Methods of evaluating fourier transforms with applications to control engineering

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METHODS OF EVALUATING

FOURIER TRANSFORMS WITH

APPLICATIONS TO CONTROL

ENGI1N EERING

TECBNISCBE UNIVERS ITEIT

Scheepshydromechani ca

Archief

Mekeiweg 2, 2628

D Deif t

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STELLINGEN

Eij het interpoleren van eon onvoldoende vaak bemonsterd

signaal kan men in bepaalde gevallexi met voordeel gebruik

maken van de in paragraaf 6.2.2.1.3 van dit proefochrif t

ontwikkelde gedachtengang. Het bemonsterde signaal wordt

daartoe in oem even en oem oneven component gesplitst die

met versohillende functies worden geconvolueerd.

Voor een lineair systeem kan met behuip van een

bemonscerings-methode overeenkomstig paragraaf 6.2.2.2 van dit proefschrif t

de responsie op een lineair oplopend signaal rechtstreeks

worden bepaald uit de frequentieresponsie.

De argumenten waarmee Lewis de methode voor benaderde

harmo-nische analyse gebaseerd op aequidistante bemonsteringen van

een signaal becritiseert zijn niet doorolaggevend.

F.1. Lewis, J. appi. Mech. 2(1935)A-137.

De fouriertransformatie gebaseerd op gewogen bemonsteringen

en de tweede reienmethode van paragraaf 6.2.3.1 van dit

proefschrift kan met eon analoog rekensysteem overeenkomstig

fig. 4 van onderstaand artikel worden uitgevoerd.

W.K. Linvill, R.E. scott en 2.A. Guillemin, I.R.E.

Trans. CT-2(1955)243.

Bij de voor kristallografische structuurbepalingen ontwikkelde

"Lipson-Beevers strips" is het door middel van een gewijzigde

opzot mogelijk de benodigde rekentijd te bekorten.

Paragraaf 6.2.3.1 van dit proefschrift.

De op de z-transformatie gobaseerde formules voor de derde

en hogere afgeleiden voorgesteld door Truxal zin in hot

algemeen oribruikbaar.

J.G. Truxal, I.R.E. Trans. CT-1(1954)49.

Bij fouriertransformatie met behuip van eon gebroken lineaire

benadering van de gegeven grootheid geef t het "principle of

equi-valent area's" in het algemeeri niet de nauwkeurigste resultaten.

R.E. Andeen, Trans. A.I.E.E. 79(1960) II (Applic. and

Industr. )332.

De voorwaarde, die Solodovnikov stelt voor eenvoudige

toepas-baarheid van eec op orthogonale ontwikkeling gebaseerde methode

voor benaderde fouriertransformatie, is niet essentieel.

V.V. Solodovnikov, Introduction to the statistical

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Het verdient

aanbeveling

na te gaan of er eenvoudige benade-ringamethoden mogelijk zijn voor de rechtstreekse bepaling van responsies in het tijdsdomein uit de amplitude of fase-karakteristiek van systemen met minimumfase eigonschappen. Indien een nformatieverwerkend systeem mechanische in en uitgangsgrootheden heef t is een pneumatische uitvoering sons

te verkj.ezen boyen een elektrische of elektronische.

Het is van technisch standpunt uit gezien over het algemeen onnodig machines te maken die met de hand geschreven schrif t kunnen lezen.

In veel gevallen geven de woorden "classificeren' of

"sorteren" duidelijker aan wat er in een "herkenningsmachine" gebeurt dan het modewoord "herkennen".

Hot wetenschappelijk werk van stafleden op de AfUeling der Technische Natuurkunde van de Technische Hogeschool te Delf t zou in sommige gevallen efficienter

kunnen

geschieden indien er meer hulppersoneel kon worden ingeechakeld voor die delen van het onderzoek waarvoor geen academische opleiding maar wol scholing en toewi.jding vereist zum.

I-let is wenselijk aan de Technische Hogeschool te Deift proever! te nemen met zg. geprogrammeerd onderwijs (teaching machines). Een vordere en systematische coördinatie van colleges, instruc-ties, practica en oefeningen zou het jongerejaars onderwijs aan de Technische Rogeechool te Deift ten goede kunneri komen. Ret oprichten van zelfstandige organisaties voor beoefenaren van allerlei min of meer gespecialiseerde gebieden van weten-schap en techniek is te veroordelen.

De invoering van een "derde programma" bij de radioomroep dient voorrang te krijgen boyen de invcering van reclame-televisie.

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METHODS OF EVALUATING

FOURIER TRANSFORMS WiTH

APPLICATIONS TO CONTROL

ENGI1N EERING

TECHSCHE UWVERSITET

Laboratorium voor

Scheepshydromechanca

Archtef

Mek&weg 2, 2628 CD Deft

eL 015-786873- Fax 015-

781B33

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METHODS OF EVALUATING

FOURWR TRANSFORMS

WITH

APPLICATIONS

rço CONTROL

ENGI1N EERING

TECHJISCHE UNIVERSITEIT

Laboratorium voor

Scheepshydromechanca

Archief

Mekeweg 2, 2628 CD Deift

eL: 015.786873.

Fax 015 781833

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR

IN DE TECHNISCHE WETENSCHAP AAN DE

TECH-NISCHE HOGESCHOOL TE DELFT OP GEZAG VAN

DE RECTOR MAGNIFICUS IR. H. j. DE

WIJS,HOOG-LERAAR IN DE AFDELING DER MIJNBOUWKUNDE,

TE VERDFDIGEN OP VRIJDAG 5 JULI 1963

DES NAMIDDAGS TE 4 UUR

DOOR

JOSEPH ZORN

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Dit proefschrift is goedgekeurd door de Promotor

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Aan nu1 n vader

Aan de naedacIitenis

van inyn weder

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5.5

The aut000rrelation function 42

3.6 Step responses distorted by noise 43

CHAPTER 4 METHODS GIVING RELATIONS BETWEEN CHARACTERISTIC FREQUENCY AND TRANSIENT RESPONSE PARANETERS

4.1 General discussion 46

4.2 Graphical relations of Chestnut and Mayer 47

4.2.1 Description of the method 47

4.2.2 Application to system 1 48

4.2.4 Conclusions 50

4.3 Relations of Jaworski, Cunningham, Ludbrook, Axelby, Bradford and De Merit, Stallard and

Horowitz 50

4.3.1 Description of the relations 50

4.3.2 Application to the systems i and 2 54

4.4 Discussion of the results 55

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a

MFTHODS APPLYING GRAPHICAL OR NUMERICAL INTE-JIATION

introduction

57

Method of Profos and Keller ₅₇

Method of HMnny 59

5.4 Method of Leonhard

₆₀

5.5

Method of Hosenbrock

₆₀

5.5.1

Theory and comouting routines ₆₁

5.5.2

Application to system 1 ₆₂

5.5.3

Application to system 2 ₆₂

5.5.4

Application to system ₃ ₆₃

5.5.5

Conclusions ₆₃

CHAPTER 6 METHODS BASED ON SAMPLING

6.1

General discussion ₆₄

6.2

_{Sampling methods applying a multiplicative or}

additive correction function ₆₅

6.2.1

_{Weighted sampling methods resulting from}

appro-ximations by elementary functions ₆₅

6.2.1.1

Introduction ₆₅

6.2.1.2

The direct Fourier transform ₆₆

6.2.1.2.1

Staircase approximation ₆₆

6.2.1.2.2

Approximation by straight-line segments ₆₇

6.2.1.2.

Approximation by higher degree functions ₆₈

6.2.1.3

The inverse Fourier transform ₆₉

6.2.2

Methods of compensating for the errors due to

the use of a sampling rate not satisfying

Shan-nont s theorem

70

6.2.2.1

The direct Fourier transform ₇₀

6.2.2.1.1 Discussion of the problem

₇₀

6.2.2.1.2

The time domain approach

₇₂

6.2.2.1.3

The frequency domain approach ₇₃

6.2.2.1.4

Summary of formulas ₇₇

6.2.2.2

The inverse Fourier transform

79

6.2.3

The numerical evaluation of the formulas

(6.19)-(6.24)

₈₀

6.2.3.1

The evaluation of trigonometric series

₈₀

6.2.3.2

The numerical evaluation of Filon's formulas

₈₇

6.2.3.3

Thbles of functions

₈₇

6.2.4

Application to system 1

₈₇

6.2.5 ₉₀

6.2.6 ₉₃

6.2.7

Conclusions

₉₇

6.2.7.1

The direct Fourier transform

₉₇

6.2.7.1.1

The selection of multiplicative or additive

correction functions ₉₇

6.2.7.1.2

The effect of truncation ₉₈

6.2.7.1.3

The choice of 8, N and M and the computing

technique

99

6.2.7.2

The inverse Fourier transform ₁₀₁

6.2.7.2.1

The selection of weight functions ₁₀₁

6.2.7.2.2

The effect of truncation ₁₀₁

6.2.7.2.3

The choice of N and M and the computing technique

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9

6.3

Method of Samulon-Tagajewskaja 101

6.3.1

Theory and computing routines 101

6.3.2

105

6.3.3 ₁₀₄

6.3.4

Application to system 5 104

6.3.5

Conclusions

₁₀₄

CHAPTER 7 METHODS BASED ON PERIODIC REPETITION OF THE INPUT SIGNAL

7.1

General discussion

106

7.2

Method of \'Jass and Hayman

107

7.2.1

Theory

107

7.2.2

Computing routines

108

7.2.3

7.2.4

7.2.5

110

7.2.6

Conclusions 111

7.3

Method of von Hrnos et al. 111

7.5.1

Description of the method 111

7.3.2

112

7.3.3

7.3.4

7.3.5

Conclusions

113

7.4

Methods of Tustin, Stallard and Rushton

113

7.4.1

Theory

113

7.4.2

The numerical evaluation of the step response

115

7.4.5

117

7.4.4

117

7.4.5

7.4.6

Conclusions

118

7.5

A recursion method based on periodic repetition

of the input signal 119

7.5.1

Evaluation of the step response from the

fre-quency response

119

7.5.1.1

Theory 119

7.5.1.2

Computing routines 121

7.5.2

Evaluation of the frequency response from the

step response

124

7.5.2.1

Theory

₁₂₄

7.5.2.2

Computing routines

124

7.5.3

Evaluation of the impulse response from the

frequency response 126

7.5.3.1

Theory 126

7.5.3.2

Computing routines

127

7.5.4

Evaluation of the frequency response from the

impulse response

128

7.5.4.1

Theory ₁₂₈

7.5.4.2

Computing routines 128

7.5.5

Related formulas 128

7.5.6

Related recursion methods

130

7.5.6.1

Wedrnore's method 130

7.5.6.2

Methods of harmonic analysis leading to

formu-las similar to (7.15),

(7.21)

or

_(7.25)

₁₃₀

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10

7.5.6.2.2 Methods of harmonic analysis which determine sums of ordinates of the integral of the func-tion to be analysed or evaluate integrals of the product of the given function and some modulating function approximating to the sine

or cosine 131

7.5.9 Application to system 3 ₁₃₅

CHAPTER 8 METHODS BASED ON APPROXIMATION BY ELEMENTARY

FUNCTIONS

8.1 Introduction 138

0.2 Staircase approximation 138

8.2.1 General discussion 138

8.2.2 Method of Russ and Donegan 139

0.2.2.1 Theory 139

8.2.2.2 Computing routines 140

8.2.2.3 The choice oft and N 140

8.2.2.4 Application to system 1 141

8.2.2.7 Conclusions 143

8.2.2.7.1 The direct Fourier transform 143

8.2.2.7.2 The inverse Fourier transform 143

8.3 Approximation by straight-line segments 144

8.3.1 General discussion 144 8.3.2 Method of Zemanian 144 8.3.2.1 Theory 144 8.3.2.2 Computing routines 146 8.3.2.5 Application to system 1 146 8.3.2.4 Application to system 2 147 8.3.2.5 Application to system 3 148 8.3.2.6 Conclusions 149 8.3.3 Method of Wait 149 8.3.5.1 Theory 149

8.3.5.2 Computing routines; applications of the method 150

8.3.3.3 Discussion of the method 151

8.3.4 Method of Hnny-Ludbrook 152

8.3.4.1 Theory 152

8.3.4.2 Computing routines 152

8.3.4.3 Comparison of formula (8.14) with formula (8.7) 153

8.5.5 Method of Floyd 153

8.3.6 Method of Solodovnikov et al. 154

8.5.6.1 Theory and computing routines 154

8.3.6.5 Conclusions 156

8.4 Approximation by higher degree functions 156 8.5 A special photographic apparatus facilitating

the numerical evaluation of some approximate

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11.2.1

11.2.2 11.2.3 11.2.4

The Fourier transform of an autocorrelation function

General discussion

The weighted sampling method The recursion method

Conclusions

Approximate frequency responses from step responses distorted by noise

General discussion

The weighted sampling method Zemanian's method

Conclusions

CHAPTER 12 GENERAL CONCLUSIONS

12.2 The direct Fourier transform 184

12.3 The inverse Fourier transform 186

APPENDIX 1 Derivation of formula (1.12) 188 11

CHAPTER ₉ METHODS BASED ON EXPANSION IN ORTHOGONAL FUNCTIONS

9.1

General discussion 163

9.2 Method of Yachter 166

9.2.1 Theory and computing routines 166

9.3 Method of Dawson 1 70 9.3.1 Theory and computing routines 170

CHAPTER 10 SPECIAL METHODS

10.2 Methods based on the characteristics of first

and second order systems 174

10.3 Methods applicable to systems having a

mono-tonic step response 174

10.5.1 Introduction 174

10.3.2 Method of Clair 175

10.3.3 Method of Thai-Larsen 175

10.3.4 Method of Ormanns 175

10.3.5 Method of Streic 175

iO..6

Method of Naslin 175

10.4 Method of moments 176

10.5 Method of Westcott 176

CHAPTER 11 APPLICATION OF SOME APPROXIMATE FOURIER TRANS-FOR METHODS TO AN AUTOCORRELATION FUNCTION AIW TO STEP RESPONSES DISTORTED BY NOISE.

178 178 178 179 180 180 180 1 80 182 183

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APPENDIX 2 Proof of the identity, for k> 1, of the multi-plicative correction functions Wk(8) and the

weight functions wk(e) 188

APPENDIX 3 Derivation of (7.21) from (7.26) 189

APPENDIX 4 Derivation of (9.5) from the sampling theorem 190

APPENDIX ₅ Tables of functions 190

REFERENCES 193

SUMMARY 198

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CHAPTER 1 INTRODUCTION AND FUNDAMENTAL

RELATIONS

i INTRODUCTION.

In many control engineering problems it is important to consider both the frequency response and the response to aperiodic input signals.

For linear systems, to which this study will be confined, the

response x0(t) to an input signal x(t) is given by

x(t)

_=f

_{x(t_t) h()d}

_(1.1)

-where h(t) is the impulse response of the system. Thus, the

re-sponse to any specified input can be determined if the impulse

response is known. In the following, the step response c(t) will frequently be used as a characteristic transient of _{a system. This} is the response to a unit step input at t=O, initial _conditions being zero. It follows from (i.i) that

c(t)

f

o

It is sometimes more convenient to _{measure the step response than} the impulse response.

The fequency response H(jL) of a linear system is _{the Fourier}

transform of the impulse responseh(t) provided the _transform ex-ists0 The Fourier transform of the step _{response does not exist} generally but if the step response approaches the _{constant value}

C as t- the Fourier transform of the transient _{error c(t)(C_c(t)} exists and equals {H(0)_H(ju)}/jw (cf.(1.12) _{and (1.23.2)) from} which H(jw) can be evaluated.

It is also possible to derive the frequency response from the response to an arbitrary input signal. Thia can be performed in two ways:

The Fourier transforms (frequency spectra) X(jw) _{and X0(jw) of} x(t) and x0(t) respectively are determined assuming that these transforms are defined). Then H(jw) follows _{from the}

equa-t ion

X0(jw) = X1(jw) H(jw) (1.2)

whoh is the Fourier transform of (1.1).

Equation (1.1) is solved for h(t) followed _{by a Fourier} trans-form of this function0

Conversely, the impulse response is the inverse Fourie'

trans-form of the frequency response; the step response can be derived from an inverse Fourier transform of {H(0)-H(j)}/jgi.

It follows from the above that the dynamic behaviour of a sys-tem can be described by the frequency _{response or by the response}

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to an aperiodic test signal e.g. impulse or step input and that

the descriptions in the frequency and tine domains are equivalent. A linear system can also be characterized by the differential

equation or by the transfer function H(s) which may be derived

from the differential equation (and vice versa) and which is the Laplace transform of the impulse response. The differential equa-tien as well as the transfer function completely describe the linear system. If one of these characteristics is known, the

frequency response can easily be derived but the determination

of the response to aperiodic input signals directly from the differential equation or from H(s) requires the solution of the

differential equation or the evaluation of the inverse Laplace

transform. The latter method implies the solution of the

charac-teristic equation (exactly or approximately); alternatively, an

approximate inverse Laplace transform method may be utilized

[E2; W2; L1,285 ff; H5; K5; X4,5; H6,700]*. Such methods will

not be considered hereafter. The calculationsinvolved in the

above techniques may be laborious unless analog or digital com-puter facilities are available.

If the differential equation of the system or its transfer function is not or only partly known analytically as is frequently

the case in the analysis of existing systems, the frequency re-sponse or the rere-sponse to sorne aperiodic input may be obtained

by direct measurement. It may be more convenient to measure the

response to an aperiodic test signal than to measure the

re-sponse of the system when it is excited by sinusoidal inputs of various frequencies and inversely. For some reasons knowledge of

the other response may, however, be desired. This illustrates the importance of methods which permit a rapid and sufficiently ac-curate evaluation of the direct and inverse Fourier transforms.

The study of such methods constitutes the main subject of this

thesis.

If the frequency response is specified or given as experimen-tal data, the evaluation of the inverse Fourier transform is not the only way of obtaining the transient response of the system. It is also possible to derive a transfer function from the

fre-quency response data. This is a problem related to a well-known approximation problem in network theory [T13,345]. The given

gain characteristic is approximated by a suitable rational

alge-braic function i.e. in terms of pois and zero locations. At the same time the phase characteristic is inspected; if necessary, a

transport delay is introduced and zeros are reflected at the ju-axis in order to obtain nonmininirr-phase characteristics. Once a suitable transfer function has been derived, the methods ferred to previously may be used for obtaining the transient re-sponse. It is felt that, generally, this procedure is of more

importance in the design stage than in the analysis of existin systems. In the design of compensating networks or the

reali-* References, indicated between square brackets, are listed at

the end of this thesis (second and following numbers indicat

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15 zation of given gain and phase characteristics it is preferable to work in the s-plane because this leads to a logical design procedure. The approximation to given gain and phase characteris-tics in the s-plane is rather tedious if an accuracy of a few percent is required and the system is complex. If one is interes-ted in the transient response only, it is often preferable, then, to apply an approximate inverse Fourier transform technique. How-ever, if the accuracy required is not high (say 10_20%) it may be expedient to derive a transfer function corresponding to the fre-quency response characteristics. Then the system can be simulated on an analog computer and the transient response recorded. This

method also seems useful if the response of one system to many

different inputs has to be determined.

Conversely, if the response of a system to a deterministic aperiodic input signal has been measured or is specified, the eva-luation of the direct Fourier transform is not the only way of ob-taining its frequency response. Other technioues may be applied

e.g. that of the Laplace transform. A number of acproximate Lapla

transform methods have been developed

[D7;E1;L6;G6,707;ic5;H6,661-E5] . These methods will not be discussed. The evaluation of the Fourier transform via the Laplace transform generally implies a multiple of the computing effort required for the majority of the

methods for evaluating Fourier transforms directly. Only in some

special cases the Laplace transform is an appropriate technique for obtaining the frequency response (cf.Ch.10).

If a system to be analysed is subject to noise or if it is not allowed to be excited by sinusoidal or aperiodic inputs, random variations of the input (already present or deliberately introduced) may be used together with the random variations of the output to derive system characteristics. If correlation tech-niques are applied, it may be suitable to transform the correla-tien functions in order to obtain power density sPectra. From the Fourier transforn of the autocorrelation function of the input and that of the crosscorrelation function of input and output the frequency response of the system is obtained by applying a formula similar to (1.2)

[s9,126J.

Since the correlation functions are presented as empirical data, it is generally most expedient to evaluate the Fourier transforms numerically or graphically.

So far responses of physical systems have only been considered.

The approximate Fourier transform techniques are, however, also useful in the study of signals whatever their origin. As has been

indicated previously, a deterministic signal and its frequency spectrum constitute a Fourier transform pair (assuming that the transform exists). The same relation links up the autocorrelation function of a stochastic signal and its power density spectrum.

The problem of evaluating the Fourier transform of a transient or a signal being substantially zero outside a finite interval of time is closely related to the problem of harmonic analysis

as will be set forth in section 1.2.6. Many papers have been

deote'1 D the study of the latter problem since the last decades

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16

Numerical or graphical methods of evaluating Fourier transforms of responses and signals have been extensively studied in the last two decades and much literature on the subject has been published. It is strikin that only few authors in the control field (e.g. Schneider [s5J and Barber [B5,12,28] seem to have noticed the close relationship between the Fourier transform and harmonic analysis as well as the high degree of perfection which the nu-merical methods of harmonic analysis had reached as early as 1905.

This appears from the publication in recent years of approximate Fourier transform methods that are absolutely inferior to those easily derived from some existing methods of harmonic analysis of empirical periodic functions. It also appears from the fact that the majority of the published approximate Fourier transform

methods were found to have their counterparts in harmonic anal-ysis which were generally published decades previously (cf. sect. 6.1, 7.5.6.2, 8.5.1 and 9.1) without being referred to by the authors on approximate Fourier transform methods.

In view of' the wealth of literature on the above subjects it

was considered useful to present a critical survey of published

approximate (direct and inverse) Fourier transform methods in-cluding methods of harmonic analysis. The purpose of such a

sur-vey was meant to be a classification of the most valuable methods

with respect to their applicability to specific control

engi-neering problems being giventhe accuracy of the data, the

distor-tion by noise, the desired accuracy, etc. Furthermore it was ex-pected that the study of existing methods would lead to the

de-velopment of new or improved methods. Many methods could be improved

indeed. Besides, some at least partly new methods will be

pre-sented. The development of one of these methods has been greatly

stimulated by a study of approximate methods of harmonic anal-ysis published many decades ago.

Some reviews of the subject have been published previously,

viz, the reports of Eggleston and Mathews (1954) [El], Buss and Donegan (1956) [R9] and Schneider (1961) [s5]. The scope of these

reviews is limited in one or more of the following aspects: The first rsport only deals with direct Fourier transforms

whereas the other two papers are confined to the evaluation

of inverse Fourier transforms.

The methods discussed are sometimes applied only to elementary systems such as first or second order systems which neither

require general methods (cf. Ch.10) nor constitute, as a rule, severe tests.

The influence of noise distorting the transient response from which the frequency response is evaluated is not estimated comparatively for the various methods.

Among the reviews of methods of harmonic analysis the publi-cations of the following authors have been considered: Grover (1913) [04], Russell (1915) [R13], Dellenbaugh (1921) [D6], Ro-bertson (1935) [R4] and Barber (1961) [B3]

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critical survey from the engineering point of view* in which all published approximate direct and inverse Fourier transform methods known to the author including the improved ones would be examined. The promising methods have been applied to a num-ber of test systems and judged in the following respects: accu-racy, speed, scope of applicability to different types of re-sponses and signals, and sensitivity to distortion of input data

by noise. An extensive literature research on approximate Fourier

transform methods and approximate methods of harmonic analysis has been made but the review does not pretend to be exhaustive.

The criteria for the accuracy of the results have been defined

as follows: error < 1% very good, 1-2% good, 2-5% satis-factory, 5-10% reasonable, >10% unsatisfactory. With Re H(j)

or Im H(jw) the percentage refers to IH(0)I f'

IReH(jw)IIH(0)I

or Im H(jw)jlH(0)I, alternatively the percentage equals the

relative error. With c(t) the percentage refers to the value Cj.

The computing times reported in the sections dealing with the application of the methods to the test systems are based on the assumption that the given response is available in graph-ical form. They have only relative value since the calculations have been performed by the author who had not much training be-cause each method was applied only a few times. Therefore the computing times above 100 min have been rounded off to multiples of 10, those between 50 and 100 min to multiples of 5. It is probable that a trained calculator will be able to reduce some of the times reported. The comparison of speed is based on the

use of simple numerical or graphical techniques, the computing

aids utilized being restricted to slide rule., desk calculator, special cursors, nomograms, graduated discs and a simple pho-tographic device. If Fourier transforms have to be evaluated only occasionally, it is useful to have methods at hand which can be performed quite rapidly using simple devices. If the cal-culation of Fourier transforms becomes a matter of routine, it

nay be economical to use a special purpose computer (generally

an analog device). Since a discussion of such devices is outside the scope of this investigation, only some references will be presented. For a survey of mechanical electrical and optical analogs the reader is referred to [s3j. The following papers on this subject are not covered by this review: {L9; R2; R3; Il;

we; ci] . If a digital computer is available, it can be used

to perform the calculations required with a number of the appro-ximate Fourier transform methods to be discussed in the following chapters (cf. sect. 6.2.7.1.5).

The systems to which the promising methods have been applied were carefully selected in order to make them more or less re-presentative of linear approximations to a group of actual systems. The following systems have been chosen: a third order closed-loop

*In the following sections the main conditions governing the

formulas utilized are indicated but full mat1ematical rigour is not claimed since it was not intended to present mathe-matical theory.

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system having duplicator characteristics as met with in the

ser-vomechanisms field, a lightly damped closed-loop system including

a transport delay in the forward path, and a nonminirnum-phase fourth order system with poor damping as may be found in the process field. Deliberately, first and second order systems and critically damped or aperiodic systems have not been chosen as test systems. As a rule, such systems do not present difficulties in the application of approximate Fourier transform methods.

Besides, some special methods have been published which permit

a rapid evaluation of the transform in these cases (cf.Ch.1O).

The application of the methods to the test systems resulted

in the selection of a number of methods which proved most

valu-able by the standards accuracy and speed. In order to test their applicability to more complex signals and responses some of

these methods have been used for the evaluation of a power den-sity spectrum from an autocorrelation function having an oscil-latory character.

It is emphasized that in the application of the methods to the three test systems mentioned above, the data - either

fre-quency or step response data - from which the "unknown" response

was evaluated, were assumed to be accurately known. This was

necessary in order to be able to test the methods with respect

to accuracy. In practice noise may interfere with the

measure-ment of the response. The influence of noise on the measuremeasure-ment

of the frequency response can be considerably reduced by special techniques [s5; C7] so that the above assumption is reasonable

for frequency response data. In measuring the step response, the distortion by noise cannot be prevented in many cases. If the bandwidth of the system is known, high-frequency noise can be filtered out but the influence of noise in the pass-band cannot be reduced without introducing statistical techniques such as the averaging of a number of step responses or the use of correlation

functions. However, if the step response is not too badly dis-torted by noise, the distortion of the frequency response

evalu-ated from the recorded step response by an approximate mathod may still be permissible. It is to be expected that the influence

of noise on the accuracy of the approximate frequency response

will be different for the various transform methods. For those methods which proved most suitable in the "noise-free" case this has been investigated by applying them to step responses of the

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1 .2. FUNDAMENTAL RELATIONS BETWEEN TIME AND FREQUENCY DOMAIN.* 1.2.1. FOURIER AND LAPLACE TRANSFORMS.

1.2.1.1. DEFINITIONS.

The Fourier-Stieltjes transform of a real function (t) will be defined by

F(v)

e_2Vt d(t)

(i.)

where V is a real number. The integral exists whenever (t) is

of bounded variation on the interval (.-; +oo).

If ,(t) is differentiable for all t we have, defining ,(t)f(t),

F(v) =ff(t)

e_2Thjvt _dt. _(1.4)

- co

F(v) is the (direct) Fourier transform of f(t). The integral on the right exists whenever

f

If(t)I -co

The inversion formula for the Fourier transform defined by (1.4) reads

* {f(t)+f(t-) =fF(v) e

2itjvt dv (1.5)

For the validity of formula (1.5) the reader is referred to the literature [T 9j. For the functions studied here the formula applies.

As a rule, the variable y will be used for frequency and po-wer density spectra of signals but for frequency responses of systems the variable

w =

2v will be introduced in most cases.

The use of w is more adequate here since the frequency response

H(jw) of a system can be most easily related to the transfer function H(s) because s =0+

jw.

The variable jw is used in lieu of w in order to preserve the same coefficients in the for-mulas of E(s) en H(jw); these coefficients are real constants for the systems considered. Furthermore, the current literature on the subject dealt with is generally written in terms of w The use of tables, cursors and other computing aids would there-fore be cumbersome if w were not used.

The formulas relating frequency and transient response can be written as one-sided Fourier transforms since transient phe-nomena can be made zero for negative values of t by a suitable

time shift and because the real and imaginary parts of the

fre-quency response of the systems considered have even and odd symmetry respectively. The formulas to be used in the following

chapters have been collected in the sections 1.2.2 to 1,2.6.

* The author is indebted to prof.dr.ir.J.W.Cohen for valuable

criticism and helpful suggestions.

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20

The Laplace transform will be used in connection with res-ponses of systems; therefore it suffices to consider the one-sided transform.

The Laplace-Stieltjes transform of (t) is defined by

00 r _-st

F(s)

= J

e d(t), Re s so (1.6)

whenever a real values of s exists for which the integral converges.

If (t) is differentiable for all t we have

00 e dt. F(s) =

f

f(t) -st o

is the Laplace transform of f(t).

The inversion formula for the Laplace transform reads

1 st (1.8)

--f

F(s)e ds = O for t < O

= f(0+) " t = O c s

C-300

= -{f(t+)+f(t_fl t, t > o

For the conditions imposed on the formulas (1.6) to (i.e) the reader is referred to the literature [W10,35]. For the functions considered hereafter the formulas apply.

1.2.1.2. THE APPLICATION 0F FOURIER AND LAPLACE TRANSFORMS.

In the application to the responses of physical systems or

physical signals the conditions for the existence of the Laplace transform are always satisfied. Those required for the existence of the Fourier transform are met in many cases; if not, some artifice may be used.

For stable systems which can be described by linear diffe-rential equations with real and constant coefficients the

re-sponse to a non-periodic input such as an impulse,astep or a ramp function is given by

N

-p t

f(t) =1 anQ(t) e

n

sin(yt_0),t > 0

(1.9)

where a , 3 , y and X are real constants; 13 O for all n;

is a polynomial of degree m 1 [G1,164] . The

num-ber N is finite for all systems which do not include a transport delay element in a closed loop; if a transport delay is present

in a closed loop, N is infinite but it may be proved that the infinite series converges.

It is concluded from the above that the Laplace transfcrms of responses of the systems considered to the above input signals, responses which can be described by (1.9), always exist. The

La-place transforms of the impulse, step and ramp functions exist

likewise.

Fourier transforms of functions given by (1.9) exist if the sum on the right does not include terms which behave like t1,kO,

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for large t. Thus the Fourier transform of the impulse response generally exists but it is non-existent for step and ramp func-tions. Furthermore the Fourier transform of the response of a system to a step or ramp input is often non-existent. In such cases it is sometimes possible to introduce an artifice leading to a converging Fourier transform. Some examples will be pre-sented:

For a discussion of this case the reader is referred to sec-tion 6.5.1.

Physical signals are generally non-zero for a finite length of time only. In such cases Fourier transforms are applicable. Periodic signals have no Fourier transform. In this case the appropriate technique is that of the Fourier series. In the following, stochastic signals will also be considered. The des-cription of such signals may be given in terms of correlation functions or power density spectra; the latter may be derived from the former by a Fourier transform and/or a Fourier series

[s 9, 971.

1.2.2. RELATIONS BETWEEN TRANSIENT AND FREQUENCY RESPONSE.

'It

= o

21

1. If the step response tends to a non-zero the transient error c(t) C-c(t) will be

constant C as t..00,

absolutely inte-grable to infinity by virtue of (1.9)

have a Fourier transform.

and consequently will

2. It may occur that f(t)-. (t-t1) as t -.

constant.

, for a a non-zero

For a physical system which can be described by a linear

differential equation with real and constant coefficients and which has a transfer function having no poles on or to the right of the jw-axis and tending to zero as s one may derive [Blo,532; L4;

J5.

H(jw)

=Jh(t)eJWt

dt (1.10) Re R(jw) =

f

h(t)cos uit dt (1.10.1) Ire H(jtv -

I

h(t) uit dt (i .10.2) 'Jo

H(jui) eJWtd c(t) (i.ii)

Re H(jw)

=f

cos uit d c(t) (1.11.1)

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22

E(iu)=C_iwf c(t)

e_Wtdt

* _(1.12)

Re H(jw)=C-

uf

c(t)

sin

ut

dt _(1.12.1)

Im H(ju)=

_wf e(t)

cos ut dt (1.12.2)

o + 00

h(t)=

₂

i

H(jw)

ejut du t O (1.13)

h(t)

=

f

Re

H(jw)

cos ut dw

t

0

_(1.13.1)

-f

Irs H(jw)sin ut du

t > O

(1.13.2)

c(t) =

f

Re i(,jw)

t

0

(1.14)

o c(t) =

ii(o) +

rRe H()-R(0)

lt u

sin

ut

du, t>O(1.14.1)

Jo

c(t) =

s(o)+

coz ut du t

0ImH(0)=O.

2f

Ici H(,iu)

lt w

0

_(1.15)

Whenever

a function

f(t)

_{is discontinuous at}

t it will be

tacitly assumed that f(t) is replaced by - _{{f(t)+f(t_fl in the}

formulas for the inverse Fourier transforms givi in the

sstiis 1.2.2 if.

1.2.3. RELATIONS BETWEEN SIGNALS AND FREQUENCY SPECTRA AND

RE-LATIONS BETWEEN CORRELATION FUNCTIONS _{AND POWER DENSITY} SPECTRA.

The two-sided Fourier transform ((1.4) and (1.5))

can be used

if the conditions referred to in section

1.2.1.1 are satisfied.

If the signal starts at t=t0

_{(j t0j <}

₎ _{to may sometimes be} chosen as a new time-zero. After this time-shift both _{direct and}

inverse Fourier transforms can be written _{as one-sided transforms,}

the formulas being similar

to

_{those relating frequency and} im-pulse response

_{((1.1O)-(1.1O.2),}

_{(1.13)-(1.13.2)). Ifsucha} tine-shift is not desirable, the signal

_nay

_{be split up into two}

signals which are even and odd functions respectively

_[33,15]

For esch of these components the yourier transforms _{can be} writ-ten as one-sided transforms.

For correlation functions and power density spectra _relations

similar to (1.4) and (1.5) are valid if the _{conditions referred} * _{See Appendia 1.}

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23

to in section 1.2.1.1 are satisfied. If the correlation function has d.c. or periodic components, these should be subtracted and analyzed harmonically; in general, the remainder will be abso-lutely integrable and then will have a Fourier transform.

Because of the symmetry of autocorrelation functions, the formulas relating these functions to power density spectra may be written

(v)

=

21 ''0(t)

cos

2tjvtdr

(1.16)

=

W(v) cos 2mjvt dv

(1.17)

It should be noticed that frequency spectra and power den-sity spectra behave differently from frequency responses when the time-unit is changed. Let t and w indicate time and frequency before the change of units and t' and w' hereafter. Since H(jw) is no function of t, it is invariant: H'(jw')=H(jw). The spectrum F(v) of a signal f(t) and power density spectra such as

(y)

are not invariant with respect to a change of the time-uni

be-cause f'(t')=f(t) corresponds to F'(v')dv'=F(v)dv and '00(t')=

(t) corresponds to '1'' 0(v')dv'= '00(v)dv 1.2.4. FOURIER TRANSFORMS INVOLVING DERIVATIVES.

A number of approximate Fourier transform methods apply f

or-mulas which constitute a relation between a function of time or

frequency and one of the derivatives of its Fourier transform

[@5; T12j. Such formulas may be derived from each of the rela-tions presented in the secrela-tions 1.2.2-1.2.3 but only those

for-mulas are considered which will be used in some of the following

chapters. Unlike the literature referred to, the following dis-cussion is confined to formulas involving first and second de-rivatives. The relations have been derived by one or two suc-cessive integrations by parts from some of the formulas given previously. Only the results will be presented here. In order to avoid repetition it is assumed that the functions of time or frequency considered in this section are twice differenti-able and that the integrals occurring in the formulas given hereafter exist.

If h(co)=O and

()=O

it follows from (1.10) that

00

H(jw)=

h(O)

1

.(O+)+J.

-jwt

-- {h

(te

utj

,w>o.

(1.18)

w

o 00

For w=O (1.10) changes into H(0) =

J

h(t)dt. Hence

H(0) = t2 ct) dt (1.18.1)

provided lin t h(t) = O and um t2 h(t) = 0

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o

provided lia t (t) = 0.

t-..

From (1.15.1) one may derive the relation

h(t)

=--

j

Re E(jw)

Re H(jwcoswt dJ t> O

o (1.20)

provided Re H(jw) and its first derivative tend to zero as w For t = 0+ we have co i 2 Re H(jw)dw h(0+) = - w aJ thu2 (1.20.1) o

_2d

provided lin w Re H(jui) = O and

lin w - Re H(jw) = 0.

_w

From (1.13.2) it follows (if Im E(0) = O and both Im H(jw) and Im H(jw) tend to zero as

w-00)

that

d co 2 h(t) =

---

f

__ Im H(jw)sin wt dw, t > 0.(1.21) o

There is no formula for h(0+)

in

terms of derivatives of Im H(jw) since (1.13.2) is not valid for t=0.

Finally, if Im E(0) and both Im H(jw)/w and its first deriva-tive tend to zero as

w-00,

one may derive from (1.15)

oc 2 ' d Im H(jw)) d In H(,lw) -Ç cos wt th ,t>0 c(t)=H(0) _-

_[Uw

_w

L0+

f

w o (1.2.) For t=0 we have 1 f' d Im H(jw) c(0)=H(0)+

J

thu w o

provided Im H(0) = 0, hm Im H(jw)=0 and

11mw -

2 d Im H(iw)

-0.

w dw w

1.2.5. ASYIIPTOTIC RELATIONS.

The final value theorem states that

him f(t) = lin s F(s) (1.23)

t.00

SO

if the Laplace transforms of f(t) and f(t) exist and s F(s) is analytic on the jw-axis and in the right-half plane [T 13,44].

(1.22.1)

24

If

() =

O it follows from (1.11) that H(jw) { (o+)

(t)e_wt _{at },} >0. (1.19)

For w = 0 (1.11) changes into H(0)

=J

_t=od c(t). Hence

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25

This and the following theorem will only be applied to impulse and

step responses of the physical systems defined in section 1.2.2. For these responses the above conditions are met.

If f(t) is identified with the impulse response h(t), (1.23) changes into

lin h(t) = 0 (1.23.1)

too

since um H(s) = H(0)<00 for the systems considered.

so

If f(t) is identified with the step response c(t) whose Laplace transform is H(s)/s, it follows that

11m c(t) = H(0) (1.23.2)

to0

The initial value theorem states that

hm f(t) =

lin s F(s) (1.24)

to

S.00

if the Laplace transforms of f(t) and f(t) and lin s F(s) exists

[T 13,46]. s

Applying this theorem to h(t) and c(t), respectively, one obtains

lin h(t) = 11m s H(s) (1.24.1)

o

s

lin c(t) =

hm H(s) = 0

(1.24.2)

tio

S-00

If H(s) is rational algebraic, it can be shown that lin s

S -. 00

11m w H(jw). The limit on the right is equal to lin wIn H(jw)

w_000

-2

since Re H(jw) 0(w )(woo) because it is an even function of w.

Hence

11m h(t)= - lin w Im H(jw) . (1.24.3)

t0

If H(s) is rational algebraic and satisfies the conditions of (1.24), a special relation nay be derived from theorem (1.174) given on page 46 of [T 13] which is quoted here:

If f(t)-. K tn_i as t 0, then F(jw)

+ n+i (i .25) (n-1)!

w-00

(3W) (3W)

where K and K' are constants.

Identifying f(t) with h(t) and F(jw) with H(jw) one may conclude:

if c(t)--

tn as t 0, then A(w) _I H(jw) -.- as

w-04

Hence,

n! n

d log c(t) t (t w

if lin

- hm

to

d log t

u

ct

- n, then

d log A(w) w d A(w)

lin - 11m

u-00

d log w A(w)

dw

- -n.

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2E

d log c(t)

_l

d log A(a)

1 26)

d log t -irn

d log w

From the above condition that H(s) be rational algebraic one may infer that (1.24.3) - (1.26) do not hold for systems including

a transport delay element.

1.2.6. RELATIONS BETWEEN FOURIER TRANSFORMS AND FOURIER COEFFICIENTS. 1.2.6.1. HARMONIC ANALYSIS OF A FUNCTION OF TIME.

When a non-periodic function

f(t)

defined on the interval [ O;T]

is expsnded on this interval in a Fourier series [L 8,3] , the

latter is a periodic function which equals

f(t+)f(t_)}/2 for Ot0.

It will be shown that the coefficients of this Fourier series are related to the frequency spectrum F(jw) of f(t). It suffices to consider impulse and step responses since the formulas applying to

impulse responses may also be used for signals (cf. sect. 1.2.3).

If the impulse response has settled to its final value (zero) at

t=T , it can be expanded in a Fourier series on the interval [O;T0]

as ollows: h(t+)+h(t-) - a + E (a cos 2m ₊ b sin 2m ,-), O t T 2 o m o m=1 o o where T 2 a

f

O h(t) eQs 2m dt m T o o brn 2

1T

h(t) sin 2m

_'4-

dt

Comparing these formulas with (1.10.1) and (1.10.2) one obtains

T 21t

(a _-

b), m

0 (1.28)

m

o

If h(O+) = h(T)=O it is preferable to expand h(t) in a Fourier

sine series [L 1,217]: h(t)

=

m1

b sin rn

4-,

OtT0, where

T 2 bm =

f

° h(t) sin m

74-

dt, n 1. o o

ing

(1.27.1) with (1.10.2) one finds

T

Im H(jm -) -

b.

o

n O (1.27)

2.

If

the step response has settled to a constant value at t=T0,

the transent error can be expanded in a Fourier series on the interval L0;T] as follows:

t

-

a + E (a cos 2m

t- +

b

sin 2mit), 0tT0,

o m n

2

(28)

where 2 T am =

J'

c(t) cos 2m dt b 7

JT

(t) sin 2m dt

Comparing these formulas with (1.12.1) and (1.12.2) one obtains

H(jrn

)=C_mn(bm+j a), n

o

(1.2)

o

If (o+) = (T0 = O it is preferable to expand «t) in a Fourier

cosine series [L 1,218]:

«t) =

a + rn1 a cos a O < t

< T,

where 2 a n T0

_J'

° e (t) cos a

4-

dt, ni O. o

Comparing this formula with (1.12.2) one finds

ImH(jm = -m

fl)

a a

T n

o

1.2.6.2. HARMONIC ANALYSIS OF A FUNCTION OF FREQUENCY.

Harmonic analysis may also be applied to frequency _{responses or} spectra if they are bandwidth-limited. Ii' the functions of

frequen-cy occurring in (i.i.i) to (1.15) and (1.17) are zero for iuQ,

they can he expanded in cosines or sines (depending _{ori the} ymme-try) on the interval L

-O;

+ The formulas relating the impulse

response and the Fourier coefficients of Re H(jw) or Im H(jw) and those relating the transient error and the Fourier _{coefficients of} Irs H(jW)/W will be presented here.

1. Re H(jw) is expanded in cosines since it is an _{even functior}

of w:

a

Re H(ja) = a + E a

cos na,

- a where o n n=1 o a

Re H(jw) cos na- da

, n O. o

Cooparing this formula with (1.13.1) one obtains

O

rt\

o

ri O O it n

o

(1.30)

3ecause In H(jw) is on odd function of _a, _{Lt is expanded in} sines:

ni

h sin -' where h -

J°

im H(ja) sin n d , n 1. 27 (i .29.1) 2 o s n o o

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28

By comparison with (1.13.2) one finds

h(n -) = - -2 b0, n 1. (1.31)

5. Im H(jw)/w is expanded in cosines since it is an even function

of w: Irs H(j)/w = a + E a cos n dw - where n=1 o 2 a = / ° Im H(jw)/w cos n d.w, n o.

oJ

o o

Comparison of this formula with (1.15) leads to

o

c(n -)=E(o)

+a , n

_n O.

o

4. If, for signals of finite duration, the origin of time is chosen at the moment the signal starts, the formulas relating h(t) and

ar or b are the same as those relating the signal and the Fourier

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29

CHAPTER 2 CLASSIFICATION AND REVIEW OF THE

AP-PROXIMATE FOURIER TRANSFORM METHODS

2.1. INTRODUCTION.

In section 2.2 a systematic classification of the methods will

be presented, followed by some remarks on the procedure adopted

in the discussions of the various methods. In section 2.3 the

methods will be reviewed on the basis of some theorems and with

a view to bringing out their mutual relationship. 2.2. CLASSIFICATION.

The methods have been classified in the following groups:

METHODS GIVING RELATIONS BETWEEN CHARACTERISTIC FREQUENCY AND

TRANSIENT RESPONSE PARAMETERS (chapter 4).

By means of relation established theoretically or empirically by a number of authors one may rapidly determine corle parameters of the unknown response.

METHODS APPLYING GRAPHICAL OR NUMERICAL INTEGRATION (chapter

5).

These methods are most readily explained by giving an example.

If one wishes to determine h(t) for t=t, n=1(1)N, from Re H(jw)

by (1.13.1) utilizing these methods, one has to evaluate each of the integrals

_J

Re H(jw)cos Wtn dw, n=1(1)N, separately. With

some of the methods of this group a number of points of the in-tegrand are computed and plotted for graphical integration of the resulting curve. Thus N integrands must be plotted and integrated. Other methods of this group apply procedures of numerical inte-gration (e.g. Simpson's rule) or use special cursors to evaluate the integrals.

The characteristic feature of all of these methods is that the N integrals are evaluated independently by approximating to the integrand as a whole as distinguished from methods

approxi-mating to the given function, e.g. Re H(jw) in the example, in

such a way as to make the Fourier transform integrable in a closed form. In the latter case the resulting expression can

rea-dily be evaluated for a number of values of the variable.

METHODS BASED ON SAMPLING (chapter 6).

In this group those methods are collected w}ich replace the

Fourier integral by a sum comprising equidistant samples of the

given response. Although theorem is of basic importance for these methods, it will be shown that on certain conditions a sampling rate significantly smaller than that required by Shan-non's theorem suffices to obtain accurate results.

METHODS BASED ON PERIODIC REPETITION OP THE INPUT SIGNAL

(chapter

7).

Though these methods may be considered as special sampling methods in so far as weighted samples from the given response

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50

enter into the formulas, they can also be interpreted in a different way: the input signal is thought to be periodically repeated so

that the output, being a periodic function too, has a line spec-trum consisting of harmonics of the fundamental frecuency. A nuin-ber of methods for evaluating inverse Fourier transforms are

based on this principle. It may be shown that with these methods the spacing of the harmonics has to be chosen in such a relation

to the settling tine of the transient response that Shannon's theorem is satisfied. ith one of the methods this is not required;

here, the concept of periodic repetition of the input signal is used to derive relations between series of weighted samples in the frequency and time domains which may serve for the evaluation

of Fourier transforms.

5.

METHODS BASED ON APPROXIMATION BY ELEMENTARY FUNCTIONS

(chapter

o).

The given response in either the time or frequency domain (in the latter case the real or imaginary part) is approximated by a weighted sum of elementary functions, i.e. functions having a

simple Fourier transform. Thus the Fourier transform is

approxi-mated by a closed expression which can be evaluated numerically

or graphically. The following approximations are met with: Staircase approximation.

Straight-line segments approximation.

Approximation by functions of higher degree such as second

and higher degree parabolas and exponentials.

Some of the methods classified as sampling methods can be con-sidered as methods of group 5. They will, however, be discussed in chapter 6 (group ) because they make use of equidistant sam-pling of the given response and differ only in the weight

func-tion utilized. It was desirable to discuss them together in order to show which type of weight function should be applied

with a given response.

6. METHODS BASED ON EXPANSION IN ORTH000NAL FUNCTIONS (chapter

9).

The given response is expanded in orthogonal functions. The Fourier transforms of these functions can be computed and tabu-lated. If the series converges rapidly, it suffices to retain

a limited number of terms.

7.

SPECIAL METHODS (chapter

io).

The methods collected in this group are less generally

appli-cable than those of the preceding groups. Some of them only apply to first or second order systems, other ones have been developed

for systems having a monotonic step response.

In the first section of each chapter (in chapter 8 in a

num-ber of subsections) a general discussion of the pertinent methods is presented. Any method that is obviously not competitive with

other ones with respect to speed or accuracy is discarded at once. The remaining methods are applied to the three test systems.

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31

In each group the methoc for evaluating both direct and

in-verse Fourier transforms are discussed first. Then the methods

approximating to the direct Fourier transform are dealt with and finally the methods for evaluating the inverse Fourier transform are considered.

In order to avoid ambiguity it was frequently necessary to

utilize a notation differing from that adopted in the

origi-nal papers.

In the classification, the methods have been described in terms of frequency and transient response in agreement with the

majority of the original publications. As indicated previously

(cf. sect. 1.2.3) a signal and its frequency spectrum may be transformed into each other by utilizing formulas similar to thoce relating impulse and frequency response. In order to

fa-cilitate the application of the methods, the formulas relating frequency to impulse response are given, if appropriate, in addition to those relating frequency to step response. As a

rule, only the latter are applied to the test systems.

The sections of the chapters 5 to ₉ dealing with the

appli-cation of the methods to the test systems have been divided

into parts A and B for those methods which were applied to both direct and inverse Fourier transforms. In part A the evalu-ation of the frequency response is discussed, in part B that of the step response (in chapter 4, A and B have a different meaning, cf. section 4.1).

2.3. REVIEW OF THE METHODS. 2.3.1. GENERAL SURVEY.

The methods of the groups 1 and 2 constitute two extremes. Those of group 1 make a maximum use of the theoretical and

ex-perimental knowledge about the relations between the frequency

and transient response of certain types of systems in order to obtain information concerning the unknown response with a minimum of computing effort. As a consequence, they are only succesfully applicable to some classes of systems and often yield rough es-timates

On the other hand, the methods of group 2 utilize only a few general properties common to responses of physical systems and cctual signals so that they are universally applicable, hut the computing effort tends to become unduly large if reasonable accuracy is required.

The methods of the groups 5 to 6 are in an intermediate po-sition. They are as universally applicable as the methods of group 2, at least theoretically. Their application to an ar-bitrary response or signal may be as laborious as that of the

methods of group 2 but for the applications considered here they

require less computing time for obtaining results of a reasona-ble accuracy.

This is because they utilize one of the following properties: Control systems or filters often have low-pass or band-pass characteristics and are bandwidth-limited to a few decades. Transients whose Fourier transform exists (cf. sect. 1.2.1.2) may be considered to be non-zero for a limited intervsl f

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32

These properties make effective the use of Shannon's sam-pling theorem and the relations between Fourier integrals and Fourier series discussed in section 1.2.6; furthermore,

they make it possible to approximate a given function by a

limited number of elementary functions.

The methods of the groups 3 to 6 will now be discussed in

more detail.

2.3.2. THE DIRECT FOURIER TRANSFORM.

The methods of group 3 evaluating the direct Fourier

trans-form are based on Shannon's sampling theorem [s7] which reads: If the frequency spectrum of a signal or the frequency response of a system is zero for wi a the signal or transient response

f(t) is completely determined by specifying an infinite series of samples of f(t) at intervals dt = z

They may be recovered by interpolation with sinc-functions

as follows:

sin (t-ndt) + sin(

t-n)

f(t)=Ef(ndt)

(t-ndt)

- n_r1

Q0t-z

. (2.1)

The Fourier transform of sin t/Qt is ,r/ = dt for w and it is zero elsewhere. That of {sin(Q0t_nz)?/(ç20t-nz) is

dt

or

u4<,

and O elsewhere.

Therefore the Fourier transform of (2.1) reads [ B8,77 _J F(jw) = dt E f(ndt)e dt o

n=-=0

_iui o

If the frequency response or spectrum tends to zero

asympto-tically as is often met with, an approximation to Q0 has to

be selected at which the magnitude of the response or spectrum has fallen off to a small fraction of its initial or peak value.

If only the transient response or the signal is known, Q has to

be estimated. A rough estimate of Q may be obtained by graphical extrapolation from wb and wd (cf. fig. 4.3 and table 4.6). A more reliable method has been presented by Danielson and Lanczos [D3,435]. These authors give some tests by which one may get an idea of the magnitude of the errors due to the selection of too a large sampling interval dt. These tests can be applied with a number of values of dt until the deviations are commen-surate with the errors in the transient response or signal. This procedure generally leads to a small value of dt so that a great number of samples has to be retained in (2.2). This implies that

the computing time may become unduly large. These tests have not

been applied hereafter because the errors could be estimated from the known frequency responses of the systems considered. Besides, it was found that it is often possible to compensate for the errors due to a departure from Shannon's theorem (cf. sect. 6.2), thus reducing the number of samples to be considered

as well as the computing time.

It follows from property 2 of section 2.3.1 that it suffices

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33

generally, to retain a finite number of terms in the sum on the right-hand side of (2.2). If a function is given by a finite number of samples, it is possible to expand the function in a

fi-nite Fourier series by a least-squares method 8;R1O,211;Z2,317] which yields the best approximation to the sample points

(tri-gonometric approximation)*.

If the number of Fourier coefficients is equal to the given num-ber of samples, the sample values are exactly represented by the Fourier series although the sampling rate may not be as high as

is required for defining the signal or response completely (tri-gonometric interpolation). The theorem valid for this case may be

stated as follows [R8]

If N equidistant samples of f(t) are given covering the interval T/N t T, N being even**, the coefficients of the Fourier

se-ries

*N-1

g(t) 4a +

E (a cos 2m + b sin 2m

m=1

+ -aN/2 cos N

which equals f(t) at the sample points t = n t = n T/N, n=1 (i )N, are defined by the formulas

N 2 n a = - E

fnt)cos 2m s

n N N 2 N n m=O(1) (2.4) b = - E f(ndt)sin 2m n n N N n=1

These formulas may be combined as follows:

= E

f(nt)e2"

itndt/T, m=O(1) a -j b n m n= 1 (2.3) (2.4.1)

where dt/T has been substituted for N1. Apart from a constant

(cf. (1.28)) the right-hand side of (2.4.1) is a special form of that of (2.2) (for w =2itm/T).

It follows from the above that formula (2.2) has meaning only

for w=2sm/T if the spacing of the (finite number of) samples does

not satisfy Shannon's theorem.

A number of authors have given computing schemes for the evalu-ation cf am and bm of (2.4)(cf.sect. 6.1). These coefficients are related to points of F(jw) at equidistant frequencies w=2nm/T. If the spacing of these points is too large, additional points may be obtained as follows: the Fourier transform of the right-hand, side of (2.3) is derived, which can be expressed in a closed

form. The resulting expression is readily evaluated for a

suffi-* The pertinent formulas have already been derived by Bessel in a different way [w9].

**This condition is not essential, but for odd values of N the formulas differ slightly frein those given here [ji].

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34

ciently large number of frequencies. The Fourier series in (2.3) is a special case of an expansion in orthogonal functions (group 6).

It has been indicated previously that it is frequently dif-ficult or undesirable to strictly satisfy Shannon's theorem in the application of sampling methods. If F(jQ) differs to some

extend from zero, (2.1) is only aoproximately valid (Q0 should

be replaced by Q). It will be proved in chapter 6 that the

appro-ximation of f(t) by sinc-functions is then, generally, not the most suitable one. The sinc-funotions can be replaced by other

types of elementary functions (group

5).

If a constsnt

inter-valwidth is used, the approximations by some types of

elemen-tary functions can be considered as weighted sampling methods (group 3). Let g1(t) be an elementary function centered at t=O and let f(t) be approximated by the series

+

I _{f(nM)g1(t-nt). The Fourier transform of this}

approxima-tion reads

-3wfldt

F(jw)= E

f(nt) G1(3w)e *

(2.5)

n=-where 01(jw) is the Fourier transform of g1(t). Comparing (2.5) with (2.2), it is seen that t is replaced by G1(juj), a weight function depending on the type of elementary function utilized.

The methods of group 4, which are for the greater part

appro-ximations to the inverse transform, will be discussed in the

following section.

2.3.3. THE INVERSE FOURIER TRANSFORM.

On definite conditions it is permissible to replace the integral constituting the inverse Fourier transform by a Fourier sum, i.e. to replace the continuous spectrum by a line spectrum, The spacing of the samples of the frequency response or spectrum required for an exact evaluation of the transient response or signal by a straightforward sampling method can be determined by applying Shannon's sampling theorem to functions

of frequency

[wii]

.

The theorem then reads as follows: If a

function f(t) is confined to the interval [ --Tc; + -T0J , the

Fourier transform F(,jw) is completely defined by specifying

an infinite series of samples of F(jw) at intervals w=2it/T0.

It may be recovered by interpolation with sinc-functions as follows:

2 sin -nit)

F(jw)

=

_00m I)

--wT-n°z (2.6)

This theorem may be proved along the same lines as followed by Shannon for bandwidth-limited signals [B8,76] , or it may

be derived from the symmetry of direct and inverse Fourier

transforms II B3,15 ]

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If (2.6)

applies, f(t) is related toF(j) by

f(t)=

jn

e3flt

jtT0

Utilizing the relation F(_jw)=F*(ju) which is valid for the responses and signals considered here, one obtains

f(t) = -F(0)+

n1

Re F(jn&)cos ndwt

-2m - E Im F(jndw).sin ndw.t,w= - , t1 <-T n=l o 2m = ,

It I<T0

o (2.7.1)

This formula may be changed into

f(t)= F(0)+

n1

F(jnd)j cosndtarg F(ini],

(2.7.2)

The right-hand sides of (2.7) to (2.7.2) contain samples of

both Re F(jw) and Im F(jw) (or F(jw)I and arg F(jw)). From sec-tion 1.2.6.1 one may derive Fourier extansions of f(t) containing samples of either Re F(jw) or Im F(jw). The formulas relating f(t) to Re F(jw) or Im F(jw) read

f(t)= *dw{ F(0)+ E Re F(jndw)cos ndwt (2.8)

f(t)=-

Im P(jn) sin ndw.t

(2.9)

where u = m/T0 and f(t)=O outside [o; T0].

The methods of group 3 approximating to the inverse Fourier trans-form are based on the above trans-formulas.

It follows from property i of section 2.3.1 that it suffices, generally, to retain a finite number of terms in the sums on the

right-hand sides of (2.7) to (2.9). If Shannon's theorem is not

satisfied and finite numbers of samples are considered, (2.8) and

(2.9) have meaning only at t=mn/0 where t0;c0] is the frequency

range covered by the samplec (cf. sect. 2.3.2 and 1.2.6.2). A different approach to the evaluation of signals or transient responses by sampling methods is the concept of periodic repeti-tion, upon which the methods of group 4 are based. If T0 is the exact settling time of the response or the duration of the signal assumed to start at t=O, the response or signal is thought to be part of a periodic response or signal which can be described in terms of a Fourier series.

For step responses the response to a square wave of a period Tf=2T0 is considered so that only odd harmonics of the

fundamen-tal frequency wf=2J/Tf = m/T0 occur; therefore the spacing of the harmonics is dw = 2m/T0, provided samples of both Re H(jw)

and Im H(j) (r from

H(jw)I and arg H(jw)) are considered.

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