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

2. Leo Lam © 2010-2013 Today’s menu Lab – detailed arrangements Homework – vacation week From yesterday (Intro: Signals) Intro: Systems More: Describing Common Signals

3. Taking a signal apart Leo Lam © 2010-2013 a0a0 T t (seconds) A+a 0 A sound signal Offset (atmospheric pressure) Frequency Amplitude

4. Frequency Leo Lam © 2010-2013 196 t (seconds)f (Hz) = time-domainfrequency-domain

5. t to f Leo Lam © 2010-2013 293.66 t (seconds) 196 440 659.26 F (Hz)

6. Combining signals Leo Lam © 2010-2013

7. Summary: Signals Signals carry information Signals represented by functions over time or space Signals can be represented in both time and frequency domains Signals can be summed in both time and frequency domains Leo Lam © 2010-2013

8. Systems A system describes a relationship between input and output Examples? Leo Lam © 2010-2013 v(t)y(t)g(t)

9. Definition: System A system modifies signals or extracts information. It can be considered a transformation that operates on a signal. Leo Lam © 2010-2013

10. Motivation: Complex systems Leo Lam © 2010-2013

11. Filters All kinds, and everywhere Leo Lam © 2010-2013

12. Surprising high pass Leo Lam © 2010-2013

13. Summary: System System transforms an input to an output System can extract information System can “shape” signals (filters) Leo Lam © 2010-2013

14. Signals: Digging in Types of signals Some “standard” signals (alphabets!) Leo Lam © 2010-2013

15. 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 © 2010-2013

16. Signal types Continuous time / Discrete time –An x-axis relationship Discrete time = “indexed” time Leo Lam © 2010-2013

17. Signals: Notations A continuous time signal is specified at all values of time, when time is a real number. Leo Lam © 2010-2013

18. Signals: Notations A discrete time signal is specified at only discrete values of time (e.g. only on integers) Leo Lam © 2010-2013

19. What types are these? Leo Lam © 2010-2013 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)

20. Analog / Digital values (y-axis) An analog signal has amplitude that can take any value in a continuous interval (all Real numbers) Leo Lam © 2010-2013 Where Z is a finite set of values

21. 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 © 2010-2013 Where Z and G are finite sets of values

22. 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 © 2010-2013

23. 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 © 2010-2011

24. Common signals Building blocks to bigger things Leo Lam © 2010-2011 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

25. Sinusoids/Decaying sinusoids Leo Lam © 2010-2011

26. Decaying and growing Leo Lam © 2010-2011

27. Generalizing the sinusoids Leo Lam © 2010-2011 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?

28. Imaginary signals Leo Lam © 2010-2011 z r a b z=a+jb real/imaginary z=re jΦ magnitude/phase  real imag Remember how to convert between the two?

29. Describing signals Of interest? –Peak value –+/- time? –Complex? Magnitude, phase, real, imaginary parts? –Periodic? –Total energy? –Power? Leo Lam © 2010-2011 0 s(t) t Time averaged

30. 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 © 2010-2011 For all integers n

31. Periodic signals: examples Sinusoids Complex exponential (non-decaying or increasing) Infinite sum of shifted signals v(t) (more later) Leo Lam © 2010-2011 x(t)=A cos(   t+  ) T0T0