- 1. 1 SIGNALS AND SYSTEMS For Graduate Aptitude Test in Engineering SIGNALS AND SYSTEMS For Graduate Aptitude Test in Engineering By Dr. N. Balaji Professor of ECE JNTUK
- 2. 2 Session: 2 Topic : Classification of Signals and Systems Date : 12.05.2020 Session: 2 Topic : Classification of Signals and Systems Date : 12.05.2020 By Dr. N. Balaji Professor of ECE
- 3. 3 Syllabus Syllabus • Continuous-time signals: Fourier series and Fourier transform representations, sampling theorem and applications; Discrete-time signals: discrete-time Fourier transform (DTFT), DFT, FFT, Z-transform, interpolation of discrete-time signals; • LTI systems: definition and properties, causality, stability, impulse response, convolution, poles and zeros, parallel and cascade structure, frequency response, group delay, phase delay, digital filter design techniques.
- 4. 4 Contents Contents Unit Impulse, Unit Step, Unit Ramp functions and their Properties Example Problem on properties of the functions. Classification of Systems Causal and Non-causal Systems Linear and Non Linear Systems Time Variant and Time-invariant Systems Stable and Unstable Systems Static and Dynamic Systems Invertible and non-invertible Systems Solved Problems of previous GATE Exam
- 5. 5 Unit Impulse Function Unit Impulse Function One of the more useful functions in the study of linear systems is an Unit Impulse Function. An ideal impulse function is a function that is zero everywhere but at the origin, where it is infinitely high. However, the area of the impulse is finite. • The unit impulse function, = undefined for t=0 and has the following special property Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 6. 6 Unit Impulse Function Unit Impulse Function • A consequence of the delta function is that it can be approximated by a narrow pulse as the width of the pulse approaches zero while the area under the curve =1. → Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 7. 7 Unit Impulse Function Unit Impulse Function ; Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 8. 8 Representation of Impulse Function Representation of Impulse Function • The area under an impulse is called its strength or weight. It is represented graphically by a vertical arrow. An impulse with a strength of one is called a unit impulse. Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 9. 9 Unit Impulse Train Unit Impulse Train • The unit impulse train is a sum of infinitely uniformly- spaced impulses and is given by Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 10. 10 • is an even signal • It is a neither energy nor power signal. • Weight/strength of impulse • Area of weighted impulse = weight of impulse Properties of an Impulse Function Properties of an Impulse Function Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 11. 11 • Eg:- • Scaling property of impulse:- = ) Scaling Property of an Impulse Function Scaling Property of an Impulse Function Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 12. 12 Multiplication property of an Impulse Function Multiplication property of an Impulse Function Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 13. 13 Multiplication property of an Impulse Function Multiplication property of an Impulse Function Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 14. 14 Sampling Property of an Impulse Function Sampling Property of an Impulse Function • The Sampling Property Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 15. 15 Example Problem based on Sampling property Example Problem based on Sampling property Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 16. 16 Proof of Sampling Property Proof of Sampling Property Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 17. 17 Example problem on Sampling Property Example problem on Sampling Property Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 18. 18 Example Problem Example Problem Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 19. 19 Example problem on Sampling Property Example problem on Sampling Property Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 20. 20 Derivatives of impulse function Derivatives of impulse function Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 21. 21 Example problem for Derivatives of an Impulse function Example problem for Derivatives of an Impulse function Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 22. 22 Unit Step Function Unit Step Function Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 23. 23 Unit Ramp Function Unit Ramp Function , 0 ram p u u 0 , 0 t t t t d t t t •The unit ramp function is the integral of the unit step function. •It is called the unit ramp function because for positive t, its slope is one amplitude unit per time. Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 24. 24 Relation among Ramp, Step and Impulse Signals Relation among Ramp, Step and Impulse Signals Acknowledgement : Lecture slides from http://DrSatvir.in
- 25. 25 Sinusoidal and Exponential Signals Sinusoidal and Exponential Signals Sinusoids and exponentials are important in signal and system analysis because they arise naturally in the solutions of the differential equations. Sinusoidal Signals can expressed in either of two ways : cyclic frequency form- A sin (2Пfot) = A sin(2П/To)t radian frequency form- A sin (ωot) ωo = 2Пfo = 2П/To To = Time Period of the Sinusoidal Wave Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 26. 26 Sinusoidal and Exponential Signals Contd. Sinusoidal and Exponential Signals Contd. x(t) = A sin (2Пfot+ θ) = A sin (ωot+ θ) x(t) = Aeat Real Exponential = Aejω̥t = A[cos (ωot) +j sin (ωot)] Complex Exponential θ = Phase of sinusoidal wave A = amplitude of a sinusoidal or exponential signal fo = fundamental cyclic frequency of sinusoidal signal ωo = radian frequency Sinusoidal signal Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 27. 27 x(t) = e-at x(t) = eαt Real Exponential Signals and damped Sinusoidal Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 28. 28 Signum Function Signum Function 1 , 0 sg n 0 , 0 2 u 1 1 , 0 t t t t t Precise Graph Commonly-Used Graph The Signum function, is closely related to the unit-step function. Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 29. 29 Rectangular Pulse or Gate Function Rectangular Pulse or Gate Function Rectangular pulse, 1 / , / 2 0 , / 2 a a t a t t a Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 30. 30 The Unit Triangle Function The Unit Triangle Function A triangular pulse whose height and area are both one but its base width is not one, is called unit triangle function. The unit triangle is related to the unit rectangle through an operation called convolution. Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 31. 31 Sinc Function Sinc Function The unit Sinc function is related to the unit Rectangle function through the Fourier Transform. It is used for noise removal in signals sin sinc t t t Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 32. 32 Sinc Function Sinc Function Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 33. 33 Sampling Function Sampling Function Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 34. 34 Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 35. 35 Introduction to System Introduction to System • Systems process input signals to produce output signals • A system is a combination of elements that processes one or more signals to accomplish a function and produces output. system output signal input signal Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 36. 36 Types of Systems Types of Systems • Causal and non-causal • Linear and Non Linear • Time Variant and Time-invariant • Stable and Unstable • Static and Dynamic • Invertible and non-invertible Systems Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 37. 37 Causal, Anti Causal and Non-Causal Signals Causal, Anti Causal and Non-Causal Signals • Causal signals are signals that are zero for all negative time(or spatial positions). • Anticausal are signals that are zero for all positive time. • Non-causal signals are signals that have nonzero values in both positive and negative time. Causal signal Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 38. 38 Causal and Non-causal Systems Causal and Non-causal Systems • Causal system : A system is said to be causal if the present value of the output signal depends only on the present and/or past values of the input signal. • Examples: 1. y[n]=x[n]+1/2x[n-1] 2. y(t) = x(t) 3. y(t) = x(t-1) 4. y(t) = x(t) + x(t-1) Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 39. 39 Causal and Non-causal Systems • Non-causal system : A system is said to be Non-causal if the present value of the output signal depends also on the future values of the input signal. • Example: 1. y[n]=x[n+1]+1/2x[n-1] 2. y(t) = x(t+1) 3. y(t) = x(t) + x(t+1) 4. y(t) = x(t-1) + x(t+1) 5. y(t) = x(t-1) + x(t) + x(t+1)
- 40. 40 Exercise on Causal and Non-causal Systems Exercise on Causal and Non-causal Systems Q) Check whether the following are casual or non-casual system. 1. y(t) = x(2t) 7. y(t) = 2. y(t) = x(-t) 8. y(t) = 3. y(t)= x(sin t) 9.y(t) = 4. y 5. y(t) = odd [x(t)] 6. y(t) = sin (t+2) x(t-1)
- 41. 41 Solution to the Problems 1. y(t) = x(2t) Substitute t=1 in the above then y(1) = x(2) Hence the given System is Non-Casual 2. y(t) = x(-t) Substitute t=1 in the above then y(-1) = x(1) Hence the given System is System is non-casual 3. y(t) = x(sin t) Substitute t= - in the above y(-Π) = x(0) (- Π = -3.14) System is non-casual
- 42. 42 Solution to the Problems Solution to the Problems 4. y substitute t= -1 y(-1) = x(-2), which is past value of input. 0, substitute t = 1 y(1) = x(0), which is past. System is Casual. 5. y(t) = odd x(t) y(t) = x(t) – x(−t) substitute t=-1 then y(-1) = x(−1) – x(1) , which is dependent on future value. Hence the given System is non-casual
- 43. 43 Solution to the Problems on Causal and non-causal System Solution to the Problems on Causal and non-causal System 6. y(t) = sin(t+2) x(t-1) put t = 1 y(1) = sin(3) x(0) Constant coefficient Past value System is casual 7. y(t) = y(t) = Present output depends on present and past values Hence, the system is casual 8. y(t) = y(t) = Present output depends on future values also Hence, the system is non-casual 9. y(t) = y(t) = Present output depends on future values also. Hence, the system is non-casual
- 44. 44 Linear and Non Linear Systems Linear and Non Linear Systems • A system is said to be linear if it satisfies the principle of superposition or if it satisfies the properties of Homogeneity and Additivity. • Consider a system where an input of x1[t] produces an output of y1[t]. Further suppose that a different input, x2[t], produces another output, y2[t]. The system is said to be additive, if an input of x1[t] + x2[t] results in an output of y1[t] + y2[t], for all possible input signals. • Homogeneity means that a change in the input signal's amplitude results in a corresponding change in the output signal's amplitude. In mathematical terms, if an input signal of x[t] results in an output signal of y[t], an input of cx[t] results in an output of cy[t], for any input signal and c is a constant. Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 45. 45 Linearity Condition Linearity Condition For the system to be linear, it should satisfy two properties then Then Additivity B Scaling (or) Homogeneity or b Super position
- 46. 46 For linearity: 1.Output should be zero for zero input. 2.There should not be any nonlinear operation Example : The functions like Sin, Cos, tan, Cot, Sec, cosec, Log, Exponential, Modulus, Square, Cube, Root, Sampling function(), sinc(), Sgn() etc.…. have nonlinear operations. Linearity Condition Linearity Condition Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 47. 47 Problem based on Linearity and Non-Linearity Problem based on Linearity and Non-Linearity 1. y(t) = x(t) + 2 If input is (t), then (t) is output If input is (t) then (t) is output If input is (t) + (t) then the output must be (t) + (t) (t) (t) = (t) + 2 (t) (t) = (t) + 2 (t) + (t) (t) + (t) (t) + 2 + (t) + 2 = (t) + (t) + 4 (t) + (t) ≠ (t) + (t) + 4 Hence the system is non-linear
- 48. 48 2. Check whether the given system is linear or nonlinear Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 49. 49 Time Invariant and Time Variant Systems Time Invariant and Time Variant Systems • A system is said to be time invariant if a time delay or time advance of the input signal leads to a identical time shift in the output signal. Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 50. 50 Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 51. 51 Time Variant and Time in Variant System Time Variant and Time in Variant System Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 52. 52 Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 53. 53 Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 54. 54 Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 55. 55 Stable and Unstable Systems Stable and Unstable Systems • A system is said to be bounded-input bounded- output stable (BIBO stable) if every bounded input results in a bounded output. Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 56. 56 Stable and Unstable Systems Contd. Stable and Unstable Systems Contd. Example y[n]=1/3(x[n]+x[n-1]+x[n-2]) 3 y[n] 1 x[n] x[n 1] x[n 2] 1 (| x[n]| | x[n 1]| | x[n 2]|) 3 x x x x 1 (M M M ) M 3 Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 57. 57 Stable and Unstable Systems Contd. Stable and Unstable Systems Contd. Example: The system represented by y(t) = A x(t) is unstable ; A˃1 Reason: let us assume x(t) = u(t), then at every instant u(t) will keep on multiplying with A and hence it will not result in a bounded value and it may tend to infinite value. Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 58. 58 Stable and Unstable Systems Stable and Unstable Systems 1. y(t) = x(t) +2 put x(t) = 10 y(t) = 10 + 2 = 12 As input is bounded value, output is also a bounded value. Hence System is Stable 2. y(t) = t x(t) put x(t) = 10 y(t) = 10t As ‘t’ can be any value between -∞ to ∞, y(t) is unbounded. Hence System is Unstable
- 59. 59 Problems on Stable and Unstable Systems Problems on Stable and Unstable Systems 3. y(t) = put x(t) = 2 y(2) = When ‘t’ is 0 and Π, then sin(t) has values of sin 0 = 0 and Sin(Π) = 0 respectively. Therefore, y(t) = i.e., y(t) is unstable as output is not bounded
- 60. 60 Static Systems Static Systems • A static system is memoryless system • It has no storage devices • Its output signal depends on present values of the input signal • For example Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 61. 61 Dynamic Systems Dynamic Systems • A dynamic system possesses memory • It has the storage devices • A system is said to possess memory if its output signal depends on past values and future values of the input signal Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 62. 62 Example: Static or Dynamic? Example: Static or Dynamic? Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 63. 63 Example: Static or Dynamic? Example: Static or Dynamic? Answer: • The system shown above is RC circuit • R is memoryless • C is memory device as it stores charge because of which voltage across it can’t change immediately • Hence given system is dynamic or memory system Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 64. 64 Examples Examples Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 65. 65 Exercise Problems Exercise Problems Check whether given system is Static or Dynamic 1. y(t) = x(t) + x(t-1) 2. y(t) = x(-t) 3. y(t) = x(sin t) 4. y(t) = x(t-1) 5. y(t) = Even [x(t)] 6. y(t) = Real [x(t)]
- 66. 66 Invertible & Non-invertible Systems Invertible & Non-invertible Systems • If a system is invertible if it has an Inverse System. Otherwise it is non-invertible system • Example: y(t)=2x(t) – System is invertible must have inverse, that is: – For any x(t) we get a distinct output y(t) – Thus, the system must have an Inverse • x(t)=1/2 y(t)=z(t) y(t) System Inverse System x(t) x(t) y(t)=2x(t) System (multiplier) Inverse System (divider) x(t) x(t) Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 67. 67 Check whether the following Systems are invertible Check whether the following Systems are invertible Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 68. 68 Check whether the following Systems are invertible Check whether the following Systems are invertible Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 69. 69 Check whether the following Systems are invertible Check whether the following Systems are invertible Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 70. 70 Previous Gate Questions Previous Gate Questions Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 71. 71 Gate 2013 question Gate 2013 question Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 72. 72 Gate 2013 solution Gate 2013 solution Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 73. 73 Gate 2011 question Gate 2011 question Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 74. 74 Gate 2011 solution Gate 2011 solution Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 75. 75 Gate 2010 question Gate 2010 question Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 76. 76 Gate 2010 solution Gate 2010 solution Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 77. 77 Gate 2008 question Gate 2008 question Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 78. 78 Gate 2008 solution Gate 2008 solution Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 79. 79 Gate 2005 question Gate 2005 question Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 80. 80 Gate 2005 solution Gate 2005 solution Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 81. 81 Gate 2004 question Gate 2004 question Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 82. 82 Gate 2004 solution Gate 2004 solution Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 83. 83 Gate 2004 question Gate 2004 question Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 84. 84 Gate 2004 solution Gate 2004 solution Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff
- 85. 85 Sources and Reference Material Sources and Reference Material Sources: i) Lecture slides of Michael D. Adams and ii) Lecture slides of Prof. Paul Cuff iii) Solved Problems from Standard Textbooks. Disclaimer: The material presented in this presentation is taken from various standard Textbooks and Internet Resources and the presenter is acknowledging all the authors. Acknowledgement : Lecture slides of Michael D. Adams and Prof. Paul Cuff