Signals & Systems Spring 2009 Instructor: Mariam Shafqat UET Taxila

1. Signals & Systems Spring 2009 Instructor: Mariam Shafqat UET Taxila

2. Today's lecture • The course • Course contents • Recommended books • Course structure • Assessments breakdown • Before we start… • Introduction to signals and systems

3. The Course • Core course • First course in Computer Engineering • A strong foundation for advanced courses and research • What the course is about • Analysis and processing of information • System design for required processing • Mathematical & theoretical • Calculus, Linear Algebra, Differential • Expectations • Extensive and tough

4. Labs Session • Performance criteria: • Performance within the lab • Lab report • Lab report submission after one week • Lab report submission only in the lab • Viva from each individual student from his/her lab report • Announcement of marks obtained by each individual students in the lab at the end of lab session.

5. Course contents • Introduction to Signals and Systems • Sinusoids • Spectrum Representation • Analysis of Periodic Waveforms • Sampling and Aliasing • Z-Transform • Convolution • Frequency response • Fourier Series and Transforms • Continuous-time & Discrete-time Systems

6. Books Signals & Systems (Second Edition)  Text Book by Alan V. Oppenheim, Alan S. Willsky, S. Hamid Nawab Signal Processing First  Reference Book by James H. McClellan, Ronald W. Schafer, Mark A. Yoder

7. Assessments Quizzes 10% Assignments 10% Mid Term 20% Labs 16% Final Exam 40% Attendence 4%

8. Signal • What is a signal • A description of how one parameter is related to another parameter. • Examples • The voltage varies with time v t

9. Signal • The Speech Signal • The ECG Signal

10. Signal • The image

11. Signal • The image

12. Signal • It is the variation pattern that conveys the information, in a signal • Signal may exist in many forms like acoustic, image, video, electrical, heat & light signal

13. System • An entity that responds to a signal • Examples • Circuit system input output

14. System • The camera • The Speech Recognition System Image Identified

15. System • The audio CD-player • Block Diagram representation of a system • Visual representation of a system • Shows inter-relations of many signals involved in the implementation of a complex system system Output Signal Input Signal

16. Mathematical Representation • A signal can be represented as a function of one or more independent variables • Examples t

17. Mathematical Representation • The image is a function of two spatial variables

18. Continuous-Time Signals • Most signals in the real world are continuous time, as the scale is infinitesimally fine. • E.g. voltage, velocity, • Denote by x(t), where the time interval may be bounded (finite) or infinite

19. Continuous-time signals • A value of signal exists at every instant of time Independent variable Independent variable

20. Discrete-Time Signals • Some real world and many digital signals are discrete time, as they are sampled • E.g. pixels, daily stock price (anything that a digital computer processes) • Denote by x[n], where n is an integer value that varies discretely.

21. Discrete-time signals • The value of signal exists only at equally spaced discrete points in time Independent variable Independent variable

22. Discrete-time signals • Why to discretize • How to discretize • How closely spaced are the samples • Distinction between discrete & digital signals • How to denote discrete signals • Is the image a discrete or continuous signal • The image is generally considered to be a continuous variable • Sampling can however be used to obtain a discrete, two dimensional signal (sampled image)

23. Analog vs Digital signals • the difference is with respect to the value of the function (y-axis). • Analog corresponds to a continuous y-axis, while digital corresponds to a discrete y-axis.

24. Notation • A continuous-time signal is represented by enclosing the independent variable (time) in parentheses () • A discrete-time signal is represented by enclosing the independent variable (index) in square brackets []

25. Important Parameters • Signal power • Signal energy

26. Continuous time Signal power • Our usual notion of the energy of a signal is the area under the curve f(t)2.

27. Some further classification of signals

28. Periodic vs Aperiodic signals • Periodic signals repeat with some period T, while aperiodic, or nonperiodic, signals do not. • We can define a periodic function through the following mathematical expression, where t can be any number and T is a positive constant: • f (t) = f (T + t) • The fundamental period of our function, f (t), is the smallest value of T that the still allows equation to be true.

29. Periodic vs Aperiodic signals

30. Causal vs. Anticausal vs. Noncausal • Causal signals are signals that are zero for all negative time, • Anticausal are signals that are zero for all positive time. • Noncausal signals are signals that have nonzero values in both positive and negative time

31. Causal vs. Anticausal vs. Noncausal

32. Even vs. Odd • An even signal is any signal f such that f (t) = f (-t) • Even signals can be easily plotted as they are vertical about the vertical axis. • An odd signal is a signal such that f(t)=-f(t).

33. Even vs. Odd

34. Deterministic vs. Random • Deterministic signal is a signal in which each value of the signal is fixed and can be determined by a mathematical expression, rule, or table. Because of this the future values of the signal can be calculated from past values with complete confidence. • Random signal has a lot of uncertainty about its behavior. The future values of a random signal cannot be accurately predicted and can usually only be guessed based on the averages of sets of signals

35. Deterministic vs. Random

36. Finite vs. Infinite Length • f (t) is a finite-length signal if it is nonzero over a finite interval • t1 < f (t) < t2 • Infinite-length signal, f (t), is defined as nonzero over all real numbers:

37. Signal Operations/Transformations • Signal operations are operations on the time variable of the signal. • Two signal operations are considered • Time shifting/Time reversal • Time scaling

38. Time Shifting • Time shifting is, the shifting of a signal in time. This is done by adding or subtracting the amount of the shift to the time variable in the function. Subtracting a fixed amount from the time variable will shift the signal to the right (delay) that amount, while adding to the time variable will shift the signal to the left (advance). • Delay x(t-2) • Advance x(t+2)

39. Time Shifting

40. Time Shifting

41. Time Scaling

43. Sinusoidal signals • x(t) = A cos(ωt + Φ) • A is the maximum amplitude of the sinusoidal signal • ω is the radian frequency •  is the phase shift

44. Unit impulse function

45. Unit step

46. Unit step

47. Unit Step

48. Continuous time unit step Discontinuous at time t=0

49. Continuous time unit impulse

50. Relation b/w unit step and unit impulse Running integral for t<0 and t>0