Topic 9: The Laplace Transform o Introduction o Laplace Transform - PowerPoint PPT Presentation

ELEC361: Signals And Systems Topic 9: The Laplace Transform o Introduction o Laplace Transform & Examples o Region of Convergence of the Laplace Transform o Review: Partial Fraction Expansion o Inverse Laplace Transform & Examples o Properties of the Laplace Transform & Examples o Analysis and Characterization of LTI Systems Using the Laplace Transform o LTI Systems Characterized by Linear Constant-Coefficient DE o Summary Dr. Aishy Amer Concordia University Electrical and Computer Engineering Figures and examples in these course slides are taken from the following sources: • A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997 • M.J. Roberts, Signals and Systems, McGraw Hill, 2004 • J. McClellan, R. Schafer, M. Yoder, Signal Processing First, Prentice Hall, 2003 1 • Web Site of Dr. Wm. Hugh Blanton, http://faculty.etsu.edu/blanton/

Introduction � Transforms: Mathematical conversion from one way of thinking to another to make a problem easier to solve solution problem in time in time inverse Laplace domain solution domain Laplace transform in transform s domain s = σ +j ω • Other transforms • Fourier Transform • z-transform 2 � Reduces complexity of the original problem

Introduction time domain x(t) y(t) linear time differential domain equation solution Laplace transform inverse Laplace transform Laplace algebra Laplace transformed solution equation 3 Laplace domain or complex frequency domain

Introduction � CT Fourier Transform: � representation of signals as linear combination of complex exponentials e st , s = j ω ∞ ∞ 1 ∫ ∫ ω = − ω = ω ω ω j t j t X j x t e dt x t X j e d ( ) ( ) ( ) ( ) π 2 − ∞ − ∞ � Laplace Transform: � Representation of signals as linear combination of e st ,s = σ +j ω � A generalization of CTFT � Can be applied in contexts where the FT cannot � Investigation of stability/instability & causality of systems 4 � Laplace transform applies to continuous-time signals

Introduction Sample in time Discrete-time Sampling period = T s Continuous-time Discrete-time Continuous-time analog sequence analog sequence analog signal analog signal x [n] x(t) x [n] C D x(t) Continuous Laplace Discrete-Time Discrete z-Transform Continuous Laplace Discrete-Time Discrete z-Transform Fourier Transform X(z) Fourier Transform Transform Fourier Transform Fourier Transform X(z) Fourier Transform Transform Fourier Transform X(f) C X(k) Ω X( Ω) = X(s) X(f) j X(k) z e X( Ω) X(s) C C ∞ ∞ s = σ +j ω ∑ π − 2 nk ∫ C N s = σ +j ω 1 π - n ∞ - j 2 ft ∑ x [n] z - j x(t) e dt ∑ ∞ Ω n x [n] e N - j x [n] e ∫ − st − ∞ x(t) e dt − ∞ n = ∞ D n = 0 n = - − ∞ ≤ ≤ ∞ ∞ ≤ ≤ ∞ z − ∞ - f ≤ ≤ − ≤ Ω ≤ π k N 0 2 0 1 ∞ ≤ ≤ ∞ - s Ω = j z e s = j ω Sample in ω =2 π f ω=2π f frequency , Ω = ω T s , 5 Ω = 2 π n/N, scale N = Length amplitude of sequence Continuous-variable Discrete-variable C D by 1/T s

Introduction ∞ ∫ − = = st x t X s x t e dt [ ( )] ( ) ( ) L − ∞ = σ + ω s j � Convert time-domain signals into frequency-domain � x ( t ) → X ( s ) t ∈ R , s ∈ C � Linear differential equations (LDE) → algebraic expression in complex plane � Graphical solution for key LDE characteristics � (Discrete systems use the analogous z-transform) 6

Introduction: Complex Exponential e -st σ + ω σ ω • = = = ω + σ + ω + σ = st j t t j t x t Ae Ae ( ) A t t Aj t t Ae e ( ) cos( ) sin( ) • σ σ t is the phase While the determines the rate of decay/grow th, ω is the frequency the part determines the rate of the osillation s ω ⇒ ω This is apparent in noticing that the is part of the argument t o the sinusoidal part 7

Introduction: the complex s-plane •Any time s lies in the right half plane, the complex exponential will grow through time; any time s lies in the left half plane it will decay − ω 1 ∠ ≡ φ = s tan σ Imaginary axis ∗ 2 2 ≡ ≡ = σ + ω s r s | | | | = σ + ω s j ω r Right Half Plane Left Half Plane φ σ Real axis − φ r Axis tells how fast e st − ω grows or decays ∗ = σ − ω Axis tells how fast s j 8 e st oscillates (higher (complex) conjugate frequency)

Outline Introduction o Laplace Transform & Examples o Region of Convergence of the Laplace Transform o Review: Partial Fraction Expansion o Inverse Laplace Transform & Examples o Properties of the Laplace Transform & Examples o Analysis and Characterization of LTI Systems Using the Laplace o Transform LTI Systems Characterized by Linear Constant-Coefficient o Summary o 9

Laplace Transform � As mentioned earlier, the response of an LTI system with impulse response h(t) to a complex exponential of the form e st is where � If we let s = jw (pure imaginary), the integral above is essentially the Fourier transform of h(t) � For arbitrary values of the complex variable s, this expression is referred to as the Laplace transform of h(t) � Therefore, the Laplace transform of a general signal x(t) is defined as � Note that s is a complex variable, which can be expressed in general as 10 s = σ +j ω

Laplace Transform � When s = j ω , we get the Fourier transform of x(t) � Therefore, the Fourier transform is a special case of the Laplace transform can be expressed as � Note: e -j ω t sinusoidal, e.g., bounded 11 � X( σ +j ω ) is essentially the Fourier transform of x(t)e - σ t � Properties of x(t)e - σ t determine convergence of X(s)

Laplace Transform � The Laplace transform X(s) for positive t ≥ 0 typically exists for all complex numbers such that Re{s} > a � where a is a real constant which depends on the growth behavior of x(t) � The subset of values of s for which the Laplace transform exists is called the region of convergence (ROC) � Note: the two sided (- ∞ < t < ∞ ) Laplace transform 12 is defined in a range a < Re{s} < b

Laplace Transform Laplace transform is often a rational of polynomial s + + N s s 2 s ( ) 2 5 2 = = = X s ( ) algebraic expression ; X(s) + + + D s s s s 2 ( ) ( 5 10 )( 2 ) • = + + + n N s a s a s a ( ) ... n 1 0 |X(s)| will be larger when it • = m + + + D s b s b s b is closer to the poles ( ) ... m 1 0 • ∋ = = ∞ s D s X s Poles (signulari ties) : ( ) 0 (So, ( ) ) • ∋ = = s N s X s Zeroes : ( ) 0 (So, ( ) 0 ) |X(s)| will be smaller when • it is closer to the zeros Poles and zeroes are complex • = X(s) m Order of is the number of poles • 13 Poles & Zeros of X(s) : Completely characteri ze the algebraic expression of X(s)

Laplace Transform: Order of X(s) N s K ( ) • = First Order + D s sT ( ) 1 N s K ( ) • = Second Order + + D s Js 2 Bs K ( ) Impulse Exponential K response + sT 1 Step Step, exponential K K - response + s s T 1 / 14 K KT KT Ramp Ramp, step, - - + s 2 s s T response exponential 1 /

Laplace Transform: Poles � The poles of a Laplace function are the values of s that make the Laplace function evaluate to infinity � The poles are therefore the roots of the denominator polynomial + s 10 ( 2 ) has a pole at s = -1 and a pole at s = -3 � + + s s ( 1 )( 3 ) � Complex poles (e.g., s=-2+5j) always appear in complex-conjugate pairs � The response of a system is determined by the location of poles on the complex plane 15

Laplace Transform: Zeros The zeros of a Laplace function are the values of s that make � the Laplace function evaluate to zero The zeros are therefore the zeros of the numerator � polynomial has a zero at s = -2 + s 10 ( 2 ) + + s s ( 1 )( 3 ) Complex zeros always appear in complex-conjugate pairs � Pole-Zero Cancellation: Do not eliminate poles as in + − s s ( 3 )( 1 ) = H s ( ) − s ( 1 ) Think about what may happen if H(s) was a transfer function 16 of a physical system where minor system (e.g., temperature) change could cause the pole or the zero to move

Laplace Transform: Visualization FT: X(j ω ) a complex valued function of purely imaginery � variable jw Visualize using 2D plot of real and imaginary part or � magnitude and phase LT: X(s) a complex valued function of a complex variable � s= σ +j ω Requires a 3D plot which is difficult to visualize or analyze � Solution: Poles (x) and Zeros (o) Plot � Example: � Poles: s=(-1-3j) 17 s=(-1+3j) s=-2