ECE04 Laplace Transforms PDF

INTRODUCTION TO LAPLACE TRANSFORM REGION OF CONVERGENCE (ROC) PROPERTIES OF CONVERGENCE POLES AND ZEROS OF LAPLACE TRANSFORMS LAPLACE TRANSFORM PAIRS FOR COMMON SIGNALS PROPERTIES OF THE LAPLACE TRANSFORM INVERSE LAPLACE TRANSFORM & THE INV...

INTRODUCTION TO LAPLACE TRANSFORM REGION OF CONVERGENCE (ROC) PROPERTIES OF CONVERGENCE POLES AND ZEROS OF LAPLACE TRANSFORMS LAPLACE TRANSFORM PAIRS FOR COMMON SIGNALS PROPERTIES OF THE LAPLACE TRANSFORM INVERSE LAPLACE TRANSFORM & THE INVERSION FORMULA USE OF TABLES OF LAPLACE TRANSFORM PAIRS POINTS FOR PARTIAL-FRACTION EXPANSION DISCUSSION ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS  LAPLACE TRANSFORM and Continuous-Time LTI System For a continuous-time LTI system with impulse response h(t), the output y(t) of the system to the complex exponential input of the form 𝑒 𝑠𝑡 is 𝒔𝒕 𝒔𝒕 ∞ −𝒔𝒕 𝒚 𝒕 =𝓣 𝒆 = 𝑯 𝒔 𝒆 where 𝑯 𝒔 = ‫׬‬−∞ 𝒉 𝒕 𝒆 𝒅𝒕. 2022 PRESENTATION ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS LAPLACE TRANSFORM and Continuous-Time LTI System Definition The function H(s) is referred to as the Laplace Transform of h(t). For a general continuous-time signal x(t), the Laplace Transform X(s)(also called the bilateral or two sided Laplace transform) is defined by as ∞ 𝑿 𝒔 = න𝒙 𝒕 −𝒔𝒕 𝒆 𝒅𝒕 −∞ the variable s is generally complex valued and is expressed as 𝑠 = 𝜎 + 𝑗𝜔. ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS LAPLACE TRANSFORM and Continuous-Time LTI System It is sometimes considered as an operator that transforms a signal 𝑥(𝑡) into a function 𝑋(𝑠) symbolically represented by 𝑋(𝑠) = 𝐿{𝑥(𝑡)} and the signal 𝑥(𝑡) and its Laplace transform pair denoted as 𝑥(𝑡) ↔ 𝑋(𝑠) For a unilateral(or one-sided) Laplace transform, it is defined as ∞ 𝑿 𝒔 =න𝒙 𝒕 −𝒔𝒕 𝒆 𝒅𝒕 𝟎 ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS REGION OF CONVERGENCE The range of values of the complex variables s for which the Laplace Transform finites or converges is called the region of convergence (ROC). In order for the Laplace transform to be unique for each signal 𝑥(𝑡), the ROC must be specified as part of the transform. ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS  1. Find the Laplace Transform and ROC of the ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS EXAMPLE signal 𝑥 𝑡 = 𝑒 𝑢 𝑡 , where 𝑎 is real. −𝑎𝑡  2. Find the Laplace Transform and ROC of the ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS EXAMPLE signal 𝑥 𝑡 = −𝑒 𝑢 −𝑡 , where 𝑎 is real. −𝑎𝑡  SHORTCUT FOR ROC 1. Compare 𝜎 (𝑅𝑒{𝑠}) with the real part of the coefficient of 𝑡 in power of 𝑒. 2. Check if the signal is left sided or right sided and decide < and >. ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS POLES AND ZEROS OF X(s) Usually, X(s) will be a rational function in s, that is, 𝒂𝟎 𝒔𝒎 + 𝒂𝟏 𝒔𝒎−𝟏 + ⋯ + 𝒂𝒎 𝒂𝟎 𝒔 − 𝒛𝟏 … 𝒔 − 𝒛𝒎 𝒔 = 𝒏 𝒏−𝟏 = 𝒃𝟎 𝒔 + 𝒃𝟏 𝒔 + ⋯ + 𝒃𝒏 𝒃𝟎 𝒔 − 𝒑𝟏 … 𝒔 − 𝒑𝒏 The coefficients 𝑎𝑘 and 𝑏𝑘 are real constants, and 𝑚 and 𝑛 are positive integers. The 𝑋(𝑠) is called a proper rational function if 𝒏 > 𝒎 and an improper rational function if 𝒏 ≤ 𝒎. The roots of the numerator polynomial, 𝑏𝑘 , are called the zeros of 𝑿(𝒔) because 𝑋(𝑠) = 0 for those values of s. ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS POLES AND ZEROS OF X(s) Similarly, the roots of the denominator polynomial, 𝑝𝑘 , are called the poles of 𝑿(𝒔) because 𝑋(𝑠) is infinite for those values of s. Therefore, the poles of 𝑋(𝑠) lie outside the ROC since 𝑋(𝑠) does not converge at the poles. The zeros, may lie inside or outside the ROC. Traditionally, an " × " is used to indicate each pole location and an " ° " is used to indicate each zero. ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS  1. Sketch the pole-zero plot with the ROC for the ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS EXAMPLE signal 𝑋 𝑠 = 𝑠2+4𝑠+3 and Re s > −1. 2𝑠+4  PROPERTIES OF ROC 1. The ROC does not contain any poles. 2. If 𝑥(𝑡) is a finite-duration signal, that is, 𝑥(𝑡) = 0 except in a finite interval 𝑡1 ≤ 𝑡 ≤ 𝑡2 (−∞ < 𝑡1 and 𝑡2 < ∞)then the ROC is entire s-plane except possible 𝑠 = 0 𝑜𝑟 𝑠 = ∞. 3. If 𝑥 (𝑡) is a right-sided signal, that is, 𝑥 (𝑡) = 0 for 𝑡 < 𝑡1 < ∞, then the ROC is of the form 𝑹𝒆 𝒔 > 𝝈𝒎𝒂𝒙 where 𝜎𝑚𝑎𝑥 equals the maximum real part of any of the poles of 𝑋(𝑠). Thus, the ROC is a half-plane to the right of the vertical line 𝑅𝑒(𝑠) = 𝜎𝑚𝑎𝑥 in the s-plane and thus to the right of all of the poles of 𝑋(𝑠). ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS  PROPERTIES OF ROC 4. If 𝑥(𝑡) is a left-sided signal, that is, 𝑥 (𝑡) = 0 for 𝑡 < 𝑡2 < −∞, then the ROC is of the form 𝑹𝒆 𝒔 > 𝝈𝒎𝒂𝒙 where 𝜎𝑚𝑖𝑛 equals the minimum real part of any of the poles of 𝑋(𝑠). Thus, the ROC is a half-plane to the left of the vertical line 𝑅𝑒(𝑠) = 𝜎𝑚𝑖𝑛 in the s-plane and thus to the left of all of the poles of 𝑋(𝑠). 5. If 𝑥(𝑡) is a two-sided signal, that is 𝑥(𝑡) is an infinite-duration signal that is neither right-sided nor left sided, then the ROC is of the form 𝝈𝟏 < 𝑹𝒆 𝒔 < 𝝈𝟐 where 𝜎1 and 𝜎2 are the real parts of the two poles of 𝑋(𝑠). Thus, the ROC is a vertical strip in s-plane between the vertical lines 𝑅𝑒 𝑠 = 𝜎1 and 𝑅𝑒 𝑠 = 𝜎2. ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS Find the Laplace Transform X(s) and sketch the pole- EXAMPLE zero plot for the following signals: −𝟐𝒕 a. 𝒙(𝒕) = 𝒆 𝒖(𝒕) ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS Find the Laplace Transform X(s) and sketch the pole- EXAMPLE zero plot for the following signals: −𝟐𝒕 −𝟑𝒕 b. 𝒙 𝒕 = 𝒆 𝒖 −𝒕 + 𝒆 𝒖(−𝒕)  Let 𝒙 𝒕 = 𝒆. Find 𝑋(𝑠) and sketch the zero-pole ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS −𝒂 𝒕 EXAMPLE plot and the ROC for 𝑎 > 0 𝑎𝑛𝑑 𝑎 < 0.  LAPLACE TRANSFORM PAIRS FOR COMMON SIGNALS 𝒙(𝒕) 𝑿(𝒔) ROC 𝒙(𝒕) 𝑿(𝒔) ROC 𝛿(𝑡) 1 All 𝑠 𝑡𝑒 −𝑎𝑡 𝑢(𝑡) 1 𝑅𝑒 𝑠 > −𝑅𝑒(𝑎) 1 𝑠+𝑎 2 𝑢(𝑡) 𝑅𝑒 𝑠 > 0 1 𝑠 −𝑎𝑡 −𝑡𝑒 𝑢(−𝑡) 𝑅𝑒 𝑠 < −𝑅𝑒(𝑎) 1 𝑠+𝑎 2 −𝑢(−𝑡) 𝑅𝑒 𝑠 < 0 𝑠 𝑠 1 cos 𝜔0 𝑡 𝑢(𝑡) 𝑅𝑒 𝑠 > 0 𝑡𝑢(𝑡) 𝑅𝑒 𝑠 > 0 𝑠 2 + 𝜔02 𝑠2 𝜔0 𝑘! sin 𝜔0 𝑡 𝑢(𝑡) 𝑅𝑒 𝑠 > 0 𝑘 𝑡 𝑢(𝑡) 𝑅𝑒 𝑠 > 0 𝑠 2 + 𝜔02 𝑠 𝑘+1 𝑠 + 𝑎 1 𝑒 −𝑎𝑡 cos 𝜔0 𝑡 𝑢(𝑡) 𝑅𝑒 𝑠 > −𝑅𝑒(𝑎) 𝑒 −𝑎𝑡 𝑢(𝑡) 𝑅𝑒 𝑠 > −𝑅𝑒(𝑎) (𝑠 + 𝑎)2 +𝜔02 𝑠+𝑎 1 𝜔0 −𝑒 −𝑎𝑡 𝑢(−𝑡) 𝑅𝑒 𝑠 < −𝑅𝑒(𝑎) 𝑒 −𝑎𝑡 sin 𝜔0 𝑡 𝑢(𝑡) 𝑅𝑒 𝑠 < −𝑅𝑒(𝑎) 𝑠+𝑎 (𝑠 + 𝑎)2 +𝜔02 ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS  PROPERTIES OF LAPLACE TRANSFORM A. LINEARITY If 𝒙𝟏 𝒕 ↔ 𝑿𝟏 𝒔 𝑹𝑶𝑪 = 𝑹𝟏 𝒙 𝟐 𝒕 ↔ 𝑿𝟐 𝒔 𝑹𝑶𝑪 = 𝑹𝟐 Then 𝒂𝟏 𝒙𝟏 𝒕 + 𝒂𝟐 𝒙𝟐 𝒕 ↔ 𝒂𝟏 𝑿𝟏 𝒔 + 𝒂𝟐 𝑿𝟐 𝒔 𝑹′ ⊃ 𝑹𝟏 ∩ 𝑹𝟐 The ROC of the resultant of Laplace Transform is at least as large as the region in the common between 𝑅1 and 𝑅2. Usually we have simply 𝑅 = 𝑅1 ∩ 𝑅2. ′ ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS  PROPERTIES OF LAPLACE TRANSFORM B. TIME SHIFTING If 𝒙(𝒕) ↔ 𝑿(𝒔) 𝑹𝑶𝑪 = 𝑹 Then 𝒙 𝒕 − 𝒕𝟎 ↔ −𝒔𝒕 𝒆 𝟎𝑿 𝒔 𝑹 ′ =𝑹 It indicates that the ROCs before and after the time shifting operation are the same. ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS  PROPERTIES OF LAPLACE TRANSFORM C. SHIFTING IN THE s-domain If 𝒙 𝒕 ↔𝑿 𝒔 𝑹𝑶𝑪 = 𝑹 Then 𝒔 𝒕 𝒆 𝒙(𝒕) 𝟎 ↔ 𝑿 𝒔 − 𝒔𝟎 𝑹 ′ = 𝑹 + 𝑹𝒆(𝒔𝟎 ) It indicates that the ROCs associated with 𝑋 𝑠 − 𝑠0 is that of 𝑋 𝑠 shifted by 𝑅𝑒 𝑠0. ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS  PROPERTIES OF LAPLACE TRANSFORM D. TIME SCALING If 𝒙(𝒕) ↔ 𝑿(𝒔) 𝑹𝑶𝑪 = 𝑹 Then 𝟏 𝒔 ′ 𝒙 𝒂𝒕 ↔ 𝑿 𝑹 = 𝒂𝑹 𝒂 𝒂 It indicates that scaling time variable 𝑡 by the factor of 𝑎 causes an inverse 1 scaling of the variable 𝑠 by as well 𝑎 𝑠 1 as an amplitude scaling of 𝑋 by. 𝑎 𝑎 ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS  PROPERTIES OF LAPLACE TRANSFORM E. TIME REVERSAL If 𝒙(𝒕) ↔ 𝑿(𝒔) 𝑹𝑶𝑪 = 𝑹 Then 𝐱 −𝐭 ↔ 𝐗 −𝐬 𝐑′ = −𝐑 It produces a reversal of both the 𝜎 and 𝑗𝜔 in the s- plane. ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS Find the Laplace Transform X(s) and and the EXAMPLE associated ROC for each of the following signals a. 𝒙 𝒕 = 𝜹 𝒕 − 𝒕𝟎 ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS Find the Laplace Transform X(s) and and the EXAMPLE associated ROC for each of the following signals −𝟐𝒕 b. 𝒙 𝒕 = 𝒆 [𝒖 𝒕 − 𝒖 𝒕 − 𝟓 ]  PROPERTIES OF LAPLACE TRANSFORM F. DIFFERENTIATION IN THE TIME DOMAIN If 𝒙(𝒕) ↔ 𝑿(𝒔) 𝑹𝑶𝑪 = 𝑹 Then 𝒅𝒙 𝒕 ↔ 𝒔𝑿 𝒔 𝑹′ ⊃ 𝑹 𝒅𝒕 It shows that the effect of differentiation in the time domain is multiplication of the corresponding Laplace transform by s. The associated ROC is unchanged unless there is a pole-zero cancellation at 𝑠 = 0. ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS  PROPERTIES OF LAPLACE TRANSFORM G. DIFFERENTIATION IN THE S DOMAIN If 𝒙(𝒕) ↔ 𝑿(𝒔) 𝑹𝑶𝑪 = 𝑹 Then 𝒅 −𝒕𝒙 𝒕 ↔ 𝑿 𝒔 𝑹′ = 𝑹 𝒅𝒔 ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS  PROPERTIES OF LAPLACE TRANSFORM H. INTEGRATION IN THE TIME DOMAIN If 𝒙(𝒕) ↔ 𝑿(𝒔) 𝑹𝑶𝑪 = 𝑹 Then 𝒕 𝟏 න 𝒙 𝝉 𝒅𝝉 ↔ 𝑿 𝒔 𝑹 ⊃ 𝑹𝟏 ∩ 𝑹𝒆 𝒔 > 𝟎 −∞ 𝒔 It shows that the Laplace transform operation corresponding to the 1 time domain integration is multiplication by. The form of 𝑅′ follows 𝑠 from the possible introduction of an additional pole at 𝑠 = 0 by the 𝟏 multiplication by. 𝒔 ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS  PROPERTIES OF LAPLACE TRANSFORM I. CONVOLUTION If 𝒙𝟏 𝒕 ↔ 𝑿𝟏 𝒔 𝑹𝑶𝑪 = 𝑹𝟏 𝒙𝟐 𝒕 ↔ 𝑿𝟐 𝒔 𝑹𝑶𝑪 = 𝑹𝟐 Then 𝒙𝟏 𝒕 ∗ 𝒙𝟐 𝒕 ↔ 𝑿𝟏 𝒔 𝑿𝟐 𝒔 𝑹′ ⊃ 𝑹𝟏 ∩ 𝑹𝟐 ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS Using the various Laplace transform properties, EXAMPLE derive the Laplace transform of the following signals from the Laplace transform of 𝑢(𝑡) 𝒂. 𝒕𝒖(𝒕) ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS Using the various Laplace transform properties, EXAMPLE derive the Laplace transform of the following signals from the Laplace transform of 𝑢(𝑡) b. 𝒕𝒆−𝒂𝒕 𝒖(𝒕)  SUMMARY OF THE PROPERTIES OF THE LAPLACE TRANSFORM PROPERTY SIGNAL TRANSFORM ROC 𝒙(𝒕) 𝑿(𝒔) 𝑹 𝒙𝟏 (𝒕) 𝑿𝟏 𝒔 𝑹𝟏 𝒙𝟐 (𝒕) 𝑿𝟐 (𝒔) 𝑹𝟐 Linearity 𝒂𝟏 𝒙𝟏 𝒕 + 𝒂𝟐 𝒙𝟐 𝒕 𝒂𝟏 𝑿𝟏 𝒔 + 𝒂𝟐 𝑿𝟐 (𝒔) 𝑹′ ⊃ 𝑹𝟏 ∩ 𝑹𝟐 Time shifting 𝒙(𝒕 − 𝒕𝟎 ) 𝒆−𝒔𝒕𝟎 𝑿(𝒔) 𝑹′ = 𝑹 Shifting in s 𝒆𝒔𝟎 𝒕 𝒙(𝒕) 𝑿(𝒔 − 𝒔𝟎 ) 𝑹′ = 𝑹 + 𝑹𝒆(𝒔𝟎 ) ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS  SUMMARY OF THE PROPERTIES OF THE LAPLACE TRANSFORM PROPERTY SIGNAL TRANSFORM ROC 𝟏 𝒔 Time Scaling 𝒙(𝒂𝒕) 𝑿 𝑹′ = 𝒂𝑹 𝒂 𝒂 Time Reversal 𝒙(−𝒕) 𝑿(−𝒔) 𝑹′ = −𝑹 𝒅𝒙 𝒕 Differentiation in t 𝒔𝑿(𝒔) 𝑹′ ⊃ 𝑹 𝒅𝒕 𝒅𝑿 𝒔 Differentiation in s −𝒕𝒙(𝒕) 𝑹′ = 𝑹 𝒅𝒔 𝒕 𝟏 Integration න 𝒙 𝝉 𝒅𝝉 𝑿(𝒔) 𝑹′ ⊃ 𝑹𝟏 ∩ 𝑹𝒆 𝒔 > 𝟎 𝒔 −∞ Convolution 𝒙𝟏 𝒕 ∗ 𝒙𝟐 (𝒕) 𝑿𝟏 𝒔 𝑿𝟐 (𝒔) 𝑹′ ⊃ 𝑹𝟏 ∩ 𝑹𝟐 ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS INVERSE LAPLACE TRANSFORM Inversion of the Laplace transform to find the signal 𝑥( 𝑡 ) from its Laplace transform 𝑋(𝑠) is called the inverse Laplace transform, symbolically denoted as −𝟏 𝒙 𝒕 =𝓛 𝑿(𝒔) ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS INVERSION FORMULA There is a procedure that is applicable to all classes of transform functions that involves the evaluation of a line integral in complex s-plane; that is, 𝒄+𝒋𝝎 𝟏 𝒔𝒕 𝒙 𝒕 = න 𝑿 𝒔 𝒆 𝒅𝒔 𝟐𝝅𝒋 𝒄−𝒋𝝎 In this integral, the real c is to be selected such that if the ROC of 𝑋(𝑠) is 𝛿1 < 𝑅𝑒(𝑠) < 𝛿2 , then 𝛿1 < 𝑐 < 𝛿2. ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS USE OF TABLES OF LAPLACE TRANSFORM PAIRS In the second method for the inversion of 𝑋(𝑠), we attempt to express 𝑋(𝑠) as a sum 𝑿 𝒔 = 𝑿𝟏 (𝒔) +... +𝑿𝒏 (𝒔) where 𝑋1 (𝑠),... , 𝑋𝑛 (𝑠) are functions with known inverse transforms 𝑥1 (𝑡),... , 𝑥𝑛 (𝑡). From the linearity property it follows that 𝒙 𝒕 = 𝒙𝟏 𝒕 + ⋯ + 𝒙𝒏 (𝒕) ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS PARTIAL-FRACTION EXPANSION If 𝑋(𝑠) is a rational function, that is, of the form 𝑵 𝒔 𝒔 − 𝒛𝟏 … 𝒔 − 𝒛𝒎 𝑿 𝒔 = =𝒌 𝑫 𝒔 𝒔 − 𝒑𝟏 … (𝒔 − 𝒑𝒎 ) a simple technique based in a partial expansion can be used for the inversion of 𝑋 𝑠. ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS  PARTIAL-FRACTION EXPANSION (a) When 𝑋(𝑠) is a proper rational function , 𝑚 < 𝑛: 1. SIMPLE POLE CASE 2. MULTIPLE POLE CASE If all poles of 𝑋(𝑠) , that is, all zeros of If 𝐷(𝑠) has a multiple roots, that is, if it 𝐷(𝑠), are simple (or distinct), then 𝑋(𝑠) contains factors of the form 𝑠 − 𝑝 𝑟 , we can be written as say that 𝑝𝑖 is the multiple pole of 𝑋(𝑠) with multiplicity 𝑟. Then the expansion 𝑐1 𝑐2 of 𝑋(𝑠) can consist of terms of the 𝑋 𝑠 = + ⋯+ 𝑠 − 𝑝1 𝑠 − 𝑝𝑛 forms 𝜆1 𝜆2 𝜆1 where coefficients 𝑐𝑘 are given by + 2 +⋯ 𝑠 − 𝑝𝑖 𝑠 − 𝑝𝑖 𝑠 − 𝑝𝑖 𝑟 𝑐𝑘 = 𝑠 − 𝑝𝑘 ) 𝑋 𝑠 𝑠=𝑝𝑘 1 𝑑𝑘 𝑟 𝜆𝑟−𝑘 = 𝑘 𝑠 − 𝑝𝑖 𝑋(𝑠) |𝑠=𝑝𝑖 𝑘! 𝑑𝑠 ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS PARTIAL-FRACTION EXPANSION (b) When 𝑋(𝑠) is a proper rational function , 𝑚 ≥ 𝑛: If 𝑚 ≥ 𝑛, by long division we can write 𝑋(𝑠) in the form 𝑵(𝒔) 𝑹 𝒔 𝑿 𝒔 = =𝑸 𝒔 + 𝑫(𝒔) 𝑫(𝒔) Where 𝑁(𝑠) and 𝐷(𝑠) are the numerator and denominator polynomial in s, respectively, of 𝑋(𝑠), the quotient of 𝑄(𝑠) is a polynomial s with degree 𝑚 − 𝑛, and the reminder 𝑅(𝑠) is a polynomial in s with degree less than 𝑁. The inverse Laplace Transform of 𝑄(𝑠) can be computed by using the transform pair 𝒅𝒌 𝜹(𝒕) 𝒌 ↔ 𝒔 𝒌 = 𝟏, 𝟐, 𝟑, … 𝒅𝒕𝒌 ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS Find the inverse Laplace Transform of the EXAMPLE following: 𝟏 a. 𝑿 𝒔 = 𝑹𝒆 𝒔 > −𝟏 𝒔+𝟏 ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS Find the inverse Laplace Transform of the EXAMPLE following: 𝟏 b. 𝑿 𝒔 = 𝑹𝒆 𝒔 < −𝟏 𝒔+𝟏 ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS Find the inverse Laplace Transform of the EXAMPLE following: 𝒔 c. 𝑿 𝒔 = 𝟐 𝑹𝒆 𝒔 > 𝟎 𝒔 +𝟒 ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS Find the inverse Laplace Transform of the EXAMPLE following: 𝒔+𝟏 d. 𝑿 𝒔 = 𝟐 𝑹𝒆 𝒔 > −𝟏 𝒔+𝟏 +𝟒 ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS Find the inverse Laplace Transform of EXAMPLE 𝐗 𝐬 = 𝟐𝐬+𝟒 with 𝑹𝒆 𝒔 > −𝟏 𝟐 𝐬 +𝟒𝐬+𝟑 ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS Find the inverse Laplace Transform of EXAMPLE 𝐗 𝐬 = 𝟐𝐬+𝟒 with 𝐑𝐞 𝒔 < −𝟑 𝟐 𝐬 +𝟒𝐬+𝟑 ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS Find the inverse Laplace Transform of EXAMPLE 𝐗 𝐬 = 𝟐𝐬+𝟒 with −3 < 𝑅𝑒 𝑠 < −1 𝟐 𝐬 +𝟒𝐬+𝟑 ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS Find the inverse Laplace Transform of EXAMPLE a. 𝐗 𝒔 = 𝒔(𝒔𝟐+𝟒𝒔+𝟏𝟑) 𝟓𝒔+𝟏𝟑 𝑹𝒆 𝒔 > 𝟎 ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS Find the inverse Laplace Transform of EXAMPLE 𝟐 𝒔 + 𝟐𝒔 + 𝟓 𝒃. 𝑿 𝒔 = 𝟐 𝑹 𝒔 > −𝟑 𝒔+𝟑 𝒔+𝟓 ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS Find the inverse Laplace Transform of EXAMPLE 𝟑 𝟐 𝒔 + 𝟐𝒔 + 𝟔 𝒄. 𝑿 𝒔 = , 𝑹𝒆 𝒔 > 𝟎 𝒔𝟐 + 𝟑𝒔 ECE04: LAPLACE TRANSFORM & THE CT-LTI SYSTEMS THANK YOU PREPARE FOR YOUR QUIZ 01 THIS FINALS!