Chapter 1 Fundamental Concepts

Transcription

1 Chaper 1 Fundamenal Conceps 1 Signals A signal is a paern of variaion of a physical quaniy, ofen as a funcion of ime (bu also space, disance, posiion, ec). These quaniies are usually he independen variables of he funcion defining he signal A signal encodes informaion, which is he variaion iself 2

2 Signal Processing Signal processing is he discipline concerned wih exracing, analyzing, and manipulaing he informaion carried by signals The processing mehod depends on he ype of signal and on he naure of he informaion carried by he signal 3 Characerizaion and Classificaion of Signals The ype of signal depends on he naure of he independen variables and on he value of he funcion defining he signal For example, he independen variables can be coninuous or discree Likewise, he signal can be a coninuous or discree funcion of he independen variables 4

3 Characerizaion and Classificaion of Signals Con d Moreover, he signal can be eiher a real- valued funcion or a complex-valued funcion A signal consising of a single componen is called a scalar or one-dimensional (1-D) signal 5 Examples: CT vs. DT Signals x () xn [ ] n plo(,x) sem(n,x) 6

4 Sampling Discree-ime signals are ofen obained by sampling coninuous-ime signals.. x () xn [ ] = x ( ) = xnt ( ) = nt 7 Sysems A sysem is any device ha can process signals for analysis, synhesis, enhancemen, forma conversion, recording, ransmission, ec. A sysem is usually mahemaically defined by he equaion(s) relaing inpu o oupu signals (I/O characerizaion) A sysem may have single or muliple inpus and single or muliple oupus 8

5 Block Diagram Represenaion of Single-Inpu Single-Oupu (SISO) CT Sysems inpu signal x () T oupu signal { } y() = T x() 9 Types of inpu/oupu represenaions considered Differenial equaion Convoluion model Transfer funcion represenaion (Fourier ransform, Laplace ransform) 10

6 Examples of 1-D, Real-Valued, CT Signals: Temporal Evoluion of Currens and Volages in Elecrical Circuis y() 11 Examples of 1-D, Real-Valued, CT Signals: Temporal Evoluion of Some Physical Quaniies in Mechanical Sysems y() 12

7 Coninuous-Time (CT) Signals Uni-sep funcion Uni-ramp funcion 1, 0 u () = 0, < 0, 0 r () = 0, < 0 13 Uni-Ramp and Uni-Sep Funcions: Some Properies x (), 0 xu () () = 0, < 0 r () = u( λ) dλ u () dr() d = 0 = (wih excepion of ) 14

8 The Recangular Pulse Funcion pτ ( ) = u ( + τ / 2) u ( τ / 2) 15 A.k.a. he dela funcion or Dirac disribuion I is defined by: δ () = 0, 0 The value δ (0) δ (0) The Uni Impulse ε ε δ( λ) dλ = 1, ε > 0 is no defined, in paricular 16

9 The Uni Impulse: Graphical Inerpreaion δ () = limp A A () A is a very large number 17 The Scaled Impulse Kδ() If K, Kδ () is he impulse wih area K, i.e., Kδ () = 0, 0 ε ε Kδ( λ) dλ = K, ε > 0 18

10 Properies of he Dela Funcion 1) u () = δ ( λ) dλ excep = 0 2) ε ε x () δ( ) d= x ( ) ε > (sifing propery) 19 Definiion: a signal wih period T, if Periodic Signals x () x ( + T) = x ( ) is said o be periodic Noice ha x () is also periodic wih period qt where q is any posiive ineger T is called he fundamenal period 20

11 Example: The Sinusoid x () = Acos( ω+ θ ), ω θ [ rad / sec] [ rad ] f = ω 2π [1/ sec] = [ Hz] 21 Time-Shifed Signals 22

12 Poins of Disconinuiy A coninuous-ime signal x () is said o be + disconinuous a a poin if x ( 0 0) x ( 0) where + and, being a 0 = 0 + ε 0 = 0 ε ε small posiive number x () 0 23 Coninuous Signals A signal x () is coninuous a he poin if + x ( ) = x ( ) If a signal x () is coninuous a all poins, x () is said o be a coninuous signal 24

13 Example of Coninuous Signal: The Triangular Pulse Funcion 25 Piecewise-Coninuous Signals A signal x () is said o be piecewise coninuous if i is coninuous a all excep a finie or counably infinie collecion of poins, i= 1,2,3, i 26

14 Example of Piecewise-Coninuous Signal: The Recangular Pulse Funcion pτ ( ) = u ( + τ / 2) u ( τ / 2) 27 Anoher Example of Piecewise- Coninuous Signal: The Pulse Train Funcion 28

15 Derivaive of a Coninuous-Time Signal A signal x () is said o be differeniable a a poin 0 if he quaniy x ( + h) x ( ) 0 0 h has limi as h 0independen of wheher h approaches 0 from above ( h > 0) or from below ( h < 0) If he limi exiss, x () has a derivaive a 0 dx() x( + h) x( ) 0 0 = lim 0 h 0 d = h 29 Generalized Derivaive However, piecewise-coninuous signals may have a derivaive in a generalized sense Suppose ha x () is differeniable a all excep = 0 The generalized derivaive of x () is defined o be dx() d + + x ( 0) x ( 0) δ ( 0) ordinary derivaive of x() a all excep = 0 30

16 Example: Generalized Derivaive of he Sep Funcion Definex () = Ku () K K The ordinary derivaive of x () is 0 a all poins excep = 0 Therefore, he generalized derivaive of x () is + K u(0 ) u(0 ) δ ( 0) = Kδ ( ) 31 Anoher Example of Generalized Derivaive Consider he funcion defined as x () 2+ 1, 0 < 1 1, 1 < 2 = + 3, 2 3 0, all oher 32

17 Anoher Example of Generalized Derivaive: Con d 33 Example of CT Sysem: An RC Circui Kirchhoff s s curren law: i () + i () = i() C R 34

18 RC Circui: Con d The v-i law for he capacior is dv () dy() C ic () = C = C d d Whereas for he resisor i is 1 1 ir() = vc() = y() R R 35 RC Circui: Con d Consan-coefficien linear differenial equaion describing he I/O relaionship if he circui dy() 1 C + y () = i () = x () d R 36

19 RC Circui: Con d Sep response when R=C=1 37 Basic Sysem Properies: Causaliy A sysem is said o be causal if, for any ime 1, he oupu response a ime 1 resuling from inpu x() does no depend on values of he inpu for > 1. A sysem is said o be noncausal if i is no causal 38

20 Example: The Ideal Predicor y() = x( + 1) 39 Example: The Ideal Delay y() = x( 1) 40

21 Memoryless Sysems and Sysems wih Memory A causal sysem is memoryless or saic if, for any ime 1, he value of he oupu a ime 1 depends only on he value of he inpu a ime 1 A causal sysem ha is no memoryless is said o have memory. A sysem has memory if he oupu a ime 1 depends in general on he pas values of he inpu x() for some range of values of up o = 1 41 Ideal Amplifier/Aenuaor RC Circui Examples y() = Kx() 1 y e x d () = C (1/ RC )( τ ) 0 ( τ) τ, 0 42

22 Basic Sysem Properies: Addiive Sysems A sysem is said o be addiive if, for any wo inpus x 1 () and x 2 (), he response o he sum of inpus x 1 () + x 2 () is equal o he sum of he responses o he inpus (assuming no iniial energy before he applicaion of he inpus) x () x () sysem y () + y () Basic Sysem Properies: Homogeneous Sysems A sysem is said o be homogeneous if, for any inpu x() and any scalar a, he response o he inpu ax() is equal o a imes he response o x(), assuming no energy before he applicaion of he inpu ax() sysem ay() 44

23 Basic Sysem Properies: Lineariy A sysem is said o be linear if i is boh addiive and homogeneous ax () () + bx sysem ay () + by () A sysem ha is no linear is said o be nonlinear 45 Example of Nonlinear Sysem: Circui wih a Diode y () R2 R + R x (), whenx () 0 = 1 2 0, when x( ) 0 46

24 Example of Nonlinear Sysem: Square-Law Device y 2 () = x () 47 Example of Linear Sysem: The Ideal Amplifier y() = Kx() 48

25 Example of Nonlinear Sysem: A Real Amplifier 49 Basic Sysem Properies: Time Invariance A sysem is said o be ime invarian if, for any inpu x() and any ime 1, he response o he shifed inpu x( 1 ) is equal o y( 1 ) where y() is he response o x() wih zero iniial energy x ( ) sysem 1 1 y( ) A sysem ha is no ime invarian is said o be ime varying or ime varian 50

26 Examples of Time Varying Sysems Amplifier wih Time-Varying Gain y() = x() Firs-Order Sysem y () + a() y() = bx() 51 Basic Sysem Properies: CT Linear Finie-Dimensional Sysems If he N-h derivaive of a CT sysem can be wrien in he form N 1 M ( N) ( i) ( i) = i + i i= 0 i= 0 y () a () y () b() x () hen he sysem is boh linear and finie dimensional To be ime-invarian a () = a and b () = b i and i i i i 52