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Sns slide 1 2011

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Sns slide 1 2011

  1. 1. DEE2363 Signals And Systems by Mr. Koay Fong Thai [email_address] ; [email_address]
  2. 2. Announcement (1) <ul><li>Schedule </li></ul><ul><li>Feedback to Lecturer (First two weeks of the semester) </li></ul><ul><ul><li>The availability of Room, Students, Lecturer. </li></ul></ul><ul><li>Plagiarism </li></ul><ul><li>7 cases in August’08 Final/Resit Exams. </li></ul><ul><li>1 expelled, 6 with Warning Letters and Re-take. </li></ul>
  3. 3. Announcement (2) <ul><li>Badge </li></ul><ul><li>Warning Letter. </li></ul><ul><li>Absent </li></ul><ul><li>3 times – Warning Letter and inform parents. </li></ul><ul><li>Less than 75% attendance  to be barred from final exam. </li></ul><ul><li>Use of Handphones and Laptops are STRICTLY PROHIBITED in Class and Lab. </li></ul>
  4. 4. Dos <ul><li>Feel free to ask questions during the class (without disturbance), or </li></ul><ul><li>my consultation hours, every Wednesday and Friday, 9am – 11am. </li></ul><ul><li>Be sensitive to find my errors. </li></ul><ul><li>Work hard to get a good grade. </li></ul><ul><li>Be on time in the class. </li></ul><ul><li>Be on time to submit coursework (take note of the format). </li></ul>
  5. 5. Don’ts <ul><li>Use of cell phones and computers in the class. </li></ul><ul><li>Play games!! </li></ul><ul><li>Sleep and/or talk in the class. </li></ul><ul><li>Copy tutorials/assignments (you will get zero if I catch you). </li></ul><ul><li>… etc, to be defined by me…. </li></ul>
  6. 6. Assessment <ul><li>Tutorial : 10% </li></ul><ul><li>Quiz : 10% </li></ul><ul><li>Assignment : 10% </li></ul><ul><li>Mid-Term Tests : 20% </li></ul><ul><li> 50% **** </li></ul><ul><li>Final Exam : 50% . </li></ul><ul><li> 100% </li></ul>Less than 25%  to be barred from final exam!!
  7. 7. Lecture and Tutorial Schedule
  8. 8. Lecture Schedule with Outcome
  9. 9. Broad Aims <ul><li>To introduce the students to the idea of signal and system along with the analysis and characterization . </li></ul><ul><li>To introduce the students the transformation methods for both continuous-time and discrete-time signals and systems. </li></ul><ul><li>To provide a foundation to numerous other courses that deal with signal and system concepts directly or indirectly, for instance, communication, control, instrumentation, and so on; as well as to students of disciplines such as, mechanical, chemical and aerospace engineering. </li></ul>
  10. 10. Objectives <ul><li>By the end of the course, you would have understood: </li></ul><ul><ul><li>Basic signal analysis (mostly continuous-time) </li></ul></ul><ul><ul><li>Basic system analysis (also mostly continuous systems) </li></ul></ul><ul><ul><li>Time-domain analysis (including convolution) </li></ul></ul><ul><ul><li>Laplace Transform and transfer functions </li></ul></ul><ul><ul><li>Fourier Series and Fourier Transform </li></ul></ul><ul><ul><li>Sampling Theorem and Signal Reconstructions </li></ul></ul><ul><ul><li>Basic z-transform </li></ul></ul>
  11. 11. Topics <ul><li>Signals and Systems in the time domain </li></ul><ul><ul><li>Impulse response, Convolution Integral/Sum, Differential/Difference Equations, Linear Constant-Coefficient Differential/Difference Equations. </li></ul></ul><ul><li>Fourier Transform </li></ul><ul><ul><li>Continuous/discrete Fourier Series and Transform, the properties and applications in signals and systems, Inverse Fourier Transform. </li></ul></ul><ul><li>Laplace Transform </li></ul><ul><ul><li>Properties, inverse transform; analysis of signals and systems. </li></ul></ul><ul><li>z-Transform </li></ul><ul><ul><li>Properties, inverse transform; analysis of signals and systems. </li></ul></ul>
  12. 12. Reference Books <ul><li>B.P. Lathi, “ Signal Processing and Linear Systems ”, 1998, Oxford University Press. </li></ul><ul><ul><li>PSDC Library : TK5102.9 Lat 1998 </li></ul></ul><ul><li>M.J. Roberts, “ Signals and Systems: Analysis Using Transform Methods and MATLAB ”, 2004, McGraw Hill. </li></ul><ul><ul><li>PSDC Library : TK5102.9 Rob 2004 </li></ul></ul><ul><li>Alan V. Oppenheim, Alan S. Willsky, S. Hamid Nawab, “ Signals and Systems ”, 2 nd Edition 1997, Pearson Education,. </li></ul><ul><ul><li>PSDC Library : QA402 Opp 1997 </li></ul></ul><ul><li>Simon S. Haykin, B. Van Veen, “ Signals and Systems ”, 2 nd Edition 2005, John Wiley & Sons. </li></ul>
  13. 13. Introduction to Signals (CT and DT) By Koay Fong Thai [email_address]
  14. 14. Topics <ul><li>Introduction </li></ul><ul><li>Size of a Signal </li></ul><ul><li>Classification of Signals </li></ul><ul><li>Some Useful Signal Operations </li></ul><ul><li>Some Useful Signal Models </li></ul><ul><li>Even and Odd Functions </li></ul>
  15. 15. Introduction <ul><li>The concepts of signals and systems arise in a wide variety of areas: </li></ul><ul><ul><li>communications, </li></ul></ul><ul><ul><li>circuit design, </li></ul></ul><ul><ul><li>biomedical engineering, </li></ul></ul><ul><ul><li>power systems, </li></ul></ul><ul><ul><li>speech processing, </li></ul></ul><ul><ul><li>etc. </li></ul></ul>
  16. 16. What is a Signal? <ul><ul><li>SIGNAL </li></ul></ul><ul><li>A set of information or data. </li></ul><ul><li>Function of one or more independent variables. </li></ul><ul><li>Contains information about the behavior or nature of some phenomenon. </li></ul>
  17. 17. Examples of Signals <ul><li>Electroencephalogram (EEG) signal (or brainwave) </li></ul>
  18. 18. Examples of Signals (2) <ul><li>Stock Market data as signal (time series) </li></ul>
  19. 19. What is a System? <ul><ul><li>SYSTEM </li></ul></ul><ul><li>Signals may be processed further by systems, which may modify them or extract additional from them. </li></ul><ul><li>A system is an entity that processes a set of signals ( inputs ) to yield another set of signals ( outputs ). </li></ul>
  20. 20. What is a System? (2) <ul><li>A system may be made up of physical components, as in electrical or mechanical systems (hardware realization). </li></ul><ul><li>A system may be an algorithm that computes an outputs from an inputs signal (software realization). </li></ul>
  21. 21. Examples of signals and systems <ul><li>Voltages (x 1 ) and currents (x 2 ) as functions of time in an electrical circuit are examples of signals. </li></ul><ul><li>A circuit is itself an example of a system (T) , which responds to applied voltages and currents. </li></ul>
  22. 22. Signals Classification <ul><li>Signals may be classified into: </li></ul><ul><ul><li>1. Continuous-time and discrete-time signals </li></ul></ul><ul><ul><li>2. Analogue and digital signals </li></ul></ul><ul><ul><li>3. Periodic and aperiodic signals </li></ul></ul><ul><ul><li>4. Causal, noncausal and anticausal signals </li></ul></ul><ul><ul><li>5. Even and Odd signals </li></ul></ul><ul><ul><li>6. Energy and power signals </li></ul></ul>
  23. 23. Signals Classification (2) – Continuous versus Discrete <ul><li>Continuous-time A signal that is specified for every value of time t. </li></ul><ul><li>Discrete-time A signal that is specified only at discrete values of time t. </li></ul>
  24. 24. Signals Classification (3) – Analogue versus Digital (1) <ul><li>Analogue, continuous </li></ul><ul><li>Analogue, discrete </li></ul>
  25. 25. Signals Classification (4) – Analogue versus Digital (2) <ul><li>Digital, continuous </li></ul><ul><li>Digital, discrete </li></ul>
  26. 26. Signals Classification (5) – Periodic versus Aperiodic <ul><li>A CT signal x(t) is said to be periodic if for some positive constant T o , </li></ul><ul><li>The smallest value of T o that satisfies the periodicity condition is the fundamental period of x(t) . </li></ul>
  27. 27. Signals Classification (6) – Periodic versus Aperiodic (2) <ul><li>Fundamental period = T o . Then, fundamental frequency is f o = 1/ T o in Hz or cycles per second. </li></ul><ul><li>or </li></ul><ul><li>Angular frequency, ω o = 2  / T o , radian per second. </li></ul><ul><li>Example, f(t) = C cos (2  f o t +  ) C : amplitude; f o : frequency;  : phase </li></ul><ul><ul><li>Rewriting f(t) = C cos ( ω o t +  ) </li></ul></ul>
  28. 28. Signals Classification (7) – Periodic versus Aperiodic (3) <ul><li>A DT signal x[n] is said to be periodic if for all positive integer N , </li></ul><ul><li>The smallest value of N is the fundamental period of x[n] . </li></ul><ul><li>Fundamental angular frequency,  is defined by  = 2  / N. </li></ul>
  29. 29. Signals Classification (6) – Periodic versus Aperiodic (4) <ul><li>For the signal, </li></ul><ul><li>Find the period and the fundamental frequency of the signal. </li></ul><ul><li>Solution (hint: sin  = cos (  -  /2) ): </li></ul>
  30. 30. Signals Classification (6) – Periodic versus Aperiodic (5)
  31. 31. Signals Classification (8) – Periodic versus Aperiodic (6) <ul><li>Aperiodic (Nonperiodic) signals? </li></ul>
  32. 32. Signals Classification (9) – Causal vs. Noncausal vs. Anticausal <ul><li>Causal ( 因果 ) signal: </li></ul><ul><ul><li>A signal that does not start before t =0. </li></ul></ul><ul><ul><li>f(t) = 0; t <0 </li></ul></ul><ul><li>Noncausal signal: </li></ul><ul><ul><li>A signal that starts before t =0, such as charge in capacitor before switch is turned on. </li></ul></ul><ul><li>Anticausal signal: </li></ul><ul><ul><li>A signal that is zero for all t  0. </li></ul></ul>
  33. 33. Signals Classification (10) – Even versus Odd
  34. 34. Signals Classification (11) – Even versus Odd (2) <ul><li>A signal x(t) or x[n] is referred to as an even signal if </li></ul><ul><ul><li>CT: </li></ul></ul><ul><ul><li>DT: </li></ul></ul><ul><li>A signal x(t) or x[n] is referred to as an odd signal if </li></ul><ul><ul><li>CT: </li></ul></ul><ul><ul><li>DT: </li></ul></ul>
  35. 35. Signal Classification (12) – Energy versus Power <ul><li>Signal with finite energy (zero power) </li></ul><ul><li>Signal with finite power (infinite energy) </li></ul><ul><li>Signals that satisfy neither property are referred as neither energy nor power signals </li></ul>
  36. 36. Size of a Signal (1) <ul><li>A number indicates the largeness or strength of the signal. </li></ul><ul><li>Such a measure must consider both amplitude and duration of the signal. </li></ul><ul><li>Measurement of the size of a human being, V with variable radius, r and height, h with assumption of cylindrical shape given by </li></ul>
  37. 37. Size of a Signal (2) <ul><li>Assuming f(t) = sin t , </li></ul><ul><ul><li>f(t) could be a large signal, </li></ul></ul><ul><ul><li>yet its positive and negative areas cancel each other. </li></ul></ul><ul><li>Then, indicates a signal of small size. </li></ul><ul><li>This can be solved by defining the signal size as the area under f 2 (t) ( f 2 (t) always > 0). </li></ul>
  38. 38. Size of a Signal, Energy (Joules) <ul><li>Measured by signal energy E x : </li></ul><ul><li>Generalize for a complex valued signal to: </li></ul><ul><ul><li>CT: DT: </li></ul></ul><ul><li>Energy must be finite, which means </li></ul>
  39. 39. Size of a Signal, Power (Watts) <ul><li>If amplitude of x(t) does not -> 0 when t -> ∞, need to measure power P x instead: </li></ul><ul><li>Again, generalize for a complex valued signal to: </li></ul><ul><ul><li>CT: </li></ul></ul><ul><ul><li>DT: </li></ul></ul>
  40. 40. Example <ul><li>Determine the suitable measures of the signals in the figure below: </li></ul>
  41. 41. Example
  42. 42. Summary <ul><li>By the end of the class, you would have understood: </li></ul><ul><ul><li>Examples of signals </li></ul></ul><ul><ul><li>Signals classification </li></ul></ul>
  43. 43. Signal Operations for CT Signals by Koay Fong Thai [email_address]
  44. 44. Signal Operations <ul><li>Signal operations are operations on the time variable of the signal, involve simple modification of the independent variable . </li></ul><ul><ul><li>Time Shifting </li></ul></ul><ul><ul><li>Time Scaling </li></ul></ul><ul><ul><li>Time Inversion (Reversal) </li></ul></ul><ul><ul><li>Combined operations </li></ul></ul>
  45. 45. Signal Operations: Time Shifting <ul><li>Shifting of a signal in time </li></ul><ul><li> adding or subtracting the amount of the shift to the time variable in the function. </li></ul><ul><li>x(t)  x(t–t o ) </li></ul><ul><ul><li>t o > 0 ( t o is positive value), signal is shifted to the right (delay). </li></ul></ul><ul><ul><li>t o < 0 ( t o is negative value), signal is shifted to the left (advance). </li></ul></ul><ul><li>x(t–2) ? x(t) is delayed by 2 seconds. </li></ul><ul><li>x(t+2) ? x(t) is advanced by 2 seconds. </li></ul>
  46. 46. Signal Operations: Time Shifting (2) <ul><li>Subtracting a fixed amount from the time variable will shift the signal to the right that amount. </li></ul><ul><li>Adding to the time variable will shift the signal to the left. </li></ul>
  47. 47. Signal Operations: Time Shifting (3)
  48. 48. Signal Operations: Time Scaling <ul><li>Compress es ( 压缩 ) and dilate s ( 膨胀 ) a signal by multiplying the time variable by some amount. </li></ul><ul><li>x(t)  x(  t) </li></ul><ul><ul><li>If  >1, the signal becomes narrower  compression. </li></ul></ul><ul><ul><li>If  <1, the signal becomes wider  dilation. </li></ul></ul><ul><li>Play audio recorded, f(t) in mp3 player at twice the normal recording speed? </li></ul><ul><ul><li>f(2t) or f(t/2) ? </li></ul></ul><ul><ul><li>f(2t) </li></ul></ul>
  49. 49. Signal Operations: Time Scaling (2)
  50. 50. Signal Operations: Time Scaling (3)
  51. 51. Signal Operations: Time Scaling (4)
  52. 52. Signal Operations: Time Inversion (Reversal) <ul><li>Reversal of the time axis, or folding/flipping the signal (mirror image) over the y-axis. </li></ul>
  53. 53. Signal Operations: Time Inversion (Reversal) (2)
  54. 54. Signal Operations: Combined Operations <ul><li>Signal f(at–b) can be realized in TWO ways: </li></ul><ul><ul><li>Time-shift f(t) by b  f(t–b) , then, time-scaled f(t–b) by a  f(at–b) </li></ul></ul><ul><ul><li>Time-scale f(t) by a  f(at) , then, time-shift f(at) by b/a  f[a(t – b/a)] </li></ul></ul><ul><ul><li>If a <0, it’s time inversion operation. </li></ul></ul>
  55. 55. Example <ul><li>Given the signal as shown in figure below. Plot </li></ul><ul><li>i. x ( t + 1) ii. x (1- t ) </li></ul><ul><li>iv. </li></ul>
  56. 56. Example (i) <ul><li>Shift to the left by one unit along the t axis. </li></ul>
  57. 57. Example (ii) <ul><li>Replace t with –t in x(t + 1)  x(-t + 1)  x(1-t). </li></ul><ul><li>It is obtained graphically by reflecting x(t + 1) about the y axis. </li></ul>
  58. 58. Example (iii) <ul><li>The signal x(3/2 t)  compression of x(t) by a factor of 2/3. </li></ul>
  59. 59. Example (iv) <ul><li>First, advance or shift to the left x(t) by 1 as shown in figure below. </li></ul><ul><li>Then, compress this shifted signal by a factor of 2/3. </li></ul>
  60. 60. Signal Operations
  61. 61. Some Useful Signal Models by Koay Fong Thai [email_address]
  62. 62. Signal Models: Unit Step Function <ul><li>Continuous-Time unit step function, u(t) : </li></ul><ul><li>u(t) is used to start a signal, f(t) at t =0  f(t) has a value of ZERO for t <0 </li></ul>
  63. 63. Signal Models: Unit Step Function (2) <ul><li>f(t) = e -at x u(t)  a causal form of e -at . </li></ul><ul><li>x = </li></ul>
  64. 64. Signal Models: Unit Step Function (3) <ul><li>Realize the rectangular pulse below: </li></ul>
  65. 65. Signal Models: Unit Impulse Function <ul><li>Continuous-Time unit impulse function,  (t ) is defined by P.A.M. Diarc: </li></ul><ul><li>We can visualize an impulse as a tall and narrow rectangular pulse of unit area . </li></ul><ul><li> When   0, the height is very large, 1/  . </li></ul>
  66. 66. Signal Models: Unit Impulse Function (2) <ul><li>A possible approximation to a unit impulse: An overall area that has been maintained at unity . </li></ul><ul><li>Multiplication of a function by an Impulse? </li></ul><ul><li>b  (t ) = 0; for all t  0 is an impulse function which the area is b . </li></ul>Graphically, it is represented by an arrow &quot;pointing to infinity&quot; at t=0 with its length equal to its area.
  67. 67. Signal Models: Unit Impulse Function (3) <ul><li>May use functions other than a rectangular pulse. Here are three example functions: </li></ul><ul><li>Note that the area under the pulse function must be unity . </li></ul>
  68. 68. Signal Models: Unit Ramp Function <ul><li>Unit ramp function is defined by: </li></ul><ul><ul><li>r(t) = t  u(t) </li></ul></ul><ul><li>Where can it be used? </li></ul>
  69. 69. Signal Models: Example <ul><li>Describe the signal below: </li></ul>
  70. 70. Signal Models: Example (2) <ul><li>x =t[u(t) – u(t-2)] </li></ul><ul><li> x = -2(t-3)[u(t-3) – u(t-2)] </li></ul><ul><li>t[u(t) – u(t-2)] - 2(t-3)[u(t-3) – u(t-2)] </li></ul>
  71. 71. Signal Models: Exponential Function e st <ul><li>Most important function in SNS where s is complex in general, s =  + j  </li></ul><ul><li>Therefore, e st = e (  + j  )t = e  t e j  t = e  t (cos  t + j sin  t) (Euler’s formula: e j  t = cos  t + j sin  t ) </li></ul><ul><li>If s  =  - j  , </li></ul><ul><li>e s  t = e (  - j  )t = e  t e - j  t = e  t (cos  t - j sin  t) </li></ul><ul><li>From the above, e  t cos  t = ½( e st + e -st ) </li></ul>
  72. 72. Signal Models: Exponential Function e st (2) <ul><li>Variable s is complex frequency . </li></ul><ul><li>e st = e (  + j  )t = e  t e j  t = e  t (cos  t + j sin  t) e s  t = e (  - j  )t = e  t e - j  t = e  t (cos  t - j sin  t) e  t cos  t = ½( e st + e -st ) </li></ul><ul><li>There are special cases of e st : </li></ul><ul><ul><li>A constant k = ke0t (s=0   =0,  =0) </li></ul></ul><ul><ul><li>A monotonic exponential e  t (  =0, s =  ) </li></ul></ul><ul><ul><li>A sinusoid cos  t (  =0, s =  j  ) </li></ul></ul><ul><ul><li>An exponentially varying sinusoid e  t cos  t ( s =   j  ) </li></ul></ul>
  73. 73. Signal Models: Exponential Function e st (3)
  74. 74. Signal Models: Exponential Function e st (4) <ul><li>In complex frequency plane: </li></ul>
  75. 75. Even and Odd Functions <ul><li>A function f e (t) is said to be an even function of t if f e (t) = f e (-t) </li></ul><ul><li>A function f o (t) is said to be an odd function of t if f o (t) = - f o (-t) </li></ul>
  76. 76. Even and Odd Functions: Properties <ul><li>Property: </li></ul><ul><li>Area: </li></ul><ul><ul><li>Even signal: </li></ul></ul><ul><ul><li>Odd signal: </li></ul></ul>
  77. 77. Even and Odd Components of a Signal (1) <ul><li>Every signal f(t) can be expressed as a sum of even and odd components because </li></ul><ul><li>Example, f(t) = e -at u(t) </li></ul>
  78. 78. Even and Odd Components of a Signal (2) <ul><li>Example, f(t) = e -at u(t)  casual? </li></ul>
  79. 79. Signal Models: Summary <ul><li>Unit step function, u(t) </li></ul><ul><li>Unit impulse function,  (t) </li></ul><ul><li>Unit ramp function, r(t) </li></ul><ul><li>Exponential function, e st </li></ul><ul><li>Even and off function </li></ul><ul><li>All these functions are used for CONTINUOUS-TIME signals!!!! </li></ul>
  80. 80. Sampling By Koay Fong Thai [email_address]
  81. 81. Sampling Theorem <ul><li>Sampling is the process of converting a continuous signal into a discrete signal . </li></ul>
  82. 82. Sampling Theorem (2) <ul><li>The sampling frequency or sampling rate, ( f s )  the number of samples per second taken from a continuous signal to make a discrete signal. </li></ul><ul><li>It is measured in hertz (Hz) . </li></ul>
  83. 83. Sampling Theorem (3) <ul><li>Sampling period or sampling time ( T )  The inverse of the sampling frequency which is the time between samples . </li></ul><ul><li>Given the sampling period T, the sampling frequency is given by </li></ul>
  84. 84. Sampling Theorem (4) <ul><li>The discrete-time signal x[n] is obtained by “taking-samples” of the analog signal x c (T) every T second. </li></ul><ul><li>x[n] = x c (nT) </li></ul><ul><li>It is measured in hertz (Hz) . </li></ul><ul><li>The relationship between the variable t of analog signal and the variable n of discrete-time signal is </li></ul>
  85. 85. Sampling Theorem (5) <ul><li>We refer to a system that implements the operation of the above equation as an ideal continuous-to-discrete-time (C/D) converter : </li></ul><ul><li>Block diagram representation of an ideal C/D Converter </li></ul>
  86. 86. Sampling Theorem (6) <ul><li>Sampling is represented as an impulse train modulation followed by the conversion into a sequence . </li></ul><ul><li>Figure below illustrates a mathematical representation of sampling with a periodic impulse train followed by a conversion to a discrete-time sequence. </li></ul>(a) Overall system; (b) x s (t) for two sampling rates. The dashed envelope represents x c (t); (c) The output sequence for the two different sampling rates.
  87. 87. Sampling Theorem (7) <ul><li>Figure below is the DT sequences of two CT signals at sampling frequency of 5 Hertz (samples/second). </li></ul>
  88. 88. Signal Operations for DT Signals By Koay Fong Thai [email_address]
  89. 89. Signal Operations <ul><li>Signal operations are operations on the time variable of the signal, involve simple modification of the independent variable . </li></ul><ul><ul><li>Time Shifting </li></ul></ul><ul><ul><li>Time Scaling </li></ul></ul><ul><ul><li>Time Inversion (Reversal) </li></ul></ul><ul><ul><li>Combined operations </li></ul></ul>
  90. 90. Signal Operations: Time Shifting <ul><li>Shifting of a signal in time </li></ul>
  91. 91. Signal Operations: Time Scaling <ul><li>Compress es and expanse s a signal by multiplying the time variable by some integers. </li></ul><ul><li>x(k)  x(  k) </li></ul><ul><ul><li>If  >1, the signal becomes narrower  compression  data losses  decimation ( 抽取 ) or downsampling ( 下降抽样 ). </li></ul></ul><ul><ul><li>If  <1, the signal becomes wider  expansion . </li></ul></ul><ul><ul><ul><li>Insert missing samples using an interpolation formula  interpolation ( 插值 ) or upsampling ( 后续采样 ) </li></ul></ul></ul>
  92. 92. Signal Operations: Time Scaling (2)
  93. 93. Signal Operations: Time Inversion (Reversal) <ul><li>Reversal of the time axis, or folding/flipping the signal (mirror image) over the y-axis. </li></ul>
  94. 94. Some Useful Signal Models By Koay Fong Thai [email_address]
  95. 95. Signal Models: Unit Step Function <ul><li>Discrete-Time unit step function/sequence, u(k) : </li></ul><ul><li>The unit step is the running sum of an impulse: </li></ul>
  96. 96. Signal Models: Unit Impulse Function <ul><li>Discrete-Time unit impulse function/sequence,  (k) : </li></ul><ul><li>The unit impulse is the first-difference of a unit step: </li></ul>
  97. 97. Signal Models: DT Exponential Function  k <ul><li>In CT SNS, CT exponential e st can be expressed in another form,  t where  t = e st . </li></ul><ul><li>However, in DT SNS, it is proven that  k is more convenient than e  k (k is the integer). </li></ul>
  98. 98. Signal Models: DT Exponential Function  k (2)
  99. 99. Signal Models : DT Exponential Function  k (3)
  100. 100. Questions and Answer <ul><li>Any questions? </li></ul>

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