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

3. Leo Lam © 2010-2013 Today’s menu From yesterday (Signals x and y relationships) More: Describing Common Signals Periodicity

4. Common signals Building blocks to bigger things Leo Lam © 2010-2013 constant signal t a 0 unit step signal t 1 0 unit ramp signal t 1 u(t)=0 for t<0 u(t)=1 for t≥0 r(t)=0 for t<0 r(t)=t for t≥0 r(t)=t*u(t) for t≥0

5. Sinusoids/Decaying sinusoids Leo Lam © 2010-2013

6. Decaying and growing Leo Lam © 2010-2013

7. Generalizing the sinusoids Leo Lam © 2010-2013 General form: x(t)=Ce at, a=σ+jω Equivalently: x(t)=Ce σt e jωt Remember Euler’s Formula? x(t)=Ce σt e jωt amplitude Exponential (3 types) Sinusoidal with frequency ω (in radians) What is the frequency in Hz?

8. Imaginary signals Leo Lam © 2010-2013 z r a b z=a+jb real/imaginary z=re jΦ magnitude/phase  real imag Remember how to convert between the two?

9. Describing signals Of interest? –Peak value –+/- time? –Complex? Magnitude, phase, real, imaginary parts? –Periodic? –Total energy? –Power? Leo Lam © 2010-2013 0 s(t) t Time averaged

10. Periodic signals Definition: x(t) is periodic if there exists a T (time period) such that: The minimum T is the period Fundamental frequency f 0 =1/T Leo Lam © 2010-2013 For all integers n

11. Periodic signals: examples Sinusoids Complex exponential (non-decaying or increasing) Infinite sum of shifted signals v(t) (more later) Leo Lam © 2010-2013 x(t)=A cos(   t+  ) T0T0

12. Periodicity of the sum of periodic signals Question: If x 1 (t) is periodic with period T 1 and x 2 (t) is periodic with period T 2 –What is the period of x 1 (t)+x 2 (t)? Can we rephrase this using our “language” in math? Leo Lam © 2010-2013

13. Rephrasing in math Leo Lam © 2010-2013 Goal: find T such that

14. Rephrasing in math Leo Lam © 2010-2013 Goal: find T such that Need:  T=LCM(T 1,T 2 ) Solve it for r=1, true for all r

15. Periodic sum example If x 1 (t) has T 1 =2 and x 2 (t) has T 2 =3, what is the period of their sum, z(t)? LCM (2,3) is 6 And you can see it, too. Leo Lam © 2010-2013 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 1 2 1 1 T 1 T 2

16. Your turn! Find the period of: Leo Lam © 2010-2013 No LCM exists! Why?

17. A few more Leo Lam © 2010-2013 Not rational, so not periodic Decaying term means pattern does not repeat exactly, so not periodic

18. Summary Description of common signals Periodicity Leo Lam © 2010-2013

19. Playing with signals Operations with signals –add, subtract, multiply, divide signals pointwise –time delay, scaling, reversal Properties of signals (cont.) –even and odd Leo Lam © 2010-2013

20. Adding signals Leo Lam © 2010-2013 1 1 123 t t + = ?? x(t) y(t) 1 t 123 x(t)+y(t)

21. Delay signals Leo Lam © 2010-2013 unit pulse signal t 0 1 1 What does y(t)=p(t-3) look like? P(t) 0 34

22. Multiply signals Leo Lam © 2010-2013

23. Scaling time Leo Lam © 2010-2013 Speed-up: y(t)=x(2t) is x(t) sped up by a factor of 2 t 0 1 1 t 0 1.5 y(t)=x(2t) How could you slow x(t) down by a factor of 2? y(t)=x(t)

24. Scaling time Leo Lam © 2010-2013 y(t)=x(t/2) is x(t) slowed down by a factor of 2 t 01 t 01 y(t)=x(t/2) 2 -2 y(t)=x(t)

25. Playing with signals Leo Lam © 2010-2013 What is y(t) in terms of the unit pulse p(t)? t 8 3 5 t 0 1 1 Need: 1.Wider (x-axis) factor of 2 2.Taller (y-axis) factor of 8 3.Delayed (x-axis) 3 seconds

26. Playing with signals Leo Lam © 2010-2013 t 8 3 5 in terms of unit pulse p(t) t 8 2 first step: 3 5 t 8 second step:

27. Playing with signals Leo Lam © 2010-2013 t 8 3 5 in terms of unit pulse p(t) t 8 2 first step: 3 5 t 8 second step: replace t by t-3: Is it correct?

28. Playing with signals Leo Lam © 2010-2013 3 5 t 8 Double-check: pulse starts: pulse ends:

29. Do it in reverse Leo Lam © 2010-2013 t Sketch 1

30. Do it in reverse Leo Lam © 2010-2013 t Let then that is, y(t) is a delayed pulse p(t-3) sped up by 3. 1 1 4/3 1 3 4 Double-check pulse starts: 3t-3 = 0 pulse ends: 3t-3=1

31. Order matters Leo Lam © 2010-2013 With time operations, order matters y(t)=x(at+b) can be found by: Shift by b then scale by a (delay signal by b, then speed it up by a) w(t)=x(t+b)  y(t)=w(at)=x(at+b) Scale by a then shift by b/a w(t)=x(at)  y(t)=w(t+b/a)=x(a(t+b/a))=x(at+b)

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