1 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 The Laplace transform  The linear system concept  The definition and the properties

2 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 The Laplace transform Continuous linear monovariable systems Continuous linear multivariable systems Impulse response Convolution Laplace transform: definition and properties Inverse Laplace transform Comments on stability

3 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 What is a linear time-invariant system ? Take the simplest example: Resistor Resistor R(  ) i(A) u(V) Ohm’s law u=Ri R is considered constant for any value of u or i Linear relationship between u and i R does not depend on time

4 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 What is a linear time-invariant system ? Take another simple example:Capacitor i(A) u(V) C is considered constant for any value of u or i Linear relationship between first derivative of u and i Relationship valid at any time Capacitor(F)

5 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 What is a linear time-invariant system ? Take another simple example:Inductor i(A) u(V) L is considered constant for any value of u or i Linear relationship between first derivative of i and u Relationship valid at any time Inductor L(H)

6 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Mass m F(t) x v m=constant Mechanical elements

7 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Action of a torque on a rotating mass F d I= moment of inertia about rotation axis Module

8 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Action of a force on a spring -F F Negligible mass x Length = Force r = represents spring stiffness Spring

9 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Thermal element Heat source f T

10 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 What is a linear time-invariant system ? Take another example: Linear amplifier A is considered constant for x(t) and y(t) within limits Linear relationship between x(t), y(t) and their derivatives Relationship valid at any time A may drift slowly, depends on power supplies Presence of noise A x(t) y(t) x y

11 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Simple example: first order system R C u(t)x(t) i(t) Differential equation Initial condition

12 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Solution Multiply by integration factor Convolution Transient

13 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 linear System One input : x(t) One output: y(t) Monovariable linear System Abstract concept Time-invariant or slowly varying characteristics Linearity if x(t) generates y(t) k.x(t) will generate k.y(t) [k real] [x1(t) + x2(t)] will generate [y1(t) + y2(t)] Described by differential and mathematical equations Causal (Effect follows cause) x(t-  ) will generate y(t-  ), x(t<  )= y(t<  )=0 No limits for x(t) and y(t)

14 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP Differential equations with time-invariant coefficients or slowly varying with respect to input variation rate Description of a linear system Linear System x(t)y(t) In many cases

15 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Mathematical model Development of a suitable model Oversimplified model may eliminate intrinsic characteristics Overcomplicated model may lead to mathematical difficulties Compromise Description approximated by ordinary differential equations Limits to nth-order ordinary differential equations An n-th order ordinary differential equation is called linear if each term of the equation contains at most only first power of the dependent variable or its derivatives. Systems described by linear differential equations are called linear systems Linear time-invariant systems and Linear time-varying systems

16 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 The Laplace transform Continuous linear monovariable systems Continuous linear multivariable systems Impulse response Convolution Laplace transform: definition and properties Inverse Laplace transform Comments on stability

17 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 R C u(t) x(t) L i(t) Simple example: second order system Can be represented in two-variable system Matrix form

18 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Vector of variablesits derivatives matrix and vector Multivariable system Matrix notation similar to first order

19 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Representation of A Bu x + +

20 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Example second order system

21 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Time-invariant multivariable systems Linear system InputsOutputs State vector Input vectorOutput vector State equation Output equation

22 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Properties of the State Transition Matrix The State Transition Matrix satisfies the homogeneous State Equation

23 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Non-homogeneous State equation If B is time-invariant

24 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 State representation of a linear system C A B D xy u A= System Matrix(n,n) B= Input Matrix (n,m) x= State Vector (n,1) u= Input Vector (m,1) C= Output Matrix (r,n) D= Direct Transmission Matrix (r,m) y= Output Vector (r,1)

25 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 The Laplace transform Continuous linear monovariable systems Continuous linear multivariable systems Impulse response Convolution Laplace transform: definition and properties Inverse Laplace transform Comments on stability

26 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Time t Unit impulse function: Dirac 0 Unit Doublet function = derivative of Unit impulse function Unit Triplet function = derivative of Unit Doublet function

27 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Causal System h(t) = 0, for t<0 t Stimulation: Dirac Impulse response h(t) h(t) t Linear system InputOutput u(t) y(t) Impulse response

28 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003  (t) t  a=1 1/   0 Practical model C R v(t) Application to an RC circuit Impulse response

29 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Initial conditions ! Dimensions

30 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 For 0<t<  Locus of v(  ) as  0

31 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 u(t) = 0 for t < 0 = 1 for t  0 Time t u(t) 0 1 Unit step function :Heaviside

32 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Ramp function Ramp(t) t

33 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Sine wave, cosine wave Sin(  t) Cos(  t)

34 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 The Laplace transform Continuous linear monovariable systems Continuous linear multivariable systems Impulse response Convolution Laplace transform: definition and properties Inverse Laplace transform Comments on stability

35 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Linear system InputOutput u(t) y(t) Delayed Impulse response h(t-t0) u(t) t Causal System h(t) = 0, for t<t0 Convolution Question: Knowing a given u(t) and h(t) can we get y(t) ? h(t) t

36 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 u(t) t y(t) t

37 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Linear system hence superposition y(t) = series of impulse responses Convolution product

38 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Distributivity Commutativity and associativity Properties of convolution

39 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 u(t)

40 G. Baribaud/AB-BDI Digital Signal Processing March 2003 DISP-2003 Physical representation of the convolution u(  ) t  h(t-  )u(  )h(t-  ) with t >  If y(t) t u(t) t t h(t)