Instrumentation and Measurement, Time dependent properties of a signal.pptx

Intrumentation and Measurements for Mechanical engineers!

1. Signal Any physical phenomena that conveys or carries some information can be called a signal. The music, speech, motion pictures, still photos, heart beat, etc. are examples of signals that we normally encounter in day-to-day life. Usually, the information carried by a signal will be a function of an independent variable. The independent variable can be time, spatial coordinates, intensity of colors, pressure, temperature, etc . The most popular independent variable in signals is time and it is represented by the letter “t”.

2. Signal

3. Signal The value of a signal at any specified value of the independent variable is called its amplitude . The sketch or plot of the amplitude of a signal as a function of independent variable is called its waveform. Mathematically, any signal can be represented as a function of one or more independent variables. Therefore, a signal is defined as any physical quantity that varies with one or more independent variables. Let us consider the following examples: (1.1) (1.2) x (1.3)  

4. Signal The functions and as defined by equations (1.1) and (1.2) represents two signals: one that varies linearly with time “t” and the other varies quadratically with time “t”. The equation (1.3) represents a signal which is a function of two independent variables “p” and “q”.  

5. Classification of Signals The signals can be classified in number of ways. Some of the ways of classifying the signals are: Depending on the number of sources for the signals . One-channel signals Multichannel signals Depending on the number of dependent variables . One-dimensional signals Multidimensional signals Depending on whether the dependent variable is continuous or discrete. Analog or Continuous signals Discrete signals

6. Classification of Signals One-channel signals Signals that are generated by a single source or sensor are called one-channel signals. The record of room temperature with respect to time, the audio output of a mono-speaker, etc. are examples of one-channel signals. Multi-channel signals Signals that are generated by a multiple source or sensors are called multi-channel signals. The audio output of two stereo speakers is an example of two-channel signal. The record of ECG (Electro-Cardio Graph) at eight different place in a human body is an example of eight-channel signal.

7. Classification of Signals One-dimensional signals A signal which is a function of single independent variable is called one-dimensional signal. The signals represented by equation (1.1) and (1.2) are examples of one-dimensional signals. The music, speech, heart-beat, etc., are examples of one-dimensional signals where the single independent variable is time. Multi-dimensional signals A signal is a function of two or more independent variables is called multidimensional signal. The equation (1.3) represents a two-dimensional signal. A photograph is an example of a two-dimensional signal. The intensity or brightness at each point of a photograph is a function of two spatial coordinates “x” and “y”, (and so the spatial coordinates are independent variables). Hence, the intensity or brightness of a photograph can be denoted by b(x,y).

8. Classification of Signals Analog or Continuous Signals When a signal is defined continuously for any independent variable, it is called analog or continuous signal. Most of the signals encountered in science and engineering are analog in nature. When the dependent variable of an analog signal is time, it is called continuous time signal. Discrete Signals When a signal is defined for discrete intervals of independent variable, it is called discrete signal. When the dependent variable of a discrete signal is time, it is called discrete time signal. Most of the discrete signals are either sampled version of analog signals for processing by digital systems or output of digital systems.

9. System Any process that exhibits cause and effect relation can be called a system. A system will have an input and an output signal. The output signal will be processed version of the input signal. A system is either interconnection of hardware devices or software/algorithm. A system is denoted by the letter H. Input signal of Excitation Output signal or Response H Representation of a system

10. Classification of System Depending on the type of energy used to operate the systems, the systems can be classified into Electrical systems, Mechanical Systems, Thermal Systems, Hydraulic Systems, etc. Depending upon the type of input and output signals, the systems can be classified into Continuous time and Discrete time systems.

11. Classification of System Continuous Time System A system which can process continuous time signal is called continuous time system, and so the input and output signals of a continuous time system are continuous time signals. The input of continuous time system is denoted as x(t0 and the output of continuous time system is denoted as y(t). The diagrammatic representation of a continuous time system is shown below:

12. Classification of System Continuous Time System (Continued………..) The operation performed by a continuous time system on input to produce output or response can be expressed as, Response, y(t) = H {x(t)} where, H denotes the system operation (also called system operator). Note: When a continuous time system satisfies the properties of linearity and time invariance then it is called LTI (Linear Time Invariant) continuous time system . Most of the practical systems that we encounter in science and engineering are LTI systems.

13. Classification of System Discrete time system A system which can process discrete time signal is called discrete time signal, and so the input and output signals of a discrete time system are discrete time signals. The input of a discrete time system is denoted as “x(n)” and the output of discrete time system is denoted as “y(n)”. The diagrammatic representation of a discrete time system is shown below:

14. Classification of System Discrete time system (Continued………..) The operation performed by a discrete time system on input to produce output or response can be expressed as, Response, y(n) = H {x(n)} where, H denotes the system operation (also called system operator). Note: When a discrete time system satisfies the properties of linearity and time invariance then it is called LTI (Linear Time Invariant) discrete time system .

15. Standard Continuous Time Signals Impulse signal The impulse signal is a signal with infinite magnitude and zero duration, but with an area of A. Mathematically, impulse signal is defined as, Impulse signal, = ∞ ; t = 0 and =A = 0; t ≠ 0 The unit impulse signal is a signal with infinite magnitude and zero duration, but with a unit area. Mathematically, impulse signal is defined as, Impulse signal, = ∞ ; t = 0 and = 1 = 0; t ≠ 0  

16. Standard Continuous Time Signals Step Signal The step signal is defined as, A ; t 0 ; t The unit step signal is defined as, u(t) = 1; t 0; t  

17. Standard Continuous Time Signals Ramp Signal The ramp signal is defined as, At ; t 0 ; t The unit ramp signal is defined as, t ; t 0 ; t  

18. Standard Continuous Time Signals Parabolic Signal The parabolic signal is defined as, ; for t 0 ; t The unit parabolic signal is defined as, ; for t 0 ; t  

19. Standard Continuous Time Signals Unit pulse signal The unit pulse signal is defined as, (t) = u u  

20. Standard Continuous Time Signals Sinusoidal Signal Case-I: Cosinusoidal signal The cosinusoidal signal is defined as: where, = 2 Angular frequency in rad/sec = Frequency in cycles/sec or Hz T = Time period in sec When = 0, When = Positive, When = Negative,  

21. Standard Continuous Time Signals Case-II: Sinusoidal signal The cosinusoidal signal is defined as: where, = 2 Angular frequency in rad/sec = Frequency in cycles/sec or Hz T = Time period in sec When = 0, When = Positive, When = Negative,  

22. Standard Continuous Time Signals

23. Standard Continuous Time Signals Exponential Signal Case-I: Real Exponential Signal The real exponential signal is defined as: where, A and b are real. Here, when b is positive, the signal x(t) will be an exponentially rising signal; and when b is negative the signal x(t) will be an exponentially decaying signal.  

24. Standard Continuous Time Signal Case-II: Complex exponential signal The complex exponential signal is defined as, where, = Angular frequency in rad/sec Frequency in cycles/sec or Hz T Time period in sec The complex exponential signal can be represented in a complex plane by a rotating vector, which rotates with a constant angular velocity of rad/sec.  

25. Standard Continuous Time Signal Complex exponential signal continue……….. The complex exponential signal can be resolved into real and imaginary parts as shown below: = A = A t ⸫ A = Real part of x(t) t = Imaginary part of x(t) From above equations, we can say that a complex exponential signal is the vector sum of sinusoidal signals of the form and t.  

26. Standard Continuous Time Signal Exponentially rising/decaying sinusoidal signal The exponential rising/ decaying sinusoidal signal is defined as: where, = 2 Angular frequency in rad/sec, = Frequency in cycles/sec or Hz , T = Time period in sec Here, A and b are real constants. When b is positive, the signal x(t) will be an exponentially rising sinusoidal signal; and when b is negative, the signal x(t) will be an exponentially decaying sinusoidal signal.  

27. Standard Continuous Time Signal Triangular pulse signal The triangular pulse signal is defined as:  

28. Standard Continuous Time Signal Signum signal The Signum signal is defined as the sign of the independent variable t. Therefore, the Signum signal is expressed as,  

29. Standard Continuous Time Signal Sinc Signal The Sinc signal is defined as,  

30. Standard Continuous Time Signal Gaussian signal The Gaussian signal is defined as,  

31. Classification of Continuous Time Signals The continuous time signals are classified depending on their characteristics. Some ways of classifying continuous time signals are: Deterministic and Nondeterministic signals Periodic and Nonperiodic (Aperiodic) signals Symmetric and Antisymmetric signals (Even and Odd Signals) Energy and Power Signals Causal and Noncausal signals

32. Classification of Continuous Time Signals Deterministic and Nondeterministic Signals The signal that can be completely specified by a mathematical equation is called a deterministic signal. The step, ramp, exponential, and sinusoidal signals are examples of deterministic signals. Examples of deterministic signals: , The signal whose characteristics are random in nature is called a nondeterministic signal. The noise signals from various sources like electronic amplifiers, oscillators, radio receivers, etc, are examples of nondeterministic signals.  

33. Classification of Continuous Time Signals Periodic and Non-Periodic (Aperiodic) Signals The periodic signal will have a definite pattern that repeats again and again over a certain period of time. Therefore, the signal which satisfies the condition, is called a periodic signal, and the signal which does not satisfy the forementioned condition are called an aperiodic or nonperiodic signal. In periodic signals, the term T is called the fundamental time period of the signal. Hence, inverse of T is called the fundamental frequency, in cycles/sec or Hz, and is called the fundamental angular frequency in rad/sec. Remember, the sinusoidal signals and complex exponential signals are always periodic with a periodicity of T, where  

34. Classification of Continuous Time Signals When a continuous time signal is a mixture of two periodic signals with fundamental time periods and , then the continuous time signal will be periodic, if the ratio of and (i.e., ) is a rational number. The periodicity of the continuous time signal will be the LCM (Least Common Multiple) of and  

35. Classwork Verify whether the following continuous time signals are periodic. If periodic, find the fundamental period. (1) x(t) = 2   (2) x(t) = 2   (3) x(t) =  

36. Classification of Continuous Time Signals Symmetric (Even) and Antisymmetric (Odd) Signals The signals may exhibit symmetry or anti-symmetry with respect to t = 0. When a signal exhibits symmetry with respect to t = 0 then it is called an even signal . Therefore, the even signal satisfies the condition, When a signal exhibits anti-symmetry with respect to t = 0, then it is called an odd signal . Therefore, the odd signal satisfies the condition Since , the cosinusoidal signals are even signals and since , the sinusoidal signals are odd signals.  

37. Classification of Continuous Time Signals Energy and Power Signals The signals which have finite energy are called energy signals. The nonperiodic signals like exponential signals will have constant energy and so nonperiodic signals are energy signals. The signals which have a finite average power are called power signals . The periodic signals like sinusoidal and complex exponential signals will have constant power and so periodic signals are power signals. The energy E of a continuous time signal x(t) is defined as, The average power P of a continuous time signal x(t) is defined as,  

38. Classification of Continuous Time Signals Energy and Power Signals (Continue.….) For periodic signals, the average power over one period will be the same as average power over an interval. For energy signals, the energy will be finite (or constant) and average power will be zero. For power signals the average power is finite (or constant) and energy will be infinite. i.e., For energy signal, E is constant (i.e., 0 ) and P = 0. For power signal, P is constant (i.e., 0 ) and E =   For periodic signals, power,  

39. Classwork Determine the power and energy for the following continuous time signals: (1) x(t) = 1.2 sin   (2) x(t) = 3   (3) x(t) = t u(t)

40. Classification of Continuous Time Signals Causal, Noncausal, and Anti-causal Signals   A signal is said to be causal, if it is defined for t Therefore, if x(t) is causal, then x(t)=0, for t < 0.   A signal is said to be noncausal, if it is defined for either t or for both t and t Therefore, if x(t) is noncausal, then x(t) 0, for t < 0.   When a noncausal signal is defined only for t it is called anti-causal signal.  

41. Classification of Continuous Time Signals Causal, Noncausal, and Anti-causal signals (Continued……) Examples of causal and non-causal signals: Step Signal, x(t) = A ; Unit Step Signal, x(t) = u(t) = 1; Exponential Signal, x(t) = A u(t) Complex exponential signal, x(t) = A u(t)   Exponential signal, x(t) = A ; for all t Complex exponential signal, x(t) = A ; for all t  

42. Harmonic Signals A function is said to be simple harmonic in terms of a variable when its second derivative is proportional to the function but of opposite sign. Any signal that follows the definition of harmonic function is called harmonic signal. In its most elementary form, simple harmonic motion is defined by the relation: ………….(2.1) where, s = instantaneous displacement from equilibrium = amplitude, or maximum displacement from equilibrium = circular frequency, rad/s, and t = any time instant.  

43. Harmonic Signals Pendulum motions of small amplitude, a mass on a beam, mass-spring system, all vibrate with simple harmonic motion, or very neatly so. By differentiation, the following relations may be derived from Equation 2.1, ………..(2.2) Following from (2.2), ………..(2.3) In the above equation, = velocity, = maximum velocity or velocity amplitude, = acceleration, = maximum acceleration or acceleration amplitude.  

44. Harmonic Signals Equation (2.3) satisfies the definition of simple harmonic motion. The acceleration is proportional to the displacement , but is of opposite sign. The proportionality factor is  

45. Periodic Series and Fourier Series Representation Any periodic non-sinusoidal signal can be expressed as a linear weighted sum of harmonically related sinusoidal signals. If the periodic function meets the Dirichlet conditions, i.e., if it has a single value, be finite, and have finite number of discontinuities, and maxima and minima in one cycle, it may be represented by Fourier series. That is, where and are called Fourier coefficients.  

46. Conditions for Existence of Fourier Series The Fourier series exists only if the following Dirichlet’s conditions are satisfied: The signal x(t) is well defined and single valued, except possibly t a finite number of points. The signal x(t) must possess only a finite number of discontinuities in the period T. The signal must have a finite number of positive and negative maxima in the period T. Note: The value of signal x(t) at is if is a point of continuity. The value of signal x(t) at is if is a point of discontinuity.  

47. Periodic Series and Fourier Series Representation Multiplying each term of Equation (2.5) by dx and integrating over any interval of 2 length, we get          

48. Periodic Series and Fourier Series Representation The factor may be determined by multiplying both sides of Equation ( 2.7 ) by , (where is any fixed positive integer), and integrating each term over the interval of :      

49. Periodic Series and Fourier Series Representation In general, there are the following terms:   and     For special case ,    

50. Periodic Series and Fourier Series Representation Hence,   ………(2.7) In like manner, if we multiply both sides of Equation ( 2.5 ) by and integrated term by term over the interval , we may obtain     ………(2.8)

51. Periodic Series and Fourier Series Representation Most complex dynamic-mechanical signals, steady state or transient, whether they are time functions of pressure, displacement, strain, or something else, may be expressed as a combination of simple harmonic components. Each component will have its own amplitude and frequency and will be combined in various phase relations with the other components. A general mathematical statement of this may be rewritten by replacing by as:     ………(2.9)

52. Periodic Series and Fourier Series Representation Equation ( 2.9 ) may be written in the two equivalent forms:     ………(2.9) ………(2.10) where the harmonic coefficients are determine by the relation     and the phase relations and are determined as follows       and ………(2.11)

53. Periodic Series and Fourier Series Representation The Fourier series is an infinite series, and in order to get a perfect reconstruction of we would have to add an infinite number of terms. Since most engineering applications do not require a perfect reconstruction of , generally, is approximated by a truncated Fourier series. Very often less than ten harmonics are adequate for most engineering applications.   After the Fourier series of a particular is found, the steady state response of any measuring instrument can be determined by frequency response technique or principle of superposition i.e. response to each harmonic component or sinusoidal component is found and then algebraically added to get the total response.  

54. Review Questions What is a Fourier Series? What is its application? Write down the expressions for the coefficients of Fourier series.

55. Examples for Fourier series Learning Objectives Make able to analyze time dependent signals. Determine Fourier series expansion of time dependent signals.

56. Examples of Fourier series Question-1 The following equation represents the time variation of a mechanical strain: = 120 + 95 sin 15t + 40 sin 30t + 18 sin45t – 55 cos 15t-24 cos45t How many harmonics are present? What is the fundamental frequency in Hz? What is the amplitude of the third harmonic component?   Answers: 3 (b) 2.3864 Hz 30

57. Examples of Fourier series Question-2: Determine Fourier series expansion for the time dependent signal shown in Figure E2.2. Answer:  

58. Examples of Fourier series

59. Examples of Fourier series Question-3: Determine Fourier series expansion for the time dependent signal shown in Figure E2.3.  

60. Examples of Fourier series

61. Fourier Coefficients of Signals With Symmetry Even Symmetry A signal, x(t) is called even signal, if the signal satisfies the condition x (-t) = x (t). The waveform of an even signal exhibits symmetry with respect to t = 0 (i.e., with respect to vertical axis) and so the symmetry of a waveform with respect to t = 0 or vertical axis is called even symmetry. For an even signal the Fourier coefficients are given by,      

62. Example of Even Signal Question-4 Answer:  

63. Fourier Coefficients of Signals With Symmetry Odd Symmetry A signal, x(t) is called odd signal if it satisfies the condition x (-t) = x (t). The waveform of odd periodic signal will exhibit anti-symmetry with respect to t = 0 (i.e., with respect to vertical axis) and so the anti-symmetry of a waveform with respect to t = 0 or vertical axis is called odd symmetry. For odd signal, the Fourier coefficients are given by,      

64. Example of Odd Signal Question-5 Answer:  

65. Gibbs Phenomenon We have observed that the trigonometric form of a periodic signal, x(t), with period T is defined as: When the signal f(t) is reconstructed or synthesized with only N number of terms of the infinite series, the reconstructed signal exhibits oscillation (or overshoot or ripples), especially in signals with discontinuities.  

66. Gibbs Phenomenon