ENGG 330

1. ENGG 330 Class 2 Concepts, Definitions, and Basic Properties

2. Quiz • What is the difference between • Stem & Plot • How do I specify a discrete sample space from 0 to 10 • How do I multiply a scalar times a matrix • How do I express e3[n]

3. Remember • Real world signals are very complex • Can’t hope to model them • Can model simple signals • Can tell a lot about systems with simple signals • Can model complex signals with, dare I say, transformations of simple signals

4. Transformations of the Independent Variable • Example Transformations • Periodic Signals • Even and Odd Signals

5. Transformations of Signals • A central concept is transforming a signal by the system • An audio system transforms the signal from a tape deck

6. Example Transformations • Time Shift – Radar, Sonar, Seismic • x[n-n0] & x(t-t0) • Notice a difference? n for D-T, t for C-T • Delayed if t0 positive, Advanced if t0 negative • Time Reversal – tape played backwards • x[n] becomes x[-n] by reflection about n = 0 • Time Scaling – tape played slower/faster • x(t), x(2t), x(t/2)

7. Time Shift t0 < 0 so x(t-t0) is an advanced version of x(t)

8. Time Reversal

9. Time Scaling

10. ? What does x(t+1) look like? Th e other way – t + 1 +1 advanced in time When t = -2 t+1 = -1 what is x(t) at –1? 0 When t = -1 t+1 = 0 what is x(t) at 0? 1 When t = 0 t+1 = 1 what is x(t) at 1? 1 When t = 1 t+1 = 2 what is x(t) at 2? 0

11. Given x(t) what would x(t-1) look like?

12. ? What does x(-t+1) look like? When t = -1 -t+1 = 2 what is x(t) at 2? 0 When t = 0 -t+1 = 1 what is x(t) at 1? 1 When t = 1 -t+1 = 0 what is x(t) at 0? 1 When t = 2 -t+1 = -1 what is x(t) at –1? 0

13. The other wayx(-t + 1) Apply the +1 time shift Apply the –t reflection about the y axis

14. ? What does x(3 /2t) look like? When t = -1 3t/2 = -3/2 what is x(t) at -3/2? 0 When t = 0 3t/2 = 0 what is x(t) at 0? 1 When t = 1 3t/2 = 3/2 what is x(t) at 3/2? ? When t = 2/3 3t/2 = 1 what is x(t) at 1? 1 Why 2/3? What is the next t that should be evaluated? 4/3 why?

15. ? What does look like? First apply the +1 and advance the signal Next apply the 3t/2 and compress the signal

16. Signal Transformations • X(at + b) where a and b are given numbers • Linearly Stretched if |a| < 1 • Linearly Compressed if |a| > 1 • Reversed if a < 0 • Shifted in time if b is nonzero • Advanced in time if b > 0 • Delayed in time if b < 0 • But watch out for x(-2t/3 + 1)

17. Periodic Signals • x(t) = x(t + T) x(t) periodic with period T • x[n] = x[n + N] periodic with period N • Fundamental period T or N • Aperiodic

18. Even and Odd Signals • Even signals • x(-t) = x(t) • x[-n] = x[n] • Odd signals • x(-t) = -x(t) • x[-n] = -x[n] • Must be 0 at t = 0 or n = 0

19. Any signal can be broken into a sum of two signals on even and one odd • Ev{x(t)} = ½[x(t) + x(-t)] • Od{x(t)} = ½[x(t) – x(-t)]

20. Exponential and Sinusoidal Signals • C-T Complex Exponential and Sinusoidal Signals • D-T Complex Exponential and Sinusoidal Signals • Periodicity Properties of D-T Complex Exponentials

21. C-T Complex Exponential and Sinusoidal Signals • x(t) = Ceat where C and a are complex numbers • Complex number • a + jb – rectangular form • Rejθ – polar form • Depending on Values of C and a Complex Exponentials exhibit different characteristics • Real Exponential Signals • Periodic Complex Exponential and Sinusoidal Signals • General Complex Exponential Signals

22. Real Exponential Signals • If C and a are real • x(t) = Ceat then called real exponential • If a is positive x(t) is a growing exponential • If a is negative x(t) is a decaying exponential • If a 0 x(t) is a constant • That depends upon the value of C • Use MATLAB to plot • e2n, e-2n , e0n , 3e0n

23. Periodic Complex Exponential and Sinusoidal Signals • If a is purely imaginary • x(t) is then periodic • x(t) = ejw0t – Plot via MATLAB • ? j is needed to make a imaginary • a closely related signal is Sinusoid

24. General Complex Exponential Signals • Most general case of complex exponential • Can be expressed in terms of the two cases we have examined so far

25. Periodicity Properties of D-T Complex Exponentials

26. Unit Impulse and Unit Step Functions • D-T Unit Impulse and Unit Step Functions • C-T Unit Impulse and Unit Step Functions

27. C-T & D-T Systems • Simple Examples

28. Basic System Properties • Memory • Inverse • Causality • Stability • Time Invariance • Linearity

29. Memory • Memoryless output for each value of independent variable is dependent on the input at only that same time • Memoryless • y(t) = x(t), y[n]= 2x[n] – x2[2n] • Memory • Y[n] = Σx[k], y[n] = x[n-1]

30. Inverse • Invertible if distinct inputs lead to distinct outputs • Think of an encoding system • It must be invertible • Think of a JPEG compression system • It isn’t invertible

31. Causality • A system is causal if the output at any time depends on values of the input at only present and past times. • See Fowler Note Set 5 System Properties

32. Stability • If the input to a stable system is bounded the the output must also be bounded • Balanced stick • Slight push is bounded • Is the output bounded

33. Time Invariance • See Fowler Note Set 5 System Properties

34. Linearity • See Fowler Note Set 5 System Properties

35. Assignment • Read Chapter 1 of Oppenheim • Generate math questions for Dr. Olson • Buck • Section 1.2 a, b, c, d • Section 1.3 a, b, c • Section 1.4 a, b • Turn in .m files • All plots/stems need titles and xy labels • Answers to questions documented in .m file with references to plots/stems