Signals and Systems.

1. CHAPTER 1 Signals and Systems. EKT 232

2. Signals and Systems. 1.1 What is a Signal ? 1.2 Classification of a Signals. 1.2.1 Continuous-Time and Discrete-Time Signals 1.2.2 Even and Odd Signals. 1.2.3 Periodic and Non-periodic Signals. 1.2.4 Deterministic and Random Signals. 1.2.5 Energy and Power Signals. 1.3 Basic Operation of the Signal. 1.4 Elementary Signals. 1.4.1 Exponential Signals. 1.4.2 Sinusoidal Signal. 1.4.3 Sinusoidal and Complex Exponential Signals. 1.4.4 Exponential Damped Sinusoidal Signals. 1.4.5 Step Function. 1.4.6 Impulse Function. 1.4.7 Ramped Function.

3. Cont’d… 1.5 What is a System ? 1.5.1 System Block Diagram. 1.6 Properties of the System. 1.6.1 Stability. 1.6.2 Memory. 1.6.3 Causality. 1.6.4 Inevitability. 1.6.5 Time Invariance. 1.6.6 Linearity.

4. 1.1 What is a Signal ? • A common form of human communication; (i) use of speechsignal, face to face or telephone channel. (ii) use of visual, signal taking the form of images of people or objects around us. • Real life exampleof signals; (i) Doctor listening to the heartbeat, blood pressure and temperature of the patient. These indicate the state of health of the patient. (ii) Daily fluctuations in the price of stock market will convey an information on the how the share for a company is doing. (iii) Weather forecast provides information on the temperature, humidity, and the speed and direction of the prevailing wind.

5. Cont’d… • By definition,signal is a function of one or more variable, which conveys information on the nature of a physical phenomenon. • A function of time representing a physical or mathematical quantities. e.g. : Velocity, e.g. : Velocity, acceleration of a car, voltage/current of a circuit. An example of signal; the electrical activity of the heart recorded with electrodes on the surface of the chest — the electrocardiogram (ECG or EKG) in the figure below.

6. 1.2 Classifications of a Signal. • There are five types of signals; (i) Continuous-Time and Discrete-Time Signals (ii) Even and Odd Signals. (iii) Periodic and Non-periodic Signals. (iv) Deterministic and Random Signals. (v) Energy and Power Signals.

7. 1.2.1 Continuous-Time and Discrete-Time Signals. Continuous-Time (CT) Signals • Continuous-Time (CT) Signals are functions whose amplitude or value varies continuously with time, x(t). • The symbol tdenotes time for continuous-time signal and ( ) used to denote continuous-time value quantities. • Example, speed of car, converting acoustic or light wave into electrical signal and microphone converts variation in sound pressure into correspond variation in voltage and current. Figure 1.1: Continuous-Time Signal.

8. Cont’d… Discrete-Time Signals • Discrete-Time Signals are function of discrete variable, i.e. they are defined only at discrete instants of time. • It is often derived from continuous-time signal by sampling at uniform rate. Ts denotes sampling period and n denotes integer. • The symbol n denotes time for discrete time signal and [] is used to denote discrete-value quantities. • Example: the value of stock at the end of the month. Figure 1.3: Discrete-Time Signal.

9. 1.2.2 Even and Odd Signals. • A continuous-time signal x(t) is said to be an even signal if • The signal x(t) is said to be an oddsignal if • In summary, an evensignal are symmetric about the vertical axis (time origin) whereas an odd signal are antisymetric about the origin. Figure 1.4: Even Signal Figure 1.5: Odd Signal.

10. Cont’d…

11. Example 1.1: Even and Odd Signals. Find the even and odd components of each of the following signals: • x(t) = 4cos(3πt) Answer: ge(t) = 4cos(3πt) go(t) = 0

12. 1.2.3 Periodic and Non-Periodic Signals. Periodic Signal. • A periodic signal x(t) is a function of time that satisfies the condition where T is a positive constant. • The smallest value of T that satisfy the definition is called a period. Figure 1.6: Aperiodic Signal. Figure 1.7: Periodic Signal.

13. 1.2.4 Deterministic and Random Signals. Deterministic Signal. • A deterministic signal is a signal that is no uncertainty with respect to its value at any time. • The deterministic signal can be modeled as completely specified function of time. Figure 1.8: Deterministic Signal; Square Wave.

14. Cont’d… Random Signal. • A random signal is a signal about which there is uncertainty before it occurs. The signal may be viewed as belonging to an ensemble or a group of signals which each signal in the ensemble having a different waveform. • The signal amplitude fluctuates between positive and negative in a randomly fashion. • Example; noise generated by amplifier of a radio or television. Figure 1.9: Random Signal

15. Energy Signal and Power Signals. A signal with finite signal energy is called an energy signal. A signal with infinite signal energy and finite average signal power is called a power signal.

16. 1.2.5 Energy Signal and Power Signals. Energy Signal. • A signal is refer to energy signal if and only if the total energy satisfy the condition; Power Signal. • A signal is refer to power signal if and only if the average power of signal satisfy the condition;

17. 1.3 Basic Operation of the Signals. 1.3.1 Time Scaling. 1.3.2 Reflection and Folding. 1.3.3 Time Shifting. 1.3.4 Precedence Rule for Time Shifting and Time Scaling.

18. 1.3.1 Time Scaling. • Time scaling refers to the multiplication of the variable by a real positive constant. • If a > 1 the signal y(t) is a compressedversion of x(t). • If 0 < a < 1 the signal y(t) is an expandedversion of x(t). • Example: Figure 1.11: Time-scaling operation; continuous-time signal x(t), (b) version of x(t) compressed by a factor of 2, and (c) version of x(t) expanded by a factor of 2.

19. Cont’d… • In the discrete time, • It is defined for integer value of k, k > 1. Figure below for k = 2, sample for n = +-1, Figure 1.12: Effect of time scaling on a discrete-time signal: (a) discrete-time signal x[n] and (b) version of x[n] compressed by a factor of 2, with some values of the original x[n] lost as a result of the compression.

20. 1.3.2 Reflection and Folding. • Let x(t) denote a continuous-time signal and y(t) is the signal obtained by replacing time t with –t; • y(t) is the signal represents a refracted version of x(t) about t = 0. • Two special casesfor continuous and discrete-time signal; (i) Even signal; x(-t) = x(t) an even signal is same as reflected version. (ii) Odd signal; x(-t) = -x(t) an odd signal is the negative of its reflected version.

21. Example 1.2:Reflection. Given the triangular pulse x(t), find the reflected version of x(t) about the amplitude axis (origin). Solution: Replace the variable t with –t, so we get y(t) = x(-t) as in figure below. Figure 1.13: Operation of reflection: (a) continuous-time signal x(t) and (b) reflected version of x(t) about the origin x(t) = 0 for t < -T1 and t > T2. y(t) = 0 for t > T1 and t< -T2. • .

22. 1.3.3 Time Shifting. • A time shift delayor advances the signal in time by a time interval +t0 or –t0, without changing its shape. y(t) = x(t - t0) • If t0 > 0 the waveform of y(t) is obtained by shifting x(t) toward the right, relative to the tie axis. • If t0< 0, x(t) is shifted to the left. Example: Figure 1.14: Shift to the Left. Figure 1.15: Shift to the Right. Q: How does the x(t) signal looks like?

23. Example 1.3: Time Shifting. Given the rectangular pulse x(t) of unit amplitude and unit duration. Find y(t)=x (t - 2) Solution: t0 is equal to 2 time units. Shift x(t) to the right by 2 time units. Figure 1.16: Time-shifting operation: • continuous-time signal in the form of a rectangular pulse of amplitude 1.0 and duration 1.0, symmetric about the origin; and (b) time-shifted version of x(t) by 2 time shifts. • .

24. 1.3.4 Precedence Rule for Time Shifting and Time Scaling. • Time shiftingoperation is performed first on x(t), which results in • Time shift has replace t in x(t) by t - b. • Time scaling operation is performed on v(t), replacing t by at and resulting in, • Example in real-life: Voice signal recorded on a tape recorder; • (a > 1) tape is played faster than the recording rate, resulted in compression. • (a < 1) tape is played slower than the recording rate, resulted in expansion.

25. Example 1.4:Continuous Signal. A CT signal is shown in Figure 1.17 below, sketch and label each of this signal; a) x(t-1) b) x(2t) c) x(-t) Figure 1.17 x(t) 2 t -1 3

26. x(-t) 2 t -3 1 • Solution: • (a) x(t-1) (b) x(2t) • (c) x(-t) x(t) x(t-1) 2 2 t t 0 4 -1/2 3/2

27. Example 1.5:Continuous Signal. A continuous signal x(t) is shown in Figure 1.17a. Sketch and label each of the following signals. a) x(t)= u(t-1) b) x(t)= [u(t)-u(t-1)] c) x(t)= d(t - 3/2) Solution: Figure 1.17a (a) x(t)= u(t-1) (b) x(t)= [u(t)-u(t-1)] (c) x(t)=d(t - 3/2)

28. x[n] 4 2 0 1 2 3 n Example 1.5: Discrete Time Signal. A discrete-time signal x[n] is shown below, Sketch and label each of the following signal. (a) x[n – 2] (b) x[2n] (c.) x[-n+2] (d) x[-n]

29. Cont’d… x[n-2] 4 2 0 1 2 3 4 5 n (a) A discrete-time signal, x[n-2]. • A delay by 2

30. Cont’d… x(2n) 4 2 0 1 2 3 n (b) A discrete-time signal, x[2n]. Down-sampling by a factor of 2.

31. Cont’d… x(-n+2) 4 2 -1 0 1 2 n (c) A discrete-time signal, x[-n+2]. Time reversal and shifting

32. Cont’d… x(-n) 4 2 -3 -2 -1 0 1 n (d) A discrete-time signal, x[-n]. • Time reversal

33. x(t) 4 0 4 t In Class Exercises . A continuous-time signal x(t) is shown below, Sketch and label each of the following signal (a) x(t – 2) (b) x(2t) (c.) x(t/2) (d) x(-t)

34. 1.4.2 Sinusoidal Signals. • A general form of sinusoidal signal is • where A is the amplitude, wo is the frequency in radian per second, and q is the phase angle in radians. Figure 1.20: Continuous-Time Sinusoidal signal A cos(ωt + Φ).

35. 1.4.5 Step Function. • The discrete-time version of the unit-step function is defined by, Figure 1.24: Discrete–time of Step Function of Unit Amplitude.

36. Precise Graph Commonly-Used Graph The Unit Step Function The product signal g(t)u(t) can be thought of as the signal g(t) “turned on” at time t = 0. Dr. Abid Yahya

37. The Unit Ramp Function Dr. Abid Yahya

38. 1.4.6 Impulse Function. • The discrete-time version of the unit impulse is defined by, Figure 1.26: Discrete-Time form of Impulse. • Figure 1.41 is a graphical description of the unit impulse d(t). • The continuous-time version of the unit impulse is defined by the following pair, • The d(t) is also refer as the Dirac Delta function.

39. Graphical Representation of the Impulse The impulse is not a function in the ordinary sense because its value at the time of its occurrence is not defined. It is represented graphically by a vertical arrow. Its strength is either written beside it or is represented by its length. Dr. Abid Yahya

40. The Unit Step and Unit Impulse The unit step is the integral of the unit impulse and the unit impulse is the generalized derivative of the unit step Dr. Abid Yahya

41. Properties of the Impulse The Sampling Property The sampling property “extracts” the value of a function at a point. The Scaling Property This property illustrates that the impulse is different from ordinary mathematical functions. Dr. Abid Yahya

42. The Unit Periodic Impulse The unit periodic impulse is defined by The periodic impulse is a sum of infinitely many uniformly-spaced impulses. Dr. Abid Yahya

43. FEW Questions from Past Exam Paper , for Practice and Use as a Tutorial

44. Q.1

45. A.1

46. Q.2

47. A.2

48. A.2

49. Q.3

50. A.3