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Chapter 1 Elementary Signals 1 26 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications b. (1) Referring to the given waveform we observe that discontinuities occur only at , , and . Therefore, and . Also, by the sampling property of the delta function and with these simplifications (1) above reduces to The waveform for is shown below. vt () e 2t u 0 t e 10t 30 + u 0 t2 20t 80 + u 0 t3 120 u 0 t5 ++ + = +1 0 t7 0 + u 0 dv dt ----- 2 e u 0 t e δ t 2e 10 + u 0 e 10t 30 + δ + = 20u 0 80 + δ 20u 0 120 δ + 10u 0 70 + δ + = = = δ t 0 = δ 0 = e 10t 30 + δ e 10t 30 + = δ = 10 δ 20t 80 + δ 20t 80 + = δ = 20 δ = 120 δ 20t 120 = δ = 20 δ = dv dt u 0 t u 0 10u 0 10 δ = 20u 0 20 δ 0 20 δ 10u 0 u 0 t u 0 [] 10 δ 10 u 0 u 0 20 δ = 10 u 0 u 0 20 δ 10 u 0 u 0 + 20 10 Vs t s 20 10 1 2 3 4 5 6 7 10 δ 20 δ t 3 20 δ t 5

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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 2 1 Copyright © Orchard Publications Chapter 2 The Laplace Transformation his chapter begins with an introduction to the Laplace transformation, definitions, and properties of the Laplace transformation. The initial value and final value theorems are also discussed and proved. It continues with the derivation of the Laplace transform of common functions of time, and concludes with the derivation of the Laplace transforms of common wave- forms. 2.1 Definition of the Laplace Transformation The two sided or bilateral Laplace Transform pair is defined as (2.1) (2.2) where denotes the Laplace transform of the time function , denotes the Inverse Laplace transform, and is a complex variable whose real part is , and imaginary part , that is, . In most problems, we are concerned with values of time greater than some reference time, say , and since the initial conditions are generally known, the two sided Laplace trans- form pair of (2.1) and (2.2) simplifies to the unilateral or one sided Laplace transform defined as (2.3) (2.4) The Laplace Transform of (2.3) has meaning only if the integral converges (reaches a limit), that is, if T L ft () {} Fs = e st dt = L 1 = 1 2 π j -------- σ j ω σ j ω + e ds = L L 1 s σ ω s σ j ω + = t tt 0 0 == L F = s t 0 e f t 0 e L 1 f = t 1 2 π j σ j ω σ j ω + e =
Chapter 2 The Laplace Transformation 2 2 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications (2.5) To determine the conditions that will ensure us that the integral of (2.3) converges, we rewrite (2.5) as (2.6) The term in the integral of (2.6) has magnitude of unity, i.e.,

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