1. 1 LTI Systems; Convolution September 4, 2000 EE 64, Section 1 ©Michael R. Gustafson II Pratt School of Engineering

2. 2 Last Time Block diagrams Series, parallel, and feedback Proving linearity Proving time invariance Using unit step and unit ramp functions

3. 3 LTI Systems LTI: Linear, Time Invariant LTI systems are linear - so superposition works LTI systems are time invariant - so their properties do not change with macroscopic shifts in time

4. 4 Properties of LTI Systems 1) For an LTI system, the output from a scaled, shifted unit impulse is simply a scaled, shifted version of the output from a standard unit impulse. 2) Any finite discrete input can be created from a linear combination of shifted and scaled unit impulse functions.

5. 5 Property 1 (Discrete) Mathematically,

6. 6 Property 2 (Discrete) Assuming a signal x[n] is finite, it can be represented:

7. 7 Conclusion (Math) Using these two properties, the following holds true:

8. 8 Conclusion (English) If the response to a unit impulse can be found, then the response to a linear combination of shifted impulses is the same linear combination of shifted impulse responses.

9. 9 Example Find the output for the given system:

10. 10 Property 1 (Continuous) Mathematically,

11. 11 Property 2 (Continuous) Assuming a signal x[n] is finite, it can be represented:

12. 12 Conclusion (Math) Using these two properties, the following holds true:

13. 13 Conclusion (English) If the response to a unit impulse can be found, then the response to a linear combination of shifted impulses is the same linear combination of shifted impulse responses.

14. 14 Example Find the output for the given system:

15. 15 Mathematical Solutions Assume the following system:

16. 16 Mathematical Solutions (cont) Multiply out to get steps multiplied together:

17. 17 Mathematical Solutions (cont) Determine the three requirements generated by the step functions and rewrite the integral:

18. 18 Mathematical Solutions (cont)

19. 19 Mathematical Solutions (cont)

20. 20 Graphical Solutions Plot the graph of the input as a function of . Plot the impulse response using t-  with  as the independent variable. Note how the graph of the impulse response "moves" as t increases. Determine the regions of overlap and proceed from there.

21. 21 Assignment Read Chapter 2 of OW. Finish the homework assignment for Wednesday.

22. 22 Next Time Properties of LTI systems.

23. 23 Questions ?