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Leo Lam 2010-2013 Signals and Systems EE235

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Leo Lam 2010-2013 Stanford The Stanford Linear Accelerator Center was known as SLAC, until the big earthquake, when it became known as SPLAC. SPLAC? Stanford Piecewise Linear Accelerator.

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Leo Lam 2010-2013 Todays menu Today: Linear, Constant-Coefficient Differential Equation Particular Solution

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Zero-state output of LTI system Leo Lam 2010-2013 4 Response to our input x(t) LTI system: characterize the zero-state with h(t) Initial conditions are zero (characterizing zero-state output) Zero-state output: Total response(t)=Zero-input response (t)+Zero-state output(t) T (t) h(t)

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Zero-state output of LTI system Leo Lam 2010-2013 5 Zero-input response: Zero-state output: Total response: Total response(t)=Zero-input response (t)+Zero-state output(t) Zero-state: (t) is an input only at t=0 Also called: Particular Solution (PS) or Forced Solution

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Zero-state output of LTI system Leo Lam 2010-2013 6 Finding zero-state output (Particular Solution) Solve: Or: Guess and check Guess based on x(t)

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Trial solutions for Particular Solutions Leo Lam 2010-2013 7 Guess based on x(t) Input signal for time t> 0 x(t) Guess for the particular function y P

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Particular Solution (example) Leo Lam 2010-2013 8 Find the PS (All initial conditions = 0): Looking at the table: Guess: Its derivatives:

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Particular Solution (example) Leo Lam 2010-2013 9 Substitute with its derivatives: Compare:

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Particular Solution (example) Leo Lam 2010-2013 10 From We get: And so:

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Particular Solution (example) Leo Lam 2010-2013 11 Note this PS does not satisfy the initial conditions! Not 0!

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Natural Response (doing it backwards) Leo Lam 2010-2013 12 Guess: Characteristic equation: Therefore:

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Complete solution (example) Leo Lam 2010-2013 13 We have Complete Sol n : Derivative:

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Complete solution (example) Leo Lam 2010-2013 14 Last step: Find C 1 and C 2 Complete Sol n : Derivative: Subtituting: Two equations Two unknowns

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Complete solution (example) Leo Lam 2010-2013 15 Last step: Find C 1 and C 2 Solving: Subtitute back: Then add u(t): y n ( t ) y p ( t ) y ( t )

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Another example Leo Lam 2010-2013 16 Solve: Homogeneous equation for natural response: Characteristic Equation: Therefore: Input x(t)

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Another example Leo Lam 2010-2013 17 Solve: Particular Solution for Table lookup: Subtituting: Solving: b=-1, =-2 No change in frequency! Input signal for time t> 0 x(t) Guess for the particular function y P

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Another example Leo Lam 2010-2013 18 Solve: Total response: Solve for C with given initial condition y(0)=3 Tada!

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Stability for LCCDE Leo Lam 2010-2013 19 Stable with all Re( j