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Bài giảng Xử lý tín hiệu số - Signal and System in Time Domain

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  1. Xử lý tín hiệu số Signal and System in Time Domain Ngô Quốc Cường ngoquoccuong175@gmail.com sites.google.com/a/hcmute.edu.vn/Ngô Quốc Cường ngoquoccuong
  2. Signal and System in Time Domain • Discrete time signals • Discrete time systems • LTI systems 2
  3. 2.1 Discrete - time signals • A discrete time signal x(n) is a function of an independent variable that is integer. 3
  4. 2.1 Discrete - time signals • Alternative representation of discrete time signal: 4
  5. 2.1 Discrete - time signals • Alternative representation of discrete time signal: 5
  6. 2.1.1 Some elementary signals 6
  7. 2.1.1 Some elementary signals 7
  8. 2.1.1 Some elementary signals 8
  9. 2.1.1 Some elementary signals 9
  10. 2.1.2 Classification of discrete time signal • Energy signals and power signal – The energy E of a signal x(n) is given: – If E is finite, x(n) is call an energy signal. 10
  11. 2.1.2 Classification of discrete time signal • Energy signals and power signal – The average power P of a signal x(n) is defined: – If P is finite (and nonzero), x(n) is called a power signal. 11
  12. 2.1.2 Classification of discrete time signal • Energy signals and power signal – Example: the average power of the unit step signal is: 12
  13. 2.1.2 Classification of discrete time signal • Periodic signals and aperiodic signals – A signal x(n) is periodic with period N (N >0) if and only if 13
  14. 2.1.2 Classification of discrete time signal • Symmetric (even) and antisymmetric (odd) signals – A real value signal x(n) is call symmetric if – A signal is call antisymmetric if 14
  15. 2.1.2 Classification of discrete time signal • Symmetric (even) and antisymmetric (odd) signals 15
  16. 2.1.2 Classification of discrete time signal – The even signal component is formed by adding x(n) to x(- n) and dividing by 2. – Odd signal component 16
  17. 2.1.3 Simple manipulations of signals • Transformation of time – A signal x(n) may be shifted by replacing n bay n-k. • k is positive number: delay • k is negative number: advance – Folding: replace n by -n – Time scaling: replace n by cn (c is an integer) 17
  18. 2.1.3 Simple manipulations of signals • Transformation of time – Find x(n-3) and x(n+2) of x(n) 18
  19. 2.1.3 Simple manipulations of signals • Transformation of time – Find x(-n) and x(-n+2) of x(n) 19
  20. 2.1.3 Simple manipulations of signals • Transformation of time – Show the graphical representation of y(n) = x(2n), where x(n) is 20
  21. 2.1.3 Simple manipulations of signals • Addition, multiplication, and scaling – Amplitude scaling – Sum – Product 21
  22. Exercises 22
  23. Exercises • x(n) is illustrated in the figure • Sketch the following signals 23
  24. 2.2 Discrete time systems • Device or algorithm that performed some prescribed operation on discrete time signal. 24
  25. 2.2 Discrete time systems • Determine the response of the following systems to the input signal 25
  26. 2.2 Discrete time systems • Block diagram representation – An adder 26
  27. 2.2 Discrete time systems • Block diagram representation – A constant multiplier 27
  28. 2.2 Discrete time systems • Block diagram representation – A signal multiplier 28
  29. 2.2 Discrete time systems • Block diagram representation – A unit delay element – A unit advance element 29
  30. 2.2 Discrete time systems • Block diagram representation 30
  31. 2.2 Discrete time systems • Classification of discrete time systems – Static versus dynamic systems – Time invariant versus time variant systems – Linear versus nonlinear systems – Causal versus noncausal systems – Stable versus unstable systems 31
  32. 2.2 Discrete time systems • Classification of discrete time systems – Static versus dynamic systems • Static: output at any instant n depends at most on the input sample at the same time – memoryless. • Dynamic: to have memory 32
  33. 2.2 Discrete time systems • Classification of discrete time systems – Time invariant versus time variant systems • Time invariant: input – output characteristics do not change with time. 33
  34. 2.2 Discrete time systems • Classification of discrete time systems – Linear versus nonlinear systems 34
  35. 2.2 Discrete time systems 35
  36. 2.2 Discrete time systems • Classification of discrete time systems – Causal versus noncausal systems • The output of the system at any time n depends only on present and past inputs but does not depend on future inputs. 36
  37. 2.2 Discrete time systems 37
  38. 2.2 Discrete time systems • Classification of discrete time systems – Stable versus unstable systems • An arbitrary relaxed system is said to be bounded input bounded output stable if and only if every bounded input produces a bounded output. 푛 ≤ < ∞ 푛 ≤ < ∞ 38
  39. 2.2 Discrete time systems 39
  40. 2.2 Discrete time systems • Interconnection of discrete time systems 40
  41. 2.3 Analysis of discrete time LTI systems • LTI: Linear Time Invariant • 2 methods: – Solve the difference equation – Decompose the input signal into a sum of elementary signals. Using the linearity property, the responses of the system to the elementary signals are added to obtain the total response of the system. 41
  42. 2.3 Analysis of discrete time LTI systems • Resolution of discrete time signal into impulses • Example: 42
  43. 2.3 Analysis of discrete time LTI systems • Response of LTI system to arbitrary input – Denote the response y(n, k) of the system to unit sample sequence at n = k by symbol h(n, k). – The response of the system to x(n) 43
  44. 2.3 Analysis of discrete time LTI systems • Response of LTI system to arbitrary input: convolution – The formula reduces to – The response at n = n0 is given as 44
  45. 2.3 Analysis of discrete time LTI systems • Response of LTI system to arbitrary input – Summarize 45
  46. 2.3 Analysis of discrete time LTI systems • Response of LTI system to arbitrary input – Example – The output is 46
  47. Exercise 47
  48. Exercise 48
  49. Exercise 49
  50. 2.3 Analysis of discrete time LTI systems • Properties of convolution and interconnection 50
  51. 2.3 Analysis of discrete time LTI systems • Causal linear time invariant system • An LTI system is causal if and only if its impulse response is zero for n<0. 51
  52. 2.3 Analysis of discrete time LTI systems • Stability of linear time invariant system – A linear time invariant system is stable if its impulse response is summable. 52
  53. • LTI system can be characterized in terms of its impulse response h(n). – Finite duration impulse response – Infinite duration impulse response • Causal FIR system 53
  54. • Practical DSP methods fall in two basic classes: – Block processing methods. – Sample processing methods. • In block processing methods, the data are collected and processed in blocks. • In sample processing methods, the data are processed one at a time. Sample processing methods are used primarily in real-time applications. 54
  55. Block processing • Block processing methods: – Direct form – Convolution table – LTI form – Matrix form – Flip-and-slide form – Overlap-add block convolution form 55
  56. Block processing • In many practical applications, we sample our analog input signal (in accordance with the sampling theorem requirements) and collect a finite set of samples, say L samples, representing a finite time record of the input signal. The duration of the data record in seconds will be: 56
  57. Block processing • The direct and LTI forms of convolution given by 57
  58. Block processing • Direct Form – Consider a causal FIR filter of order M with impulse response h(n), n = 0, 1, . . . , M. It may be represented as a block: – For the direct form 58
  59. Block processing • Direct Form – Consider the case of an order-3 filter and a length-5 input signal. 59
  60. Block processing • Direct Form 60
  61. Block processing • Direct Form 61
  62. Block processing • Convolution table 62
  63. Block processing • Convolution table – Folding the table, 63
  64. Block processing • LTI form – The input signal – X can be written by 64
  65. Block processing • LTI form – The effect of the filter is to replace each delayed impulse by the corresponding delayed impulse response 65
  66. Block processing • LTI form 66
  67. Block processing • Matrix form – The convolutional equations can also be written in the linear matrix form – The filter matrix H must be rectangular with dimensions 67
  68. Block processing • Matrix form – There is also an alternative matrix form written as follows: – the data matrix X has dimension: 68
  69. Block processing • Flip and Slide form 69
  70. Block processing • Overlap-Add block 70
  71. Block processing • Overlap-Add block – The input is divided into the following three contiguous blocks – Convolving each block separately with h = [1, 2, −1, 1] 71
  72. Block processing • Overlap-Add block – aligning the output blocks according to their absolute timings and adding them up 72
  73. Problems • Compute the convolution, y = h ∗ x, of the filter and input • Using the following three methods: – (a) The convolution table. – (b) The LTI form of convolution, arranging the computations in a table form. – (c) The overlap-add method of block convolution with length-3 input blocks. Repeat using length-5 input blocks. 73
  74. 2.4 Discrete time systems described by difference equations • The practical implementation of the IIR system is impossible since it requires an infinite number of memory locations, multiplications, and additions. • Practical and computationally efficient means: difference equations 77
  75. 2.4 Discrete time systems described by difference equations • Recursive and non-recursive system – Compute the cumulative average of a signal x(n) defined in the interval 0 ≤ k ≤ n – In the different way – Hence Recursive system 78
  76. 2.4 Discrete time systems described by difference equations • Recursive and non-recursive system – Non-recursive system: depends only on the present and the past inputs. 79
  77. 2.4 Discrete time systems described by difference equations • Recursive and non-recursive system 80
  78. 2.4 Discrete time systems described by difference equations • The general form: • N: the order of the difference equation = the order of the system. 81
  79. 2.4 Discrete time systems described by difference equations • Solution of linear constant coefficient difference equation – Direct method – Indirect method (z - transform) • The direct solution method assumes that the total solution is the sum of two parts: – yh(n): homogeneous solution – yp(n): particular solution 82
  80. 2.5 Structure for the realization of LTI systems • Consider the 1st order system • This realization uses separate delays for both input and output, called Direct Form 1 structure. 83
  81. 2.5 Structure for the realization of LTI systems • Interchange the order of the recursive and non-recursive systems. 84
  82. 2.5 Structure for the realization of LTI systems • Two delay elements can be merged into one delay Direct Form 2 structure 85
  83. 2.5 Structure for the realization of LTI systems • Direct Form 1 – M+N delays – M+N+1 multipliers 86
  84. 2.5 Structure for the realization of LTI systems • Direct Form 2 87
  85. 2.5 Structure for the realization of LTI systems 88