Troubles at the Origin Consistent Usage and Properties of the Unilateral Laplace Transform Kent H Lundberg Haynes R Miller and David L Trumper Massachusetts Institute of Technology The Laplace transform is a standard tool associated with the analysis of signals models and control systems and is consequently taught in some form to almost all engineering students The bilateral and unilateral forms of the Laplace transform are closely related but have somewhat different domains of application The bilateral transform is most frequently seen in the context of signal processing whereas the unilateral transform is most often associated with the study of dynamic system response where the role of initial conditions takes on greater signi cance In our teaching we have found some signi cant pitfalls associated with teaching our students to understand and apply the Laplace transform These confusions extend to the presentation of this material in many of the available mathematics and engineering textbooks as well The most signi cant confusion in much of the textbook literature is how to deal with the origin in the application of the unilateral Laplace transform That is many texts present the transform of a time function f t as L f t f t e st dt 1 0 without properly speci ying the meaning of the lower limit of integration Said informally does the integral include the origin fully partially or not at all This issue becomes signi cant as soon as singularity functions such as the unit impulse are introduced While it is not possible to devote full attention to this issue within the context of a typical undergraduate course this skeleton in the closet as Kailath 8 called it needs to be brought out fully into the light Our purpose in writing this article is to put forward a consistent set of Laplace transform de nitions and properties which allow the correct analysis of dynamic systems in the presence of arbitrary initial conditions and where the system is driven by functions which include singularities We also present reasonable mathematical support for these properties as well as a consistent treatment of singularity functions without becoming fully enmeshed in the theory of generalized functions which quickly becomes too far divorced from applications 1 To properly learn about and apply the unilateral Laplace transform students need to be taught a consistent set of properties that correctly handle problems with arbitrary inputs and initial conditions The proper form of the unilateral Laplace transform fully includes the origin L f t 0 f t e st dt 2 as indicated by the 0 notation Thus the integral includes interesting events which happen at t 0 such as impulses or higher order singularity functions steps and the beginnings of other transients Some texts refer to this form as the L transform However since we regard this as the only correct usage of the unilateral transform in the context of dynamic systems we will omit any additional notation and use the symbol L to represent the transform as de ned in 2 Following from this de nition of the transform is the time derivative rule L f t sF s f 0 3 whereby the initial conditions existing before any t 0 transient are brought into the analysis Note that the 0 indicates that the response will be calculated in terms of what we term the pre initial conditions Also associated with this de nition is the the initial value theorem lim sF s f 0 s 1 4 where the notation 1 indicates that the limit is taken along the positive real axis It is interesting to note here that the value calculated by 4 is associated with the post initial values at t 0 This form of the initial value theorem is the correct result and it is also the desired one since we are mostly interested in the initial value after any discontinuities at t 0 More generally if F s is able to be written as a polynomial plus a function F s converging to zero as s 1 then lim sF s f 0 s 1 5 These properties 3 4 5 and some extensions are developed more fully in the appendices With this overview in hand the remainder of the paper is organized as follows We rst motivate the discussion with two simple dynamic system examples such as might be presented in a sophomore engineering course The responses of these systems are calculated via the Laplace transform de nition and properties presented above Next we discuss the application of the Laplace transform to abstract signals independent of any dynamic systems context to clarify the need for the consistent de nitions presented above Finally the article concludes with appendices which introduce and develop the transform 2 C vO 1 t 0 0 t 0 vI t R Fig 1 Schematic of a high pass electrical lter driven by an unstep The initial state of this system is the capacitor voltage vC 0 1 and thus the initial output voltage is vO 0 0 properties with reasonable mathematical support In order to do this we need to think carefully about how singularity functions are de ned how these are combined with regular functions to form generalized functions and how required mathematical operations on these generalized functions can be consistently de ned I A PPLICATION E XAMPLES In this section we present a pair of example problems drawn from electrical and mechanical engineering respectively To correctly calculate the transient response of these systems requires care in applying the transform 2 the set of properties presented above 3 4 and 5 yield the correct answer A First Order High Pass Filter Driven by an Unstep First consider the high pass lter shown in Figure 1 which is driven by an unstep function vI t 1 t 0 0 t 0 The Laplace transform of this input is L vI t Vi s 0 which certainly seems uninteresting We also specify the initial condition vO 0 0 and thus the capacitor is initially charged to vC 0 1V To nd the total system response to the initial state and this input we start with the differential equation C d vO 0 vI vO dt R or dvO vO dvI dt RC dt 3 Input Voltage vI t 1 0 8 0 6 0 4 0 2 0 0 2 1 0 5 0 0 5 1 1 5 2 2 5 3 1 5 2 2 5 3 Output Voltage vO t 0 2 0 0 2 0 4 0 6 0 8 1 1 Fig 2 0 5 0 0 5 1 Response of the high pass lter driven by an unstep The Laplace transform of this differential equation using the correct form of the derivative rule is sVo s vO 0 Vo s sVi s vI 0 RC which reduces to Vo s sVi s vI 0 vO 0 s 1 RC The associated pre initial values are vI 0 1 and vO 0 0 and we have already calculated Vi s 0 Thus the expression simpli es to Vo s 1 s 1 RC 6 Inverting this transform gives