

an Optimum Value of the Time-Constant Parameter

Taizo Iijima

Soken Institute Inc., Tanashi, Japan 188-0014

Mamoru Iwaki

Japan Advanced Institute of Science and Technology, Hokuriku, Ishikawa, Japan 923-1292

SUMMARY

The natural observation transform is a new formu-

lation of the time waveform as a transform, when a

waveform is observed in the short-term (instantaneous)

region specified by the time-constant parameter at an

arbitrary time. The basic observed value sequence not

only includes the considered time, but also the informa-

tion extracted from the waveform in the region specified

by the time-constant parameter. Based on the above

property of the natural observation transform, the object

of the transform is not the input waveform itself, but the

waveform appearing in the region specified by the time

and the time-constant parameter. This paper notes this

situation, and discusses the uncertainty that exists be-

tween the behavior of the waveform, which is the object

of the normal natural observation transform in the whole

time region, and the aspect in which the basic observation

values are distributed in the sequence for the whole time

region. Based on the result of investigation, the optimal-

ity criterion for the time-constant parameter is proposed

using the equivalent time-constant, and a method of

determining the optimal value is shown. © 2000 Scripta

Technica, Electron Comm Jpn Pt 3, 83(8): 30�38, 2000

Key words: Theory of natural observation; normal

type; uncertainty; natural observation parameter.

1. Introduction

The Fourier transform is one of the transforms that

observe the waveform uniformly in the whole time region

and transform the data all at once. It is typical of such

transforms. Consequently, the result of the transform does

not contain time�essentially. In contrast, the natural ob-

servation transform [1�15] is a new formulation of the time

waveform transform in a short-term (instantaneous) region

in regard to an arbitrary time t and specified by the time-

constant parameter s. Consequently, the basic observation

value sequence obtained by the transform contains not only

the considered time t, but also the information extracted

from the waveform in the region specified by s.

Based on the above property of the natural observa-

tion transform, the object of the transform is not simply the

input waveform itself, but the waveform that appeared in

the region specified by the time-constant parameter s. The

time t, however, changes without waiting for the process-

ing, and the basic observed value sequence also changes

with the time.

© 2000 Scripta Technica

Electronics and Communications in Japan, Part 3, Vol. 83, No. 8, 2000Translated from Denshi Joho Tsushin Gakkai Ronbunshi, Vol. J81-A, No. 4, April 1998, pp. 743�750

30

This paper discusses the uncertainty relation that

exists between the behavior of the waveform, which is the

object of the normal natural observation transform, in the

whole time region, and the distribution aspect of the basic

observed value in the sequence for the whole time region.

Based on the results, the optimality criterion for the time-

constant parameter s is proposed using the equivalent time-

constant, together with the method for determining the

optimal value.

2. Preparation

This section outlines the M-th order natural observa-

tion theory, and prepares the characteristic parameters for

the normal natural observation transform. In the natural

observation theory, the primitive operation G and the ad-

joint primitive operator L are given, respectively, as

The M-th order normal natural observation theory is

as follows. Using the m-th degree basic observation opera-

tor derived from the above expressions

the waveform f�t� is transformed into the basic observed

value sequence {nm�M��t�}

Based on this relation, the theory provides the basis for the

instantaneous waveform analysis.

The basic observed value sequence {nm�M��t�} satisfies

where

As the parameter representing the behavior of the

waveform f�t� in the time region, the equivalent time-con-

stant Gt is known:

Let the Fourier transform of the waveform f�t� be

It is also written as

Using the time average,

the variance of the waveform is defined as

It is related to the equivalent time-constant as

As the next step, the parameter representing the dis-

tribution of the basic observed value sequence is prepared.

Using the average degree

the average degree index is calculated as

It is the average degree which is normalized in regard to the

order M.

(8)

(1)

(2)(where I is the identity operator)

(3)

(4)

(5)

(6)

(7)

(9)

(10)

(11)

(13)

(14)

(15)

(12)

31

For the precise expression for L and the variance of

the degree

in relation to the average degree index, the following lem-

mas hold [15]:

[Lemma 1]

[Lemma 2]

where A and B are constants given by

and there holds

In the following, the uncertainty of the normal natural

observation transform and the optimal value of the time-

constant parameter s are discussed.

3. Uncertainty of Normal Natural

Observation Transform

This section analyzes the uncertainty included in the

normal natural observation transform. Using the primitive

function

let the waveform which is obtained by observing an arbi-

trary waveform f�t� be g�t�. Then,

Letting the equivalent time-constant of g�t� be Gct, the

following theorem applies.

[Theorem 1] There exists the following relation to

the equivalent time-constant Gt of f�t�:

In addition, �Gct�2 and L are inversely proportional:

(Proof) According to Schwarz� inequality, there

holds in general

Modifying the above relation,

(16)

(17)

(18)

(19)

(20)

(23)

(21)

(22)

(24)

(25)

(26)

32

Dividing both sides by the right-hand side,

Letting the Fourier transform of g�t� be G�iZ�,

Letting the equivalent time-constant of g�t� be Gct,

Using Eqs. (10) and (30), Eq. (28) is written as

Rearranging both sides, Eq. (24) is obtained. Comparing

Eqs. (30) and (17), Eq. (25) is obtained.

(end of proof)

Theorem 1 indicates that the equivalent time-constant

and the average degree index are in an inversely propor-

tional relation. It is also seen that Eqs. (25) and (18) indicate

the relation between �Gct�2 and �'m�2 through the average

degree index L. Those relations represent the spreads on the

observation degree axis and the time axis produced by the

normal natural observation transform, that is, the uncer-

tainty principle in the normal natural observation trans-

form.

By the above reasoning, the following properties are

derived as the general properties of the normal natural

observation transform.

[Theorem 2]

(1) �Gct�2 / s2 o f and L o 0 �mBB

o 0� are equiva-

lent.

(2) �Gct�2 / s2 o 0 and L o f �mBB

o M� are equiva-

lent.

(3) �Gct�2 / s2 1 and L 1 �mBB

M/2� are equiva-

lent.

[Theorem 3]

Those relations are summarized in Fig. 1.

4. Discussion of Uncertainty in Normal

Natural Observation Transform

This section discusses the uncertainty of the normal

natural observation transform. The uncertainty is consid-

ered in the previous section for the waveforms

g�t� �Gf��t� observed once in the primitive form. A ques-

tion may arise as to why g�t� is discussed and not f�t�. The

reason is that the basic observation transform essentially

considers the waveform g�t� �Gf��t�, not the input wave-

form f�t� (this point is discussed in detail in the appendix).

Each value in the basic observed value sequence {nm�M��t�}

is an extraction of the information of f�t� in the neighbor-

hood specified by s, and is not the extraction of the infor-

mation of f�t� in the whole time region. In the previous paper

by the authors, the insight into this point was insufficient,

due to lack of knowledge of the situation.

The above situation is also seen in recognition of the

image. The usual way in image recognition is to set the

viewpoint and the view-field first, to limit the information

(28)

(29)

(30)

(31)

Fig. 1. Relation between �Gct / s�2 and �'m /M�2.

(27)

33

from unnecessary regions. Then, the object of recognition

is not the whole of the presented image, but the image after

the above suppression (i.e., the image in the considered

region). Considering the recognition of the time waveform

in analogy to the above, the present time t corresponds to

the viewpoint, and the time-constant parameter s corre-

sponds to the spread of the view-field. The waveform

corresponding to the image after suppression is g�t�, and not

f�t�, which represents the whole waveform.

By such reasoning, the reason for using Gct as the

basis for discussing the uncertainty is made obvious. This

paper discusses the relation between the actual waveform

f�t� and the waveform g�t� after suppression, from the

standpoint of the basic observed value sequence.

The result in this paper is compared with that in the

previous paper [15] as follows. The previous paper tried to

represent the uncertainty using the equivalent time-constant

of f�t�, but some aspects which were not clear enough were

left in the theory. For the case such that 1 d L2 d s2 / �Gt�2,for example, the behavior is not clear. In contrast, in this

paper, the whole theory is made clear by establishing the

theory based on the above standpoint.

The expression of the uncertainty of the normal natu-

ral observation transform in this paper differs from the

uncertainty in the Fourier transform. If considered in anal-

ogy to the Fourier transform, the following parameter may

be introduced concerning the product of the equivalent

time-constant and the variance of the degree:

The above is the relation which xm�M� determined by arbitrar-

ily given s and M satisfies for the waveform f�t�. Conse-

quently, when f�t� changes, P also changes in the range

0 d P d 1.

Equation (32) apparently seems to indicate that

�Gct�2 and �'m�2 are in an inversely proportional relation,

but the matter is actually not simple. When �Gct�2 o 0, for

example, there holds �'m�2 o 0. Then, P o 0. This implies

that the relation is not simply inversely proportional. Be-

cause of such a situation, the relation between Gct and L, as

well as between L and 'm, are represented in this paper as

the relation through L.

It should also be noted that the representation of the

uncertainty of the normal natural observation transform

contains inevitably the time-constant parameter s. This is

due to the observation of the waveform by the natural

observation transform in the instantaneous region specified

by s. In other words, when the waveform information is

extracted into the basic observed value sequence by the

natural observation transform, the time-constant parameter

s should be determined considering the above property.

5. Optimal Natural Observation Parameter

Based on the result up to the previous section, the

optimality criterion for the time-constant parameter s is

proposed using the equivalent time-constant. Then, a

method for determining the optimal value is presented

following the idea.

5.1. Optimal value of time-constant parameter

In the M-th order normal natural observation theory,

the input waveform is represented using (M + 1) basic

observed values. For this representation to be meaningful

as an M-th order observation, the ideal situation is that the

information of the waveform is uniformly distributed in the

basic observed value sequence. This situation is represented

by the condition mBB

M/2.

It follows in this case from Eq. (15) that L = 1.

Consequently, it is seen from Eq. (25) that there must be the

following relation between s and Gct:

From such a viewpoint, it is considered that the optimal

value for the time-constant parameter s for the natural

observation is given by Eq. (33). In this case, �'m�2 almost

takes the maximum value.

By the definition, �Gct�2 is given by

Considering the relation

it follows from the condition of Eq. (33) that

This is equivalent to the relation

(32)

(33)

(34)

(35)

(36)

34

When s o 0, we have g�t� o f�t�. When s o f, we have

g�t� o 0. The optimal value of s is an appropriate value

between those.

The significance of the optimization condition for the

time-constant parameter is considered as follows. The

equivalent time-constant Gct of g�t� satisfies, by its defini-

tion,

Consequently, the following theorem is obtained.

[Theorem 4] When s Gct, there holds

(Proof) The Laplace transform of the waveform

g�t� is given by

Let the basic observed value of g�t� be n̂m�M��t�. Its Laplace

transform is

Consequently, it is represented as

Thus,

It follows from

that

Consequently,

Substituting Eqs. (43) and (46) into Eq. (38), and

rearranging using the relation s Gct,

Then, Eq. (39) is obtained.

(end of proof)

Equation (39) should apply to any M. Letting espe-

cially M = 0,

is obtained. Then, the following corollary is derived.

[Corollary 1]

The above corollary indicates that the optimization

condition for s is that the power component of the primitive

observed waveform is the same as that of the adjoint primi-

tive observed waveform.

5.2. Calculation of optimal value

It is not easy in practice to calculate the optimal value

of s. It is practical to use Eq. (36). F�iZ� is calculated as the

Fourier transform of f�t�. Substituting it into Eq. (36), the

problem is reduced to the solution of a transcendental

equation for s2.

The following function is defined:

Then, the solution O satisfying

(37)

(38)

(39)

(40)

(41)

(42)

(43)

(44)

(45)

(46)

(47)

(48)

(49)

(50)

35

is given as O s2, where s is the solution of Eq. (36).

The equation is solved as follows. An appropriate

initial value O0 and an appropriate constant P are given.

Calculating successively the sequence {On} by

then the solution is obtained as its convergence value [16].

5.3. Example

Let the input waveform f�t� be

Its Fourier transform is

Since

O�! 0� is the solution of the equation

By Newton�s method, it is seen that

Consequently, the optimal value of s is given by

This is larger than the equivalent time-constant

Gt �CC3 / W of f�t�.

6. Conclusions

This paper has considered the situation where a wave-

form is observed by the normal natural observation trans-

form, and has derived the uncertainty relation between the

behavior of the waveform in the whole time region and the

distribution aspect of the basic observed value in the se-

quence. The relation is characterized by the property of the

natural observation transform that it is the transform repre-

senting the observation of the waveform in a short-term

(instantaneous) region specified at an arbitrary time t and

by the time-constant parameter s.

Based on the result of the investigation, the optimality

criterion for the time-constant parameter s is proposed

using the equivalent time-constant, together with the

method for determining the optimal value. From the stand-

point of analyzing the instantaneous characteristics of the

waveform, it will be meaningful to consider a similar rela-

tion for the instantaneous (short-term) observation, not for

the whole time region. This is left for future study.

REFERENCES

1. Iijima T. Basic theory for natural observation of

waveform. Trans IEICE 1984;J67-A:951�958.

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3. Iijima T. Analysis of instantaneous waveform by

natural observation. Trans IEICE 1986;J69-A:1540�

1546.

4. Iijima T. Recognition theory for structure of instan-

taneous waveform. Trans IEICE 1988;J71-A:854�

859.

5. Iijima T. A theoretical study of waveform reconstruc-

tion by natural observation filter. Trans IEICE

1991;J74-A:430�434.

6. Iijima T. Natural observation filter that accepts al-

most-periodic waveform. Trans IEICE 1991;J74-

A:435�441.

7. Iijima T. Acceptance and generation of polynomial

waveform by natural observation filter. Trans IEICE

1991;J74-A;442�447.

8. Iijima T. Instantaneous frequency analysis by natural

observation. Trans IEICE 1991;J74-A:907�912.

9. Iijima T. Basic theory of natural observation trans-

form. Trans IEICE 1993;J76-A:1620�1626.

10. Iijima T. A study concerning origin of natural obser-

vation theory (canonical form). Trans IEICE

1993;J76-A:1627�1630.

11. Iijima T. Relation between neighboring and equilib-

rium types in natural observation theory. Trans.

IEICE 1994;J77-A:538�542.

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form by natural observation filter. Trans IEICE

1995;J78-A:722�727.

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to reconstruct waveform by a finite sum�develop-

(51)

(52)

(53)

(54)

(55)

(56)

(57)

(58)

(59)

36

ment of normal natural observation theory. Trans

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APPENDIX

Representation of Basic Natural

Observation Operator by Primitive

Operators

Consider the operator Xm�M� that represents the basic

observation transform. Its Laplace transform is given by

The expression is factorized as

Then, it is seen that

Consider also

Forming the Laplace transform, and calculating the deriva-

tive by Leibniz� formula,

This is nothing but the Laplace transform of the normal

natural observation transform Xm�M�. Consequently,

It is seen from the above calculation that Xm�M� operates

first the primitive operator G, independently of the degree.

In other words, Xm�M� is represented by G and the following

operators.

(A.1)

(A.2)

(A.3)

(A.4)

(A.5)

(A.6)

37

AUTHORS (from left to right)

Taizo Iijima (honorary member) graduated from Tokyo Institute of Technology in 1948 and joined the Electrotech. Lab.

He has engaged in research on theory of electromagnetic field, theory of pattern recognition, image processing and speech

recognition, as well as R&D of OCR. He served as head of Iijima Spec. Lab.; professor (1972) and professor emeritus (1986)

Tokyo Institute of Technology; professor (1986) and professor emeritus (1991) Tokyo University of Engineering; professor

(1991), vice-president (1992), and professor emeritus (1997) Japan Advanced Institute of Science and Technology, Hokuriku;

president (1997) Soken Institute Inc. He served as editor, survey and general secretary, auditor, advisor and chairman of Comm.

Pattern Recognition and Learning, chairman of Tokyo chapter and vice-president, IEICE. Honorary member, 1991. Four Paper

Awards, Publ. Award, and Achievement Award 1976, and Meritorious Award 1989 IEICE. Purple medal, 1989. He holds a

D.Eng. degree.

Mamoru Iwaki (member) graduated from the University of Tsukuba in 1989 and completed the doctoral program in

1994. He holds a D.Eng. degree. He then became a research associate in the School of Information Sciences, Japan Advanced

Institute of Science and Technology, Hokuriku. His research interests are digital signal processing as well as speech analysis,

synthesis, and recognition. He is a member of IEEE.

38