1. EE3010_Lecture2 Al-Dhaifallah_Term332 1 2. Introduction to Signal and Systems Dr. Mujahed Al-Dhaifallah EE3010: Signals and Systems Analysis Term 332

2. EE3010_Lecture2Al-Dhaifallah_Term3322 Dr. Mujahed Al-Dhaifallah د. مجاهد آل ضيف الله Office: Dean Office. E-mail: muja2007hed@gmail.com Telephone: 7842983 Office Hours: SMT, 1:30 – 2:30 PM, or by appointment

3. EE3010_Lecture2Al-Dhaifallah_Term3323 Rules and Regulations  No make up quizzes  DN grade == 25% unexcused absences  Homework Assignments are due to the beginning of the lectures.  Absence is not an excuse for not submitting the Homework.

4. Grading Policy Exam 1 (10%), Exam 2 (15%) Final Exam (60%), Quizzes (5%) HWs (5%) Attendance & class participation (5%), penalty for late attendance Note: No absence, late homework submission allowed without genuine excuse. EE3010_Lecture2Al-Dhaifallah_Term3324

5. Homework Send me e-mail Subject Line: “EE 3010 Student” EE3010_Lecture2Al-Dhaifallah_Term3325

6. EE3010_Lecture2Al-Dhaifallah_Term3326 The Course Goal To introduce the mathematical tools for analysing signals and systems in the time and frequency domain and to provide a basis for applying these techniques in electrical engineering.

7. EE3010_Lecture2Al-Dhaifallah_Term3327 Course Objectives 1. Identify the types of signals and their characterization. 2. Use the Fourier series representation. 3. Differentiate between the continuous and discrete- time Fourier transforms. 4. Grasp the fundamental concepts of the Laplace and Z transforms. 5. Characterize signals and systems in the frequency domain. 6. Apply signals and systems concepts in various engineering applications.

8. EE3010_Lecture2Al-Dhaifallah_Term3328 Course Syllabus 1.Signal and Systems : Introduction, Continuous and discrete-time signals, Basic system properties. 2.Linear Time-Invariant (LTI) Systems: Convolution, LTI systems properties, Continuous and discrete-time LTI causal systems. 3.Fourier series Representation of Periodic Systems: LTI system response to complex exponentials, Properties of Fourier series, Applications to filtering, Examples of filters.

9. EE3010_Lecture2Al-Dhaifallah_Term3329 Course Outlines 4.Continuous-Time Fourier Transform: Fourier transform of aperiodic and periodic signals, Properties, Convolution and multiplication properties, Frequency response of LTI systems. 5.Discrete-Time Fourier Transform: Overview of Discrete-time equivalents of topics covered in chapter 4.

10. EE3010_Lecture2Al-Dhaifallah_Term33210 Course Outlines 6.Laplace transform (Laplace transform as Fourier transform with convergence factor. Properties of the Laplace transform 7.z transform. Properties of the z transform. Examples. Difference equations and differential equations. Digital filters.

11. Signals & Systems Concepts Specific Objectives: Introduce, using examples, what is a signal and what is a system Why mathematical models are appropriate What are continuous-time and discrete-time representations and how are they related EE3010_Lecture2Al-Dhaifallah_Term33211

12. Recommended Reading Material Signals and Systems, Oppenheim & Willsky, Section 1 Signals and Systems, Haykin & Van Veen, Section 1 EE3010_Lecture2Al-Dhaifallah_Term33212

13. What is a Signal? Signals are functions that carry information. Such information is contained in a pattern of variation of some form. Examples of signal include: Electrical signals – Voltages and currents in a circuit Acoustic signals – Acoustic pressure (sound) over time Mechanical signals – Velocity of a car over time Video signals – Intensity level of a pixel (camera, video) over time EE3010_Lecture2Al-Dhaifallah_Term33213

14. How is a Signal Represented? Mathematically, signals are represented as a function of one or more independent variables. For instance a black & white video signal intensity is dependent on x, y coordinates and time t f(x,y,t) In this course, we shall be exclusively concerned with signals that are a function of a single variable: time EE3010_Lecture2Al-Dhaifallah_Term33214 t f(t)f(t)

15. Example: Signals in an Electrical Circuit EE3010_Lecture2Al-Dhaifallah_Term33215 +-+- i vcvc vsvs R C The signals v c and v s are patterns of variation over time Note, we could also have considered the voltage across the resistor or the current as signals Step (signal) v s at t=1 RC = 1 First order (exponential) response for v c v s, v c

16. Continuous & Discrete-Time Signals EE3010_Lecture2Al-Dhaifallah_Term33216 Continuous-Time Signals Most signals in the real world are continuous time, as the scale is infinitesimally fine. Eg voltage, velocity, Denote by x(t), where the time interval may be bounded (finite) or infinite 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 Sampled continuous signal x[n] =x(nk) – k is sample time x(t)x(t) t x[n]x[n] n

17. Signal Properties In this course, we shall be particularly interested in signals with certain properties: Periodic signals: a signal is periodic if it repeats itself after a fixed period T, i.e. x(t) = x(t+T) for all t. A sin(t) signal is periodic. Even and odd signals: a signal is even if x(-t) = x(t) (i.e. it can be reflected in the axis at zero). A signal is odd if x(-t) = -x(t). Examples are cos(t) and sin(t) signals, respectively EE3010_Lecture2Al-Dhaifallah_Term33217

18. Signal Properties Exponential and sinusoidal signals: a signal is (real) exponential if it can be represented as x(t) = Ce at. A signal is (complex) exponential if it can be represented in the same form but C and a are complex numbers. Step and pulse signals: A pulse signal is one which is nearly completely zero, apart from a short spike,  (t). A step signal is zero up to a certain time, and then a constant value after that time, u(t). These properties define a large class of tractable, useful signals and will be further considered in the coming lectures EE3010_Lecture2Al-Dhaifallah_Term33218

19. What is a System? Systems process input signals to produce output signals Examples: A circuit involving a capacitor can be viewed as a system that transforms the source voltage (signal) to the voltage (signal) across the capacitor A CD player takes the signal on the CD and transforms it into a signal sent to the loud speaker EE3010_Lecture2Al-Dhaifallah_Term33219

20. Examples A communication system is generally composed of three sub-systems, the transmitter, the channel and the receiver. The channel typically attenuates and adds noise to the transmitted signal which must be processed by the receiver EE3010_Lecture2Al-Dhaifallah_Term33220

21. How is a System Represented? A system takes a signal as an input and transforms it into another signal In a very broad sense, a system can be represented as the ratio of the output signal over the input signal That way, when we “multiply” the system by the input signal, we get the output signal This concept will be firmed up in the coming weeks EE3010_Lecture2Al-Dhaifallah_Term33221 System Input signal x(t) Output signal y(t)

22. Continuous & Discrete-Time Mathematical Models of Systems EE3010_Lecture2Al-Dhaifallah_Term33222 Continuous-Time Systems Most continuous time systems represent how continuous signals are transformed via differential equations. E.g. circuit, car velocity Discrete-Time Systems Most discrete time systems represent how discrete signals are transformed via difference equations E.g. bank account, discrete car velocity system First order differential equations First order difference equations

23. Properties of a System In this course, we shall be particularly interested in systems with certain properties: Causal: a system is causal if the output at a time, only depends on input values up to that time. Linear: a system is linear if the output of the scaled sum of two input signals is the equivalent scaled sum of outputs EE3010_Lecture2Al-Dhaifallah_Term33223

24. Properties of a System Time-invariance: a system is time invariant if the system’s output signal is the same, given the same input signal, regardless of time of application. These properties define a large class of tractable, useful systems and will be further considered in the coming lectures EE3010_Lecture2Al-Dhaifallah_Term33224

25. How Are Signal & Systems Related (i)? EE3010_Lecture2Al-Dhaifallah_Term33225 How to design a system to process a signal in particular ways? Design a system to restore or enhance a particular signal – Remove high frequency background communication noise – Enhance noisy images from spacecraft Assume a signal is represented as x(t) = d(t) + n(t) Design a system to remove the unknown “noise” component n(t), so that y(t)  d(t) System ? x(t) = d(t) + n(t) y(t)  d(t)

26. How Are Signal & Systems Related (ii)? How to design a system to extract specific pieces of information from signals – Estimate the heart rate from an electrocardiogram – Estimate economic indicators (bear, bull) from stock market values Assume a signal is represented as x(t) = g(d(t)) Design a system to “invert” the transformation g(), so that y(t) = d(t) EE3010_Lecture2Al-Dhaifallah_Term33226 System ? x(t) = g(d(t))y(t) = d(t) = g -1 (x(t))

27. EE3010_Lecture2Al-Dhaifallah_Term33227 How Are Signal & Systems Related (iii)? How to design a (dynamic) system to modify or control the output of another (dynamic) system – Control an aircraft’s altitude, velocity, heading by adjusting throttle, rudder, ailerons – Control the temperature of a building by adjusting the heating/cooling energy flow. Assume a signal is represented as x(t) = g(d(t)) Design a system to “invert” the transformation g(), so that y(t) = d(t) dynamic system ? x(t)x(t) y(t) = d(t)

28. EE3010_Lecture2Al-Dhaifallah_Term33228 Lecture 2: Exercises Read SaS OW, Chapter 1. This contains most of the material in the first three lectures, a bit of pre-reading will be extremely useful! SaS OW: Q1.1 Q1.2 Q1.4 Q1.5 Q1.6 In lecture 3, we’ll be looking at signals in more depth.

29. EE3010_Lecture2 Al-Dhaifallah_Term332 A1. Review of Complex Numbers 29

30. EE3010_Lecture2Al-Dhaifallah_Term332 Complex Numbers Complex numbers: number of the form z = x + j y where x and y are real numbers and x: real part of z; x = Re {z} y: imaginary part of z; y = Im {z} 30

31. EE3010_Lecture2Al-Dhaifallah_Term332 Complex Numbers 31

32. EE3010_Lecture2Al-Dhaifallah_Term332 Representing Complex numbers Rectangular representation Complex-plane (s-plane) s 1 =x + j y Imaginary axis real axis Imaginary Part y x Real part 32

33. EE3010_Lecture2Al-Dhaifallah_Term332 Representing Complex numbers Polar representation Complex-plane (s-plane) Imaginary axis real axis Imaginary Part y x Real part ρθρθ 33

34. EE3010_Lecture2Al-Dhaifallah_Term332 Conversion between Representations Example Imaginary axis real axis 4 3 θ 34

35. EE3010_Lecture2Al-Dhaifallah_Term332 Euler Formula 4 3 θ 35

36. EE3010_Lecture2Al-Dhaifallah_Term332 Complex Numbers Addition /Subtraction 36

37. EE3010_Lecture2Al-Dhaifallah_Term332 Complex Numbers Multiplication/Division 37

38. EE3010_Lecture2Al-Dhaifallah_Term332 Operations Examples 38

39. EE3010_Lecture2Al-Dhaifallah_Term332 More Examples 39

40. EE3010_Lecture2Al-Dhaifallah_Term332 Conjugate Complex-plane (s-plane) Imaginary axis real axis 40

41. EE3010_Lecture2Al-Dhaifallah_Term332 Conjugate 41

42. EE3010_Lecture2Al-Dhaifallah_Term332 Conjugate 42

43. EE3010_Lecture2Al-Dhaifallah_Term332 Conjugate real/imaginary part 43

44. EE3010_Lecture2Al-Dhaifallah_Term332 Operations Polar coordinate Multiplication/Division 44

45. EE3010_Lecture2Al-Dhaifallah_Term332 More Examples 45

46. EE3010_Lecture2Al-Dhaifallah_Term332 Keywords Conjugate Modulus Real part Imaginary part Polar coordinates Complex plane Imaginary axis Pure imaginary 46