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

2. An e x and a Constant were… …walking down the street; when a Differentiator walked up to them. Constant started running away, and e x asked him, “what are you doing?!” Constant replied, “If I meet a Differentiator, I will disappear!” e x said, proudly, “I don’t care, I am e x !”, and walked up to the Differentiator. “Hi I am e x,” he said, thumbing his nose… “Hi,” said the Differentiator, “I’m d/dy.” Leo Lam © 2010-2012

3. Today’s menu Textbook Chapter 1, Schaum’s Chapter 1 To do: –Sign up to Facebook Group –Bookmark our website From yesterday: definitions End of hand-waving Describing Common Signals –Type of signals –Some standard signals Periodicity

4. Signals: A signal is a mathematical function –x(t) –x is the value (real, complex)  y-axis –t is the independent variable (1D, 2D etc.)  x-axis –Both can be Continuous or Discrete –Examples of x… Leo Lam © 2010-2012

5. Signal types Continuous time / Discrete time –An x-axis relationship Discrete time = “indexed” time Leo Lam © 2010-2012

6. Signals: Notations A continuous time signal is specified at all values of time, when time is a real number. Leo Lam © 2010-2012

7. Signals: Notations A discrete time signal is specified at only discrete values of time (e.g. only on integers) Leo Lam © 2010-2012

8. What types are these? Leo Lam © 2010-2012 1)90.3 FM radio transmitted signal 2)Daily count of orcas in Puget Sound 3)Muscle contraction of your heart over time 4)A capacitor’s charge over time 5)A picture taken by a digital camera 6)Local news broadcast to your old TV 7)Video on YouTube 8)Your voice (c) ((c)) (c) (continuous) (c) (d) (discrete)

9. Analog / Digital values (y-axis) An analog signal has amplitude that can take any value in a continuous interval (all Real numbers) Leo Lam © 2010-2012 Where Z is a finite set of values

10. Analog / Digital values (y-axis) An digital signal has amplitude that can only take on only a discrete set of values (any arbitrary set). Leo Lam © 2010-2012 Where Z and G are finite sets of values

11. Nature vs. Artificial Natural signals mostly analog Computers/gadgets usually digital (today) Signal can be continuous in time but discrete in value (a continuous time, digital signal) Leo Lam © 2010-2012

12. Brake! X-axis: continuous and discrete Y-axis: continuous (analog) and discrete (digital) Our class: (mostly) Continuous time, analog values (real and complex) Clear so far? Leo Lam © 2010-2012

13. Common signals (memorize) Building blocks to bigger things Leo Lam © 2010-2012 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

14. Sinusoids/Decaying sinusoids Leo Lam © 2010-2012

15. Decaying and growing Leo Lam © 2010-2012

16. Generalizing the sinusoids Leo Lam © 2010-2012 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?

17. Imaginary signals Leo Lam © 2010-2012 z r a b z=a+jb real/imaginary z=re jΦ magnitude/phase  real imag Remember how to convert between the two?

18. 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 © 2010-2012 For all integers n

19. Periodic signals: examples Sinusoids Complex exponential (non-decaying or increasing) Infinite sum of shifted signals v(t) (more later) Leo Lam © 2010-2012 x(t)=A cos(   t+  ) T0T0

20. 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 © 2010-2012

21. Rephrasing in math Leo Lam © 2010-2012 Goal: find T such that

22. Rephrasing in math Leo Lam © 2010-2012 Goal: find T such that Need:  T=LCM(T 1,T 2 ) Solve it for r=1, true for all r

23. 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 © 2010-2012 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 1 2 1 1 T 1 T 2

24. Your turn! Find the period of: Leo Lam © 2010-2012 No LCM exists! Why? Because LCM exists only if T 1 /T 2 is a rational number

25. A few more Leo Lam © 2010-2011 Not rational, so not periodic Decaying term means pattern does not repeat exactly, so not periodic

26. Summary Description of common signals Periodicity Leo Lam © 2010-2011