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Signals and Systems Fall 2003 Lecture #18 6 November 2003 Inverse Laplace Transforms Laplace Transform Properties The System Function of an LTI System Geometric Evaluation of Laplace Transforms and Frequency Responses

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Inverse Laplace Transform Fix σ ROC and apply the inverse Fourier But s = σ + jω (σfixed) ds =jdω

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Inverse Laplace Transforms Via Partial Fraction Expansion and Properties Example: Three possible ROCs corresponding to three different signals Recall

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ROC I:Left-sided signal. ROC II:Two-sided signal, has Fourier Transform. ROC III:Right-sided signal.

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Properties of Laplace Transforms Many parallel properties of the CTFT, but for Laplace transforms we need to determine implications for the ROC For example: Linearity ROC at least the intersection of ROCs of X1(s) and X2(s) ROC can be bigger (due to pole-zero cancellation) ROC entire s-

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Time Shift

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Time-Domain Differentiation ROC could be bigger than the ROC of X(s), if there is pole-zero cancellation. E.g., s-Domain Differentiation

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Convolution Property x(t) y(t)=h(t) x(t) h(t) For Then ROC of Y(s) = H(s)X(s): at least the overlap of the ROCs of H(s) & X(s) ROC could be empty if there is no overlap between the two ROCs E.g. x(t)=e t u(t),and h(t)=-e -tu (-t) ROC could be larger than the overlap of the two.

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The System Function of an LTI System x(t) y(t) h(t) The system function characterizes the system System properties correspond to properties of H(s) and its ROC A first example:

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Geometric Evaluation of Rational Laplace Transforms Example #1:A first-order zero

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Example #2:A first-order pole Still reason with vector, but remember to "invert" for poles Example #3:A higher-order rational Laplace transform

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First-Order System Graphical evaluation of H(jω)

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Bode Plot of the First-Order System

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Second-Order System complex poles Underdamped double pole at s = ω n Critically damped 2 poles on negative real axis Overdamped

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DemoPole-zero diagrams, frequency response, and step response of first-order and second-order CT causal systems

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Bode Plot of a Second-Order System Top is flat when ζ= 1/2 = 0.707 a LPF for ω < ω n

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Unit-Impulse and Unit-Step Response of a Second- Order System No oscillations when ζ 1 Critically (=) and over (>) damped.

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First-Order All-Pass System 1.Two vectors have the same lengths 2.

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Laplace Transform (1).

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