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Analyzing Linearity and Time Invariance in RC Circuits

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This resource explores the properties of linearity and time invariance in systems, specifically through the example of an RC circuit. It explains that a system is linear if it conforms to specific input-output relationships, and it demonstrates time invariance using formulas and assumptions. By contrasting linear and time-varying systems using signals and examples like amplitude modulation, the text elucidates key concepts in system theory, including causality, memory, and the distinctions between causal and non-causal systems.

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Analyzing Linearity and Time Invariance in RC Circuits

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  1. y x Recall: RC circuit example + + R x(t) y(t) C _ _ assuming

  2. y x Properties of Input-Output Systems Linearity. The system is linear if Example: RC circuit is linear. Follows from equation

  3. Time Invariance Property: If , then x In words, a system is T.I. when: given an input-output pair, if we apply a delayed version of the input, the new output is the delayed version of the original output. t y t t t

  4. + + R x(t) y(t) C _ _ Time invariance of RC circuit Seems intuitive based on physical grounds Let’s prove it using the formula Assume all signals are zero for t < 0. Introduce the notation

  5. Now, apply the delayed input x(u) = 0 for u < 0 Remark: proof works for any h(t)!

  6. Another example: y(t) = t x(t) x(t) 1 Linear? Yes, easy to show. Time invariant? No: 1 y(t) 1 x(t-1) 2 T[x(t-1)] They are different! Compare for a particular x(t)  1 Time-varying system 1 y(t-1) t 1 2

  7. y x Example: amplitude modulation + y(t) x(t) Used in AM radio!

  8. Modulator Again, this is a linear system. Is it time invariant? Only equal if Therefore, it is a time varying system Notation: LTI = linear, time invariant LTV= linear, time varying

  9. Causality and memory • A system is causal if y(t_0) depends only on x(t), t <=t_0 • (present output only depends on past and present inputs) • A system is memoryless if y(t_0) depends only on x(t_0) • (present output only depends on present input). • Causal, not memoryless: we say it “has memory”. Examples: Delay system is causal, and has memory. Backward shift system is non-causal: output anticipates the input. Non-causal systems are not physically realizable

  10. Recap: properties of main examples

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