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

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

3. Time Invariance Property: If, then 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 t t t x y

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

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

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

8. Again, this is a linear system. Modulator 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 EXAMPLE RC CircuitModulator Linear?YY Time Invariant? YN Causal?YY Memoryless?NY