1. Leo Lam © 2010-2013 Signals and Systems EE235 Leo Lam

2. Leo Lam © 2010-2013 Today’s menu From Wednesday: Manipulating signals How was Lab 1? To Do: Really memorize u(t), r(t), p(t) Today: More of that! Even and odd signals Dirac Delta function

3. People types There are 10 types of people in the world: Those who know binary and those who don’t. Leo Lam © 2010-2013

4. Playing with time Leo Lam © 2010-2013 t 1 What does look like? 2 1 -2 Time reverse of speech: Also a form of time scaling, only with a negative number

5. Playing with time Leo Lam © 2010-2013 t 1 2 Describe z(t) in terms of w(t) 1 -213 t

6. Playing with time Leo Lam © 2010-2013 time reverse it: x(t) = w(-t) delay it by 3: z(t) = x(t-3) so z(t) = w(-(t-3)) = w(-t + 3) t 1 2 1 -21 3 x(t) you replaced the t in x(t) by t-3. so replace the t in w(t) by t-3: x(t-3) = w(-(t-3))

7. Playing with time Leo Lam © 2010-2013 z(t) = w(-t + 3) t 1 2 1 -21 3 x(t) Doublecheck: w(t) starts at 0 so -t+3 = 0 gives t= 3, this is the start (tip) of the triangle z(t). w(t) ends at 2 So -t+3=2 gives t=1, z(t) ends there

8. Summary: Arithmetic: Add, subtract, multiple Time: delay, scaling, shift, mirror/reverse And combination of those Leo Lam © 2010-2013

9. Even and odd signals Leo Lam © 2010-2013 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:

10. Even and odd signals Leo Lam © 2010-2013 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:

11. Even and odd signals Leo Lam © 2010-2013 Euler’s relation: What are the even and odd parts of Even part Odd part

12. Summary: Even and odd signals Breakdown of any signals to the even and odd components Leo Lam © 2010-2013

13. Delta function δ(t) Leo Lam © 2010-2013 “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”)

14. Delta function δ(t) Leo Lam © 2010-2013 Each rectangle has area 1, shrinking width, growing height ---limit is (t)

15. Dirac Delta function δ(t) Leo Lam © 2010-2013 “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 :

16. Dirac Delta example Evaluate Leo Lam © 2010-2013 = 0. Because δ(t)=0 for all t≠0

17. Dirac Delta – Your turn Evaluate Leo Lam © 2010-2013 = 1. Why? Change of variable: 1

18. Dirac Delta – Another one Evaluate Leo Lam © 2010-2013

19. Is this function periodic? If so, what is the period? (Sketch to prove your answer) Slightly harder Leo Lam © 2010-2013 Not periodic – delta function spreads with k 2 for t>0 And x(t) = 0 for t<0