1. 16.362 Signal and System I Analysis and characterization of the LTI system using the Laplace transform Causal ROC associate with a causal system is a right-half plane for t<0 Right-side Converse Causal ROC: right half plane Causal ROC: right half plane Unless rational

2. 16.362 Signal and System I Converse Causal ROC: right half plane Unless rational Example ROC: s>-1 Not causal

3. 16.362 Signal and System I Analysis and characterization of the LTI system using the Laplace transform Causal ROC associate with a causal system is a right-half plane for t<0 Right-side then converse Causal ROC: right half plane Causal ROC: right half plane If rational

4. 16.362 Signal and System I Example ROC: -1<s<1 Not causal ROC is not to right of the rightmost pole.

5. 16.362 Signal and System I Stability Bounded output for EVERY bounded input. System stable If and only if If and only if the ROC of H(s) contains entire j  axis, i.e. Re(s) = 0. If the system stable, then ROC contains the j  axis.

6. 16.362 Signal and System I Stability Bounded output for EVERY bounded input. System stable If and only if If and only if the ROC of H(s) contains entire j  axis, i.e. Re(s) = 0. for any 

7. 16.362 Signal and System I Example 2 2 2

8. 16.362 Signal and System I A causal LTI system with rational H(s) is stable if and only if all poles of H(s) lie in the left-half of the s-plane, i.e. all poles have negative real parts If: all poles of H(s) lie in the left-half of the s-plane ROC associate with a causal system is a right-half plane ROC includes Re(s) = 0Stable Only if: all poles have negative real parts Stable ROC includes Re(s) = 0+ causal No poles in the right-half s-plane

9. 16.362 Signal and System I Example Not stable 2

10. 16.362 Signal and System I c1 c2 Not stable

11. 16.362 Signal and System I LTI Systems characterized by linear constant-coefficient differential equations CasualROC to the right most pole ROC Re(s) >-3

12. 16.362 Signal and System I LTI Systems characterized by linear constant-coefficient differential equations Always rational Zeros: Poles: Initial rest conditionCausalROC: to the rightmost pole Causal and stable:ROC: to the rightmost pole & include the Re(s) = 0.

13. 16.362 Signal and System I Example Causal and stable:ROC: to the rightmost pole & include the Re(s) = 0. Re(s)<0 R L C

14. 16.362 Signal and System I Example Causal and stable:ROC: to the rightmost pole & include the Re(s) = 0. Re(s)<0

15. 16.362 Signal and System I Example: an LTI system 1. causal 2. H(s) is rational and has two poles, at s = -2, and s = 4 3. If x(t) =1, then y(t) = 0. 4. The value of the impulse response at t = 0+ is 4.

16. 16.362 Signal and System I Example: an LTI system 1. Causal & stable 2. H(s) is rational and has one pole, at s = -2, and doesn’t have a zero at the origin. 3. The locations of other poles and zeros are unknown. converge   ROC contain -3 is the impulse response of a causal and stable system Same ROC Stable ROC contains j  axis Stable

17. 16.362 Signal and System I Example: an LTI system 1. Causal & stable 2. H(s) is rational and has one pole, at s = -2, and doesn’t have a zero at the origin. 3. The locations of other poles and zeros are unknown.  Has pole at s = 2Not stable ROC contains whole s-plane No sufficient info. contains at least one pole in its Laplace transform has finite duration 

18. 16.362 Signal and System I Butterworth filters Restrict the impulse response of the Butterworth filter is real Roots:

19. 16.362 Signal and System I Butterworth filters Roots: Poles appear in pairs Restrict B(s) causal and stable Poles are in the left-half plane

20. 16.362 Signal and System I Butterworth filters Roots: N=1 Restrict B(s) causal and stable Poles are in the left-half plane x x N=2 x x x x

21. 16.362 Signal and System I Roots: N=3 x x x x x x N=4 x x x x x x x x