Signals and Systems: Chapter 5 Laplace Transform

Department of Electrical Engineering University of Arkansas EE 2000 SIGNALS AND SYSTEMS ELEG 3124 SYSTEMS AND SIGNALS Ch Laplace Transform Dr Jingxian Wu wuj@uark.edu (These slides are taken from Dr Jingxian Wu, University of Arkansas, 2020.) OUTLINE • Introduction • Laplace Transform • Properties of Laplace Transform • Inverse Laplace Transform • Applications of Laplace Transform INTRODUCTION • Why Laplace transform? – Frequency domain analysis with Fourier transform is extremely useful for the studies of signals and LTI system • Convolution in time domain ➔ Multiplication in frequency domain – Problem: many signals not have Fourier transform x(t ) = exp( at )u(t ), a  x(t ) = tu(t ) – Laplace transform can solve this problem • It exists for most common signals • Follow similar property to Fourier transform • It doesn’t have any physical meaning; just a mathematical tool to facilitate analysis – Fourier transform gives us the frequency domain representation of signal OUTLINE • Introduction • Laplace Transform • Properties of Laplace Transform • Inverse Lapalace Transform • Applications of Fourier Transform LAPLACE TRANSFORM: BILATERAL LAPLACE TRANSFORM • Bilateral Laplace transform (two-sided Laplace transform) + X B ( s) =  x(t ) exp( − st )dt , − s =  + j – s =  + j is a complex variable – s is often called the complex frequency – Notations: X B ( s ) = L[ x(t )] x (t )  X B ( s ) • Time domain v.s S-domain – x(t ) : a function of time t → x(t) is called the time domain signal – X B (s ) : a function of s → X B (s ) is called the s-domain signal – S-domain is also called as the complex frequency domain LAPLACE TRANSFORM • Time domain v.s s-domain – x(t ) : a function of time t → x(t) is called the time domain signal – X B (s ) : a function of s → X B (s ) is called the s-domain signal • S-domain is also called the complex frequency domain – By converting the time domain signal into the s-domain, we can usually greatly simplify the analysis of the LTI system – S-domain system analysis: • Convert the time domain signals to the s-domain with the Laplace transform • Perform system analysis in the s-domain • Convert the s-domain results back to the time-domain LAPLACE TRANSFORM: BILATERAL LAPLACE TRANSFORM • Example – Find the Bilateral Laplace transform of x(t ) = exp( −at )u(t ) • Region of Convergence (ROC) – The range of s that the Laplace transform of a signal converges – The Laplace transform always contains two components • The mathematical expression of Laplace transform • ROC LAPLACE TRANSFORM: BILATERAL LAPLACE TRANSFORM • Example – Find the Laplace transform of x(t ) = − exp( −at )u(−t ) LAPLACE TRANSFORM: BILATERAL LAPLACE TRANSFORM • Example – Find the Laplace transform of x(t ) = exp( −2t )u(t ) + exp(t )u(−t ) 10 LAPLACE TRANSFORM: UNILATERAL LAPLACE TRANSFORM • Unilateral Laplace transform (one-sided Laplace transform) + X ( s) =  − x(t ) exp( − st )dt – 0− :The value of x(t) at t = is considered – Useful when we dealing with causal signals or causal systems • Causal signal: x(t) = 0, t < • Causal system: h(t) = 0, t < – We are going to simply call unilateral Laplace transform as Laplace transform 38 INVERSE LAPLACE TRANSFORM • High-order repeated linear factors AN A1 A2 B X ( s) = = + ++ + N N ( s − a ) ( s − b) s − a ( s − a ) ( s − a) s −b  d N −k N ( ) Ak = s − a X ( s) N −k ( N − k )! ds B = (s − b )X ( s) s =b  k = 1,, N s =a 39 OUTLINE • Introduction • Laplace Transform • Properties of Laplace Transform • Inverse Lapalace Transform • Applications of Laplace Transform 40 APPLICATION: LTI SYSTEM REPRESENTATION • LTI system – System equation: a differential equation describes the input output relationship of the system y ( N ) (t ) + aN −1 y ( N −1) (t ) +  + a1 y (1) (t ) + a0 y(t ) = bM x( M ) (t ) +  + b1 x (1) (t ) + b0 x(t ) N −1 y (N) M (t ) +  an y (t ) =  bm x ( m ) (t ) (n) n =0 m =0 – S-domain representation  N N −1 n  M m  s +  an s Y ( s) =   bm s  X ( s) n =0    m =0  – Transfer function M Y (s) H (s) = = X (s) b m =0 m sm N −1 s +  an s n N n =0 41 APPLICATION: LTI SYSTEM REPRESENTATION • Simulation diagram (first canonical form) Simulation diagram 42 APPLICATION: LTI SYSTEM REPRESENTATION • Example – Show the first canonical realization of the system with transfer function s − 3s + H (S ) = s + 6s + 11s + 43 APPLICATION: COMBINATIONS OF SYSTEMS • Combination of systems – Cascade of systems H ( S ) = H1 ( s ) H ( s ) – Parallel systems H ( S ) = H1 ( s ) + H ( s ) 44 APPLICATION: LTI SYSTEM REPRESENTATION • Example – Represent the system to the cascade of subsystems s − 3s + H (S ) = s + 6s + 11s + 45 APPLICATION: LTI SYSTEM REPRESENTATION • Example: – Find the transfer function of the system LTI system 46 APPLICATION: LTI SYSTEM REPRESENTATION • Poles and zeros H (s) = – Zeros: – Poles: ( s − z M )( s − z M −1 )  ( s − z1 ) ( s − p N )( s − p N −1 )  ( s − p1 ) z1 , z ,, z M p1 , p2 ,, pN 47 APPLICATION: STABILITY • Review: BIBO Stable – Bounded input always leads to bounded output  + − | h(t ) | dt   • The positions of poles of H(s) in the s-domain determine if a system is BIBO stable H (s) = AN A1 A2 + +  + s − s1 ( s − s2 ) m s − sN – Simple poles: the order of the pole is 1, e.g s1 sN – Multiple-order poles: the poles with higher order E.g s 48 APPLICATION: STABILITY • Case 1: simple poles in the left half plane (s −  k ) + k = ( s −  k + jk )( s −  k − jk ) p1 =  k − j k hk (t ) =  + − k k  p2 =  k + jk exp( k t ) sin( k t )u (t ) hk (t ) dt = Impulse response • If all the poles of the system are on the left half plane, then the system is stable 49 APPLICATION: STABILITY • Case 2: Simple poles on the right half plane (s −  k ) + k = ( s −  k + jk )( s −  k − jk ) p1 =  k + jk hk (t ) = k k  p2 =  k − jk exp( k t ) sin( k t )u (t ) Impulse response • If at least one pole of the system is on the right half plane, then the system is unstable 50 APPLICATION: STABILITY • Case 3: Simple poles on the imaginary axis (s −  k ) hk (t ) = + k k = ( s −  k + jk )( s −  k − jk ) k = sin( k t )u (t ) • If the pole of the system is on the imaginary axis, it’s unstable 51 APPLICATION: STABILITY • Case 4: multiple-order poles in the left half plane m k  stable hk (t ) = t exp( k t ) sin( k t )u (t ) k • Case 5: multiple-order poles in the right half plane m hk (t ) = t exp( k t ) sin( k t )u (t ) unstable k  k • Case 6: multiple-order poles on the imaginary axis hk (t ) = k t m sin( k t )u (t ) k  unstable k  52 APPLICATION: STABILITY • Example: – Check the stability of the following system H ( s) = 3s + s + s + 13 ... expression of Laplace transform • ROC 8 LAPLACE TRANSFORM: BILATERAL LAPLACE TRANSFORM • Example – Find the Laplace transform of x(t ) = − exp( −at )u(−t ) LAPLACE TRANSFORM: BILATERAL LAPLACE TRANSFORM. .. OUTLINE • Introduction • Laplace Transform • Properties of Laplace Transform • Inverse Laplace Transform • Applications of Laplace Transform INTRODUCTION • Why Laplace transform? – Frequency domain... Example – Find the Laplace transform of x(t ) = exp( −2t )u(t ) + exp(t )u(−t ) 10 LAPLACE TRANSFORM: UNILATERAL LAPLACE TRANSFORM • Unilateral Laplace transform (one-sided Laplace transform) + X