Characteristics of a Linear System

1. Characteristics of a Linear System 2.7

2. Topics • Memory • Invertibility • Inverse of a System • Causality • Stability • Time Invariance • Linearity

3. Memory • A system has a memory if its output at to depends on the input on values of the input other than x(to)

4. Memoryless Circuit

5. Invertibility • A system is said to be invertible if distinct inputs results in distinct outputs • Y(t)=x2(t) is not invertible • Output of 4V is attributable to +2 V and -2V. • Y(t)=x5(t) is invertible

6. Inverse of a System • The inverse of a system is a second system that, when cascaded with T, yields the identity system. Gain of 5 Gain of 1/5

7. Causality • A system is causal if the output at any time to is dependent on the input only for t≤to • Y(t)=x(t-2) is causal. The present output is equal to the input of 2 s ago • Y(t)=x(t+2) is not causal since the output at t=0 is equal to the input at t=2s. • All physical real-time systems are causal because we can not anticipate the future!

8. Non-Causal signal

9. Continuous-Time Systems - Stability • Stability has Many different definitions • Bounded-input-bounded-output • Example: • An ideal amplifier y(t) = 10 x(t)  B2=10 B1 • Square system: y(t)=x2(t)  B2=B12 • A system can be unstable or marginally stable • If a system is used responsibly (the input is bounded), the system will behave predictably.

10. Time Invariance • A system is time invariant if a time shift in the input signal results only in the same time shift in the output signal

11. Test for Time Invariance • A system is time invariant if a time shift in the input signal results only in the same time shift in the output signal The system is time invariant if

12. Example 1 The system is time invariant.

13. Example 2 The system is time varying.

14. Example 3: Time Reversal

15. Continuous-Time Systems – Linearity • A linear system must satisfy superposition condition (additive and homogeneity