Minimum phase in sampled-signal theory

lfU\8

Bibliotheek TU Delft

M I N I M U M P H A S E

I N

S A M P L E D - S I G N A L T H E O R Y

PROEFSCHRIF

BI2LIOTHttK

DER

TECHNJSCHE HOGESCHOOL

DELFT

ter verkrijging van de graad van doctor in de technische

wetenschappen aan de Technische Hogeschool Delft, op gezag

van de rector magnificus Dr. Ir. C . J . D . M . Verhagen,

hoog-leraar in de afdeling der Technische Natuurkunde, voor een

commissie uit de senaat te verdedigen op woensdag 4 februari

1970 te 16 uur

door

AUGUSTINUS JOHANNES BERKHOUT

Elektrotechnisch Ingenieur geboren te Den Helder

1

Aan mijn Ouders

Aan mijn Vrouw

Production Laboratory, Rijswijk. The author is very grateful to Shell Research N. V. for being permitted to use the results of the work as the basis of this thesis.

In particular he wishes to express his sincere thanks to Dr. W. L. Scheen for his encouragement and his most helpful comments, and to Dr. Ir. L. Ong-kiehong for many stimulating discussions. Thanks are also due to Mr. P.H. Muusze for his programming assistance.

CONTENTS 5 -page Glossary 7 Summary 13 Samenvatting 15 I. Introduction 17 II. Definitions and reappraisal of basic theorems 22

A. Basic properties of band-limited and sampled time functions 22

B. z-transform 35 C. Minimum-phase function 57

m . Minimum phase and signal length 80 IV. Minimum phase and least-sqiiare inverse filtering 95

V. Formulations of the necessary and sufficient condition for the

minimum-phase property 128 VI. Minimum-phase criterion 147

A. Minimum-phase criterion for signals of finite duration 147 B. Minimum-phase criterion for signals of infinite duration 187 VII. Minimum-phase criterion and stability constraints in systems

theory 199 A. Minimum-phase criterion and the Routh-Hurwitz problem 199

B. Minimum-phase fcriterion and the Schur-Cohn problem 210

Vin. Minimum phase and minimum complexity 221

IX. Conclusions 229 References 231 Figures 237

GLOSSARY

In our notation, time functions are used with different indications. For example the time function f(t) may be encountered as follows:

"'f--(t)

and its Fourier transform as

— I V —

f - ( v ) and its z-transform as

~ f " ( z )

The symbol written above left is used for labelling purposes, the symbol written above right denotes manipulation and the symbol below right gives

1 T

information about the function. For example, f. (t) denotes the transpose of

2'^jf

the first sampled time function with sampling interval A and f (v) denotes the conjugate complex of the second periodic Fourier transform. Formulae are referred to by page numbers. For example (100c) denotes the third formula on page 100.

List of symbols

A, a constants

a(t),a (t), a.(t) time functions which represent desirable information (chapter IV)

time functions which represent an analytic signal (chapter VIII) (p denotes that the time function is periodic, A indicates that

the time function is sampled with sampling interval A) B, b constants

b.(t) sampled time function with sampling interval A which represents noise

8

-D leading principal minor determinant of the correlation matrix

% x(o)/x®(o)

d. 'Schur determinant' of the order k (k = 1, 2 , . . , N) E total energy of a signal

diagonal matrix containing predicted tail energies

E predicted tail energy of a leastsquare p r e -diction filter with duration nA

^min^°^ energy of the difference signal 6(t) - y^(t), y.(t) being the output of a least-square inverse filter with duration (n+l)A

F triangular matrix containing samples of least-square inverse filters with different duration f(t), g(t), h(t) non-sampled time functions; a time function may represent a signal o r the unit-impulse response of a linear time-invariant system f.(t), SAi), li.(t) sampled time functions of any duration with

sampling interval A; a sampled time function may represent a sampled signal or the unit-impulse response of a linear time-invariant discrete (digital) system

f„(t), g»j(t), h„(t) one-sided sampled time functions with dura-tion (N+1)A

^-NT N (*)• S_vr vr (t). h_vr u (t) two-sided sampled time functions ranging

— iN.,JN„ iN.,iN(^ JN.JJNQ

from (-N^A) to (+N2A)

f(n), g(n), h(n) nth sample of the sampled time functions f^(t), g^(t) and h^(t), respectively

f (n), g>j(n), h (n) nth sample of the sampled time functions

i^{t), gj^(t) and hj^(t), respectively

f (t), gA (t), h (t) transposes of the time functions f^W, gA(t) and h (t), respectively

+ + +

Ï A W , S A W . h (t) sampled time functions the z-transforms of which have no singularities for | z | < 1 sampled time functions the z-transforms which have no singularities for | z | > 1 ^«(t). ëA^), h.(t) sampled time functions the z-transforms of

f®(t), g®(t), h®(t)

f?(t). gf(t). h®(t)

jnin,,. min... .min... V <*)• ^A (*>' '^A (*)

O * ) ' %"(*)• '^"<t)

fp(s). gp(s), hp(s) T ( v ) , ^ ( v ) , Sp(v) fN(v). g^C^). ^ N M f(z),. i ( z ) , h(z) ? N ( Z ) . gN^"")' "^^""^ H{f^(t)} j K. k, L, 1 L(z) M, m, N, n o^(t) P ^A^t) R (t) '^A'^A

energy-bounded sampled time functions the z-transforms of which have no singularities and no zeros for | z | < 1 (generalised minimum-phase time functions)

energy-bounded sampled time functions the z-transforms of which have no singularities and no zeros for | z| > 1

minimum-phase sampled time functions maximum-phase sampled time functions Laplace transforms of the sampled time functions fA(t), g.(t) and h.(t), respectively Fourier transforms of the sampled time functions f^(t), g^(t) and h.(t), respectively

(p denotes that the Fourier transforms are periodic)

Fourier transforms of the sampled time functions f^(t), gvf(t) and h„(t), respectively z-transforms of the sampled signals fA(t), g.(t) and h (t), respectively

z-transforms of the sampled signals f^(t), g„(t) and h„(t), respectively

Hilbert transform of the sampled time funct • / ^ 0, +1, +2 CO 2 f(n)z'^ n=N+l 0, +1. +2

sampling operator with sampling interval A subscript for periodic functions, e.g. f (v), a (t)

entire sampled time function + 0O

function f (t)

J X*(T - t)X^(T)dT

R (t)

da(t) dt 10

\ . y / ) |"xJ(T-t)y,(T)dT

— QO R correlation matrix f > ) / f „ ( o ) s Laplace variable: s = o + J2TTV sign V +1 for v > O, - 1 for v < O SA(t) sampled signal without noise S.(t) recording without noise

T(s) transfer function of a linear time-invariant system

T(z) transfer function of a linear time-invariant discrete (digital) system

t time variable

u.(t), v.(t), w.(t) sampled time functions with sampling interval A

U(t) unit-step function (Heaviside function) X triangular matrix

x (t) sampled time function with sampling interval A, input for least-square inverse filters ^x^(t) x^(t + T)U(t)

y (t) sampled time function with sampling interval A, output of least-square inverse filters y (t) sampled time function with sampling interval

A, desired output of least-square inverse filters

-sA

z e

z zero of f(z)

2' ?J^®h..?ZS^2L?.

a,,3,Y constant phase angles

a(t) complexity instantaneous frequency

6(t) unit impulse (Dirac delta function) 6 Kronecker symbol

m , n

6, € Q r\ e V § a T Cf cp(v), Hv) cPp(v), 'l'p(v) min, , ,i,min, ^ cpp (V), tp (V) max, , .max, , cp®(v), t®(v) 3 • Pi^5r_ syïï.'^oi? ^ 1 £[t°3 "i?

a(--)

+

e

small positive numbers variable in the z-plane X + i2nt

constant phase angle real constant

frequency variable

variable on the frequency axis

real part of the Laplace variable: s = a + j2nv £ { ( t - £ [ t l } ) 2 ]

£ { ( v - e { v ^ ] ) ^ }

variable on the time axis triangular matrix

phase spectra of non-sampled time functions phase spectra of sampled time functions

(p denotes that the phase spectra are periodic) phase spectra of sampled time functions with duration (N+1)A

phase spectra of minimum-phase sampled time functions

phase spectra of maximum-phase sampled time functions

phase spectra of generalised minimum-phase sampled time functions

unit circle in the z-plane

nth-order moment of a sampled time function 1. 2

is used before an integral and denotes that the principal value should be taken

variation of the quantity within brackets

see see see see see see see f^(t) f^(t) '>) ' > t(S) $(v) f(z)

13 -SUMMARY

A theoretical investigation has been carried out on the minimum-phase aspects of sampled signals. The signals may be complex and of infinite duration.

The first part is devoted to basic properties of band-limited and sampled time functions, and to properties of the z-transform. The definition of the minimum-phase property of a signal is given by considering both a signal and

*)

its frequency inverse '. It is proved that there exists at most one minimum-phase signal for a given amplitude spectrum. Properties of the minimum-minimum-phase signal are derived.

A quantitative measure of signal length is defined by means of a set of moments relative to the time origin. In particular the length of minimum-phase signals is investigated. It is shown that among all one-sided signals**) with identical amplitude spectra, the minimum-phase signal has the smallest length.

The minimum-phase signal plays a unique role in the least-square inverse-filtering theory**"*^. The consequences of this are more closely investigated by an extension of the Wiener-Hopf technique.

With the aid of the least-square inverse-filtering theory, we have derived a minimum-phase criterion for signals of finite duration and signals of infinite duration. As an immediate consequence, a new interpretation of Schur and Cohn's stability constraints has been given.

Finally, the minimum-complexity signal****) is introduced. Owing to the similarity of complexity in the time domain and phase in the frequency domain, properties of the minimum-phase signal are translated into properties for the minimiun-complexity signal.

*) The time function x(t) is called the frequency inverse of a signal f(t) if the convolution of x(t) with f(t) yields the unit impulse 6(t).

**) A signal, represented by the time function £(t), is called one-sided if fl[t) = 0 for t < 0 .

***) A least-square inverse filter converts a signal optimally into an (the unit) impulse according to a minimum-energy criterion.

SAMENVATTING

Een theoretisch onderzoek is gedaan naar de minimum-fase aspecten van bemonsterde signalen. De signalen mogen complex zijn en van oneindige duur.

Allereerst worden fundamentele eigenschappen behandeld van tijdfuncties met beperkte bandbreedte, bemonsterde tijdfuncties en eigenschappen van de z-transformatie.

De definitie van de minimum-fase eigenschap van een signaal wordt gege-ven door zowel het signaal als zijn frekwentie inverse*) te beschouwen. E r wordt bewezen dat er maximaal één minimum-fase signaal bestaat met een ge-geven amplitude spectrum. Eigenschappen van het minimum-fase signaal worden afgeleid.

Een kwantitatieve maat voor signaallengte wordt gedefinieerd door middel van een aantal momenten ten opzichte van de tijdoorsprong. In het bijzonder wordt er bewezen dat van alle éénzijdige signalen**) met identieke amplitude spectra het minimum-fase signaal minimale lengte heeft.

Het minimum-fase signaal speelt een belangrijke rol in de theorie van het inverse füteren volgens de methode van de kleinste kwadraten***). De konsekwenties hiervan worden nader onderzocht met een uitbreiding van de Wiener-Hopf techniek.

Met behulp van de theorie van het inverse filteren volgens de methode van de kleinste kwadraten wordt een minimum-fase kriterium afgeleid voor signalen van eindige en oneindige duur. Een onmiddellijk gevolg hiervan is dat een nieuwe interpretatie wordt gegeven van Schur's- en Cohn's stabiliteitsvoorwaar-den.

Tenslotte wordt het signaal van 'minimale gecompliceerdheid' geïntrodu-ceerd. Door de overeenkomst van het begrip 'gecompliceerdheid' ****) in het tijdgebied en fase in het frekwentiegebied worden eigenschappen van het mini-mum-fase signaal vertaald in eigenschappen voor het signaal met 'minimale gecompliceerdheid' .

*) De tijdfunctie x(t) is de frekwentie inverse van een signaal f(t) indien de convolutie van x(t) met f(t) de eenheidsimpulse oplevert.

**) Een signaal, gerepresenteerd door de tijdftuictie f(t), is éénzijdig indien f(t) = O voor t < O. ***) Een inverse filter volgens de methode van de kleinste kwadraten transformeert een signaal

optimaal in de (eenheids)impulse volgens een minimum-energie kriterium.

17 -I. INTRODUCTION

In the echo techniques used with sonar, r a d a r and in seismology for the measurement of location and velocity, the incoming total response from the measuring pick-ups is called a 'recording'. It consists of a sequence of similar reflections, called signals, with arrival times that contain the desired information. To avoid the masking of one signal by the tail of another, the signals have to be short, or in other words a good recording has to have a high 'resolution'. The length of the signals will depend on the length of the emitted test signal and on the distortion produced by the transmitting and reflecting medium. In terms of the Fourier theory, the amplitude spectrum of each signal should be as wide and flat as possible, while the phase spectrum has to be as small as possible. However, if a signal starts at a prescribed time t ( | t | < OD), the phase cannot become smaller than a certain value. This value depends on the amplitude spectrum. Consequently, for a certain amplitude spectrum there must exist a signal which has the smallest possible phase for each frequency. This signal is called the minimum-phase signal. It has the favourable property that its energy is concentrated as far as is possible towards its origin. For this reason this signal is also called a dela.y signal. Thus minimum-phase signals are desirable for high resolution in the recording.

The resolution of a recording can be improved by a process which flattens the amplitude spectrum. To maintain a minimum-phase signal, a change in the amplitude spectrum has to be combined with a change in the phase spectrum. The procedure for flattening the amplitude spectrum and minimising the corresponding phase is referred to as inverse filtering or deconvolution. The application of the inverse-filtering process to the minimum-phase signal has the unicpie property that its time function need not be known. This is of great importance since in many cases the shape of the received signals is not precisely known.

Because of the unique properties of minimum-phase signals on the one hand, and the increasing importance of digital processing on the other, a theoretical investigation of the minimum-phase aspects of sampled signals has been carried out. The starting points of the investigation can be distinguished into five p a r t s :

Definition and properties of the minimum-phase signal Minimum phase and signal length

Minimum phase and least-square inverse filtering Minimum-phase criterion

1. Definition and properties of the minimum-phase signal

In the existing literature it is customary to define the minimum-phase signal in terms of zeros and singularities of its z-transform. We follow an approach which considers both the signal and its frequency inverse. Taking into account the two signals (which may be complex and of infinite duration), a general definition of the minimum-phase property is obtained. In this way, a simple formulation can be given which presents the most essential property (enei^y-bounded one-sided frequency inverse) in a direct way. Moreover, no ambiguity exists if the z-transform has zeros on the unit circle (|z| = 1).

We have not found any publication which gives a general treatment to basic properties of minimum-phase sampled signals. In chapter II-c, we prove that for a certain amplitude spectrum there exists at most one minimum-phase signal. The constraints for the existence of the minimum-phase signal are

related to Kolmogorov's conditions. We also prove that the minimum-phase signal has minimum phase, minimum phase derivative and minimum tail energy. It is shown that for signals of finite duration a maximum-phase signal can also be defined.

2. Minimum phase and signal length

Since signal length is considered to be an important property, we have investigated the length of minimum-phase sampled signals. For this, we have used a set of normalised moments of the sampled signal relative to the time origin:

2 (nA)''|f(n)|^

£ r t ' ' ] = I i ^ i k = 0, 1 2 |f(n)|2

n=0

f(n) being the nth sample of the one-sided sampled signal f^(t) and A being the sampling interval. The idea of u s i i ^ moments for a quantitative measure of signal length was introduced by Gabor (1946). With the aid of a second-order moment in the time and frequency domain, he derived the uncertainty principle and introduced the elementary signal.

Until now the use of signal moments for these types of proofs has not received much attention. As is shown in this work, the application of signal

1 9

-moments can be an extremely powerful tool. In chapter HI we prove that of all one-sided signals with a certain amplitude spectrum, the minimum-phase signal has a minimum kth-order moment (k = 0, 1 ). We also prove that the normalised first-order moment (ejqjressed in sampling intervals) of a signal with unit amplitude spectrum is equal to the number of zeros of its z-transform

inside the unit circle.

3. Minimum phase and least-square inverse filtering

In chapter IV, least-square inverse filtering is defined as a linear physically-realisable filter process which converts a signal optimally into the unit impulse according to a minimum-energy criterion:

" i2

Z I 1 - y(n) I is minimal

n = 0

y(n) being the nth sample of the output of the least-square inverse filter. There is a great deal published work on digital least-square (inverse) filtering, the most important contributions originating from Wold (1938), Kolmogorov (1941), Robinson (1954), Rice (1962) and Treitel (1964). They concentrate mainly on the derivation and properties of least-square prediction filters (Wold, Kolmogo-rov, Robinson) and least-square inverse filters (Robinson, Rice, Treitel).

In this contribution, properties of the output of least-square inverse filters are also investigated. Chapter IV starts with the relationship between least-square inverse filters and least-square prediction filters. The first theorems (which are closely related to theorems formulated by Robinson, 1963) are devoted to the minimum-phase property of digital least-square inverse filters of finite and infinite duration. With the aid of an extension of the Wiener-Hopf technique, the relationship between the factorisation problem*) and digital square inverse filtering is discussed. By investigating the outputs of least-square inverse filters, it is shown that the number of zeros of the z-transform x(z) inside the unit circle (|z| < 1) is given by the first-order moment of the output of the least-square inverse filter for the signal x(t). This property has been used as the starting point for the second minimum-phase criterion

(chapter VI-B).

*) In the factorisation problem, a time function x» (t), with x (z) 4 0 for N "^ l i is constructed such that a given autocorrelation function R (t) can be written as R (t) = 3r''(t) « HXA ("*)] •

4. The minimum-phase criterion

A minimum-phase criterion for sampled signals of finite duration is equivalent to necessary and sufficient conditions for the zeros of a polynomial

{= z-transform of the signal) to be larger*) than unity in absolute value.

Similar conditions are important in many other fields (e.g. stability of linear systems, stability of difference equations) and a great number of papers have been published on this subject. The most important contribution is that of I. Schur (1917, 1918), in which he derives a criterion for the convergence of a power s e r i e s :

T(z) = 2 f(n) z° for | z | < 1 n=0

From this criterion Schur formulates necessary and sufficient conditions for the polynomial

~ N f (z) = 2 f(n) z""

^ n = 0

to have all its zeros inside the unit circle (Schur's Conditions). In the existing literature this criterion is wrongly referred to as the Schur-Cohn conditions. The criterion requires the evaluation of N determinants of the order 2n (n = 1, 2 N). Schur also introduces a Hermitian form H [f„(z)} and shows that H {f„(z)} is positive definite if and only if f^(z) has all its zeros inside the unit circle. Cohn (1922) extends this result by showing that the number of zeros of f„(z) outside the unit circle can also be determined from this

N^ '

Hermitian form. However, he does not indicate how this result should be incorporated in Schur's criterion (this was done by Marden, 1949). Cohn also derives a set of stability constraints (Cohn's Conditions) which are most simple in comparison with Schur's conditions. Moreover, Cohn's criterion determines the number of zeros of i..rlz) outside the unit circle. Cohn's criterion can be

N^ '

derived directly from Schur's convergence criterion for power s e r i e s . However, Cohn's derivation is based on Rouché's Theorem. We have noticed that Cohn's powerful and simple criterion is unknown to many authors. - S ^ S A

*) z has b e e n defined by z = e . I f t h e definition z = e is used (as is c o m m o n i n systems t h e o r y ) , t h e word 'larger* should be r e p l a c e d by ' s m a l l e r ' .

4

21 -Because of the great number of computations involved with Schur's criterion, a great deal of effort has been made to simplify the stability determinants. The most important contribution comes from Jury (1962). He considers polynomials with real coefficients and derives, from Schur's criterion, constraints which require the evaluation of 2N determinants of the order n (n = 1, 2 N). However, the Jury-Schur stability constraints are still very complicated in comparison with Cohn's conditions. Moreover, we show (chapter VII-B) that Schur's stability determinants can be simplified considerably more than Jury suggests.

With the aid of least-square inverse filtering, we have derived a

minimum-phase criterion for signals of finite duration (chapter VI-A) and signals of infinite duration (chapter VI-B). As an immediate consequence, a new

interpretation of Schur's and Cohn's stability constraints can be given. In addition, the relationship between both stability criteria becomes obvious. 5. Minimum phase and minimum complexity

In the chapters II-VII, minimum-phase sampled signals are discussed by investigating properties of one-sided sampled time functions. In chapter Vm, we limit our attention to one-sided sampled frequency functions. The inverse Fourier transforms of these frequency functions yield complex periodic time functions. In the literature [ J . R. Carson and T. C, Fry (1937), D. Gabor (1946), J. Ville (1948), P . M . Woodward (1953), J. Dugundji (1958)], time functions with one-sided frequency spectra are referred to as analytic signals. The analytic signal a(t) is written as a product of two components:

a(t) = I a(t) I e^^^'>

dixit)

In the existing literature, | a(t) I and ,^ ' are called the envelope and

instantaneous frequency, respectively. We have used the name 'complexity' for the angle a(t).

In chapter VIH, the concept of an analytic signal is approached from a different point of view. Owing to the similarity of complexity in the time domain and phase in the frequency domain, properties of the minimum-phase signal are translated into properties for the 'minimum-complexity signal'.

II. DEFINITIONS AND REAPPRAISAL OF BASIC THEOREMS A. Basic properties of band-limited and sampled time functions

In this chapter, the sampling operator 0.(t) and the sampled time function f^(t) are defined. It is shown that the nonsampled time function f(t) can be reconstructed from the sampled time function f (t) by convolution only if f(t) is band-limited. It is also shown that for band-limited time functions, integrals such as the Fourier integral and the convolution integral can be written as s e r i e s .

Finally, the Fourier transform and the inverse Fourier transform are derived for sampled time functions. An expression for the energy of f (t) is given in terms of the Fourier transform. Moments in the time domain are related to derivatives in the frequency domain,

We will describe equidistant sampling in t of a time function f(t) by

f^(t) = f(t) [A E 5 (t - nA)] (22a)

6(t) being the unit impulse (Dirac's delta function). The periodic function

o (t) = A 2 6 ( t - nA) ^ n = - o °

is called the sampling operator, and A the sampling interval. Any realisation of the sampling operator will be called a sampling system, A sampling system has a channel input corresponding to the operand and produces a one-channel output,

With the definition f(n) = Af(nA) (22a) is written as + 00 (22b) + 00 f^(t) = 2 f(n) 6(t - nA) n = - ' »

(22b) is the basic notation for sampled time functions.

23

-+ 00

f (t.T) = f ( t - T) JA 2 6 ( t - n A ) 1

Sampling and shifting are in general not commutative. Only if T write

f(t - T ) 0^(t) = f(t - T) 0^(t - T) or

f^(t. T) = f^(t - T)

A system is defined as linear if the superposition principle holds for its iiput and output, A system is defined as time-invariant if a time shift of the input causes an equal time shift of the output only,

Theorem n - 1

The samplii^ system is a linear time-variant system. Proof

a. Consider the time function h(t) = Cjf(t) + C2g(t) For h,(t) we can write

A ^ '

h^(t) = h(t) 0^(t) = [c^f(t) + C2g(t)]o^(t)

= c^f(t)0^(t) + C2g(t)0^(t)

= ^l^A^*) ^ '=2gA<*) Thus the sampling system is linear, b. According to (22a) and ^ 3 a ) , we may write

(23a)

f^(t - T) = f(t - T) 0 ^ ( t - T)

and

f^(t, T) = f(t - T) 0^(t)

Hence only if T = kA (k = O, + 1, + 2 ) , may we write

f^(t, T) = f^(t - T)

Thus the sampling system is time-variant, Corollary

If a time function x(t) is formed from N identical time functions f(Q with different time shifts T, " ,

N

x(t) = 2 f(t - T.) \ * +^' + 2A k = l '^ ^

then the sampled time function x.(t) will consist of N different sampled time functions + 00 X (t) = A 2 x(t) 6(t - nA) n=-oo N = 2 f (t, T ) k = l ^ '^ Theorem II-2

If f(t) is a band-limited time function, ^ ( v ) = 0 for | v | ^yr, which can be represented by a Fourier integral, then f(t) can be reconstructed from f.(t) by convolution.

Proof

For the Fourier transform of (22ai we can write:

25 -?p(^^ = ^(v) je Op(v)

= r(v) «TA T e-J2^^°^1

1— n — — OD - 1 +00 = r ( v ) » 2 6 ( v - S ) n=-oo " S f (V - -7) n=-oo " (25a)

This result shows that the spectrum of a sampled signal is periodic with — , Note that if we sample with O. (t - T) (T ^ 0, + A, + 2A, . . . . ) , the spectrum does not become periodic:

m 0^(t - T) « » T(V) X [ Op(v) - j 2 n v T

<—4 T r(v-S)e-J2"'^(^/^)

n = -° With the constraint

f(V) = 0 for _{1^1 ^ 2 A}

we can write, according to (25a),

f'(v) =Tp(v) [ U ( v + i ) - U ( v - i ) ] _(25b)

U(V) being the unit-step function (Heaviside function). The inverse Fourier transform of (25b)yields

sin n t f(t) = f,(t) « ^

^' A TTt

Thus, the band-limited time function f(t) can be reconstructed from f.(1) by convolution.

Corollary.

"^ 1 1 1

If f(t) is a band-limited time function, f(v) = 0 for | v | 5 ÖA > which can be represented by its Fourier integral, then f(t) can be written as a weighted sum of sine functions:

• H*

f(t) = 2 " f(nA) ^ ^ ^ (26a) n=co i

-Without proof, we state:

Lemma II-1

10 o K Q « H — 1 Ï Y V I ï f o H fiTV»o f i m o f i r ^ T l ft\i\ = rt fr^T» I M I 2^

2A

If f(t) is a band-limited time function, f(v) = 0 for | v | ^-^, then the time function g(t) = f^(t) is also band limited:

g(V) = 0 for | v | ^ - ^ (k = 1, 2, . . . . )

Theorem II-3a

If the Fourier transform of the time function f(t) has the property "(v) = 0 for | v | =\ ( k = 1, 2 ) then f(t) dt can be written as a s e r i e s :

— 00

+ >" + 00

[ f(t)dt = A 2 f (HA) (26b)

Proof

According to (25a) it follows with the constraint on f(v) that

27 -Thus + CO + CO f(t)dt = f ( t ) d t + CO = 2 f(n) n=-oo + 00 A 2 f(nA)

which represents an infinite s e r i e s . Theorem II-3b

If f(t) is a band-limited time function, f (v) = 0 for | v | > "iTT + CO

(k = 1, 2, . . . . ) , then f^(t) dt can be written as a s e r i e s : — 00 r"^"" k "^" k f ( t ) d t = A 2 f (nA) (27a) '' n = -oo — QO Proof Define g(t) = t\t)

With the constraint on 7{v) and lemma H - l , it follows that '(V) = 0, for | v ] > |

Consequently, according to (25a), we may write

g(o) = gp(o)

+ co + CO

J

g(t)

dt = J

g^(t)

dt

+ 00 , +00 f^{1) dt = 2 g(n) n=-°° A 2 g(nA) n = - o s + •" k A 2 f (nA) n = -oo which represents an infinite s e r i e s . Corollary;

' * ' r I 1

If f(t) is a band-limited time function, f(v) = 0 for | v | ^ r r r , then 2kA

+ 00 f> ^ 2 k |f(t)| dt can be written as a s e r i e s : + 00 | f ( t ) p ' ' d t = A 2 | f ( n A ) r (28a) •^ M = - m where k = 1, 2 Theorem II-4

If the functions f(t) and g(t) are band-limited time functions, f(v) = 0 and -w , , 1

g(v) = 0 for IV I ^ "ÖT, then the convolution integral can be written as a s e r i e s :

+ C0 • + •

-f(t - T) gCO dT = A 2 f(t - nA) g(nA) (28b)

Proof

- 29

h(t) = f(t) « g(t)

The Fourier transform of this equation yields

h(V) = f'(V)|(V)

Because both f(t) and g(t) are band limited, we may write

+ CD

h(v) = ^(V) 2 g ( v - - 5 ) n=-°°

= ?'(v)gp(v)

The inverse Fourier transform of this equation yields

h(t) = f(t) * g^(t) . + °° = J f(t - T) g^(T) dT + CO + 0 0 J f(t- T)g(T)[^A 2 6(T -nA)] dT + oo A 2 f(t-nA)g(nA) n=-°o which is an infinite s e r i e s . C o r o l l a r ^ l

If f(t) and g(t) a r e two band-limited time functions, f (v) = 0 and g(v) = 0 for | v | ^"öT» sampling and convolution a r e commutative:

+ 00

= r 2 f(n - m) g(m)1 6 (t - nA) '-m = -°° -^

= f^(t) ie g (t) (30a)

Corollar^_2

The cross-correlation function of two sampled band-limited time functions is equal to the sampled cross-correlation function of those time functions

R, (t) = f* (-t) K g,(t)

A ' ^ A ^

[f*(-t) »g(t)]o^(t)

= \ g(t) 0^(t) (30b)

provided f (v) = 0 and g(v) = 0 for | v | ^ -g^ .

Theorem II-5

If f(t) is a band-limited time function, f(v) = 0 for | v ] > - ^ , then ^(v) can be written as a series

r(v) = A r f(nA)e-J2"^'^^ (30c)

1 ' 1 ^ for | v | <-oT» provided f(v) exists*).

Proof Define g(t) Then for g(v) = f(t) e' we can •J2nvQt write

^"^ 1^1

<-h

g(v) = r ( v + v^) with |v^| < ^ + CO 2

31

According to (25a), it follows with the constraints on T(v) and v that

Thus ^(o) = gp(o)

r(v^=j""f(t)e-J2nVot

_dt

= J g(t) dt

. + 00 g^(t) dt A 2 g(nA) + m r/ Av - j 2 n v , . , n A A 2 f(nA) e •" o _{^°^ 1^1 <2A} and f(v ) = 0 ^ o' ^°^ l^ol ^ 2 A

Without proof, we formulate Theorem n - 6

The Fourier transform of a sampled time function f (t) can be written as a series T (V) = 2 f(n)e' y Il=_co • j 2 n v n A (31a) provided f (v) exists*), + 00

*) f ( V ) exists (in the sense of convergence in the mean) if 2 1 ^n) I < °^ ,

Corollary

a. The Fourier transform of an even, sampled time function can be written as a series of cosine functions:

CO

T (V) = f(o) + 2 2 f(n) cos 2 n v n A (32a)

P n= 1

b. The Fourier transform of an odd, sampled time function can be written as a series of sine functions;

f ' ( v ) = - 2 j 2 f(fl)sin2nVnA (32b) P n= 1

c. The Fourier transform of a sampled time function of finite duration can be written as a series with a finite number of t e r m s ;

^^(V) = ? f ( : ^ e - ^ 2 ^ ^ ° ^ (320 1 P n = - N , with N, 2 f (t) = 2 f(n) 6(t-nA) n = N

-To derive an expression for f(n) in terms of the Fourier transform f (V), consider the following integral:

r r , s j 2 n v n A _dv P" • 2A ^^ - + " • j 2 n v m A n j 2 n v n A ^^ A f [ 2 f ( m ) e - ^ 2 n v m A n • ' l L i n = _co J 2A + =° ,"'"2A A E f(m) f e - J 2 ^ ^ ( ' ^ - ' ^ ^ d v in=-oo J j ^ "2A

33

-2 f(m) 6 m = -oo m. n

(6 being the Kronecker symbol)

= m

Hence the inverse Fourier transform of a sampled time function can be written as

(33a)

+ 00

Equation (33a) is valid for every n if 2 | f(i^ | < oo

To derive an expression for the energy of f (t) in terms of the Fourier transform T (v), consider the cross-correlation function of two sampled time functions:

^f s (*> = \ (-*) * S.(t)

+ ">

= r 2 f** (n-m)g(n)"| 6 (t - nA)

'- n = - o o -J

For the Fourier transform of R„ (t), we can write

A'^A

R, (V) = ?* *(v) g (V) ^A'^A P ^ ' ^P^ ' Combining this with (33a), we obtain

+ <» „ 2A 2 f'^ (n- m) g(n) = A ? * (v) f (v) e^ n=-oo "Jj^ P P "2A 2 n v m A dv or for m = 0:

+ 00 H

2 f" (n) g(n) = A I ^^ 7 * (v) IT (v) dv

"2A

(34a)

(34a) f o r m u l a t e s P a r s e v a l ' s t h e o r e m for s a m p l e d t i m e functions, If we take f (t) = g . (t), (34a) b e c o m e s 2 | f ( n ) | 2 = n = ~co p"^2A „

A i?;(v)rdv

J 1 p 2A 1 (34b) + 00

2 | f ( n ) | being defined a s the e n e r g y of f . ( t ) ,

The following d e r i v a t i o n r e l a t e s m o m e n t s in the t i m e domain to d e r i v a t i v e s in the frequency d o m a i n . U s i n g (3la), we can w r i t e t h e k d e r i v a t i v e of f (v) (provided it exists) a s t h d f„(V) d v + '^ p ' ' , . „ A.k „ k - , , - j 2 n v n A ^— = ( - J 2 T T A ) 2 n f(n) e •* n = - 0 0 (34c) th "^

Note t h a t t h e k d e r i v a t i v e of f (v) e x i s t s (in the s e n s e of c o n v e r g e n c e in the mean) if + ^ 2 | n ' ' f ( n ) | ^ <oo n = -o° (34c) i s absolutely c o n v e r g e n t for e v e r y v if + 0O 2 | n ' ^ f ( n ) | < A c c o r d i n g to (34a) and (34c), we m a y w r i t e k + m „ * , , , 2 n f(n)g(n) = n = -oo (-i)"^A r"2A d'-rp*(v) d " ^ ( v ) j2nA)'^-*-°^ J l dv"^ dv°^ (j2nA)'^-*-°^ M 2A dv

and if f(n) = g(n), 35 k + m 1^, , i 2 2 n | f ( i ^ | n=-oo

l-«°A {'' tll^ '11^ <,,

„..A^"» J ^ dv^ av» ,3^^,

and if k = m, + "• 2k 2 A r^2A D 1 J^(°) 1 9k n=-oo (2nA)''"- ^ 1 2A d''?p(v) dv'^ 2 dv (35b) B. z-transform

We begin by defining the z-transform f(z) of the sampled time function f.(t). It is shown that if all samples of f.(t) are finite, f(z) can be written as the sum of a 'plus function' f (z) which is analytic inside the unit circle C. and a 'minus function' f (z) which is analytic outside C If additional constraints on f(z) are added, relationships can be derived between f (z), f ~(z) and 7(z).

Definitions are given for the one-sided sampled time function, the widely-bounded and strictly-widely-bounded sampled time functions, the widely-stable and strictly-stable sampled time functions. Properties are derived for the z-transforms of these functions.

It is shown that for a strictly-stable and strictly-bounded sampled time function, the number of zeros of the z-transform inside 0 is only

determined by the first finite number of samples of the time function. Finally, the relationship between stable one-sided nonsampled time functions and stable one-sided sampled time functions is considered.

Consider the two-sided Laplace transform of the sampled time function fA(t) =

, + 00

-St

p + oo _ = J f(t) 0^(t) e"^' dt If V f ( n ) e - ^ ' ^ (36a) n = - 00 „ I , , , - s n A | 2 . 2 |f(n) e I < 00 for s = Qj^ + j 2 n v and n=0 2 I f(n) e ^ I < CO for s = a^ + j 2 TT v^ Il=oo "2 • " " 2

then f (s) converges absolutely and uniformly on and inside each contour within the strip determined by

a^ < Re{s] < a^ (a^ < a^)

Moreover, f (s) exists (in the sense of convergence in the mean) on the boundary of the strip.

Consider the transformation*)

z = e "SA (36b)

The s-plane is then mapped on the z-plane in the following way:

The imaginary axis of the s-plane (Re{s3 = 0) is mapped onto the unit circle C^ of the z-plane ( | z | = 1), the right-half s-plane (Re{s] > 0) is mapped into the interior of C ( | z | < 1) and the left-half s-plane is mapped onto the exterior of C. ([z] > 1).

We will define the two-sided z-transform of the sampled time function f^(t) by

37

-f(z) = 2 f(n)z'^ (37a)

I l = - o o

We have therefore written f(z) as a Laurent series about z = O which converges absolutely and uniformly on and inside each contour within the annulus

- a A -a,A e 2 < |z| < e 1

Moreover, f(z) exists (in the sense of convergence in the mean) on the

boundary of the annulus. From (37a) and Cauchy's integral formula (Titchmarsh, p . 80) follows the inversion formula

f('^) = 2ÏÏJ f ^ dz (37b)

0 "^

for each contour in the annular ring, including the boundary. If the unit circle (7- belongs to the annular ring, then with z = e •' ej5)ression (33a) is

obtained: fe^) = 2ÏÏJ !^ ^ dz ^ 1 ^

4 •

y , , j 2 n v n A ,

= AJ^ r(v)eJ dv

" I A

Let us assume that 0 belongs at least to the boundary of the annular ring*), Then we can define on CL

i{^) = r + ( z ) + ^ " ( z ) (37c)

with

+ 00

*) This is t h e case if 2 I f(n) I <

a, r"^(z) = 2 f(i^z'^

n=0 for Izl < 1 I I > . f (z) is analytic for | z | < 1, According to (37b) we can write

^''^> = „fjï^ i ' ^ * ] ^ " 1^1

_{^ . «} z ^ 1

^"'4 .= 0 c

- ? m P

Hi)

£

^?7-r

z

dC

n + l o r

'^<">-i^ f f ^ «

f (z) = 2 f(n)z n = - t o for I z I < 1 |z| ^ 1 (38a)

f (z) is analytic for Izl > 1 and lim f (z) = 0

I z l —» 00

According to (37b), we can write

''^''\LA^>^P''V

1

^

1 *•

' ^ , [ " 2 ^ f ?(C)C°-^dClz-° |z| i l

n = 1 /v -^ °° j - n - 1 2nj r ^v-' ^ , — T '^^ •" n = l z

= 2%

F(S)

a-,

hi 5 1

or

39

-If f(z) satisfies the Holder condition on 0

|f'(Vi) -^(^i)i ^^IVi-^il""

for |z > 1 (39a)

A > 0, H > 0 then we may write (Noble, p , 145):

or

?"^(z.) = lim r'^(z)

' z >z. 1 Izl < 1, Iz, = 1 (39b) f "(z.) = lim f (z) z—»z. 1 z| > 1, | z . | = 1 (39c) Definitions

l a . The time function f (t) is called one-sided if f(n) = 0 for n < 0.

b, A linear system is called physically realisable if its impulse response is one-sided,

2a, The time function f (t) is called bounded in the strict sense if 0.. f(z) is of bounded variation on 0 •

?l^(Vi) -^(^i>i <^i

1^1 = 1

for each subdivision.

P. f(z) satisfies the Holder condition on 0 z

|f(Vi) -^(==i)l < A 2 l V i - ^ l

A^ > 0, M > 0 b. A linear system is called strictly bounded if its impulse response is

C o r o j l a r ^ l If

| ? ( z . ^ ^ ) - ? ( z . ) | < A | z . ^ ^ - z . | | z . | = 1 for each subdivision, f (t) is strictly bounded.

A

Corollaix_2

If f (t) is strictly bounded, f (z) is continuous for | z | $ 1 and f (z) is continuous for [z] ^ 1 .

3a, The time function f (t) is called bounded in the wide sense if | f(n) | < A < co for every n,

b, A system is called widely bounded if its impulse response is widely bounded.

If f^(t) is called bounded, f (t) is taken to be bounded at least in the wide sense.

4a, The time function f (t) is strictly stable of the order k if

+ " k

2 |n | | f ( n ) | < - k = 0, 1, 2, , . . . n = - ° o

b. A system is called strictly stable of the order k if its impulse response is strictly stable of the order k,

5a, The time function f (t) is called widely stable of the order k if

+ 00

2 | n ' ^ | | f ( n ) | ^ < CO k = 0 , 1 , 2

n=-°=

b, A linear system is called widely stable of the order k if its impulse response is widely stable of the order k,

If a linear system is called stable its impulse response is taken to be at least widely stable of the order zero.

Theorem n - 7

41 -Proof

The time function f (t) is one-sided and hence we may write

A

^(z) = 2 f(n)z° n=0

Since f(z) is bounded, we have |f(n)| < A. Takii^ this into account we may write, for a point inside C ,

GO |f(z)| = I 2 f(n)z°| n=0 ^ 2 | f ( n ) | | z | ° n = 0 < A 2 | z | ° n = 0

< T ^ l^l<i

< 00 | z | < 1

We have written f(z) as a power series which is absolutely and uniformly convergent on and inside every contour within 0^. Hence f (z) is analytic for Izl < 1.

If f^(t)

analytic for | z | < 1

If f (t) is a stable one-sided time function, then its z-transform is

Any bounded time function has a z-transform which can be written as the sum of a 'plus function' and a 'minus function':

+ CD

^(z) = 2 f(n)z^ n=-oo

- 1

= 2 f(n)z'^ + 2 f(n)z° n=0 n = - « '

7+(z) + F"(z)

• ' + ,

f (z) being analytic for | z | < 1, f (z) being analytic for | z | > 1 with lim ?"(z) = 0.

| z | — » o o

Note that the relationships (38a) and (39a,b,c) are only valid if additional constraints on f(z) are introduced,

Theorem II-8

If the time function f. (1) is strictly bounded, f(n) = 0(—) . Proof

f (z) is continuous on C. [definition (2a), p , 39] so according to (37b) we may write, ^ ( n ) = 2 ^ ^ ^ ^ ^ ^1 r / V j 2 n v n A , A [ ? (V) e^ 2A "^2A _ ^ 2 A A I Re{f'(v)]cos 2 n v n A d v - A [ Im{f' (v)] s i n 2 n v n A d v J 1 P J J P 2A 2A ."'2A ^ +2A + jA Imff ( v ) } c o s 2 n v n A d v +j A Re{f (v)] sin 2 n v n A d v

J -. " P J -i p

"2A "2A (42a) f(z) is of bounded variation on C. [definition (2a), p, 39]:

or

Hence

43

-? l y v i ) - w i < A

f ([K^)(vi)} - ^n^p(^i)}]'" M^^^vi)} - K^p(^i)}]')'

^K^(vi)} - i^K^p^v)

< A j < A

K^)(vi)}-M^(^i)}

< Ag ^ A

Thus, if f (V) is of bounded variation, R e j ? ' ( v ) | and ImJ f ( v ) | are also of bounded variation.

According to (42a) and the Riemann-Lebesque lemma (Whittaker and Watson, p, 172), it follows that f(n) = 0 ( - )

Corollary.

If f (t) is a strictly-bounded time function, f (t) is also stable,

Theorem n - 9

If f (t) is a one-sided time function with

2 nf(i^z n = l

n-1

for all z on C „ we may write

|f(Vl) - % l < ^ 2 l V l - ^ i l

for each z. , and z. on C,.

1+1 1 1 Proof

f ( z ) = 2 f(n)z" n = 0 ^

hil =1

and

f ( W = ^/(^)<+i

n=0 ^ + l l = 1 From these e^qjressions it follows

h v i )

-

^(^i)

2 f(n)z° - 2 f(n)z° n=0 ^ ^ n=0 ^ 2 f(n) (z. - z ) n=0 * -^ z . , , - z. 1+1 1 „ f, . , n-1 , n-2 , n - 1 , 2 f(i^ ( V l + V i + 1 + • • • • ^ ) n= 1 z . , 1 - z . _{1+1 1} z. ₁ n = l \ 1+1 n - 1 ' z. n-1

V l .

< A ^ o ^i+1 - h 2 nf(n)z. n - 1 n-1 i+1 < A 2 | Z i ^ l - z . | . which proves the theorem,

Corollar;^

If f.(t) is a one-sided time function with

A

2 nf(n)z n = l

n-1 _{< A}

45

-Theorem H-IO

If f (t) is a strictly-bounded one-sided time function, its z-transform can be written as a power series which converges uniformly inside and on C... Proof

f (t) is a bounded one-sided time function and hence, according to theorem n - 7 , the power series of f (z) converges uniformly inside and on every

form:

every contour within 0 . Let us write this power series in a somewhat different

f'(z) = 2 f(i^z° | z ] ^ l - 6 , 0 < 6 < 1 n=0 = 2 f(n)S°(|)° I d = 1 n=0 ^ 2 g(n) ^\^ (45a) with n = 0 g(n) = f(n)c''

'C'

If we write

z = |xk

with IXI < 1, then (45a) becomes

h(X) = 2 g ( n ) | X | °

n = 0

f (z) is continuous inside and up to C. (corollary 2, p, 40) and we may write lim7(z) = f(C) | z | < 1

z—»C

lim ^f(\.) = ^f(l) X | - * l _

We have proved (theorem n - 8 , p. 42) that

m =

0(i)

or

g(n) = 0(-i)

We can hence conclude, using Littlewood's theorem, (Titchmarsh, p, 233) that lim h(K) = 2 g(n) | X | _ ^ 1 _ n=0 or {{Q = 2 f(n)C° Kl = 1 n=0 or y , . V c, ^ ~i 2n vnA f„(v) = 2 f(n)e •• P n=0

Thus, the real part and the imaginary part of f (v) can be written as convergent Fourier s e r i e s :

00

Rej?' (v)j = 2 rRe|f(n)| cos 2nvnA + Im|f(n)j sin 2TTVnA'|

CO

Imj?' ( v ) | = 2 rim|f(n)j cos 2nvnA - Re|f(n)| sin 2nvnA"l

Since Re-^f (v) j- and Imjf ( v ) | are periodic, continuous and of bounded

variation, it follows from the theory of Fourier series (Whittaker and Watson, p. 179) that Re|f (v) j- and Imjf (v) V can be represented by uniformly convergent

47

-Fourier s e r i e s . Hence the power s e r i e s of f (z) converges uniformly on C . Now consider

^L(z) = 2 f(n)z° | z | <: 1 n=N+l

L(z) is analytic for | z| .^ 1-6 (0 < 6 ^ 1) so in the region | z | .$ 1-5, | L(z) | has its maximum for | z | = 1-5. Moreover, | L(z) | is continuous inside and up to C,. Consequently, in the region | z | $ 1, | ^L(z) | will be a maximum for Izl = 1 . Since f(z) is uniformly convergent for | z | = 1, we have

I^L(z)| < %

^^o

for each N > N and for all z on C,. _{o — 1 ^}

However, in the region | z | $: 1, | ^L(z)| has its maximum for )z| = 1, and thus we also have

| % z ) | < e M

for every N > N and for all z inside and on C Thus f(z) converges uniformly inside and on C ,

From the definition of strict stability [definition (4a), p. 40 ] and the moment-derivative relation (34c), there follows:

Theorem II-11a

If f.(t) is strictly stable of the order k, the k derivative of f (z) is absolutely and uniformly convergent on C...

i t . . i ^^

If f (t) is also one-sided, the k derivative of f (z) is absolutely and uniformly convergent inside and on 0

Coroll^ar^

If f (z) is analytic on (7., f (t) is strictly stable of any order, From the definition of wide stability [definition (5a), p. 40] and the moment-derivative relation (35a), there follows:

Theorem I I - l i b

If f (t) is widely stable of the order 2k, the k derivative of f(z) is quadratically integrable (and thus also absolutely integrable) on 0 Moreover, if f (t) is one-sided, we also have the relationship (Kolmogorov, 1941);

1 -2A 2A d*" 7 (V) d V d v > - 00 Corollary

If f (z) is analytic on 0 , f (t) is widely stable of any order. Theorem n-12a

The convolution of two strictly-stable time functions is strictly stable. Proof

Assume that f (t) and g (t) are two strictly-stable time functions and that h .(t) is their convolution. For 2 | h(n) | , we can write

n = - o o + 00 + 1 » + QO 2 |h(n)| = 2 I 2 f(m)g(n-m)| n=-oo n = - ' ° m = - ' " + 00 + 0 0 ^ 2 2 |f(m)||g(n-m)| n = - o o m = _03

g (t) is strictly stable and so we may write

From this + 03 2 n = -we + °= 2 n = -1 g(n) -1 00 obtain 1 h(ii) 1 00 < < «A S A + 00 2 i n = -oo

49

-+ 0O

2 I f (m) I < ^A m = -c°

Hence we finally obtain

+ 00

2 I h(n) I < ^A *A

n=-Thus h (t) is strictly stable.

Theorem n-12b

The convolution of two strictly-bounded time functions is strictly bounded, Proof

Assume f (t) and g (t) a r e two strictly-bounded time functions. If h (t) is the convolution result of f (t) and g^(t), we can write ïi(z) as

h(z) = ?(z) i(z)

a. Firstly, investigate the Holder condition on 0^:

|h(z.^j) -h(z.)| = |?(z.^^)ï(z.^j) -r(z.)g(z.)|

= pz.^^) -?(z.)]g(z.^^) + p(z.^^) -ï(z.)]?(z.)

^ Ig(Zi+l) i I^'(Zi+i) - U^) I + \U^i) I Ig(Zi+i) - g(Zi) < g A B ^ | z . ^ ^ - z / l + f A B 2 | z . ^ ^ - z . | ' 2

i+1 i Thus h(z) satisfies the Holder condition on C Secondly, consider the variation of h(z) on C,

2 |f'(z.^j)i(z.^^) - f'(z.)i'(z.) I =

2 |[?(z.^^) - F ( z . ) ] i ( z . ^ j ) + [g(z.^j) - i ( z . ) ] ? ( z . ) < 2 g(z.^l) 1 I f ( Z i ^ l ) - f ( Z j ) + 2 i f(Zi) g(Z.^j) - g(z.) < ^A 2 i f(z.^^) - f (z.) + ^A 2 g(z.^^) - g(z.) < B <

Thus h(z) is also of bounded variation on C Theorem 11-13

The convolution of a strictly-stable time function with a bounded time function is bounded.

Proof

Assume that f (t) is strictly stable, g (t) is bounded and h (t) is the convolution of f (t) with g (t). For | h ( n ) | , we can write

+ 00 |h(n)| = I 2 f(m)g(n-m)| m= + 00 < 2 |f(m)||g(n-m)| < ^A 2 |f(m)| m < B < Thus h.(t) is bounded, Theorem 11-14

The convolution of two stable time functions is bounded. Moreover, its

_1_ J^

2A* 2A

51

Proof

a. If f (t) and g (t) are two stable time functions and h.(t) is the convolution of f^(t) with g (t), we can write |h(n)| as

-)-00

|h(n)|2 = I 2 f(m)g(n-m)|2 m=-°o

According to the Schwartz inequality, we obtain | h ( n ) | 2 ^ 2 |f(m)|2 2 ig(n-m)|2 m m < Rf f (o) Rg g (o) A' A ^A'^A < B < 00 Thus h.(t) is bounded. A

b. For the Fourier transform of h (t) we may write ïp(v) = fp(v) gp(v)

With the Schwartz inequality we obtain

+i

+J_

+J-r +J-r 2A ^ , - 2 +J-r ^^ , - , 2 +J-r ^^ , - ,2

|_A J

| r ( v ) gp(v)

I

dvj^

$ A J I

fp(v)

r dv A J I

gp(v)

I dv

"2A ~2A ~2A < Rf f (o) R (o)

^A' A ^A'^A

< B <

-V 1 1 T Thus n (V) is absolutely integrable on [-'öT. "ÖA-I

Theorem n - 1 5

If f (t) is a strictly-bounded or strictly-stable one-sided time function, there exists an N such that _o

2 f(n)z n = N + l

N

2 f(n)z'^ n=0

for all z on C, and for every N > N ,

1 o Proof

If f (t) is a strictly-bounded o r strictly-stable one-sided time function, its z-transform is uniformly convergent inside and on 0 We can therefore split f(z) into two s e r i e s :

f(z) = 2 f(n)z° + 2 f(i^z n = 0 n = N + l with = fjj(z) + ^L(z) N

rL(z)| < |e„ (z)|

for every N ^ N and for all z on C If f(z) has no zeros on C., we can chose N so that, for instance, e (z) = Tfr f(z) on C If so,

o |NL(Z)| < ^ | ? ( Z ) | and

luz)! = \m - ^L(z)| >^\m\

w

10 for all z on C..,

Thus we see that in the case of a strictly-bounded or strictly-stable one-sided time fimction we can always find an N such that

53

-| y z ) -| > -|^L(z)-|

for all z on C , provided f(z) has no zeros on (7... If z is a zero of f(z) on (7.,, we have

^N<V + ^L(z^ = 0

or

\ W I = I ''L(Z„) I

Consequently, we can say that for a strictly-bounded or strictly-stable one-sided time function we can always find an N such that

\ï^(z)\ ^ | N L ( Z ) | | Z | = 1

The equals sign applies at the zeros of f(z) on C . Theorem n-16a

The z-transform of a strictly-bounded or strictly-stable one-sided time function has no zeros inside and on C- if and only if the z-transform of the truncated time function

~ N f (z) = 2 f(n)z''

n=0

has no zeros inside C,, N has to be chosen so that

|f (z)| > I 2 f(n)z°| (53a) n = N + l

for all z on (7.. Proof

a. According to Rouché's theorem (Titchmarsh, p. 116), using condition (53a), it follows that f|j(z) and f(z) have the same number of zeros inside 0 and no zeros on 0 Consequently, if i^{z) has no zeros inside 0-, and (53a) applies, then f(z) has no zeros inside and on C'

b. Assume f(z) has zeros on 0.. Then, if z is a zero of f(z) on C „ we may write

I W I = l''^(z^l

which is in contradiction with (53a).

Assume f(z) has zeros inside C... Since f.(t) is strictly bounded or strictly stable, we can always find an N such that (Theorem n-15)

|?j^(z)| > |^L(z)|

According to Rouché's theorem, it follows that f(z) and f>r(z) have the same number of zeros inside C.,, Thus f,.(z) also has zeros inside C,, But this

1 «.- N^ ' 1 contradicts the condition that f^(z) has no zeros inside 0 .

Theorem II-16b

The z-transform of a stable one-sided time function has no zeros inside C. if the z-transform of the truncated time function

f (z) = 2 f(n)z° n = 0

has no zeros inside and on C. and if

00

I ? (z)| ^ I 2 f(n)z°| (54a) ^ n = N + l

for all z on C.. Proof

If t^{z) has no zeros inside and on C „ condition (54a) becomes, for every | c| > 1,

I c ^ z ) ! > | ^ L ( z ) | | z | = 1 Therefore, according to Rouché's theorem, the function

55

also has no zeros inside and on C., for every | c | > 1. Since "f(z) converges uniformly to f(z), we may conclude from a theorem of Hurwitz (Titchmarsh, p, 119) that f(z) has no zeros inside C,.

Definition

The time function x (t) is called the frequency inverse*) of the time function f (t) if the convolution of x (t) with f (t) yields the unit impulse

f^(t) KX^(t) = 6(t) or in terms of the z-transform,

x(z) = : ^ f(z) Corollary

The phase spectra of x (t) and f (t) are identical with respect to their absolute values,

Remarks

1. If f(t) is a stable time function, f (t) need not be stable. We must add the condition that f(t) is also bounded,

2, According to the Paley-Wiener criterion (Paley and Wiener, 1934), a stable one-sided time function cannot be band-limited. Hence, if f(t) is sampled (with sampling interval A), aliasing must occur:

y v ) f ?(v) for |v I < ^ Thus f(t) cannot be recovered from f (i) by the convolution (25c):

t

s i n TT —

m +

f,(t)

«

-

^

If we do convolve f (t) with

t S i n n

-the nonsampled time function ^f(t) = h(t) X f^(t) will become two-sided. Since

f^(t) = f(t)0^(t)

= ^f(t)0^(t)

1 k

f(t) must have zeros for t = - - ^ (k = 1, 2 ), Consequently, there exists a one-to-one correspondence between the one-sided sampled time function f (t) and the two-sided band-limited time function f(t):

A f^(t) = ^f(t)0^(t) (56a) and 1 s i ° " A f(t) = f ^(t) « - ^ ^ (56b) ^ 1

If f (t) is stable, the amplitude spectrum | f(v) | of f(t) satisfies the conditions r ^'^ I i r , , 1 2 , ^ a. I f (V) I dv < oo

-L

" 2 A

4A

r In I •'•f(v) I dv > - 00 •2A

57

-C. Minimum-phase function

In this chapter a definition is given of the minimum-phase sampled time function f~ (t). It is derived that f (z) has no singularities and no zeros inside C.. The maximum-phase function r (t) is also defined for time-functions of finite duration.

It is proved that for a certain amplitude spectrum there exists at most one minimum-phase time function. An expression is derived for the phase spectrum of i (t) in terms of the amplitude spectrum.

It is shown that of all the time functions with identical amplitude spectra, the minimum-phase time function has, algebraically, the smallest phase, the smallest phase derivative and the smallest tail energy.

Finally it is shown that of all the time functions of finite duration with identical amplitude spectra, the maximum-phase time function has,

algebraically, the largest phase, the largest phase derivative and the largest tail energy.

Definition

The time function f .(t) is called a minimum-phase function if both f (t) and

This implies that if f (t) is a minimum-phase time function, its frequency its frequency inverse are stable one-sided time functions

This implies that if f (t) is a minimum-p! inverse is also a minimum-phase time function. Theorem II-17 a

If f (t) is a minimum-phase time function, f(z) has no singularities and no zeros inside C...

Proof

f (t) is a stable one-sided time function, so that according to theorem II-7,

A

f(z) is analytic inside C... The frequency inverse of f (t) is also a stable

one-••• 2 A ~

sided time function. Hence is also analjrtic inside C . Consequently, f(z) f(z)

has no singularities and no zeros inside C

As long as f (t) and its frequency inverse are stable, f(z) may have A

singularities and zeros on C... However, the minimum-phase time function of finite duration has no singularities and no zeros inside and on 0

Remark

Rather than giving constraints on the singularities and zeros of f(z) (as is common in the existing literature), we give the definition of 'minimum phase' in terms of both the time function and its frequency inverse. By doing so, a

simple formulation is obtained which mentions the most essential prc^erty (stable one-sided frequency inverse) in a direct way. Moreover, no ambiguity exists if f(z) has zeros on 0

Theorem II-17b

If f (t) is a minimum-phase time function,

2A

l n | r ( v ) | d v

2A

Proof

f (t) is a stable one-sided time function, so that according to Kolmogorov A

(1941) we may write*)

2A

J ln|fp(v)| dv>

2A

The frequency inverse of f (t) is also a stable one-sided time function and hence we may write

2A In •2A f (V) p^ ' dv > or

59 -(. 2A J In |fp(v)| dv 2A s o p r o v i n g the t h e o r e m . T h e o r e m 11-18 If f (t) i s a s t r i c t l y - b o u n d e d m i n i m u m - p h a s e t i m e function and if f (v) =(= 0, A P then its frequency i n v e r s e x (t) is a l s o a s t r i c t l y - b o u n d e d m i n i m u m - p h a s e t i m e

function.

P r ^ f

a. f(z) is a n a l y t i c i n s i d e 0 and h a s no z e r o s inside 0 s o that x(z) i s a l s o a n a l y t i c inside C

b . f(z) is continuous inside and on C . , and h a s no z e r o s i n s i d e and on C . , s o that x(z) i s a l s o continuous inside and on 0 and h a s no z e r o s inside and on

0 M o r e o v e r , x(z) s a t i s f i e s the Holder condition on C..:

1 1 |x(z.^j) - x ( z . ) | f(^i+l) f(^i) f(^i+l) - ^(^i) f(Zi^l)?(z.) = | x ( z . ^ ^ ) x ( z . ) | | f ( z . ^ ^ ) - f(z.)| < V I f ( z . ^ j ) - r(z.) I X . 2 .nl I (J < A B z. , - z. r^ ' 1+1 i ' I A being t h e u p p e r l i m i t of x^(t)j | z . | = 1 , B > 0, M > 0 c. x(z) is of bounded v a r i a t i o n on C • 2 f;^(z.^^) - x(z.) I = 2 f(Zi^l) f(z.)

2 i ^(^+l) - 'N f(Zi^l)f(z.) 2|x(z.^^)x(z.)I |f(z.^^) - f(z.)|

< V2]?(z.^^) -r(z.)|

i

1^1 = 1

< B < =

From a, b and c it follows that x (t) is also a strictly-bounded minimum-phase time function.

Definition

T

The transpose f (t) of the time-function f (t) can be found by replacing A ~ ' ^ 1 ,'*'r , '»' , all analytic zeros z (z # 0) of f(z) by the zeros so that | f ^ ( v ) | = |f ( v ) | .

z P P n

Corollary_

If f (t) is a time function of finite duration with f(o) ^^ 0, we can write f^,(z) = z ^ f* (—) z N = z ^ 2 f*(n)z-° n=0 Thus N = 2 f*(N - n)z" n=0 T ^ ^ f^(t) = 2 f"(N - n) 6(t - nA) n=0

61

-Hence if we m i r r o r the time function f (t) relative to the origin, take the conjugate complex and shift the result by NA, then the time function f (t) is obtained:

fJ^(t) = f^(-t + NA) provided f(o) # 0

Note that f^(t - nA) = rfj^(t - nA)"]

Assume that the transpose of the time function g.(t) is equal to the minimum-phase time function f (t);

gj(t) = f .(t) A A Then also

[g^(t).\^(t)]'' = f^(t)

with \(z) = ^ ^ ^ |z I < 1 ^ ' s 1 I ni

Hence there exist an infinite number of time functions with identical amplitude spectra of which the transposes are equal to f (t). However, if we take a minimum-phase time function of finite duration f»j(t), then there exists only one time function of the same duration (theorem 11-22) of which the transpose is equal to tf^{t).

Definition

If f (t) is a bounded one-sided time function of finite duration and if f(z) has no zeros outside C „ then f (t) is called a maximum-phase time function. Corollarj^

a. If f„(t) is a maximum-phase time function with f(o) ^ 0 and with ^ ( v ) ^ 0, T

then f (t) is a minimum-phase time function.

b. The transpose of a minimum-phase time function of finite duration is equal to a maximum-phase time function.

Theorem 11-19

Amoi^ all stable time functions with identical amplitude spectra there exists at most one minimum-phase time function with zero phase for V = 0.

Proof

If f (t) is stable, its autocorrelation function exists and we may write R. . (t) = f (t) « f^(-t) or in the z-plane, Rj J (z) = f ( z ) ? * ( i ) A' A z and on C , Rf f (V) = |f (v)| A' A ^

The minimum-phase function is one-sided, so |f C^)| niust satisfy Kolmogorov's condition.

If this is so, f.(t) may considered to be one-sided and f(z) is analytic inside 0 Hence f(z) has only a finite number of zeros inside 0 and we may write

~ N

f(z) = w(z) n (z^ + z) I ZQI < 1 n = l

Using this equation we can write R, , (z) as follows:

A' A ~ N ^ N R (z) =w(z) n (z^ + z ) w ( - ) n ( - + z^ ) A' A n = l z n = l N ~ N z w(z) n (1 + z^*z) w* (^) n (-^ + 1) n = 1 z n= 1 -®,i - »

-r<'>K'7']

f®(z) f®(z)

63 -or in the time domain,

®,.s „ r^©, - *

R (t) = f®(t) X rf®(-t)]

^A' A ^ - ^ ^ = f®(t) X f®(t)

We see, therefore, that the zeros of f(z) can be brought outside C. without changing |T(v)|. The time function f (t) has no singularities and no zeros inside C . If we assume that the singularities and zeros of f(z) on C. are such that the frequency inverse of f (t) is also stable, then f (t) is a minimum-phase time function for the amplitude spectrum | f ( v ) | : r (t) = f (t). But f^ (t)e is also a minimum-phase function with amplitude spectrum | f ( v ) | . We will chose a so that the phase spectrum of i~ (t) is zero for v = 0.

Now consider a second minimum-phase time function g (t) with

amplitude spectrum |f (v) | . From f^ (t) and g (t), a new function h (t) is constructed having a z-transform

h(z) = f ^ ^

h(z) has no singularities and no zeros inside C . On C. we have |h(z)| = 1

Consider the function

•^h(z) = lnh(z)

h(z) is analytic inside (? . On C, we have

{ h(v)| = 0

| ^ h ( v ) | = ^cp(v) - ^cp(v) Im

'h (t) = j 3 6(t)

A (3 being a real constant)

or

h(z) = j e

Hence we can write h(z) = e i3 and

~min, , jïnin, , - j t g (z) = r " (z) e ••

Conclusions

la. There exists at most one minimum-phase time function with amplitude spectrum |f (v)| and with zero phase for v = 0.

b. If f (t) is a stable one-sided time function for which the amplitude spectrum | i (v)| satisfies the condition

'2A 1 2A 1 ^p(^

J r -

d v <

then the minimum-phase function f (t) with amplitude spectrum If (v) | exists and is constructed by bringir^ all zeros of f(z) outside C,, such that a zero z (|z | < 1) of f(z) becomes a zero of f"^^°(z).

n ^1 ni ' * z _n

2. If f (t) is a stable one-sided time function then there exists one time

A 0 ^ function f (t) which can be constructed by bringing all zeros of f(z) outside

^ -"^ 1 ''^iss

C.., so that a zero z (|z | < 1) of f(z) becomes a zero of f®(z). We z

will call the stable one-sided time function f (t) the generalised minimum-A m

phase time function. Note that the inverse of f (t) need not be stable.

65

-3. Any stable one-sided sampled time function with amplitude spectrum f (v) can be represented as follows (canonical representation for sampled time functions);

?(z) =11 ^ ^ 7%)ei^

n zz + 1 n

or in the time domain, f(t) = a(t) K f®(t)

z being the zeros of f(z) inside C (must be a finite number), f (t) being the (generalised) minimum-phase time function with amplitude spectrum

I f (v)|and a(t) being a one-sided time function with amplitude spectrum P ©

| ï (v) I = 1. Note that if f. (t) is a minimum-phase function, the inverse of f. (t) (which will be two-sided) is also stable.

Q

We will derive an expression for the phase spectrum of f. (t) by factorising Rf f (z).

For this, let us write

I n R j ^ (z) = TlnRj ^ (z)1 + TlnR^ ^ (z)1 A ' A L A ' A •- A ' A z = 1 and + TlnR (z)1 f *^(z) = A e ^ ^ | z | 5: 1 and flnRf ^, (z) • J . - A A ^ ^ ( z ) = A " ^ e " ^' ^ | z | ^ 1 If f (t) is strictly bounded and if f(z) has no zeros on C.,, we may write,

A ^