Chapter 1 Slides

Signaux et Systmes

Signals and SystemsInstructor: Dr. Rizwan AkbarPhD (EE), France

Electrical Engineering DepatmentFall 2013

12Course Outline Signals and Systems Linear Time Invariant Systems (LTI) Fourier Series Continuous time Fourier Transform (CTFT) Discrete time Fourier Transform (DTFT) Discrete Fourier Transform (DFT) Time Frequency characterization of signals and systems Laplace transform Z Transform Sampling23I. Signals and Systems1 Continuous-Time and Discrete-Time signals

2 Transformations of the independent variable

3 Exponential and Sinusoidal signals

4 Unit step and Unit impulse Functions

5 Continuous-Time and Discrete-Time Systems

6 Basic System Properties

34I.1. Continuous time and Discrete time signalsA) Signal examples and their mathematical representationsignal = any entity which carries informationExamples:Acoustic WaveMusic, Speech,...Light WaveLight source(stars gas, )...Electric current delivered by microphoneElectric current delivered by spectrometerSequence of numbersPhysical measurements

Images...

4Les signaux peuvent dcrire une trs grande varite de phnomnes physiques. Le signal est porteur d'information et correspond gnralement aux variations au cours du temps d'une grandeur physique: lectrique, acoustique, lumineuse...

Exemples de signaux:- Les variations au cours du temps de la tension aux bornes d'une capacit- vairiation d'un force f appliqu un vehicule et sa vitesse rsultante- enregistrement de la voix par un microphonequi capte les variations de la pression acoustique et qui les convertit en un signal lectrique.Dans une image monochromatique, le signal est reprsent par les variations de luminosit.5Continuous time signals:Signal = function of one or more independent variables:ex: (Voice)Acoustic pressure= f(time) (Image)Brightness= f(x,y:spatial variables) In the following: 1 independent variable= timeMathematical representation:The indpendent variable is continuous tex: Voice as a function of time, Atmospheric pressure as a function of altutudeDiscrete time signals:Defined only at discrte instantsThe indpendent variable is a set of discrete values nex: Dow-Jones stock market indexDemographic studies...5Exmples de Signaux discrets:Etudes dmographiques pour lequel de nombreux attributs tq le budget moyen, le taux de criminalitsont tabules en fonction de variables discrtes : taille des famille, population totale...6Remarks:Exemples:a) CT signal x(t) b) DT signal x[n]:x[n] is defined only for integer values of n. x[n] : DT signal or DT sequence.

2 types of discrete signals:a) Signals representing a phenomenon for which independent variable is discreteb) Signals obtained through the operation of sampling:x[n] represents the successive samples of a phenomenon for which the independent variable is continuous

6Un signal Temps Discret x[n] peut reprsenter un phnomne pour lequel la variable indpendante est discrte de manire inhrente: donnes dmographiquesUne autre catgorie de signaux Temps Discret provient de lchantillonnage de signaux Temps continu. Dans ce cas, le signal x[n]reprsente les chantillons successif dun phnomne sous-jacent pour lequel la variable indpendante est continue.

A notre poque, les traitements numriques ont supplante les traitement analogiques grace aux performances des processeurs numriquesen terme de rapidit,de puissance de calculs et de flexibilit...Les systmes numriques sont prsents dans de tres nombreux equipements allant du pilotage automatique d'un avion un systme numrique audio. De tels systmes necessite l'utilisation de sequences temps-discret representant des versions echantillonnes des signaux temps-continu (ex: position d'une avion, vitesse , voix...73 Types of signals:B) Energy and Power of a signalEnergy

Average Power

CTDT

- Signals of Finite energy- Signals of Finite average power- Signals of infinite energy and power7Un signal Temps Discret x[n] peut reprsenter un phnomne pour lequel la variable indpendante est discrte de manire inhrente: donnes dmographiquesUne autre catgorie de signaux Temps Discret provient de lchantillonnage de signaux Temps continu. Dans ce cas, le signal x[n]reprsente les chantillons successif dun phnomne sous-jacent pour lequel la variable indpendante est continue.

A notre poque, les traitements numriques ont supplante les traitement analogiques grace aux performances des processeurs numriquesen terme de rapidit,de puissance de calculs et de flexibilit...Les systmes numriques sont prsents dans de tres nombreux equipements allant du pilotage automatique d'un avion un systme numrique audio. De tels systmes necessite l'utilisation de sequences temps-discret representant des versions echantillonnes des signaux temps-continu (ex: position d'une avion, vitesse , voix...8- Finite Energy Signals- Signals with infinite energy and average powert011- Finite Average Power Signals0......4nt11

8Ex=1

Px=4

9I.2. Transformations of the independent variableA) Examples Time Shifting

t 0 < 0 : ADVANCEn 0 > 0 : DELAY10Time Scaling

Time Reversal

11B) Periodic signals

Remarks:

T0 = fundamental period= minimum value of T

11Singnaux priodique: inchangs par un dcalage temporel deT12C) Even and Odd signalsEvenOdd

Property:Any signal can be decomposed into a sum of:- An even signal xeven(t), and- An odd signal xodd(t)

13I.3. Exponential and Sinusoidal signalsA) In continuous time Real exponential signals: Periodic complex exponential and sinusoidal signals:

14Remarks :- Set of harmonically related exponentials =Set of complex periodic exponentials with commom period T0 :

General complex exponential signals:

15B) In discrete timeReal exponential signals:

16Complex exponential and sinusoidal signals:

Periodicity property in DT:

0 < 0 < 2 0 < f0 < 11) Signals are not distinct!...

The rate of oscillations ofdoes not increase with 0 !Low frequencies

High frequencies

17DT sinusoidal sequences for several different frequencies

182)Periodicity: Not always!...

Therefore

Fundamental frequencyPeriodic if 0 / 2 is a rational number

Non periodic!

periodicperiodic non periodic19General complex exponential signals:

3) Harmonically related exponentials

Only N distinct exponentials ...

20I.4. Unit Impulse and Unit Step functionsA) In DT

Unit Impulse:01nUnit Step:

10n...

Relations:

21

22

23

B) In CTUnit Impulse:Unit Step:

u(t)t

We want:Problem!...

Pulse Signal

24

25

26

27I.5. Continuous time and Discrete time systemsContinuous time systemx(t)y(t)x[n]y[n]x(t) y(t)x[n] y[n]Exemples:- Relationship between the in put and output voltage in a RC circuit- Relationship between the speed of automobile and applied force- Bank account balanace evolution Linear differential equation of 1st order:

Discrete time system28Interconnection of systemsIdea:Complex systems can be constructed using an interconnection of simple subsystems ..Cascade interconnectionParallel interconnectionFeedback interconnectionSystem 1System 2InOutSystem 1System 2+System 1System 2+InOutOutIn29I.6. Propoerties of systemsMemoryless systems: The output y at an instant t or n depends only on the input x at that instantInvertible systems: Distinct inputs produce distinct outputsSystemSystem inversex[n]y[n]w[n]=x[n]System causal: The output at any instant depends only on the current and prior values of the input

30Stable system: A bounded input: |x(t)| M t corresponds to a bounded output |y(t)| N t Time-invariant system:

Systemx[n-n0]y[n-n0]Systemx(t-t0)y(t-t0)Linear system: Superposition principle

if

then

A temporal shift in the input signal corresponds to same temporal shift on the output signal