Pierre Bremaud - Mathematical Principles of Signal Processing

8/17/2019 Pierre Bremaud - Mathematical Principles of Signal Processing

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Mathematical Principles ofSignal Processing


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Springer-Verlag Berlin Heidelberg GmbH


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Pierre Bremaud

Mathematical Principles ofSignal Processing

Fourier and Wavelet Analysis

, Springer


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ToMarion


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Contents

Preface

A Fourier Analysis in L

Introduction

Al Fourier Transforms of Stable SignalsA 1·1 Fourier Transform in L

Al·2 Inversion Formula . . . . . . . .

A2 Fourier Series of Locally Stab le Periodic Signals

A2·1 Fourier Series in }oc .A2·2 Inversion Formula . . . . . . . . . .

A3 Pointwise Convergence of Fourier SeriesA3·1 Dini s and Jordan s Theorems.A3·2 F6jer s TheoremA3·3 The Poisson FormulaReferences . . . . . . .

B Signal ProcessingIntroduction

BI FilteringB 1·1 Impulse Response and Frequency ResponseBl·2 Band-Pass Signals .

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viii Contents

B2 SamplingB2·1 Reconstruction and Aliasing .

B2·2 Another Approach to SamplingB2·3 Intersymbol Interference .B2-4 The Dirac Formalism . . . . .

B3 Digital Signal ProcessingB3·1 The DFf and the f AlgorithmB3·2 The Z-Transform .B3·3 All-Pass and Spectral Factorization

B4 Subband Coding

B4·1 Band Splitting with Perfect Reconstruction .B4·2 FIR Subband FiltersReferences. . . . . . . . . . .

C Fourier Analysis in L 2

Introduction

C l Hilbert SpacesC · Basic Definitions.

C ·2 Continuity PropertiesC ·3 Projection Theorem .

C2 Complete Orthonormal SystemsC2· Orthonormal Expansions .C2·2 Two Important Hilbert Bases

C3 Fourier Transforms of Finite-Energy SignalsC3· Fourier Transform in L 2

C3·2 Inversion Formula in L 2 • • • • • • • • •

C4 Fourier Series of Finite-Power Periodic SignalsC4· Fourier Series in Toc .C4·2 Orthonormal Systems of Shifted FunctionsReferences. . . . . . . . . . . . . . . . . . . . .

D Wavelet Analysis

Introduction

D1 The Windowed Fourier TransformD1·1 The Uncertainty Principle .D1·2 The WFf and Gabor s Inversion Formula .

D2 The Wavelet TransformD2·1 Time-Frequency Resolution of Wavelet TransformsD2·2 The Wavelet Inversion Formula .

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3 Wavelet Orthonormal ExpansionsD3·1 Mother Wavelet . . . . . . .D3·2 Mother Wavelet in the Fourier DomainD3·3 Mallat s Algorithm. . . . . . . . . .

4 Construction of an MRAD4·1 MRA from an Orthonormal System .D4·2 MRA from a Riesz BasisD4·3 Spline Wavelets . . . . . . . . . . .

5 Smooth MuItiresolution AnalysisD5·1 Autoreproducing Property of the Resolution SpacesD5·2 Pointwise Convergence Theorem . . .D5·3 Regularity Properties ofWavelet BasesReferences . . . . . . . . . . . . . . . . . . .

Appendix

The Lebesgue IntegralReferences. . . .

Glossary of Symbols

Index

Contents ix

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Preface

Fourier theory is one of the most useful tools in many applied sciences, particularly, in physics, economics, and electrical engineering. Fourier analysis is awell-established discipline with a long history of successful applications, and therecent advent of wavelets is the proof that it is still very alive. This book is anintroduction to Fourier and wavelet theory illustrated by applications in communications. I t gives the mathematical principles of signal processing in such a waythat physicists and electrical engineers can recognize the familiar concepts of theirtrade.

The material given in this textbook establishes on firm mathematical ground thefield of signal analysis. I t is usually scattered in books with different goals, levels,and styles, and one of the purposes of this textbook is to make these prerequisitesavailable in a single volume and presented in a unified manner.

Because Fourier analysis covers a large part of analysis and finds applicationsin many different domains, the choice of topics is very important if one wantsto devise a text that is both of reasonable size and of meaningful content. Thecoloration of this book is given by its potential domain of applications-signalprocessing. In particular, I have included topics that are usually absent from thetable of contents of mathematics texts, for instance, the z-transform and the discreteFourier transform among others.

The interplay between Fourier series and Fourier transforms is at the heart ofsignal processing, for instance in the sampling theory at large (including multiresolution analysis). In the classical Fourier theory, the formula at the intersection of theFourier transform and the Fourier series is the Poisson formula. In mathematicallyoriented texts, it appears as a corollary or as an exercise and in most cases receiveslittle attention, whereas in engineering texts, it appears under its avatar, the formula


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xii Preface

giving the Fourier transfonn of the Dirac combo For obscure reasons, it is believedthat the Poisson sum fonnula, which belongs to classic analysis, is too difficult,

and students are gratified with a result of distributions theory that requires fromthem a higher degree of mathematical sophistication. Surprisingly, in the appliedliterature, whereas distribution theory is implicitly assumed to be innate, the basicproperties of the Lebesgue integral, such as the dorninated convergence and theFubini theorem, are never stated precisely and seldom used, although these toolsare easy to understand and would certainly answer many of the questions that alertstudents are bound to ask. In order to correct this unfortunate tradition, which hasa demoralizing effect on good students, I have insisted on the fact that the c1assicalPoisson fonnula is all that is needed in signal processing to justify the Diracsymbolism, and I have devoted some time and space to introduce the Lebesgueintegral in a concise appendix, giving the precise statements of the indispensabletools.

The contents are organized in four chapters. Part A contains the Fourier theoryin L up to the c1assical results on pointwise convergence and the Poisson sumfonnula. Part B is devoted to the mathematical foundations of signal processing.Part C gives the Fourier theory in L 2 . Finally, Part D is concemed with the timefrequency issue, inc1uding the Gabor transfonn, wavelets, and multiresolutionanalysis. The mathematical prerequisites consist of a working knowledge of theLebesgue integral, and they are reviewed in the appendix.

Although the book is oriented toward the applications of Fourier analysis, themathematical treatment is rigorous, and I have aimed at maintaining a balancebetween practical relevance and mathematical content.

Acknowledgments

Michael Cole translated and typed this book from a French manuscript, and Claudio Favi did the figures. Jean-Christophe Pesquet and Martin Vetterli encouragedme with stimulating discussions and provided the illustrations of wavelet analysis. They also checked and corrected parts of the manuscript, together with GuyDemoment and Emre Telatar. Sebastien Allam and e a n F r a n ~ o i sGiovanelli werealways there when TEX tried to take advantage of my incompetence. To all of them,I wish to express my gratitude, as well as to Tom von Foerster, who showed infinitepatience with my prornises to deliver the manuscript on time.

Gif sur Yvette, FranceMay 2,2001

Pierre Bremaud


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Part A

Fourier Analysis in L1


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ntroduction

In 1807 Joseph Fourier (1768-1830) presented a solution ofthe heat equation l

ae a2e- = K -

t a2xwhere e x, t) is the temperature at time t and at loeation x of an infinite rod, andK is the heat eonduetanee. The initial temperature distribution at time 0 is given:

e x,O) = f x) .

(The solution of the heat equation is derived in Seetion A 1·1.)

In fact, Fourier eonsidered a cireular rod of length, say, 21T whieh amounts toimposing that the funetions x -- f x ) and x -- e x, t) are 21T-periodie. He gavethe solution when the initial temperature distribution is a trigonometrie series

f t ) = L cne t .neZ

Fourier claimed that his solution was general beeause he was eonvineed that alI21Tperiodie funetions ean be expressed as a trigonometrie series with the eoefficients

1

1lT

-intCn = cn f) = - f t )e dt.21T 0

lThe definitive form of his work was published in Theorie Analytique de la Chaleur,Finnin Didot ed., Paris, 1822.


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4 Part A Fourier Analysis in L I

Special cases of trigonometric developments were known, for instance, byLeonhard Euler (1707-1783), who gave the formula

1 . 1 . 1 .x = sm(x) - 2 sm(2x) 3 sm(3x) - ' ,

true for - l < x < +l . But the mathematicians of that time were skeptical aboutFourier s general conjecture. Nevertheless, when the propagation of heat in solidswas set as the topic for the 1811 annual prize of the French Academy of Sciences,they surmounted their doubts and attributed the prize to Fourier s memoir, with theexplicit mention, however, that it lacked rigor. Fourier s results that were in any casetrue for an initial temperature distribution that is a finite trigonometric sum, and beit only for this, Fourier fully deserved the prize, because his proof uses the generaltricks (for instance, the differentiation rule and the convolution-multiplicationrule) that constitute the powerful toolkit of Fourier analysis.

Nevertheless, the mathematical problem that Fourier raised was still pending,and it took a few years before Peter Gustav Dirichlet 2 could prove rigorously,in 1829, the validity of Fourier s development for a large class of periodic functions. Since then, perhaps the main guideline of research in analysis has been theconsolidation of Fourier s ingenious intuition.

The classical era of Fourier series and Fourier transforms is the time when themathematicians addressed the basic question, namely, what are the functions adrnitting a representation as a Fourier series? In 1873 Paul Dubois-Reymond exhibiteda continuous periodic function whose Fourier series diverges at O For almost onecentury the threat of painful negative results had been looming above the theory.Of course, there were important positive results: Ulisse Dini 3 showed in 1880 thatif the function is locally Lipschitz, for instance differentiable, the Fourier seriesrepresents the function. In 1881, Carnille Jordan 4 proved that this is also true forfunctions of locally bounded variation. Finally, in 1904 Leopold Fejeii showedthat one could reconstruct any continuous periodic function from its Fourier coefficients. These results are reassuring, and for the purpose of applications to signalprocessing, they are sufficient.

However, for a pure mathematician, the itch persisted. There were more andmore examples of periodic continuous functions with a Fourier series that divergesat at least one point. On the other hand, Fejer had proven that if convergence istaken in the Cesaro sense, the Fourier series of such continuous periodic functionconverges to the function at all points.

S ur la convergence des series trigonometriques qui servent a epresenter une fonetionarbitraire entre des limites donnees, J reine und angewan. Math., 4 157-169.3 Serie di Fourier e altre rappresentazioni analitiche delle funzioni di une variabile reale,

Pisa, Nistri, vi 329 p.4S ur la serie de Fourier, CRAS Paris, 92, 228-230; See also Cours d'Analyse de l'lfcole

Polytechnique, I, 2nd ed., 1893, p. 99.5Untersuchungen über Fouriersehe Reihen, Math. Ann., 51-69.


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Introduction 5

Outside continuity, the hope for a reasonable theory seemed to be completelydestroyed by Nikola i Kolmogorov, 6 who proved in 1926 the existence of a periodiclocally Lebesgue-integrable function whose Fourier series diverges at alt pointsI t was feared that even continuity could foster the worst pathologies. In 1966 JeanPierre Kahane and yitzhak Katznelson 7 showed that given any set of null Lebesguemeasure, there exists a continuous periodie function whose Fourier series divergesat all points of this preselected set.

The case of continuous functions was far from being elucidated when LennartCarleson 8 published in the same year an unexpected result: Every periodic locallysquare-integrable function has an almost-everywhere convergent Fourier series.This is far more general than what the optirnistic party expected, since the periodiccontinuous functions are, in particular, locally square-integrable. This, togetherwith the Kahane-Katznelson result, completely settled the case of continuousperiodic functions, and the situation finally tumed out to be not as bad as the 1873result of Dubois-Reymond seemed to forecast.

In this book, the reader will not have to make her or his way through a jungleof subtle and difficult results. Indeed, for the traveler with practical interests, thereis a path through mathematics leading directly to applications. One of the mostbeautiful sights along this road may be Simeon Denis Poisson s9 sum formula

L J n) = L jen),n Z n Z

where J is an integrable function (satisfying some additional conditions to bemade precise in the main text) and where

j v ) = L(t)e-2irrvt dtis its Fourier transform, where is the set of real numbers. This striking formulafound very nice applications in the theory of series and is one of the theoreticalresults founding signal analysis. The Poisson sum formula is the culrninating result

of Part A, which is devoted to the classical Fourier theory.

6Une serie de Fourier-Lebesgue divergente partout, CRAS Paris, 183, 1327-1328.7S ur les ensembles de divergence des series trigonometriques, Studia Mathematica, 26,

305-306.8Convergence and growth of partial sums of Fourier series, Acta Math., 116, 135-157.9S ur la distribution de la chaleur dans les corps solides, J Ecole Polytechnique, 1geme

Cahier, XII, 1-144, 145-162.


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Al

Fourier Transforms of Stable Signals

A 1·1 Fourier Transform in LI

This first chapter gives the definition and elementary properties of the Fouriertransform of integrable functions, which constitute the specific calculus mentionedin the introduction. Besides linearity, the toolbox of this calculus contains thedifferentiation rule and the convolution-multiplication rule. The general problemof recovering a function from its Fourier transform then receives a partial answerthat will be completed by the results on pointwise convergence of Chapter A3.

We first introduce the notation: N, Z, Q, ~ , C are the sets of, respectively,integers, relative integers, rationals, real numbers, complex numbers; N+ and ~ +

are the sets of positive integers and nonnegative real numbers.In signal theory, functions from ~ to C are called (complex) signals. We shalluse both terminologies (function, or signal), depending on whether the context istheoretical or applied.

We denote by L ~ ~ )(and sometimes, for short, LI) the set offunctions f( t) 10from ~ into C such that

Lf(t)1 dt < 0 0 .In analysis, such functions are called integrable. In systems theory, they are calledstable signals.

IOWe shall often use this kind of loose notation, where a phrase such as "the functionf(t) means "the function f : lR c. We shall also use the notation "I" or f O witha mute argument. For instance, f ( · - a) is the function t --+ f ( t - a).

P. Brémaud, Mathematical Principles of Signal Processing © Springer Science+Business Media New York 2002


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8 Al. Fourier Transforms of Stable Signals

Let A be a subset of IR. The indicator function of A i s the function lA : IR 1-+{O, 1} defined by

1 if t E A,lA(t) = o i f t ~ A.The function I( t) is called locally integrable if for any closed bounded interval[a, b] C IR, the function I(t)l[a,bj(t) is integrable. We shall then write

I E L ~loe(lR)

or, for short, I E Lloe'The set of functions I (t) from IR into C such that

L/(t)1 2 < 0 0is denoted by L ~ I R ) .I t is the set of square-integrable functions. A signal I( t) inthis set is said to have a finite energy

E = L/(t)1 2 dt.The function I( t) is called locally square-integrable if for any closed boundedinterval [a, b] C IR, the function I(t)l[a,bj(t) is square-integrable. We shall thenwrite

I E L ~ , l o e l R )

or, for short, I E L toeWe recall that in L ~ I R )or ~ I R )two functions are not distinguished if they are

equal almost everywhere with respect to the Lebesgue measure.

EXERCISE AI.I. Give an example 01a function that is integrable but not 01finiteenergy. Give an example 01a function that is 01finite energy but not integrable 01finite energy. Show that

A function I : IR 1-+ C is said to have bounded support if there exists a boundedinterval [a, b] c IR such that I( t) = 0 whenever t ~ [a, b].

I f the function I ( t ) is n times continuously differentiable (that is, it has derivatives up to order n, and these derivatives are continuous), we say that it is in Cn .I f it is in Cn for all n E N, it is said to belong to COO • The kth derivative of thefunction I (t), if it exists, is denoted jCk)(t). The Oth derivative is the function itself:I(O)(t) = I(t); in particular, CO is the collection of continuous functions from IRto C. The set of continuous functions with bounded support is denoted by C ~ .


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10 Al. Fourier Transfonns of Stable Signals

and the cardinal sine

show that (see Fig. Al.2)

1

-T/2 o

reCT(t)

sin(Jrx)

sinc(x)

= rx

recT(t) ~ Tsinc (vT).

+T/2

T

1 2 3T T T

Tsinc(vT) = recT(v)

Figure A 1.2. Fourier transfonn of the rectangle function

We will show that the Gaussian pulse is its own Fr, that is,

(3)

(4)

In order to compute the corresponding Fourier integral, we use contour integrationin the complex plane. First, we observe that i t is enough to compute the F r s( v)for v 2: 0, since this F r is even (see Exercise A1.4). Take a 2: v (eventually, awill tend to 00).

Consider the rectangular contour Y in the complex plane (see Fig. A1.3),

Y = Yl + Y2 + Y3 + Y4,where the Yi 's are the oriented line segments

')'4

-a

Yl : ( -a , 0) -+ (+a, 0),

Y2 : (+a, 0) -+ (+a, v),

Y3 : (+a, v) -+ ( -a , v),

Y4: ( - a , v) -+ ( - a , 0).

v

+a

Figure A1.3. The integration path in the proof of (4)


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A 1·1 Fourier Transform in L I 11

We denote by - Y i the oriented segment whose orientation is opposite that of Yi.Wehave

i -rrz2 dz = 11 + /z + h + 14,

where l i is the integral of e - rr Z2 along Yi. Since the latter integrand is a holomorphicfunction, by Cauchy's theorem (see, for instance, Theorem 2.5.2, p. 83, of [Al],or Theorem 2.2, p. 101, of [A6]),

i - rrz2 dz = 0,and therefore,

h + /z + h + 14 = O.We now show that

lim /z = im 14 = o.~ o o a - oo

For /Z, for instance, if we parameterize Y2 as folIows,

Y2 = {a + it;O,:::: t .:::: v},then

/z = l v e-rr(a+itf i dt = l v e-rr(a2-t2)e-2irrat i dt.Therefore, since v :::: a,

l/zl :::: l a e-rr(a-t)(a+t) dt ::::l a e-rra(a-t) dt2 a 1 2= e - rra errat dt = - 1 _ e - rra ),

o J ra

where the last quantity tends to 0 as a tends to +00. A sirnilar conclusion holdsfor 14, with sirnilar computations. Therefore,

im (h + h ) = 0,a - H X l

that is,

(5)

Using for YI the obvious parameterization

im { = lim j + a e - rrt2 dt = { e - rrt2 dt = 1.a--+oo J l a--+oo - a JJ I .

Parameterizing -Y3 as folIows,

-Y3 = {iv + t; - a :::: t :::: +a},


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12 Al. Fourier Transforms of Stahle Signals

wehave

1= l + a e-n(iv+tf dt

- ) 3 - a

Therefore,

Going back to (5), we obtain

which gives the announced result. •EXERCISE A1.6. Deduce Jrom (4) that, Jor all Cl > 0,

The F r of the Gaussian pulse can be obtained by other means (see ExerciseA 1.16). However, in other cases, contour integration is often necessary.

Using contour integration in the complex plane, we show that, for a > 0,

at F r A 1s(t) = e - IlR+(t) ~ s(v) = .

a + 2 m v(6)

First observe that

s(v) = e - 2znvte - at dt = . e-2znvt-a\2inv + a) dt10 . 1 10 .o 2mv + a 0= 1 1 - Z dz.

2inv + a Y(The reader is refered to Fig. A1.4 for the definition ofthe lines y, YJ. Y2, and Y3.)

Therefore, it suffices to show that

i - z dz = 1.By Cauchy's theorem,

1 -Z dz +1 - Zdz +1 - Zdz = 0.Yl Y2 ) 3


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A 1·1 Fourier Transform in L I 13

2i7f1/

Figure A1.4. The integration path in the proof of (6)

The limit as A t 0 0 of I Y l e - Z dz is I y e - Z dz, and that of I Y3 e- Z dz = IoA e - t dtis 1. I t therefore remains to show that the limit as A t 0 0 of J, e- Z dz is 0, and

Y2

this foIlows from the bound

I{e-zdzl ::: e - A IY21,where IY21 = K x A is the length of Y2.

EXERCISE AI.7. Deducefrom (6) that

Convolution Multiplication Rule

•

THEOREM AI.I. Let h(t) and x(t) be two stable signals. Then the right-hand sideoJ

y(t) = 1 h(t - s)x(s) ds (7)

is defined Jor almost all t and defines almost everywhere a stable signal whose FTis y(v) = h(v)x(v).Proof By ToneIli's theorem and the integrability assumptions,

f1xIR Ih(t - s)llx(s)1 dt ds = ( l ' h ( t ) ' d t ) ( l ' X ( t ) ' dt) < 0 0 .

This implies that, for almost aIl t,

l ' h ( t - s)x(s)1 ds < 0 0 .

The integral IIR h(t - s )x(s) ds is therefore weIl defined for almost aIl t. Also,

1 , y ( t ) ' dt = 111h(t - s)x(s) dsldt::: 1 1 ' h ( t - s)x(s)1 dt ds < 0 0 .


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Therefore, y(t) is stable. By Fubini's theorem,

L ( L h ( t - S)X(S)dS) e-2irrvt dt

= L Lh( t - s)e-2irrv(t-s)x(s)e-2irrvs ds dt

= h(v)x(v).

The funetion y(t), the convolution of h(t) with x(t), is denoted by

y(t) = (h *x)(t).We therefore have the convolution-multiplication rule,

F r A(h *x)(t) --+ h(v)x(v).

•

(8)

EXAMPLE AI.L The convolution of he rectangular pulse reeT (t) with itself is thetriangular pulse of base [ - T, + T] and height T,

TriT(t) = (T - Itl)1[-T,+T](t).

By the convolution-multiplication rule,

TriT(t) ~ (Tsine ( v T ) f (9)

(see Fig. Al.5).

EXERCISE ALS. Let x(t) be a stable complex signal. Show that its autoeorrelationfunetion

c(t) = L x(s + t)x*(s) ds

is well defined and integrable and that its FT is Ix(v)12 •

T

AT +TTriT(t)

T 2

&II

C > ~ I ~ C >i 1 j- ~ - ~ - ~ 0 ~ ~ ~

Figure A 1.5. FT of the triangle funetion


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Al·l Fourier Transform in L I 15

EXERCISEAl.9. Showthatthenthconvolutionpoweroff(t) = e - a t l t ~ o t ) ,wherea > 0, is

t n - 1fM(t) = e - at I t>o(t).(n - I) -

/*3 = f * * , etc.) Deducefrom this the FTofs( t) = n e - a t l t ~ o t ) .

Riemann Lebesgue Lemma

The Riemann-Lebesgue lemmal is one of the most important technical toolsin Fourier analysis, and we shall use it several times, especially in the study ofpointwise convergence of Fourier series (Chapter A3).

THEOREM Al.2. The FT o f a stable complex signal s(t) satisfies

lim Is v)1= O. (10)Ivl-+oo

Proof' The F r of a rectangular pulse s(t) satisfies Is v)1 ::; K/lvl [see Eq. (3)].Hence every signal s(t) that is a finite linear combination of indicator functionsof intervals satisfies the same property. Such finite combinations are dense inL b l ~ )(Theorem 28 of the appendix), and therefore there exists a sequence sn(t)of integrable functions such that

lim (Isn(t) - s(t)1 dt = 0n-+oo JIR

andA Kn

ISn(v)1 ::; ~

for finite numbers Kn • From the inequality

Is(v) - sn(v)1 ::;

Ls(t) - sn(t)1 dt,

we deduce that

K n i:; - + Is(t) - sn(t)1 dt,lvi IRfrom which the conclusion follows easily. •

The following uniform version of the Riemann-Lebesgue lemma will be neededin the sequel.

11 Riemann, B., (1896), Sur la possibilite de representer une fonetion par une serietrigonometrique, Oeuvre Math., p. 258.


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THEOREM Al.3. Let f(t) be a 2:rr-periodic locally integrable function, and letg : [a,b] f - CbeinC I , where [a,b] ~ [-:rr, +:rr]. Then

lim lb

fex - u)g(u) sin(Au) du = 0....... 0 0 a

uniformly in x.

Proof For arbitrary E > 0, choose a 2:rr-periodic function h(t) in Cl such that

r:rrIf(x) - h(x)1 dx < E(Theorem 29 of the appendix). Integrating by parts yields

/(A) = l b hex - u)g(u) sin(Au) du

COS(AU) Ib l b , COS(AU)= - hex - u)g(u) + [hex - u)g(u)] du.A a a A

Since h E Cl and is periodic, h and h' are uniformly bounded. The same is true ofg, g' (g is in Cl). Therefore,

Now,

lim /(A) = 0 uniformly in x .. . . . . . . 0 0

11b fex - U)g(U)Sin(AU)dUI

:S I/(A)I + l b Ih(x - u) - fex - u)llg(u)1 sin(Au) du

:S I/(A)I +~ ~

Ig(U)ll

b

Ih(x - u) - fex - u)1 sin(Au) du

:S I/(A)I + max Ig(u)IE:.a: Ou: Ob

The conc1usion then follows because E is arbitrary.

AI· 2 Inversion Formula

•

EXERCISE Al.lO. Show that the FT of a stable signal is uniformly bounded anduniformly continuous.

Despite the fact that the FT of an integrable signal is uniformly bounded anduniformly continuous, it is not necessarily integrable. For instance, the FT of therectangular pulse is the cardinal sine, a non-integrable function. When its FT isintegrable, a signal admits a Fourier decomposition.


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AI· 2 Inversion Fonnula 17

THEOREM At.4. Let set) be an integrable complex signal with the Fouriertransform s(v). Under the additional condition

Ls(v)1 dv < 0 0 , (11)the inversion formula

set) = L(v)e+2iJrvt dv (12)holdsfor almost all t . l f set) is, in addition to the above assumptions, continuous,equality in (12) holds for all t.

(Note that the exponent ofthe exponential ofthe integrand is +2irrvt.)

EXERCISE At.l l . Check that the above result is true for the signal

(a E lR, a > 0, a E C).

Proof' We now proceed to the proof of the inversion formula. (lt is rather technical and can be skipped in a first reading.) Let set) be a stable signal and considerthe Gaussian density function

with the F r

We first show that the inversion formula is true for the convolution (s *h u )(t).Indeed,

(s *hu)(t) = { s(u)hu(u)e...L u (t)du,R 2u 2 ';;2 (13)

and the F r ofthis signal is, by the convolution-multiplication formula, s(v)hu(v).Computing this F r directly from the right-hand side of (13), we obtain

s(v)hu(v) = ( s(u)hu(u) ( { e I • u (t)e-2iJrvt dt) duJ ~ J ~~ ; ; r

= { s(u)hu(u)e I u (v) du.J ~ ~ , ; ; rTherefore, using the result ofExercise A1.11,

{ s(v)hu(v)e2iJrvt dv = { { s(u)hu(u)e I U (v) dU) e2iJrvt dvJ ~ J ~ J ~ ~ ;;r

= { s(u)hu(u)e I -'


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18 Al. Fourier Transforms of Stahle Signals

Thus, we have

(s *hu)(t) =

L(v)hu(v)e2i1rvt dv,

and this is the inversion formula for (s *h u )(t).

(14)

Since for all v E IR, limu-l-o v t hu(v) = 1, it follows from Lebesgue's dominated convergence theorem that when u ..I- 0 the right-hand side of (14) tendsto

L(v)e2i1rvt dvfor all t E IR. I f we can prove that when u ..I- 0 the function on the left-hand side of(14) converges in L ~ I R )to the function s(t), then, for almost all t E IR, we havethe announced equality (Theorem 25 of the appendix).

To prove convergence in L ~ I R ) ,we observe that

L(s *hu)(t) - s(t)1 dt = LLs(t - u) - S(t))hu(U)dul dt (15)(using the fact f r r ~hu(u)du = 1), and therefore, defining f(u) = iJRIs(t - u )s(t)1 dt,

Ls *hu(t) - s(t)1 dt :::::L(u)hu(u) du.Now, If(u)1 is bounded (by 2 iJRIs(t)1 dt). Therefore, iflimu-l-o f(u) = 0, then, bydominated convergence,

lim [ f(u)hu(u) = lim [ f(uu)ht(u)du = o.uwk uwk (16)Toprovethatlimu-l-O f(u) = 0, we begin with thecasewheres t)iscontinuous withcompact support. In particular, it is uniformly bounded. Since we are interested ina limit as u tends to 0, we may suppose that u is in a bounded interval around 0,and in particular, the function t -+ Is(t - u) - s(t)1 is bounded uniformly in u byan integrable function. I t follows from the dominated convergence theorem thatlimu-l-o f(u) = o.

Now, let s(t) be only integrable. Let {snO}n>t be a sequence of continuousfunctions with compact support that converges i;' L ~ I R )to sO (Theorem 27 ofthe appendix). Writing

f(u) :::::d(s(· - u), sn(' - u)) + L

Sn(t-

u)-

sn(t)1dt + d(s(·),

snO),

where

d(s(· - u), sn(' - u)) = Ls(t - u) - sn(t - u)1 dt,the result easily follows.


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Al·2 Inversion Fonnula 19

Suppose that, in addition, set) is continuous. The right-hand side of (12) definesa continuous function because s(v) is integrable. The everywhere equality in (12)follows from the fact that two continuous functions that are almost everywhereequal are necessarily everywhere equal (Theorem 8). •

The Fourier transform characterizes a stable signal:

COROLLARY ALL lf two stable signals SI (t) and S2(t) have the same Fouriertransform, then they are equal almost everywhere.

Proof' The signal set) = Sl(t) - S2(t) has the FT s(v) = 0, which is integrable,and thus by (12), set) = ° or almost all t. •EXERCISE

AI.12. Give theFT of

set) = 1/A(a 2

+ t2).

Deduce from this the valueo f he integral

l ( t ) = f ~ d U ,J .R. t+u

t > 0.

EXERCISE Al.13. Deduce from the Fourier inversion formula that

LS i ~ t )Yt = Jr.Exercise 1.14 is very important. It shows that for signals that cannot be called

pathological, the version of the Fourier inversion theorem that we have in thischapter is not applicable, and therefore we shall need finer resuIts, which are givenin Chapter A3.

EXERCISE AI.14. Let set) be a stable right-continuous signal, with a limit fromthe left at all times. Show that ifs(t) is discontinuous at some time to, its FT cannotbe integrable.

Regularization Lemma

In the course of the proof of Theorem A1.4, we have used a special case of theregularization lemma below, which is very useful in many circumstances.

DEFINITION AI.2. A regularizing function is a nonnegative function h a : IR ---+ IRdepending on a parameter a > ° nd such that

La(u) du = 1, forall a > 0,l +alim ha(u) du = 1,

a \-O - a

limha(u) = 1,a \-O

forall a > 0,

forall u E IR.

LEMMA ALl. Let h a : IR ---+ IR be a regularizing function. Let set) be in Lb(IR).Then

lim f I(s *ha )(t) - s(t)1 dt = 0.a \-O JIR


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20 Al. Fourier Transforms of Stable Signals

Proof We ean use the proof of Theorem A1.4, starting from (15). The onlyplaee where the speeifie form of h (a Gaussian density) is used is (16). We must

therefore prove that

lim ( J(u)h,,(u) = 0-1-0 JIR

independently. Fix e > O. Sinee limuto J(u) = 0, there exists a = aCe) such that

1 ~ e l ~ e(u)h,,(u) du :s - h,,(u) du :s - .- a 2 - a 2

Sinee J(u) is bounded (say, by M),

( J(u)h,,(u) du :s M ( h,,(u) du.JIR\[ -a.+a] JIR\[ -a.+a]

The last integral is, for suffieiently small a, less than e12M. Therefore, forsuffieiently small a,

( e eJIR J(u)h,,(u) du :s 2 + 2 = e. •

The funetion h is an approximation of the Dirae generalized funetion o(t) inthat, for all


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with the initial condition

e ~ ,0) = F ( H

where F(O is the F r of fex). The solution is

e ~ ,t) = ~ ) e - 4 n 2 K ~ 2 f .Since x 1-+ (4JrKt)-1/2 e (4Kf)-1/2 x2 has the F r ~ 1-+ e - 4 n 2 K ~ \the convolutionmultiplication formula gives

1 ( 2

8(x, t ) = (4JrKt)-Z J ~ fex - y)e-i'ii dy,

or

8(x, t) = ..JrrLex - 2-JKiy)e-yl dy •As we mentioned earlier, Fourier considered the finite rod heat equation, which

receives a similar solution, in terms of Fourier series rather than Fourier integrals(see Chapter A2). The efficiency of the Fourier method in solving differential orpartial differential equations of mathematical physics has been, after the pioneeringwork of Fourier, amply demonstrated 12 .

12See, for instance, the classic text of 1. N. Sneddon, Fourier Transfonns, McGraw-Hill,1951; Dover edition, 1995.


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A

Fourier Series o Locally StablePeriodic Signals

A2·1 Fourier Series in loc

Fourier Coefficients

A periodic signal is neither stable nor of finite energy unless it is almost everywherenull, and therefore, the theory of the preceding Chapter is not applicable. Therelevant notion is that of Fourier series. (Note that Fourier series were introducedbefore Fourier transforms, in contrast with the order of appearance chosen in thistext.) The elementary theory of Fourier series of this section is parallel to theelementary theory of Fourier transforrns of the previous section. The connection

between Fourier transforrns and Fourier series is made by the Poisson sum formula,of which we present a weak (yet useful) version in this chapter.

A complex signal s(t) is called periodic with period T > ° or T -periodic) if,for all t E ~

s(t + T) = s(t).A T -periodic signal s(t) is locally stable, or locally integrable, if s(t) E Lb [O, Tl),that is,

lT Is(t)1 dt < 0 0 .

A T -periodic signal s(t) is locally square-integrable if s(t) E L ~ [ O ,Tl), that is,

l T Is(t)1 2 dt < 0 0 .P. Brémaud, Mathematical Principles of Signal Processing © Springer Science+Business Media New York 2002


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24 A2. Fourier Series of Locally Stable Periodic Signals

One also says in this case that s(t) hasfinite power, since

lim.. ..

{AIs(t)12

=.. ..(T Is(t)12 dt < 0 0 .

A-+oo A 10 T 1As the Lebesgue measure of [0, T] is f i n i t e , L ~ [ O ,Tl c Lt [O, Tl). (See Theo

rem 19 of the appendix.) In particular, a finite-power periodic signal is also locallystable.

We are now ready for the basic definition.

DEFINITION A2.I. The Fourier transform {sn}, n E Z, of the locally stable Tperiodic signal s(t) is defined by theformula

I l T 2 nSn = - s(t)e- l J r T I dt,T 0 (19)and Sn is ca lIed the nth Fourier coefficient of the signal s(t).

EXERCISE A2.I. Compute the Fourier coefficients of he T -periodic function s(t)such that on [0, T), s(t) = t.EXERCISE A2.2. Let s(t) be a locally stable T -periodic signal. Defining

ST(t) = s(t)I[O,TJ(t),show that the nth Fourier coefficient Snofs(t) and the FT i:;(v) OfST(t) are linkedby

(20)

EXERCISE A2.3. Compute the Fourier coefficients of the T -periodic signal s(t)such that on [ - T 2, + T 2), s(t) = 1 [ - a ~ , + a ~ l t ) ,where a E 0, 1).

EXERCISE A2.4. Let s(t) be a T -periodic locally stable signal with nth Fouriercoefficient Sn. Show that limlnltoo Sn = O.

One often represents the sequence {Sn}nEZ of the Fourier coefficients of a Tperiodic signal by "spectrallines" separated by 1/ T from each other along thefrequency axis. The spectralline at frequency n / T has the complex amplitude Sn(see Fig. A2.1). This is sometimes interpreted by saying that the FT of s(t) is

s v) = I)nÖ V- f)'nEZ

where ö t) is the Dirac generalized function (see Section B2-4).

EXERCISE A2.5. Let s(t) be a T -periodic locally stable signal with nth Fouriercoefficient Sn. What is the nth Fourier coefficient of s(t - a), where a E llV Whatcan you say about the period and the Fourier coefficients of the signal s(t / a),where a > O?


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A2·1 Fourier Series in Lloc 25

Figure A2.l. From the Fourier transform to the Fourier coefficients

Convolution MuItiplication Rule

THEOREM A2.1. Let x(t) be a T -periodie locaily stable signal, and let h(t) be astable signal. The signal

y(t) = L(t - s)x(s)ds (21)is almost everywhere weil defined, T -periodie, and locaily stable. Its nth Fouriercoefficient is

A A(n) AYn = h T xn,

where h v) is the FT of h(t) (see Fig A2.2).

Proof· We have

L1h( t - s)llx(s)1 ds = laT IhT(t - s)llx(s)1 ds,where

hT(u) = L h(u + nT).nEZ

Now

x v)

I I I I I I I- ~ -q, - ~ 0 2 3T T T v

Figure A2.2. Filtering aperiodie signal

fI v)

(22)

,


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and hence by the usual argument (see the proof of Theorem A 1.1),

Lh t - s)llx(s)1 ds

<0 0

for almost aIl t E lR. Thus, y t) is almost everywhere weIl defined by (21). Also,

y t + T) = L t + T - s)h s)ds= L t - s)h s)ds,

which shows that y t) is periodic with period T. The same argument as in the proofof Theorem A U shows that y t) is locally stable. FinaIly,

1 l T Z nYn= - y t)e- l 1 ' j ' t dtT 0

1 l T1 - Z' n- hT t - s)x s)e- l 1 ' j ' t dt dsT 0 0

•

A2·2 Inversion Formula

The Poisson Kernel

In the proof of the Fourier series inversion formula, the Poisson kernel will playa role similar to that of the Gaussian pulse in the proof of the Fourier transforminversion formula of the previous seetion.

The Poisson kernel is the family of functions Pr : lR f-+


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A2·2 Inversion Formula 27

and therefore,

Pr(t) :::: O. (24)

Also,

1 I + T / 2- Pr(t) dt = 1.T -T /2

(25)

In view of the above expression of the Poisson kernel, we have the bound

1 [ 1 - r 2 )- Pr(t)dt <T [- t ,+t] \[-e,+s] - 11 _ e 2i1ry 12 '

and therefore, for all e > 0,. 1 [hm - Pr(t)dt = O.

r t l T [-t,+t]\[-e,+e](26)

Properties (24)-(25) make of the Poisson kernel a regularizing kernel, and inparticular,

1 tim -1 cp(t)Pr(t) dt = cp(O),

r t l T - t

for all bounded, continuouscp

:ffi. -+


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28 A2. Fourier Series of Locally Stable Periodie Signals

function LnEZ sne+2irr(n/T)t, pointwise and in ~ [ O ,T]). The result then followsfrom Theorem 25.

The statement in the case where set) is continuous is proved exactly as thecorresponding statement in Theorem A1.4. •

As in the case of stable signals, we deduce from the inversion formula theuniqueness theorem.

COROLLARY A2.1. Two locally stable periodic signals with the same period Tthat have the same Fourier coefficients are equal almost everywhere.

EXERCISE A2.6. Compute

using the expression ofthe Fourier coefficients ofthe 2-periodic signal set) suchthat

fort E [ -1 ,+1] .

EXERCISE A2.7. Let x(t) be a T -periodic locally stable signal with nth Fouriercoefficient x n such that

L InlPlxnl < 00.nEZShow that x(t) is p times differentiable and that if the pth derivative is locallyintegrable, its nth Fourier coefficient is (2i7T .f)P Xn.

The Weak Poisson Formula

The Poisson sum formula takes many forms. The strong version is

(30)

This aesthetic formula has a number of applications in signal processing (see PartB).

The next result establishes the connection between the Fourier transform andFourier series, and is central to sampling theory. I t is a weak form of the Poissonsum formula (see the discussion after the statement of the theorem).

'THEOREM A2.3. Let set) be a stable complex signal, and let 0 < T < 00 befixed. The series LnEZ set + nT) converges absolutely almost everywhere to aT -periodic locally integrable function t),the nth Fourier coefficient of which is( l /T)s(n/T).

We paraphrase this result as follows: Under the above conditions, the function

t):= L set + nT) (31)nEZ


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A2·2 Inversion Forrnula 29

is T -periodie and locally integrable, and its formal Fourier series is

Sj(t) = ~

I)( ..)e2inIfI.

T nEZ T (32)

(We speak of a "formal" Fourier series, because nothing is said about its convergence.) Therefore, whenever we are able to show that the Fourier series representsthe function at t = 0, that is, if 0) = S j(O), then we obtain the Poisson sumformula (30).

For now, we are saying nothing about the convergence of the Fourier series.This is why we talk about a weak Poisson's formula. A strong Poisson's formulacorresponds to the case where one can prove the equality everywhere (and in

particular at t = 0) of t)and of its Fourier series. We shall say more about thePoisson formula and, in particular, give strong versions of it in Seetion A3·3. Theversion we have here, and that we shaH proceed to prove, is the one we need in theShannon-Nyquist sampling theorem (Chapter B2).

Proof: We first show that t)is weH defined:

{T L Is(t + nT)1 dt = L (T Is(t + nT)1 dt1 nEZ nEZ 1

l


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= ~ { T LS(t+ kT)e- 2i1C ',f(t+kTl dtT Ja kEZ

1 { 2· n 1 n )= T JJRs(t)e- l1C'it dt = TS T . •We have a function as weH as its formal Fourier series. When both are equal

everywhere, we obtain the strong Poisson sum formula. The next exercise givesconditions for this.1t will be improved by Theorem A3.12.

EXERCISE A2 S Let set) be a stable signal with the FT s v), and suppose that

(a) L n E Z set + nT) is a continuous function, andb) L n E Z Is n/T)I < 00 .

Show that, for all t E lR.,

Ls(t+nT)= L s(f)e 2i1C ',ft.nEZ nEZ


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A3

Pointwise Convergenceo Fourier Series

A3·1 Dini s and Jordan' s Theorems

The inversion formula for Fourier series obtained in Chapter A2 requires a ratherstrong condition of summability of the Fourier coefficients series. Moreover, thiscondition implies that the function itself is almost everywhere equal to a continuousfunction. In this seetion, the dass of functions for which the inversion formula holdsis extended.

Recall Kolmogorov's negative result (see the Introduction):

THEOREMA3.1. There exists a locally integrable 2rr -periodic unction f : cefor which the Fourier se ries diverges everywhere.

This result challenges one to obtain conditions that a locally integrable 2rrperiodic function f must satisfy in order for its Fourier series to converge tof. Recall that the Fourier series associated with a 2rr-periodic locally integrablefunction f is the formal Fourier series

where cn(f) is the nth Fourier coefficient

cn(f) = _1 j Jr f(u)e-inu du.2rr - J r

(33)

(34)

The series (33) is calledformal as long as one does not say something about itsconvergence in some sense (pointwise, almost everywhere, in LI, etc). If one has



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32 A3. Pointwise Convergence ofFourier Series

no more than the condition that f is 27T-periodic and locally integrable, the worstcan happen, as Kolmogorov's theorem shows.

The purpose of this section is to find reasonable conditions guaranteeingconvergence as n -+ 0 0 of the truncated Fourier series

+nsI (x) = L ck(f)e ikx . (35)

k=-n

We have to specify (1) in what sense this convergence takes place and (2) whatthe limit iso Ideally, the convergence should be pointwise and to fitself. The nextexercise gives a simple instance where this is true.

EXERCISE A3.1. Assume that the trigonometric series

+nSn(t) = L Ckeikt

k=-n

converges uniformly to some function f(t) . Show that in this case, for all k E Z,

Ck = ck(f) .

Dirichlet s Integral

We will first express the truncated series sI in a form suitable for analysis. Forthis we write

+n { 1 j+Jr . }sI (x) = L - f ( s )e - ,ks ds e'kxk=-n 27T - J r

1 j+Jr I n I- L eik(x-s) f (s)ds .27T - J r k=-n

Elementary computations give

+n sin((n + -2 )t)L e ikt =k=-n sin(t /2)

(36)

(the function in the right-hand side is called the Dirichlet kerne ) and therefore,

f 1 j+Jr sin((n + i ) (x - s))Sn (x) = - 2 . (( _ )/2) fes) ds.

7T - J r sm x s

Performing the change of variable x - s = u and taking into account the fact thatf a n d the Dirichlet kernel are 27T-periodic, we obtain

f 1 j+Jr sin((n + )u)Sn (x) = - 2 . / 2 ) fex + u)du.

7T - J r sm u(37)

The right-hand side of (37) is called the Dirichlet integral.


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A3·1 Dini's and Iordan's Theorems 33

I f we let f( t ) = in (35), we obtain 1; on substituting this in (37),1 j+Jr sin((n + )u) _

- d u - I .27f - J r sin(u/2) (38)

Therefore, for any real number A,

I j+Jr sin((n + )u)S (x) -A=- . 2 ( f ( x + u ) - A ) d u

27f - J r sm(u/2)(39)

or, equivalently,

1 i r sin((n + )u)S (x) - A = - . 2 {fex + u) + fex - u) - 2A}du. (40)27f 0 sm(u/2)

Therefore, in order to show that, for a given x E IR, S (x) tends to A as n -+ 00,it is neeessary and suffieient to show that the Diriehlet integral in the right-handside of (39) eonverges to zero as n -+ 00.

The localization principle states that the eonvergenee of the Fourier series is aloeal property. More preeisely:

THEOREM A3.2. l f f a n d gare two locally integrable 27f -periodic complex-valued

functions such that, for a given x E IR and some 8 > 0, it holds that f (t) = g(t)whenever t E [x - 8, x + 8], thenlim{S (x) - S (x)} = O.ntoo

Proof" Using (39) we have

I j+Jr fex + u) - g(x + u)s (x) - S (x) = - 2 sin((n + )u) Ilul:::8 . ( /2) du7f - J r sm u

1 j+Jr= - sin((n + )u) w(u) du,27f - J r

where

fex + u) - g(x + u)w(u) = l lul>8 . ( /2)- sm u

is integrable over [0, 27f]. The last integral therefore tends to zero as n -+ 0 0 bythe Riemann-Lebesgue lemma. •

We now state the general pointwise convergence theorem.

THEOREM A3.3. Let f be a locally integrable 27f-periodic complex-valuedfunction, and let x E IR and A E IR be given. Then

lim S (x) = Antoo


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34 A3. Pointwise Convergence of Fourier Series

if,for some ° 8 :s Ti,

. 1. ~ ( u )

11m sm«n + )u) du = 0,ntoo 0 uj2 (41)where

~ u )= f (x + u) + f (x - u) - 2A. (42)

Proof" Taking g a constant equal to A, we have Sn(g) = A, and therefore we arelooking for a sufficient condition guaranteeing that Sn(f) - Sn(g) tends to ° s ntends to 00 . By the localization principle, it suffices to show that

lim [8 sin«n + )u) . ~ u )du = 0.ntoo 10 Sln(uj2) (43)

The two integrals in (41) and (43) differ by

1 sin«n + )u) v(u)du, (44)where

v(u) = u ){ U ~ 2- S i n ~ j 2 ) }

is integrable on [0, 8]. Therefore, by the Riemann-Lebesgue lemma, the quantity(44) tends to zero as n --+ 00 . •

Dini s Theorem

THEOREM A3.4. Let f be a 2Ti-periodic locally integrable complex-valuedfunction and let x E IR. If or some ° 8 :s Ti and some A E IR, the function

t --+f (x + t) + f (x - t) - 2A

is integrable on [0, 8], then

lim S (x) = A.ntoo

Proof" The hypothesis says that the function ~ u ) j u ,where is defined in (42),is integrable, and therefore condition (41) ofTheorem A3.3 is satisfied (RiemannLebesguelemma). •

We shall give two corollaries ofDini's result.

COROLLARY A3.1. If a 2Ti -periodie locally integrable complex-valued functionf( t) is Lipschitz continuous of order IX > ° bout x E IR, that is,

If(x + h) - f(x)1 = O(lhIO ) as h --+ 0,

then limntoo S (x) = f(x).


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A3·1 Dini's and Jordan's Theorems 35

Proof· Indeed, with A = f(x),

I(x + t) + f (x - t) - 2A I < K _ 1 _

t - ItI1 - 0 :

for some constant K and for all t in a neighborhood of zero, and 1/ltI 1- 0 : isintegrable in this neighborhood, because I - ex< 1. Dini's theoremA3.4 concIudesthe proof. •

COROLLARY A3.2. Let f( t ) be a 21T-periodic locally integrable complex-valuedfunction, and let x E lR be such that

f (x + 0) = lim f (x + h) and f (x - 0) = lim f(x - h)hW hW

exist and are finite, and further assume that the derivatives to the left and to theright at x exist. Then

I. S I ( ) _ f (x + 0) + f ( x - 0)1m n X - •ntoo 2

Prao " By definition, one says that the derivative to the right exists if

lim f (x + t) - f (x + 0)t tO t

exists and is finite, with a similar definition for the derivative to the left. Thedifferentiability assumptions imply that

lim f (x + t) - f (x + 0) + f (x - t) - f (x - 0)t tO t

exists and is finite and therefore that

tjJ(t) f (x + t) + f (x - t) - 2A

is integrable in a neighborhood of zero, where

2A = f (x + 0) + f (x - 0).Dini 's theorem A3.4 concIudes the proof. •EXAMPLE A3.1. Apply the previous theorem to the 21T-periodic function definedby

f( t ) = t when 0 < t :::; 21T.Onefinds

sin(nt)t = 1T - 2 when 0 < t < 21T.nEZ nn#O

For t = 0, we can directly check that the sum of he Fourier series is( f (O+)+f (O- ) ) = (0+21T)=1T,


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as announced in the last corollary. For t = n /2, we obtain the remarkable identityn l 1 1

4 = 1 - 3 + : 5 - 7 + · · · ·Jordan s Theorem

Jordan 's convergence theorem features funetions of bounded variation.

DEFINITION A3.1. A real-valued function q; : lR f-+ lR is said to have boundedvariation on the interval [a, b] C lR i f

n - l

sup L q;(Xi+l) - q;(Xi) < 00,'D i=O

(45)

where the supremum is over all subdivisions D = {a = Xo < Xl < . . . < Xn = b}.We quote without proof the fundamental result on the strueture of bounded

variation funetions.

THEOREM A3.5. A real-valued unction q; has bounded variation over [a, b] i f andonly ifthere exist two nondecreasing real-valuedfunctions q;l, q;2 such that,for allt E [a, b],

q;(t) = q;l (t) - q;2(t). (46)In partieular, for all X E [a, b), q; has a limit to the right q;(x + 0); for all

X E (a, b], it has a limit to the left q;(x - 0); and the diseontinuity points of q;(t)in [a, b] form a denumerable set, and therefore a set ofLebesgue measure zero.

THEOREM A3.6. Let f be a 2n-periodic locally integrable real-valuedfunctionof bounded variation in a neighborhood of a given X E lR. Then

lim st(x) = f (x + 0) + f (x - 0)ntoo 2

(47)

•The proof is omitted.

EXERCISE A3.2. Let f E L ~ l R ) .Show that, for any B > 0,

I+ B j(v)e2irrvt dv = 2B { f ( t + s)sine (2Bs) ds,-B JRand

use this to study the pointwise convergence o f he left-hand side as B tends toinfinity, along the lines of the current chapter.

The funetion

2B sine (2Bt)

is also ealled Dirichlet's kernei.


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A3·1 Dini's and Jordan's Theorems 37

EXERCISE A3.3. Let t and h be the 2rr -periodic functions defined on (-rr, +rr]by

t (x) = x,

Compute their Fourier coefficients, and use this to compute

L (_ l )n ,n ~ l n

Integration o Fourier Series

Let f(t) be a real-valued 2rr-periodic locally integrable function. Denoting byCn the nth Fourier coefficient of f(t), we have C n = c ~ because f( t) is real.Therefore, the Fourier series of f(t) can be written as

0 0

ao + L { a n cos(nx) + bn sin(nx)}, (48)n=l

where, for n ::: 1,

1 1Jran = - f(t)cos(nt)dt,rr 0

1 1Jrbn = - f(t)sin(nt)dt.rr 0

Of course, the series in (48) is purely formal when no additional constraints areput on f(t) in order to guarantee its convergence. Now, the function F(t) definedfor t E [0, 2rr) by

F(t) = Iat(f(X) - ao)dx (49)

is 2rr-periodic, is continuous (observe that F O) = F(I) = 0), and has boundedvariation on finite intervals.

Therefore, by Jordan's theorem its Fourier series converges everywhere, and forall x E lR,

0 0

F(x) = A o + L { A n cos(nx) + Rn sin(nx)},n=l

where, for n ::: 1,

1 1JrAn = - F(t) cos(nt) dtrr 0

1 [ sin(nx) ]2Jr= F ( x ) - -

rr n 01 1Jr- ( f ( t ) - ao) sin(nt) dtnrr 0

1 1Jr b- - f(t)sin(nt)dt = _. . : ,nrr 0 n


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and, with a similar computation,

B

n= .. .. (2n F(t) sin(nt) dt = an

n 10 nTherefore, for all x E lR,

1 {an . bn }F(x) = zA o + -;; sm(nx) - -;; cos(nx) . (50)

The constant Ao is identified by setting x = 0 in (50):

1 ~ b nzA o = L...- .

n = l n

(51)

Since Ao is finite we have shown, in particular, that ~ l bn/n converges for anysequence {bn }n2:1 of the form

bn = .. .. (2n J(t) sin(nt) dt,n 10

where, J(t) is areal function integrable over [0, 2n].

Gibbs Overshoot Phenomenon

We dose this section by mentioning a phenomenon typical of the behavior of aFourier series at a discontinuity of the function. Gibbs' overshoot phenomenonhas nothing to do with the failure of the Fourier series to converge at a point ofdiscontinuity of the corresponding function. It concems the overshoot of the partialsums at such a point of discontinuity. An example will demonstrate this effect.

Consider the 2n -periodic function defined in the interval ( - n, + n] by

J(x) =

I ;x2 2i fx > 0,

if x < 0,if x < O.

The partial sum of its Fourier series is

f _ sin(nx)Sn (x) - L...- .

k=l n

By Dini's theorem, the partial sum sI (0) converges pointwise to (1/2)( / (0+) +J(O-» = n /2. However, we shall see that for some A > n /2 and sufficientlylarge n,

S ( ~ ):::A. (52)Therefore, there exist a constant c > 0 and a neighborhood No of 0 such that

IsI(x) - sI (0)1 ::: c whenever x E No - {O}. This constitutes Gibbs' overshoot


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A3·2 Fejer's Theorem 39

phenomenon, which can be observed whenever the function has a point of discontinuity. The proof of (52) for this special case keeps most of the features of thegeneral proof, which is left for the reader. In this special case,

f l x sin n + ) t ) xSn (x) = dt - .o 2 sin( t) 2

Now,

l x sin n + ) t )1 dto 2 sin('it)

-_lX(sin(nt)Cos( t) 1 )-----;----"-- + -2 cos nt) dto 2sin( t )

l x sin nt) l x , ( cos( t) 1)- - dt + sm nt) 1 - -t dto t 0 2 sin('it)ll X+ - cos(nt)dt.2 0

The last two integrals converge uniformly to zero (by the uniform version of theRiemann-Lebesgue lemma). Also,

1 sin(nt) l n sin t) : r r : r rdt = dt ::::: 1.18 - > - .o t t 2 2•

A3·2 Fejer's Theorem

The Fourier sI (t) series of a 2:rr-periodic locally integrable function I convergesto I ( t ) for a given t only under certain conditions (see the previous section).However, Cesaro convergence of the series requires much milder conditions. Fora 2:rr -periodic locally stable function I, Fejer's sum

(53)

behaves more nicely than the Fourier series itself. In particular, for continuousfunctions, it converges pointwise to the function itself. Therefore, Fejer's theoremis a kind of inversion formula, in that it shows that for a large dass of periodicfunctions (see the precise statement in Theorem A3.11 ahead), the function can berecovered from its Fourier coefficients.


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Fejer's Kernel

Take the imaginary part of the identity

n - l 1 _ inu' i(k+l/2)u = e iu / 2 e .

1 - e lUk=O

to obtain

Starting from Dirichlet's integral expression for sI(t) [cf, Eq. (37)], we obtain, inview of the identity just proven, Fejer's integral representation of a (x),

{+Jr (+Jra (x) = LJr K n ( u ) f ( x - u ) d u = LJr K n ( x - u ) f ( x ) d u ,

where

I s i n 2 ( ~ n t )Kn(t) = 2 1

2n:rr sin ('it)

is, by definition, Fejer's kernel. I t has the following properties:

and [letting f( t ) = 1 in (54)], i: rKn(u) du = 1.Also (the proofs are left as an exercise),

and, for all e S :rr,

lim Kn(t) = 1,ntoo

j +Clim Kn(u) du = 1.ntoo -6

(54)

(55)

(56)

(57)

(58)

(59)

The last four properties make ofFejer 's kernel a regularization kerneion [ -:rr, +:rr]

(by definition of a regularization kernel).

Cesaro Convergence for Fourier Series of Continuous Functions

We first treat the case of continuous functions, because the result can be obtained from the basic principles of analysis, in particular, without recourse tothe Riemann-Lebesgue lemma.

THEOREM A3.7. Let f( t ) be a 2:rr-periodic continuousfunction. Then

lim sup la (x) - f(x)1 = O. (60)ntoo XE[-Jr,+Jr]


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A3·2 Fejer's Theorem 41

Proof' From (54) and (56), we have

la x) - l(x)1 ::s i:n

KnCu) I / (x - u) - l(x)1 du

18 [= + = A + B .- 8 [ -n ,+n] \ [ -8 ,+8]

(61)

For a given 8 > 0, ehoose 8 sueh that I / (x - u) - l(x)1 ::s 8/2 when lul ::s 8.Note that I is uniformly eontinuous and uniformly bounded (being aperiodie andeontinuous funetion), and therefore 8 ean be ehosen independently of x. We have

81+8 8A::s 2 - 8 Kn(u)du ::s 2and, ealling M the uniform bound of I,

B ::s 2M [ Kn(u) du.J [ - n , n ] \ [ 8,+8]

By (57) and (59), B ::s 8/2 for n sufficiently large. Therefore, for n suffieientlylarge, A + B ::s 8. •

Fejer's theorem for eontinuous periodie funetions is the key to important approximation theorems. The first one is for free. We eall a trigonometrie polynomialany finite trigonometrie sum of the form

+n

p(x) = L Ck eikx .-n

THEOREM A3.8. Let I (t) be a 2n -periodic continuous unction. Select an arbitrary8 > O. Then there exists a trigonometric polynomial p(x) such that

sup I/(x) - p(x)1 ::s 8 .tE[ -n ,+n]

Proof' Use Theorem A3.7 and observe that a (x) is a trigonometrie polynomiill. •

From this, we obtain the Weierstrass approximation theorem.

THEOREM A3.9. Let I : [a, b] 1-+ C be a continuousfunction. Select an arbitrary8 > O. There exists a polynomial P(x) such that

sup I/(x) - P(x)1 ::s 8 .tE[a,b]

Jf, moreover, I is real-valued, then P can be chosen with real coefficients.

Proof' First, suppose that a = 0, b = 1. One ean then extend I : [0, 1]] 1-+ Cto a funetion still denoted by I, I : [ - n , + n ] ] 1-+ C, that is eontinuous andsueh that I H n ) = I ( - n ) = O. By Theorem A3.8, there exists a trigonometrie


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polynomial p(x) such that

esup If(x) - p(x)l:s sup If(x) - p(x)1 :s -.

tE[D, I] tEl -rr,+rr] 2Now replace each term e ikx in p(x) by a sufficiently large portion of its Taylorseries expansion, to obtain a polynomial P(x) such that

Then

esup IP(x) - p(x)1 :s -.

tE[D,I] 2

esup I f (x) - P(x)1 :s -.

tE[D, I] 2To treat the general case f : [a, b] f-+ C, apply the result just proven to cp[0, 1] f-+ cedefined by cp(t) = f(a + (b - a)t) to obtain the approximatingpolynomial rr(x), and take P(x) = rr«x - a)j(b - a».

Finally, to prove the last statement of the theorem, observe that

If(x) - Re P(x)1 :s If(x) - P(x)l· •Fejer s Theorem

We shall first obtain for the Fejer's sum the result analogous to Theorem A3.3.First, from (54), we obtain

1 irr sin 2 ( nu)a (x ) = - . 2 2 1 { f ( x + u ) - f ( x - u ) } d u ;

2nrr D sm Czu)

therefore, for any number A,

1 irr sin 2 ( nu)a ( x ) - A = - . 22 1 { f ( x + u ) + f ( x - u ) - 2 A } d u .

2nrr D sm (:zu)

THEOREM A3.10. For any x E IR.and any constant A,

l ima (x ) = Antoo

if, for some 8 > 0,

. 1 1 . 4J(u)11m - sm 2( nu) - 2 - du = 0,ntoo n D u

where

4J(u) = f (x + u) + f (x - u) - 2A.Proof The quantity

I r sin 2( nu) 4J(U) du l < r 14J(u)1 dun 18 sin 2 ( u) - n 18 sin 2 ( u)

(62)

(63)

(64)

(65)

(66)


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A3·3 The Poisson Formula 43

tends to 0 as n t 00 . We must therefore show that

11 sin 2( nu)_ 2 fjJ(u) dun 0 sin 2( u)

tends to 0 as n t 00 . However, (65) guarantees this because

118 (1 1'S - . 2( 1 ) - 1 2 IfjJ(u)1 dun 0 SIll ZU ZU

tends to 0 as n t 0 0 (the expression in curly brackets is bounded in [0, 8], andtherefore the integral is finite). •

THEOREM A3.11. Let f (t) be a 2JT-periodic locally integrable unction and assumethat, for some x E ~ the limits to the right and to the left (respectively, f (x + 0)and f (x - 0»), exist. Then

lim u (x) = f (x + 0) + f (x - 0) . (67)2

Proof" Fix 8 > O. In view of the last result, it suffices to prove (65) with

fjJ(u) = {f (x + u) - f (x + O)} + {f (x - u) - f (x - O)}.Since fjJ(u) tends to 0 as n 00 , for any given c > 0 there exists rJ = rJ(c),o < rJ :'S 8, such that IfjJ(u)1 :'S c when 0 < u :'S rJ. Now,

1

118 sin2( nu) 1- ; fjJ(u) dun 0 u

c 1rysin 2( nu) 1 18 IfjJ(u)1< - du + - du.n 0 u 2 n u 2

The last integral is bounded; and therefore, the last term goes to 0 as n t 00 . Asfor the penultimate term, it is bounded by Ac, where

10 sin2( v)A = 2 dv < 00 .o v •

A3·3 The Poisson Formula

The following corollary of F6jer's theorem will play the key role for the proof ofthe Poisson sum formula (Theorem A3.l2).


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COROLLARY A3.1. Let f be a 2rc -periodie locally integrable function and supposethat, for some x E lR.,

(a) thefunction f is continuous at x, and

(b) its Fourier se ries sI(x) converges to some number A.Then A = fex).

Proof' From (b) we see that

lim 0 1(x) = A.ntoo

From F6jer's theorem and (a),

lim 0 1(x) = fex).ntoo •

We have already given a weak version of the Poisson sum formula in SectionA2·2. A most interesting situation is when the function cI>(t) defined by (31) isequal to its Fourier series for all t E lR., that is,

L s ( t + nT) = ~ L S ( f ) e2i n ft for all tE R (68)nEZ nEZ

The next theorem extends the result in Exercise A2.8.

THEOREM A3.12. Let set) be a stable complex signal, and let 0 < T < 00 befixed. Assume in addition that

(1) L n E Z set + n T) converges everywhere to some continuous function,

(2) L n E Z s f) e2i n ft converges for all t.

Then the strang Poissonformula (68) holds.

Proof' The result is an immediate consequence of both the weak Poisson summation result (Theorem A2.3) and the corollary of F6jer's theorem in SectionA3·2. •

Here are two important cases for which the strong Poisson sum formula holds.

COROLLARY A3.1. Let set) be a stable complex signal, and let 0 < T < 00 befixed. If, in addition, L set + nT) converges everywhere to a continuousfunctionthat has bounded variation, then the Poissonformula (68) holds.

Praof' We must verify conditions (1) and (2) ofTheorem A3.12. Condition (1)is part of the hypothesis. Condition (2) is a consequence of Iordan's theoremA3.6. •

EXAMPLE A3.1. If set) is continuous, has bounded support, and has boundedvariation, the Poisson sumformula (68) holds.


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A3·3 The Poisson Formula 45

COROLLARY A3.2. If a stable continuous signal s(t) satisfies

s(t) = 01+\t l" ' ) as Itl ~ 00 ,s(v) = oe+llvl"') as lvi ~ 00 ,

for some a > 1, then the Poisson formula (68) holds for alt ° T < 00.Proof The result is an immediate corollary of Theorem A3.12.

Convergence Improvement

(69)

•

The Poisson formula can be used to replace aseries with slow convergence by onewith rapid convergence, or to obtain some remarkable formulas. Here is a typicalexample. For a > 0,

s(t) = e-2naltl ~ s(v) = an(a 2 + v2 ) .

Since

L s ( t +n) = Le-2nalt+nlnEZ nEZ

is a continuous function with bounded variation, we have the Poisson formula, thatis,

The left-hand side is equal to

2- 1 ,

1 - e- 2na

and the right-hand side can be written as

Therefore,

1 n 1 + e- 2na 1L a 2 + n 2 = 2a 1 - e- 2na 2a 2 .nO":1

Letting a 0, we have

The general feature of the above example is the following. We have aseriesthat is obtained by sampling a very regular function (in fact, C OO ) but also slowly


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h(t-2T)

1

[ j U [ j U [ j [ j [ j ~-3T -2T -T 0 T 2T 3T

-3T -2T -T O T 2T 3T

Figure A3.1. Radar return signal

decreasing. However, because of its strong regularity, its F r has a fast decay. Theseries obtained by sampling the F r is therefore quickly converging.

Radar Return Signal

Let s(t) be a signal ofthe form

s(t) =

(I>(t-nT») f(t).

nE'L

(70)

(We may interpret h(t - nT) as a return signal of the nth pulse of a radar afterreftection on the target, and f (t) as a modulation due to the rotation of the antenna. )The F r ofthis signal is (see Fig. A3.1)

s(v) = ~ Lh( .-)f(v - .-).T nE'L T T

(71)

EXERCISE A3.4. Show that if( 1) f t)is integrable, (2) LnE 'L h(t - n T) is integrableand continuous, and (3) LnE 'L h(n/T) < 0 0 , then (71) holds true. Find otherconditions.

References

[Al] Ablowitz, M.J. and Jokas, A.S. (1997). Complex Variables, Cambridge UniversityPress.

[A2] Bracewell, R.N. (1991). The Fourier Transform and Its Applieations, 2nd rev. ed.,

McGraw-Hil1; New York.[A3] Gasquet, C. and Witomski, P. (1991). Analyse de Fourier et Applieations, Masson:Paris.

[A4] Helson, H. (1983). Harmonie Analysis, Addison-Wesley: Reading, MA.[A5] Katznelson, Y. (1976). An Introduetion to Harmonie Analysis, Dover: New York.[A6] Kodaira, K. (1984). Introduetion to Complex Analysis, Cambridge University Press.[A7] Körner, T.W. (1988). Fourier Analysis, Cambridge University Press.


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References 47

[A8] Rudin, W. (1966). Real and Complex Analysis, McGraw-Hill: New York.[A9] Titchmarsh, E.C. (1986). The Theory of Funetions, Oxford University Press.[AlO] Tolstov, G. (1962). Fourier Series, Prentice-Hall (Dover edition, 1976).[All] Zygmund, A. (1959). Trigonometrie Series, (2nd ed., Cambridge University Press.


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57/262

Introduction

The Fourier transform derives its importance in physics and in electrical engineering from the fact that many devices mapping an input signal x t) into an outputsignal y t) have the following property: I f the input is a complex sinusoid e2invt,the output is T v)e2invt, where T v) is a complex function characterizing the device. For example, when x t) and y t) are, respectively, the voltage observed at theinput and the steady-state voltage observed at the output of an e circuit (see Fig.BO.I), the input-output mapping takes the form of a linear differential equation:

y t) + RCy t) = x t),

and it can be readily checked that

IT v) =

1 2irrvRC

The e circuit is one of the physical devices that transform a signal into anothersignal, that satisfy the superposition principle, and that are time-invariant. Moreprecisely:

R1 ~ 1x t) C y t)

l ,1, ,igure BO.I. The e circuit


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52 Part B Signal Processing

(a) If Yl t) and Y2 t) are the outputs corresponding to the inputs Xl t) and X2 t),then AlYl t) A2Y2 t) is the output corresponding to the input AlXl t) A2X2 t);

(b) I f y t) is the output corresponding to x t) , then y t - -r) is the outputcorresponding to x t - -r).

Such physical devices are caIled (homogeneous linear) filters.A basic example is the convolutional filter, for which the input-output mapping

takes, in the time domain, the form

y t ) = 1 t - s)x s)ds,where h t) is caIled the impulse response, because it is the response of the filterwhen the Dirac pulse 8 t) is applied at the input. Indeed,

h t) = 1 t - s)8 s) ds.I f the impulse response is integrable, the output is weIl defined and integrable, aslong as the input is integrable. Then, by the convolution-multiplication rule, theexpression of the input-output mapping in the frequency domain is

y v) = T v)x v),

where T v) is the frequency response, that is, the F r of the impulse response:

T v) = 1 t)e-2i1Cvt dt.Observe that if the input is x t ) = e-2i1Cvt the output is weIl defined and equal to

1 s)x t - s)ds =1 s)e- i CV t-s)ds = T v)e-2i1Cvt,in accordance with what was said in the beginning of this introduction.

In the particular case of the RC filter, the solution of the differential equationwith arbitrary initial condition at 0 0 is indeed of the convolution type, with theimpulse response

The RC filter is a convolutional filter, and it contains the typical features ofthe more general filters. In the general case, since a filter is a mapping, we shallhave to define its domain of application. Depending on this domain, the inputoutput mapping takes different forms. In the above informal discussion of theRC circuit, there are a frequency-domain and a time-domain representation andalso a representation in terms of a linear homogeneous differential equation. Thelatter is not general. In fact, when it is available, the transmittance is a rationalfunction ofthe frequency v. The corresponding filters, caIled rational ilters, forman important class, and Chapter BI gives the basic concepts concerning analog(that is, continuous-time) filters.


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Introduction 53

In addition to filtering, there are two fundamental operations of interestin communications systems: frequency transposition and sampling. Frequencytransposition is a basic technique of analog communications. It has two main applications, the first of which is transmission. Indeed, the Hertzian channels are inthe high-frequency bands- in fact, much higher than the one ofbrute signals suchas the electric signals carrying voice, for instance-and consequently, the latterhave to be frequency-shifted. The second reason is resource utilization and is related to frequency multiplexing, a technique by which signals initially occupyingthe same frequency band are shifted to nonoverlapping bands and can then be simultaneously transmitted without mutual interference. From a mathematical point ofview the theory of frequency transposition (or, equivalently, of band-pass signals,to be defined in Chapter BI is not difficult. I t remains interesting because of thespecial phenomena associated with this technique, such as cross-talk in quadraturemultiplexing and dispersion phenomena.

In digital communications systems, an analog signal s t ) is first sampled, and theresult is a sequence of sampies s n T), n E Z. It is important to identify conditionsunder which the sampie sequence faithfully represents the original signal. Thecentral result of Chapter B2 is the so-called Shannon-Nyquist theorem, whichsays that this is true if the signal s t ) is stable and continuous and if the supportof its F r s v) is contained in the interval [ 1 / T + I/Tl. The original signal canthen be recovered by the reconstruction formula:

s t ) = L s nT) sinc f - n ) .nEZ

The theory of sampling is an application of the results obtained in Part A, andin particular of the Poisson sum formula. The above reconstruction formula hasmany sourees, 1 and its importance in communications was fully realized by ClaudeShannon and Harald Nyquist.

The Shannon-Nyquist sampling theorem is the bridge between the analog(physical) world and the discrete-time (computational) world of digital signal processing. The reader will find in the main text abrief discussion of the interest ofdigital communications systems. Therefore, a large portion of this Part B is devotedto discrete-time signals (Chapters B2-B4).

As we have already mentioned, the Poisson sum formula is the key to the sampling theorem. It also plays a very important role in the numerical analysis ofthe discrete Fourier transform considered as an approximation of the continuous Fourier transform (see Chapter B3) and also in the intersymbol interferenceproblem (see Chapter B2).

The study of the interaction between discrete time and continuous time is notlimited to the sampling theorem. For instance, we prove that filtering and sampling

ISee J.R. Higgins, Five short stories about the cardinal series, Bult. Amer. Math. Soc.,12, 1985,45-89.


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54 Part B Signal Processing

cmumute for base-band signals. This is not a difficult result, but it is of course afundamental one because in signal processing, one first sampies and then performs

the filtering operation in the sampled domain, since oneof

the advantagesof

digitalprocessing comes precisely from the difficulty of making analog filters.

One advantage of analog processing is that it is instantaneous. To maintaincompetitivity, the signal processing algorithms have to be fast. For instance, thediscrete Fourier transform is implemented by the so-called fast Fourier transform,an algorithm whose principle we briefty explain in this Part. Subband coding alsohas a fast algorithm associated with it. I t is a data compression technique. Thesignal is not directly quantized, but instead, it is first analyzed by a filter bank, andthe output of each filter bank is quantized separately. This allows one to dispatch

the compression resources unequally, with fewer bits allocated to the subbandsthat are less informative (see the discussion in Chapter B4). Subband coding isthe last topic of Part Band introduces the sections on multiresolution analysis inPart D.


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BI

Filtering

B 1·1 Impulse Response and Frequency Response

Convolutional Filter

We introduce a particular and very important dass of filters.

DEFINITION Bl.l. The transformation fram the stable signal x(t) to the stablesignal y(t) defined by the convolution

y(t) = 1 (t - s)x(s)ds, (1)where h(t) is stable, is ca lied a convolutional filter. This filter is called a causalfilter if h(t) = 0 for t < O.

The signal y(t) is the output, whereas the signalx(t) is the inputofthe linear filterwith impulse response h(t). Informally, if x(t) is the Dirac generalized function8(t) (an impulse at time 0), the output is (see Fig. Bl.1)

1 (t - s)8(s) ds = h(t),whence the terminology.

A causal filter responds only after it has been stimulated. For this reason, itis sometimes also called a realizable filter (Fig. B 1.1. features a causal impulseresponse). For such filters, the input-output relationship (1) becomes (note the



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56 BI. Filtering

8(t) I>

f \ h t ~

>0 0 Vimpulse impulse response

Figure B 1.1. Impulse and impulse response

upper limit of integration)

y(t) = [too h(t - s)x(s) ds. (2)DEFINITION B1.2. The Fourier transform ofthe (stable) impulse response h(t),

T(v) = L(t)e-2i1Cvt dt, (3)is ca lied the frequency response.

I f the input is the complex sinusoid x(t) = e2i1Cvt, by (1), the output isy(t) = T(v)e2i1Cvt. (4)

(Note that the output is weH defined by the convolution formula, even though inthis particular case the input is not integrable.)

EXERCISE Bl.1. Let y(t) be the output o f a stable and causal convolutional filterwith impulse response h(t) [see (2)]. Let

z(t) = 1 h(t - s)x(s)ds, t ~ 0,be the output o f he same filter, when the input x(t) is applied only from time t = 0on. Show that

lim Iz(t) - y(t)1 = O.tt+oo

A More General Definition

Convolutional filters are only a special dass of filters. A more general definitionis as foHows. Denote by C the set of functions of R into C.

DEFINmON Bl.3. Let D(12) be a set o f unctions from lR into C with the twofollowing properties:

(a) It is closed under linear operations;(ß) it is closed under translation.

12 : D(12) 1-+ C is ca lied a homogeneous linear filter with domain D(12) if:

(i) 12 is linear, and(ii) 12 is time-invariant.


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B 1·1 Impulse Response and Frequency Response 57

The meaning ofproperties (a) and (ß) is the following: (a) XI (t), X2(t) E D( c),

AI, A2 E C ===} AIXI (t) + A2X2(t) E D( c); and (ß) x(t) E D( c), T E lR ===}x(t - T) E D( c).

The meaning of properties (i) and (ii) is the following: (i) XI (t), X2(t) E D( c),c c C

AI, A2 E C , X I ( t ) ~ ·YI(t),X2(t) ~ Y2(t) ===} AIXI(tHA2X2(t) ~ AI(t)YI(tH

A2Y2(t); (ii) x(t) E D( c), T E lR, x(t) y(t) ===} x(t - T) y(t - T).

EXERCISE B1.2. Show that ife2i :rcvt E D( c), then2 · C 2·e mvt ~ T(v)e mvt (5)

for some complex number T(v).

The function T(v) is called thefrequency response ofthe filter. Every frequencyresponse is of the form

T(v) = G(v)eiß(v),

where G(v) = IT(v)1 is the amplitude gain and ß(v) = Arg T(v) is the phase.EXAMPLE Bl l et

D( c) = {x(t) :Lx(t)1 dt < oo},or

D( c) = {x(t) + e(t) : Lx t)1 dt < 00 and e(t) E E},

(6)

where E is the set of complex finite linear combinations of complex exponentials.For any signal in D( c), the right-hand side of (1) is welt defined, and we cantherefore define the filter ,C with domain D( c) by the input-output relationship(1). Thefrequency response, as defined by (5), is then the FT ofh(t).

EXAMPLE B1.2. Let h(t) E L ~ ( l R ) .We shalt see in Part C that the FT h(v) = T(v)of h(t) can be defined and that it is in L ~ ( l R ) .We take D( c) = LUlR) and define,C by the input-output relationship

y(t) = L(v)x(v)e 2i :rcvt dv, (7)where xCv) is the FT ofthe input x(t) E L ~ ( l R ) .The right-hand side of(7) has a

meaning since T(v) and xCv) being in L ~ l R )implies that T(v)x(v) is in Lt(lR)(see Theorem 20 of he appendix).

EXAMPLE B 1.3. IfT (v) is an arbitrary function, not necessarily in L ~(lR), one canalways define a filter ,C by the input-output relation (7), provided one chooses fordomain D( c) the set of signals x(t) such that the right-hand side has a meaning.


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58 B 1. Filtering

1

-B o +BLow-pass (B)

~ B ----7 ~ B ----7

- V Q oBand-pass vQ, B)

+ V Q

Figure B 1.2. Low-pass and band-pass frequency responses

Low-Pass Band-Pass and Hilbert Filters

The low-pass and band-pass filters (see Fig. B1.2) that we now define belong tothe category of Example B 1.2.

One calls low-pass (B) a filter with frequency response

T(v) = 1[-B,+B](v), 8)

where B is the cut-offfrequency. One calls band-pass (B, va), where 0 < B < va,a filter with frequency response

T(v) = 1[-vo-B,-vo+Bj(v)+ l[vo-B,vo+Bj(v),where Va is the center frequency, and 2B is the bandwidth.

9)

Hilbert's filter (see Fig. B 1.3) belongs to the category of Example B 1.3. I t is thefilter with frequency response

T(v) = i sgn (v), where T(O) = O. (10)

One possible domain for Hilbert's filter is the set of stable (resp., finite-energy)signals whose FT has compact support. The amplitude gain of Hilbert's filter is 1(except for v = 0, where the gain is zero), and its phase is

Ir /2 if v > 0,ß(v) = 0 if v = 0,- J r /2 if v < O.

(11)

+i r-

,0- - - - - - - , i

Hilbert filter

Figure B1.3. Hilbert frequency response


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B 1·1 Impulse Response and Frequency Response 59

There is no function admitting the frequency response (10). There is, in fact,a generalized function (in the sense of the theory of distributions) with F r equal

toT(v).

However, in signal theory, the Hilbert filter is used only in the theoryof

band-pass signals (see Section BI·2). For such signals the Hilbert filter coincideswith a bona fide convolutional filter:

EXERCISE Bl.3. Show that the output y(t) ofthe Hilbertfilter, corresponding toa stable signal x(t) having an FT x(v) that is null outside the frequency band[ - B, + B], can be expressed as

1 2 sin 2 (n Bs)y(t) = - x(t - s) ds.R. n s

Differentiation and Integration as Filters

Let D( c) be the set of signals

x(t) = L(v)e2iJrvt dv,where Lx(v)1 dv < 0 0 and Lvllx(v)1 < 00 .Such signals are continuous and differentiable with derivative

~ x(t) = [(2inv)x(v)e2iJrvt dv.dt JR

(12)

(13)

(Apply the theorem of differentiation under the integral sign; 15 ofthe appendix).

The mapping x(t) dx(t)/dt is a linear filter, called the differentiating filter, ordifferentiator, with frequency response

T(v) = 2inv.

Let D( c) be the set of signals of the form (12), where

The signal

~ Ix(v)1dv < 0 0 and [lx(v)1 dv < 00 .JIR JIR lvi

y(t) = [ x.(v) e2iJrvt dvJIR2 m v

is in the domain of the preceding filter (the differentiator), and therefore,

~y(t) = [ x(v)e2iJrvt dv = x(t).dt JIR

(14)

The transformation x(t) y(t) is a homogeneous linear filter, which is called theintegrating filter, or integrator, with frequency response

1T(v) = . .

2 m v(15)


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60 BI. Filtering

y(t) .c2 *.c1 series

x(t) y(t) .c2 +.c l parallel

x(t)~ - - - - , - - - - - : ~y (t)

Figure BlA. Series, parallel, and feedback configurations

Series Parallel and Feedback Configurations

We now describe the basic operations on filters (see Fig. B1.4). Let C] and C2be two convolutional filters with (stable) impulses responses h]( t ) and h2 (t) andfrequency responses T](v) and T2(V),respectively.

The series filter C = C 2 *C] is, by definition, the convolutional filter withimpulse response h(t) = (h] *h2)(t) and frequency response T(v) = T](v)T2(V).I t operates as folIows: The input x(t) is first filtered by Cl, and the output of C] isthen filtered by C2 , to produce the final output y(t).

The parallel filter C = C] + C 2 is, by definition, the convolutional filter withimpulse response h(t) = h]( t ) + h2(t) and frequency response T(v) = T](v) +T2(V). I t operates as folIows: The input x(t) is filtered by Cl, and "in parallel," itis filtered by C2 , and the two outputs are added to produce the final output y(t).

The feedback filter C = CI/(1 - C] *C 2 ) is, by definition, the convolutionalfilter with impulse response frequency re

Pierre Bremaud - Mathematical Principles of Signal Processing

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