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this work.

Wiley-VCM Vcrlag Cimbli
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<&xiisland 4064. Australia
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Rnhcratein. Alaxanrlei. Stengcr

p. ctn.
Includes ivhliographical Efcrences &nd iudex
E R N 0 471 iikXOO 6
I , Systenr theoiy. I. Rnhcnstein. Iludolf. [I. Stciigcr, Alexmdcr. Ill Title.
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reface
1 I n
1.1
1.2
I .3
1.3

xv

~ r ~ ~ u c t ~ ~ ~ i
Sigrrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Systcm:, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
OvPrvicw of the Book . . . . . . . . . . . . . . . . . . . . . . . .
Exerciscs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Is of Continuous LTI-Systems
2.1 DifferenLia.l Eqi ions . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Block Diagrmis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3 State-Space Description of LTI-Systcms . . . . . . . . . . . . . . .
2.4 Higher-orcler Differentia.L Equations. Rloclc Diagrams and the Si
ivlodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Equivdcnl. StiLt.6?-SI’ii(.e. R.epresoiit,atioris . . . . . . . . . . . . . . .
2.6 C‘ontrolla.hle arid C)lnserva.ble Systems . . . . . . . . . . . . . . . . .
2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.
1
5
13
13

17
17

19
29

~ - ~ y sint the
e ~~ ~e ~ u e r i c y - ~ o ~ ~ ~ i ~ i
3 .I Coniplex I+equt:rrcies . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Kxercism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Laplace ~
~
~
s
~

o
r
~
4. L ‘l’he Eigenhtiuictiorr Fornuilatiori . . . . . . . . . . . . .
4.2 Definition of 1. he Laplac(: ‘1i.ansforrn . . . . . . . . .
4.3 Unilateral aiitl Bilntcral Lit.place Transforms . . . . .
4.4 E.xamples of Lapliice ‘lYarisforms . . . . . . . . . . .
4.5 kgicin Of CollVt?J‘g(Tlceof the! LC?l>liLCt. ’&msforrn . .
4.6 Existence and linic~ucnessof thc Lapla.ce Tra.iisform

61

.......
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .

. . . . . . . .

61
61
63
64
66
72


1.7 Propcrtic\s of the Laplnce Transform . . . . . . . . . . . . . . . . . 75
4.8 Exertisw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83
5.1
5.2
5..j
5.4
5.5
5.6

lex Analysis and the Pnversc? Laplace Transform
87
Path Iiitc?gra.Is in tsheCorllylrx Pliuie . . . . . . . . . . . . . . . . . 87’
The Main Principle of Complex Aiialysis . . . . . . . . . . . . . . . 811
Circular Iiitegrals that. F,nclose Siugiilarities . . . . . . . . . . . . . 89
Ca.uchy Irit.egrids . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
lnvcrse Lap1;tce ‘1Y;l.asforrrt . . . . . . . . . . . . . . . . . . . . . . .
95
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
103

Analysis of ContinuousI-Systems with the La
~ a ~ $ f o r ~ ~
6.1 Syst.ern R.~sponscto Bilatcral Iriptit. Signals . . . . . . . . . . . . .
6.2 Finding ihe Syst.em Fmx%ion . . . . . . . . . . . . . . . . . . . . .
6.3 Poles ;ind Xcros of tlie Systelrr Functioii . . . . . . . . . . . . . . .
6.4 Detcwrtiuing the Syst.em Funct.ioii from Ilif€ererit3iadEcpat.ions . . .
6.5 Surnrrin.risiiigExample . . . . . . . . . . . . . . . . . . . . . . . . .
6.(i Combining Simple LTI-Systems . . . . . . . . . . . . . . . . . . . .
6.7 Comhinirtg LXI-Systems with Multiple Iirput.s arid Output.s . . . .
6.8 Arialysis of St.nte-Spacc Descriplioris . . . . . . . . . . . . . . . . .

6.9 Exercisw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

105
107

110
112
113
115
118
121

120

nitial C o ~ i ~ i t i Problcrns
oi~
with the aplace Transform 12
7.1 Firsts.Ortler. Systc?ms . . . . . . . . . . . . . . . . . . . . . . . . . .
125
7.2 Second-Ordcr Systems . . . . . . . . . . . . . . . . . . . . . . . . .
135
7.3 I-Iiglier-Ordcr Sy ms . . . . . . . . . . . . . . . . . . . . . . . . .
137
nt of the Proct:thires for Solving Initial Condition Problenis 1.18
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.19
~ ~ o ~ i v o l ~ tand
i o r Impulse
i
153

153
8.1 Motivttt.ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Tirne Uc!liriviviour of ;in RCXircuit . . . . . . . . . . . . . . . . . . . 154
8.3 ‘I’he Delta tmpulse . . . . . . . . . . . . . . . . . . . . . . . . . . .
157
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
8.4 Convolution
16’7
8.5 -Applicntions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
187
8.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
191

The Fourier Transfarm
195
9.1 Review of the Lap1
Tra.nslorm . . . . . . . . . . . . . . . . . . . :1.95
9.2 Definit.ion of tlic Fciuriclr Transform . . . . . . . . . . . . . . . . . . 196
tj.3 Similaait.iw wid Differcnccs hntwccn Fonrit.r a.iltZ L ~ ~ > l i i (‘Ika.risf(mns
X!
198
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
9.4 Examples of the Fouricr Transform
200


vii

['ontents

9.5 Syininctries oi' the Fourier Trtmsform . . . . . . . . . . . . . . . . .
Fourier 73rm:ifc)rm . . . . . . . . . . . . . . . . . . . . . . .
9.7 Yrop(!rf.i(?s of the Fouricr Trrziisforrri . . . . . . . . . . . . . . . . .
9.8 Pimcval's Theorerii . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.9 Corrc3lzLtion of' Deterrniuist.ic Signals . . . . . . . . . . . . . . . . .
9.1.0 l'iruc-Wnndwicl(,h Product . . . . . . . . . . . . . . . . . . . . . . .
9.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
PO Bode Plots
10.1 Introtfticiioxi . . . . . . . . . . . . . . . . . . .
10.2 Chr&ribution of' Iu&ivi(hd Polcs mid Zeros
10.3 Rode Plots for Multiple I'oles a r i d Zeros . .
10.4 R d c s for Bode Plots . . . . . . . . . . . . . .
10.5 Compltx Pairs of 'Poles t ~ . n t l%(>rt):i. . . . . .

208
213

4

7 -1

225
227
232
236
2411

. . . . . . . . . . . .

241

. . . . . . . . . . . . . 2/12

. . . . . . . . . . . . .
............
. . . . . . . . . . . . .
................................

2/16
248
2/19
255

11 Sampling and Perie,
261
11 1 Irit roduetion . . .
. . . . . . . . . . . . . . . . . . . .
261
11.2 Delta lrrryulse Trttiri arid Periodic F'uiictioiis . . . . . . . . . . . . . 262
11.3 Sernpliiig
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
269
11.4 Exer
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2bG

295

. . . . . . . . . . . . . . . . . . . 295
12.2 Sorrre Siniplc Sequences . . . . . . . . . . . . . . . . . . . . . . . .
297
3 2.3 Discrc+te-'l'inic Fourier Traxisfomi

. . . . . . . . . . . . . . . . . . . 301

12.4 Ya.mpling Continuous Siqinls . . . . . . . . . . . . . . . . . . . . .
12.5 Properlies of tlic .FATraiisforiii . . . . . . . . . . . . . . . . . . . .
12.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

:$OH
308

312

315

Exmplcs . . . . . . . . . . . . . . .
13.2 Hcgion of' Corivm-grrice of the z-Transforim . . . .
13.3 R.c.lzttionships 10 Ot,lic.l, 'I 'x.;iiisfc)rmal-itjiis . . . . .
13.4 Theorcrtis of Lhe 2 -Trttrisform . . . . . . . . . . . .
Tran&)rrn . . . . . . . . . . . . . . . . .
Diagrams in t.Iw z-Plaiie . . . . . . . .

.........
315
. . . . . . . . . . 320
.
.
.
.

. . . . . . . . .
........
. . . . . . . .
. . . . . . . . .

................................
iscrete-Time ~
~
~
~
~
14.1 Iril.rotliiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.2 Lincnritv arid 'I'imcb-invariance . . . . . . . . . . . . . . . . . . .

14.3 Limw Diikitwe Eqmiictns ufith CoristntitcC'ocficients . . . . . . .

322
326
328
332
334
339
339
339
340

y


...

Contents

Vlll

14.4 Charactrristic Seyuerices iilld Svsteni Functions of Discxcte LTTSysteins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.5 I3loc.lr 13iagrams Z L I IState-Spacc
~
. . . . . . . . . . .......
14.6 Discrete Convohxtion and Impulse Rwponse . . . . . . . . . . . . .
14.7 Excrciscs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

341
346

350
362

15 Causality arid the Hilbert Trarisfovin

15.1
15.2
15.3
15.4

307
Causal Syytems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
367
C‘a~isalYigirnls . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
370
Signals wilh a One-Sirlcd Spec;trttrri . . . . . . . . . . . . . . . . . . 374
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
378

16 Stability and Feedback Systems
383
16.1 BIBO. Impulse Response ancl Fkcyiieiicy R e s p ~ s Cimw
t~
. . . . . 383
16.2 Causal Stable LTI-Systems . . . . . . . . . . . . . . . . . . . . . .
388
16.3 FwdhiLck Systems . . . . . . . . . . . . . . . . . . . . . . . . . . .
394
16.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
400

17 Describing Random Signals
403
17.1 Inlrotiuction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
303
17.2 Ij;xpect?edValues . . . . . . . . . . . . . . . . . . . . . . . . . . . .
405
17.3 Sta(ionltry Random Processes . . . . . . . . . . . . . . . . . . . . .
41.1
17.4 Correlat.ion Functioiis . . . . . . . . . . . . . . . . . . . . . . . . .
416
17.5 Power Density Spectra . . . . . . . . . . . . . . . . . . . . . . . . .
425
17.6 1)escribing Discrete I<.antloin Signals . . . . . . . . . . . . . . . . . 4.30
17.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
432
18 Random Signals and LTI-Systems
437
18.1 Conhining Random Signals . . . . . . . . . . . . . . . . . . . . . .
437
18.2 Response of Url-Systems to Random Signals . . . . . . . . . . . . 441
18.3 Signal Estimal ion IJsiiig the Wirucr Filtcr . . . . . . . . . . . . . . 352
18.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
458

Appendix A Solutions to the Exorcises

465

Appendix B Tables of Transformations

563
13.1 I3ilater;tI I, ap1ac.e ‘IYiIrlnfornl E’nirs . . . . . . . . . . . . . . . . . . . 5631
13.2 Properties of the Bilateral Laplace Transfornl . . . . . . . . . . . . 562
13.3 Fouriei ‘I’raiisforni Pairs . . . . . . . . . . . . . . . . . . . . . . . .
563
13.4 Properiirs of the 1~‘oiuirr‘I’raiisfcrrm . . . . . . . . . . . . . . . . . 564
B.5 ‘Two-sided z- Transform Pairs . . . . . . . . . . . . . . . . . . . . .
565
€3.6 Propc‘rtics of the z-Trmsforni . . . . . . . . . . . . . . . . . . . . .
566
B.7 Discr cte-Tinie Fourier TPmnsformPairs . . . . . . . . . . . . . . . . 567
B.8 Properties of tlie DiscreLe-Timc>Fcruricr ’I’r:uisfor.m . . . . . . . . . 568


Conteuts

ix


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Analysing arid tlesigriing systcms with the help of suitable malht?niatic:al i,ools is
extxaordinarily important for engineers. Accordingly, systems theory is a part of
the core curriciilinn of modern clcctrical engineering and ser
assthe foimdatian
of a hrge niimhcr ol. subtfistiplincs. Iiid~rd.access to special
cry of systeriis fh(JIy.
engirireriiig tlcnlilntis a
ins tlirorv logically begins w i t h the

An introduction to
tion: linear. tirnc-irivariant systems. We find applications of such systems everywhere. and their theory has attaiiied advanced maturity md elegance. For
stiiderits who are confronted with the theoi y of lincw, tirnc-invariant syst rius for
tlric fii st time. [,lie siibjcct iitifortunately can prow c-iifficiilt, and, il the rcynired
tmd drhrrvrd neadtmlic progress does riot Inaterialise, the su1)jec.t inight
right unpopular. ‘lhis cot1ltl he diie to the abstract natriirr of the sii
coupled with the deductive arid u n c k r presentation in some lectures. IIowevw,
since failure to Icwn the fundamentals of systems theory oiroultl 1i:we catastrophic
reper(*iiss”iowfar niariy 5iibsPquont sul:, jc
, tlir stutlriit must persewrr.
this book as an easily ilccrssiljlct introduct iori to
rical rngiiwcring. The content itself is nothing I
has already btvn dcmibed in o h r books. What is new is how we deliwr the
material. By meam of small,
explanatory steps, we aim to present the
abstract conccpts and in(erconr
ns of systems tlreor o simply as to make
learning easy m t l fun. 9:tturu
rlv the retder citn
ss whet her we have
achievcvl our goal.
To aid untler81,sriding. we generally use i ~ tinductive
i
appro~wh,stmting with ail
example and thcn generalising from it. Additional c.xamples then illustrate further
aspects of an idea. Wherever a piclure or a figure can enrich t h e text, we provide
onc. Furtlierniow, as the text progresses, we con1,iiiuously order the statrments
of systcwis theory in their overall contrxt. Accor(liugly, iii this book it rliwiwion
of thc irnportancc of a uiaLhriri;i tical ioirririla or a tlicorem 1 altc+ pr~~ceclence
over

its proof. While we might omit the tlcrivation of an eqiiation, we nwcr ncy,lect
a discussion of its appii(*atiorisand consequeritcs! The iiuineroiis exercises at thc
end of each chapter (with detailed solutions in the appendix) help to reinftorce the
reader’s knowlcdgc.


xii

Prefuce

Although we have written this book prirriarily for students, we are convinced
that it will also be nscfnl for practitioners. An engineer who wants to brush
up quickly on some subj
will apprvciatc the easy rcadability of this text, its
practice-orient etl. prt ntation, and its rnaiiy examples.
This book evolved out of a course on systems theory arid the corresponding
laboratory excwises at the F1ieclric.11Alexander University in Erlcuzgeri-~~rriherg..
The course is compulsory for students of electrical engineering iii 1he fifth semester.
A s such, the rnt31mhl in this book can be worked through complet ely i n about 50
hoixrs of lecture8 and 25 hours of exercises. ?Ve do assume knowledge of the f h dumerit als of crigiricering iiiathornatics (diEerentia1 anti iiitcgral ral~ulub,liircnr
algebra) and basic luio&cdge 01electrical circuits. Assuming that, t?hismathematical knowledge hi^ been ucquired earlier, the material is also suitable for iise in the
third or fourth semester. An engineering curriculum often encompasses complex
funct ion 1heoi.y and probability theory as well: dt hough thesc fields are lrelpfid,
we do not assume familiarity with thern.
This book is altio siiitablc for self-study. Assimiing full-time. conwntrated
work. the materiiil can be covned in four to six weeks.
Our presentation bcgins with continuous signals arid systems. Contrary to
some other books that first introduce detailed forms of description for signals and
only rnuch l a t c ~add systems, wF treat sigiials and s y s t e m in parallel The purpose
of dcscribiiig sigiials by meaiis of their Lup1tix:e or Pouricr Lr;zrisforxnat,ioIis becomcs

evident o n l ~through the characteristics of lincar, tinie-inwriant syst,t>ms. T n our
presentation we ernphabisc tkic clcar concept of Rigtm functions. whose form is not
changed by systems. To take into account initial staks, we use state space descriptions. which elegantly allow us to couple XI external and an interrial cornponcnt of
the systcw response. After covering sampling. we int,roduce timc-discrete signals
arid svstenis iLrld so cxtorid (,liec.oncopts fariiiliar from thc conlinuous c asc" r T l ~ r w aftcr discrete and contimioils sigials and systems are treated together. Finally. we
discuss random signals, which tire very important today.
To avoid the arduous m d seldom perfect step of correctiiig camera-ready copy,
wti liaridlcd the layout of the book ourselves a(, the university. All formulas ilritl
most of the figures were typeset in LaTeX and then transferred onto overhead slidox
I W We itre most g r a ~
eful
that mere used h r two y e t m in the?
nte criticism Inelped us to
to some 200 registered students wh
debug the presentation and the typcset equations. T n addition, one yea's students
read thc first version of the manuscript and suggested diverse irnprovernents. Finally iiiinie~oiiswaders of the Cerrnaii wrsion r cpnrl,ed typographic erioib arid
sent comments by e-mail.
Our student, assistimts Lutz and Alexander Larnpe. Sl,q)h~mGiidde. A'Iarioxi
Schahert, Stcfan von der Mark a i d Ilubert Itubcnbauer derrionstraled trernvndous
commitment in typesettiiig and correcting the book as well as the solutions to the
rcises. We thank Ingrid Biirtsch, who typed and corrected a. large portion of
text, R S well as Susi Kosdiiiy, who pxodiiced niuny figures.


Bcrnd Girocl

Rri d olf RalJenstein

Alexaader Stcnger

This Page Intentionally Left Blank


+-0

I

I

I

100

I----

200

time (ms)

,

I

I

I

300

400

i@=


2

1. Zntrodiictioii

would hme ic different temperature c u ~ v c , Iri contrast to Figiire 1.1. t,irne hew
is; ti pnrrtrrwter of a family of ciirvcs: thc indcpcndcnt coiitiiiuous variable is the
location in t,hp wall.

10

8

6
4
2
0

a

0.02

0.04

0.06

0.08

0.1

0.12

0.14

location [m]
FigitrP 1.2: Temperature curvv for n hoiw nali

2500

i

Jan 5. 1996

June 28,1996


3

1.1. Simals

averngr stock index) arc ~ v l ~ onumbers
lc
and so likewise discrete. Iri this case bolh
the independent arid tbc dependent variables arc: discrcte.

$

15

0,

Y

10

L-

5

0
1.0 1.3 1.7 2.0 2.3 2.7 3.0 3.3 3.7 4.0 4.3 4.7 5.0

mark
Figure 1.4:Frequency of e m w d marks for a test in systems theory

The signals u7e have ronsidcrcd thus far have been ryiiant,iticas that clcpend on a
m q l e irrtlepcdcmt variable. how eve^, tliezp arc yuantitics with t~t?i)c?ntlciic.it.s
on
two or more variables. The grcyscalos of Figure 1.5 rlepeird on both thc .E and the
y co-ordinates. Here both axes represent independent variables. The dependent
variable s(x.y) is entered along one axis, but, is n greyscale value between the
extremp values black wid white.
Wlieii W P adcl mot,iori to pict,ures, wc h:me c2 depcntleiicy on t k r w iiidrpendcrit
vruid~les(Pignrtt 1.6): two co-ordinates arid thnc. %'c citll these two- 01- thrccdirnenyional (or gcnerallp multidiineiisio~~a1)
signals. When greyscaln values cliaiige
continuously over space or over space and time, these are continuous signals.

All (mr examples have shown parameters (voltage, Z,miperatiirc, stotk index,
frequencies. greyscale) that change in rda tion to values of the iridependent variahles. 'L'lierc~by thc?y tr;znurnit c*rrtaininformation. In this booh we define it signal
BS follou\s:

L-

_I____

__

Definition I: Signal

-II

A signal as a fil,nctaon o r sequence of val,ires th,at represenls .~ri$!orwcutzon.
_______

ding exaxxiples hil\rp shown that, 4gaaJs caii a ~ s i t ~ctifftwnt
m
iorms. Signals can l x c*l:wsifit.d according tJo various rritmia, 1lie inost important o f which
%resnmunarimi in Table L.1.




In tliis gcricral form we cair iriixgirie cl
with t tie outside world via varioui, sigii
rstshlihhcs n relation&ip amoirg the si

gure 1.7tlep1c.ts

I__

X
I
.

X

sllrtl

a

sp


1.2. systcllx

-~

7

1.
111s t heorv clocs not ciiconipass the implerrretit at i w ot a
relatiorirhips Clrat, the system iinposcs bet wtwi its sigonrrits. hi,with
nals. Svstems lheory I Pscrit s a powrrfid xnathcwiaticaJ tool for tlw stud\ and
mentation details h l p s to inniritaiii
bccttusc ontitting the i
c~ebigrror
s t b c ~ r ythc focus is 011 the formal

an overview of the ovrrall systcixi. In s
ny sprcialisation for specific applinat me of the irit erc or111
ions. ‘I’hi.; allrrvvs systeiris theory a, unilorm rvprescJrital ioir of procc
on tlonieiiis (P g., physics, enginecrinp,, econoinics, biology) arid
iisciyliriarv view.
The high t l c y g w ot abstraction b gs the advcirrtages of learriiiig ccwioniy m d
clar its.. 1,cai ning econorriy c~isiir:,k
n n u l d ~ din griirial form. Clarity rcsults
bcmnsc separating the ctrt>iilproblems froin the gciieral relat,iorisliips i s clcvatcd
to n. priiiriplc. However. this i s countcirtvl by t h c drawback ot a cer l a i r 1 ; ~ r r i o u n t of
unclcariiess tliat enciurihri~sinitial leavriirig iir systeiiis tliror y.

ear,
Iln important sihfirld of 4
ms thcor! is t Ire 1 l i ~ o r vof liiicar, time-invariarit
systems. ‘I his repicsmts tl
assical core ciomain of s y s k r m t Iicw J- nncl i h -cvell
develtq~xl,elrgatit arid clear. This throry also prows suitable for describiiig I ~ I
tcms Iliat caii bc Iiiitw
small signal aiiiplitiitlcs bysi eins theory
tiiiic-in\iariarrt syst
froin tlie practical pirjt)l(ws ol electrical
rnghwring over i~iorcthaii a t c
. Important a p p l i c a h n dolX*iLiri\ h r thr
thcory of liiieai, time-invaria
1 cleclxicitl erigirr
irig totlav inc hide.
Aiinlysis a i d dcisign of clcctrical circuits
e

Digital sigml processing

Clorrrmuriiw t ion?

To tlefiire thr tcrm liucazfty, kit us considcr the systcni iri Figiirc 1.9. Ir respoiitl.;
to an iriput signal . r t ( t ) with tlic output signal $11( t )ant1 t o the input signal .cZ(1)

-


In gcrieval wc canriot inakc this step>but fbr inany relationship bet.itiwn iripul
arid output parmwters, from ( I.I) [lie output sigiid follows as

F:xannpleu iiiclrtde kh(. rc4atioriship between currcnt a i d voltagc on a iwistor
given bv Ohm's r,a\y. bet,werii charge and voltage in a capacitor. and l ~ i ~ e ~ n
force aiid stretching of a spring according tu Hook's Law.

I

Figure 1.9: Definitiori of

t~ linear

linear

i

system ( & I13
, are arbitrary comnplc?x constants)

The relat.ionship expressed in (1.1).(1.2) is called t,he supcper~o.srtzonyrrrt.aple.
It, (:an be defined more generally as follows:
on of mput szgnals always
the mdimdual outpzlt sagnde, tiicn
_ll"

r

Due to the grwbt importaxice of su(b systems, we also use a more tangible term:

____

Definition 4: Linear systems
Systern.s fw whrch the supe~ipostlmipr.mr:zplc upplics arc called 1inea.r s y s t m s .

1
-

I


1.2. Syslenis

9

FOIthe s y s t m i iri Yigurc X .9 tlie superposition principle is

This cir6iiitaionvan be geiicralised for systmis with multiple inpiits and outputs

1. Ini roductioxi

10

[I!)].In thr definition me could emhaiigc ii drlaycd aiid a noiidrlay~dsignal and
recognise that the delay in Figure 1.10 could assuirit~positive or negative values.

1.2.3.3

LTI-Systems

Linear
m s iii c iiot gciierally t , i ~ i i ( ~ - i i ~ ~Likcwisc,
~ ~ i i ~ i tt,imc-invariaiii,
,
systrrrls
need not be linear. However, ils already mentioned, systems that are both linear
arid tiirie-iiivarirmt play a przrtkularly important rolc in systems tAieorq. l’lley have
been assiguetl the acronym LTI.
6:

LTI-system

A system that zs both tLme-rrcvarscmt and linear 2s termed u n LTI-systern
Lancu7, 2”1rrl,f~Tn uci,rinnt sgs.tern).
The clirzracteristicsof LTI-systems and the tools for their analysis arc the sitbjects
of subsct~rlcntchapters.

les of Systems
12.4.1 Electrical Circuits

A i an exunplc of the description of’ RTI electrical circiiit iis a wstem, we crnploy
the branching circuit in Figure 1.11. The time-dependent voltages u l ( t )and u ~ ( t )
zq)resciit contirnwus signals. siicli n s the. voiw signal in Figiir
establishes relationship betwccn two signals aiid is t h s il :>y
the electrical nature of the inner workings and the enclosed coniporlents
regial:irities, we I I ~ O VLO~ tJir represent >ition : i;m
~ input/outpzlt systcm
1.8. A s long as we have rio furtlicr inforrna1,ion on ttrr origin of thcse
random. The laws ol
signals, the assipmerit of inpiit signals to output s
arid rripacitors) allow
circ-uii. iheory for t he idealised coniporicwl,s ( i t l e d r
us to reprevenl this circuit as a linear and time-invariant system (LTI-system).