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 e 3[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-n 0 ] & x(t-t 0 ) Notice a difference? n for D-T, t for C-T –Delayed if t 0 positive, Advanced if t 0 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 t 0 < 0 so x(t-t 0 ) is an advanced version of x(t)

8. Time Reversal

9. Time Scaling

10. ? What does x(t+1) look like? 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 Th e other way – t + 1 +1 advanced in time

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 way x(-t + 1) Apply the +1 time shift Apply the –t reflection about the y axis

14. ? What does x( 3 /2 t) 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? Next apply the 3t/2 and compress the signal First apply the +1 and advance 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) = Ce at where C and a are complex numbers –Complex number a + jb – rectangular form Re jθ – 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) = Ce at 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 –e 2n, e -2n, e 0n, 3e 0n

23. Periodic Complex Exponential and Sinusoidal Signals If a is purely imaginary –x(t) is then periodic x(t) = e jw 0 t – 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] – x 2 [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