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MIT6_003S10_lec05

# MIT6_003S10_lec05 - .003 Signals and Systems Laplace...

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Unformatted text preview: .003: Signals and Systems Laplace Transform February 18, 2010 ˙ x ( t ) x ( t ) X A X oncept Map: Continuous-Time Systems ultiple representations of CT systems. + 1 + 1 2 X − − Block Diagram System Functional Y Y 2 A 2 = X 2 + 3 A + A 2 Impulse Response h ( t ) = 2( e − t/ 2 − e − t ) u ( t ) Differential Equation System Function 2¨ y ( t ) + 3 ˙ y ( t ) + y ( t ) = 2 x ( t ) Y ( s ) 2 X ( s ) = 2 s 2 + 3 s + 1 ˙ x ( t ) x ( t ) X A X oncept Map: Continuous-Time Systems Relations among representations. + 1 + 1 2 X − − Block Diagram System Functional Y Y 2 A 2 = X 2 + 3 A + A 2 Impulse Response h ( t ) = 2( e − t/ 2 − e − t ) u ( t ) Differential Equation System Function 2¨ y ( t ) + 3 ˙ y ( t ) + y ( t ) = 2 x ( t ) Y ( s ) 2 X ( s ) = 2 s 2 + 3 s + 1 oncept Map: Continuous-Time Systems Two interpretations of . X A X + 1 + 1 2 X − − Block Diagram System Functional Y Y 2 A 2 = X 2 + 3 A + A 2 Impulse Response x ˙ ( t ) x ( t ) h ( t ) = 2( e − t/ 2 − e − t ) u ( t ) Differential Equation System Function 2¨ y ( t ) + 3 ˙ y ( t ) + y ( t ) = 2 x ( t ) Y ( s ) 2 X ( s ) = 2 s 2 + 3 s + 1 ˙ x ( t ) x ( t ) X A X oncept Map: Continuous-Time Systems Relation between System Functional and System Function. + 1 + 1 2 X − − Block Diagram System Functional Y Y 2 A 2 = X 2 + 3 A + A 2 Impulse Response A→ 1 s h ( t ) = 2( e − t/ 2 − e − t ) u ( t ) Differential Equation System Function 2¨ y ( t ) + 3 ˙ y ( t ) + y ( t ) = 2 x ( t ) Y ( s ) 2 X ( s ) = 2 s 2 + 3 s + 1 ˙ x ( t ) x ( t ) X A X heck Yourself How to determine impulse response from system functional? + 1 + 1 2 X − − Block Diagram System Functional Y Y 2 A 2 = X 2 + 3 A + A 2 Impulse Response h ( t ) = 2( e − t/ 2 − e − t ) u ( t ) Differential Equation System Function 2¨ y ( t ) + 3 ˙ y ( t ) + y ( t ) = 2 x ( t ) Y ( s ) 2 X ( s ) = 2 s 2 + 3 s + 1 heck Yourself How to determine impulse response from system functional? Expand functional using partial fractions : Y 2 A 2 A 2 2 A 2 A X = 2 + 3 A + A 2 = (1 + 2 1 A )(1 + A ) = 1 + 1 2 A − 1 + A Recognize forms of terms: each corresponds to an exponential. Alternatively, expand each term in a series : Y 1 1 1 = 2 A 1 − 2 A + 4 A 2 − 8 A 3 + −··· − 2 A 1 −A + A 2 −A 3 + −··· X Let X = δ ( t ) . Then Y = 2 1 − 1 1 2 − 1 3 + −··· u ( t ) − 2 1 − t + 1 2 − 1 3 + −··· u ( 2 t + 8 t 48 t 2 t 3! t = 2 e − t/ 2 − e − t u ( t ) ˙ x ( t ) x ( t ) X A X heck Yourself How to determine impulse response from system functional? Block Diagram System Functional Y Y 2 A 2 = X 2 + 3 A + A 2 + 1 + 1 2 X − − series partial fractions Impulse Response h ( t ) = 2( e − t/ 2 − e − t ) u ( t ) Differential Equation System Function 2¨ y ( t ) + 3 ˙ y ( t ) + y ( t ) = 2 x ( t ) Y ( s ) 2 X ( s ) = 2 s 2 + 3 s + 1 ˙ x ( t ) x ( t ) X A X oncept Map: Continuous-Time Systems Today: new relations based on Laplace transform....
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MIT6_003S10_lec05 - .003 Signals and Systems Laplace...

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