Appendix A The Laplace Transform The Laplace transform was discovered originally by Leonhard Euler (1707–1783), the great 18th-century Swiss mathematician and physicist, but is named in honor of a French mathematician and astronomer Pierre-Simon Laplace (1749–1827), who used the transform in his work on probability theory. He was such a genius not only in mathematics that the great mathematician Simeon Poisson (1781–1840) labeled him the Isaac Newton of France, but also in politics that he could serve three regimes in revolutionary France – the republic, the empire of Napoleon, and the Bourbon restoration, having been bestowed a count from Napoleon and a marquis from Louis XVIII. The Laplace transform is a very useful tool in solving differential equations and much further, it plays an important role in dealing with linear time-invariant systems. A.1 Definition of the Laplace Transform The (unilateral or one-sided) Laplace transform is defined for a function x(t)ofa real variable t (often meaning the time) as ∞ X(s) = L{x(t)}= x(t)e−stdt (A.1) 0− where s is a complex variable, the lower limit, t−, of the integration interval is the instant just before t = 0, and x(t) is often assumed to be causal in the sense that it is zero for all t < 0. A.2 Examples of the Laplace Transform A.2.1 Laplace Transform of the Unit Step Function The unit step function is defined as 385 386 Appendix A δ lim 1 T T d (t ) ==lim rT (t) → (us(t + )–us (t – )) = us(t) T→0 T 0 T 2 2 dt T lim rT (t) T→0 us(t) 1 δ(t ) T t t t 00 0 (a) Unit step function (b) Rectangular pulse (c) Unit impulse function Fig. A.1 Unit step, rectangular pulse, and unit impulse functions 1fort ≥ 0 us (t) = (A.2) 0fort < 0 which is depicted in Fig. A.1(a). We can use Eq. (A.1) to obtain the Laplace transform of the unit step function as ∞ ∞ . L{ } (A=1) −st (A=2) −st (D=33) 1 −st∞ = 1 − = 1 us (t) us (t)e dt e dt e 0 (0 1) 0 0 −s −s s This Laplace transform pair is denoted by L 1 u (t) ←→ (A.3) s s A.2.2 Laplace Transform of the Unit Impulse Function The unit impulse function can be defined to be the limit of a rectangular pulse func- tion rT (t) with the pulsewidth T → 0 (converging to zero) or simply, the time derivative of the unit step function as 1 T T d δ(t) = lim rT (t) = lim us t + − us t − = us (t)(A.4) T →0 T →0 T 2 2 dt which is depicted in Fig. A.1(c). We can use Eq. (A.1) to obtain the Laplace transform of the unit impulse function as ∞ L{δ(t)}= δ(t)e−stdt = 1 0− This Laplace transform pair is denoted by L δ(t) ←→ 1(A.5) Appendix A 387 A.2.3 Laplace Transform of the Ramp Function The Laplace transform of the unit ramp function tus (t) is obtained as ∞ ∞ ∞ (A.1) (A.2) (D.36) t ∞ 1 (D.33) 1 L{tu (t)} = tu (t)e−stdt = te−stdt = e−st + e−stdt = s s 0 2 0 0 −s s 0 s This Laplace transform pair is denoted by L 1 tu (t) ←→ (A.6) s s2 A.2.4 Laplace Transform of the Exponential Function −at The Laplace transform of the exponential function e us (t) is obtained as ∞ ∞ ∞ . −at (A=1) −at −st (A=2) −at −st −(s+a)t (D=33) 1 L{e us (t)} e us (t)e dt e e dt = e dt 0 0 0 s + a This Laplace transform pair is denoted by L 1 e−atu (t) ←→ (A.7) s s + a A.2.5 Laplace Transform of the Complex Exponential Function Substituting σ + jω for a into (A.7) yields the Laplace transform of the complex exponential function as L −(σ+ jω)t (D.20) −σt e us (t) = e (cos ωt − j sin ωt)us (t) ←→ 1 s + σ ω = − j s + σ + jω (s + σ )2 + ω2 (s + σ )2 + ω2 L s + σ e−σt cos ωtu(t) ←→ (A.8) s (s + σ )2 + ω2 L ω e−σt sin ωtu(t) ←→ (A.9) s (s + σ )2 + ω2 A.3 Properties of the Laplace Transform Let the Laplace transforms of two functions x(t) and y(t)beX(s) and Y (s), respectively. 388 Appendix A A.3.1 Linearity The Laplace transform of a linear combination of x(t) and y(t) can be written as L α x(t) + βy(t) ←→ αX(s) + βY (s) (A.10) A.3.2 Time Differentiation The Laplace transform of the derivative of x(t) w.r.t. t can be written as L x (t) ←→ sX(s) − x(0) (A.11) Proof ∞ −st ∞ (A.1) dx (D.36) ∞ (A.1) L{ } = = −st − − −st = − x (t) e dt x(t)e 0 ( s) x(t)e dt sX(s) x(0) 0 dt 0 Repetitive application of this time differentiation property yields the Laplace trans- form of n th-order derivative of x(t) w.r.t. t as L x(n)(t) ←→ sn X(s) − sn−1x(0) − sn−2x (0) −···−x(n−1)(0) (A.12) A.3.3 Time Integration The Laplace transform of the integral of x(t) w.r.t. t can be written as t L 1 1 0 x(τ)dτ ←→ X(s) + x(τ)dτ (A.13) −∞ s s −∞ t This can be derived by substituting −∞ x(τ)dτ and x(t)forx(t) and x (t)into Eq. (A.11) as L t 0 x(t) ←→ X(s) = sL x(τ)dτ − x(τ)dτ ∞ −∞ Repetitive application of this time integration property yields the Laplace trans- form of n th-order integral of x(t) w.r.t. t as t t t 0 0 0 0 L ·· x(τ)dτ n ←→ s−n X(s) + s−n x(τ)dτ +···+ ·· x(τ)dτ n −∞ −∞ −∞ −∞ −∞ −∞ −∞ (A.14) Appendix A 389 A.3.4 Time Shifting – Real Translation The Laplace transform of a delayed function x(t) can be obtained as follows. ∞ . t1 (A=1) −st −st L {x(t − t1)} x(t − t1)e dτ + x(t − t1)e dτ −∞ t1 0 ∞ t−t1=τ − − τ − − τ = e st1 x(τ)e s dτ + e st1 x(τ)e s dτ; − t1 0 0 L if x(t)=0 ∀ t<0 −st1 −sτ −st1 x(t − t1), t1 > 0 ←→ e X(s) + x(τ)e dτ → e X(s) −t1 (A.15) τ τ τ = t τ = t t t 0 (a) (b) Fig. A.2 Switching the order of two integrations for double integral A.3.5 Frequency Shifting – Complex Translation L s1t e x(t) ←→ X(s − s1) (A.16) A.3.6 Real Convolution The (real) convolution of two (causal) functions g(t) and x(t) is defined as ∞ t if g(t)=0andx(t)=0 ∀ t<0 g(t) ∗ x(t) = g (τ) x (t − τ) dτ = g (τ) x (t − τ) dτ −∞ causality 0 (A.17) The Laplace transform of the convolution y(t) = g(t) ∗ x(t) turns out to be the product of the Laplace transforms of the two functions as L y(t) = g(t) ∗ x(t) ←→ Y (s) = G(s)X(s) (A.18) 390 Appendix A Proof ∞ ∞ t (A.1) (A.17) L {g(t) ∗ x(t)} = g(t) ∗ x(t)e−stdt = g(τ)x(t − τ)dτ e−stdt 0 0 0 ∞ ∞ Fig.A.2 = g(τ)e−sτ x(t − τ)e−s(t−τ)dt dτ 0 τ ∞ ∞ (t−τ=v) = g(τ)e−sτ x(v)e−svdv dτ 0 0 ∞ (A.1) (A.1) = g(τ)e−sτ dτ X(s) = G (s) X (s) 0 This property is used to describe the input-output relationship of a linear time- invariant (LTI) system which has the input x(t), the impulse function g(t), and the output y(t), where the Laplace transform of the impulse function, i.e., G(s) = L{g(t)}, is referred to as the system function or transfer function of the system. A.3.7 Partial Differentiation Given the Laplace transform of a function having a parameter a, that is L{x(t, a)}= X(s, a), the Laplace transform of the derivative of the parameterized function x(t, a) w.r.t. a equals the derivative of its Laplace transform X(s, a) w.r.t. a. ∂ L ∂ x(t, a) ←→ X(s, a) (A.19) ∂a ∂a A.3.8 Complex Differentiation The Laplace transform of −tx(t) equals the derivative of L{x(t)}=X(s) w.r.t. s. L d tx(t) ←→ − X(s) (A.20) ds This can be derived by differentiating Eq. (A.1) w.r.t. s. Example A.1 Applying Partial/Complex Differentiation Property −at To find the Laplace transform of te us (t), we differentiate Eq. (A.7) w.r.t. a and multiply the result by −1 to get L 1 te−atu (t) ←→ (A.21) s (s + a)2 Appendix A 391 Multiplying t by the left-hand side (LHS) and applying the complex differentia- tion property (A.20) repetitively, we get L 2! t2e−atu (t) ←→ (A.22) s (s + a)3 ..........................