1. Leo Lam © 2010-2012 Signals and Systems EE235

2. People types There are 10 types of people in the world: Those who know binary and those who don’t. Leo Lam © 2010-2012

3. Today’s menu From Friday: –Manipulation of signals –To Do: Really memorize u(t), r(t), p(t) Even and odd signals Dirac Delta Function

4. How to find LCM Factorize and group Your turn: 225 and 270’s LCM Answer: 1350 Leo Lam © 2010-2012

5. Even and odd signals Leo Lam © 2010-2012 An even signal is such that: t Symmetrical across the t=0 axis t Asymmetrical across the t=0 axis An odd signal is such that:

6. Even and odd signals Leo Lam © 2010-2012 Every signal  sum of an odd and even signal. Even signal is such that: The even and odd parts of a signal Odd signal is such that:

7. Even and odd signals Leo Lam © 2010-2012 Euler’s relation: What are the even and odd parts of Even part Odd part

8. Summary: Even and odd signals Breakdown of any signals to the even and odd components Leo Lam © 2010-2012

9. Delta function δ(t) Leo Lam © 2010-2012 “a spike of signal at time 0” 0 The Dirac delta is: The unit impulse or impulse Very useful Not a function, but a “generalized function”)

10. Delta function δ(t) Leo Lam © 2010-2012 Each rectangle has area 1, shrinking width, growing height ---limit is (t)

11. Dirac Delta function δ(t) Leo Lam © 2010-2012 “a spike of signal at time 0” 0 It has height = , width = 0, and area = 1 δ(t) Rules 1.δ(t)=0 for t≠0 2.Area: 3. If x(t) is continuous at t 0, otherwise undefined 0 t0t0 Shifted to time instant t 0 :

12. Dirac Delta example Evaluate Leo Lam © 2010-2012 = 0. Because δ(t)=0 for all t≠0

13. Dirac Delta – Your turn Evaluate Leo Lam © 2010-2012 = 1. Why? Change of variable: 1

14. Scaling the Dirac Delta Proof: Suppose a>0 a<0 Leo Lam © 2010-2011

15. Scaling the Dirac Delta Proof: Generalizing the last result Leo Lam © 2010-2011

16. Multiplication of a function that is continuous at t 0 by δ(t) gives a scaled impulse. Sifting Properties Relation with u(t) Summary: Dirac Delta Function Leo Lam © 2010-2011