1. Linear Sequential Machines1 LINEAR SEQUENTIAL MACHINE AND REDUCTION OF LINEAR SEQUENTIAL MACHINE

2. Linear Sequential Machines2 (I) Time Domain (2) Frequency Domain

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8. 8 Module 2 Operation (0 and 1) and so on…..

9. Linear Sequential Machines9 Note: A is always square matrix and gives number of delays Number of columns in B gives number of inputs Number of rows in C gives number of output Order of A and C should be compatible with present state and order of B and D should be compatible with input

10. Linear Sequential Machines10 Example 1:

11. Linear Sequential Machines11 State Space Representation:

12. Linear Sequential Machines12 Note: Y’s represent Next State y’s represent Present State z’s represent Output State Here, we have 3 delays, one input and two outputs

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14. Linear Sequential Machines14 Representation:

15. Linear Sequential Machines15 Example 2: Determine Characterizing matrices A,B,C and D from the given linear circuit diagram Hint: From fig.,A=2 X 2,B=2 X 1,C=1 X 2

16. Linear Sequential Machines16 Y 1 = y 2 + x Y 2 = y 1 + x z = y 1 + x From the figure,

17. Linear Sequential Machines17 Solution:

18. Linear Sequential Machines18 Reduction of Linear Sequential Machines Reduction of linear sequential machines means reducing or minimizing the number of delays

19. Linear Sequential Machines19 Example 3: Minimize (reduce) the linear machine given by following characterizing matrices A,B,C,D (Example Kohavi Page-580)

20. Linear Sequential Machines20 Reduction: The rank of the diagnostic matrix is 3, and hence the dimenstion of given linear machine CANNOT be reduced

21. Linear Sequential Machines21 Example 3: Minimize the linear machine given by following characterizing matrices A,B,C,D (Example Kohavi Page-582)

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26. Linear Sequential Machines26 A* = TAR B* = TB C* = CR D* = D

27. Linear Sequential Machines27 Note: In reduction number of Inputs and Outputs do not change and only the number of delays change

28. Linear Sequential Machines28 Example 4: Minimize the linear machine given by following characterizing matrices A,B,C,D (Example Kohavi Page-584)

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34. Linear Sequential Machines34 A* = TAR B* = TB C* = CR D* = D

35. Linear Sequential Machines35 Example 5: Minimize the linear machine given by following characterizing matrices A,B,C,D (Example Kohavi 15 - 19)

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40. Linear Sequential Machines40 A* = TAR B* = TB C* = CR D* = D

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43. Linear Sequential Machines43 Example 6: Minimize the linear machine given by following characterizing matrices A,B,C,D (Example Kohavi 15 - 20)

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48. Linear Sequential Machines48 A* = TAR B* = TB C* = CR D* = D

49. Linear Sequential Machines49 Example 7: Minimize the linear machine given by following characterizing matrices A,B,C,D (Example Kohavi 15 - 21)

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52. Linear Sequential Machines52 Determine R such that TR = I (Here R=T _1 )

53. Linear Sequential Machines53 A* = TAR B* = TB C* = CR D* = D

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55. Linear Sequential Machines55 Realisation:

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58. Linear Sequential Machines58 Realisation

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60. Linear Sequential Machines60 Note: Original and reduced system will have same transfer function after cancellation of common terms (if there) in the numerator and denominator of the Delay Transfer Function Matrix. Also, the number of columns in DTF matrix gives number of inputs and number of rows in DTF matrix gives the number of inputs

61. Linear Sequential Machines61 Analogy of Control theory and Sequential Circuits but for this course, we restrict ourselves purely to sequential circuits

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63. Linear Sequential Machines63 Example 8: Find the DTF matrix of a linear machine given by following characterizing matrices

64. Linear Sequential Machines64 Here D is the delay operator

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66. Linear Sequential Machines66 Thus, the DTF matrix is given by,

67. Linear Sequential Machines67 Identification of Sequential Machines There are two methods to solve these problems With Assumption With Out Assumption

68. Linear Sequential Machines68 Example 1: Determine whether the machine given in the following table is linear and thus find A,B,C,D

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79. Linear Sequential Machines79 Example 1: Determine whether the machine given in the following table is linear and thus find A,B,C,D (Example Kohavi Page 591)

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87. Linear Sequential Machines87,

88. Linear Sequential Machines88 Example 2: Determine whether the machine given in the following table is linear and thus find A,B,C,D (Example Kohavi Page 592)

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98. Linear Sequential Machines98 Example 2: Determine whether the machine given in the following table is linear and if it is, show a linear realization (Problem Kohavi 15-25(b))

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104. Linear Sequential Machines104 Y h 00, Y g 00 A= 0 0 1 0 0 1 0 1 1

105. Linear Sequential Machines105 C= 0 0 1 0 0 1

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115. Linear Sequential Machines115 Example 3: Determine whether the machine given in the following table is linear and if it is, show a linear realization (Problem Kohavi 15-25(a))

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124. Linear Sequential Machines124 Example 3: Test the given machine for linearity.In particular, determine if the state transistions are linear and if the outputs are linear (Problem Kohavi 15-26)

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134. Linear Sequential Machines134 Output and State Equations:

135. Linear Sequential Machines135 Verifying whether state transitions and Outputs are linear

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144. Linear Sequential Machines144 Note: Case (viii) is not true according to the given table Thus, THE GIVEN MACHINE IS STATE LINEAR BUT NOT OUTPUT LINEAR

145. Linear Sequential Machines145 HUFFMAN’S MELEY PROCEDURE FOR REDUCTION OF STATES OF SEQUENTIAL MACHINES

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147. Linear Sequential Machines147 Group the states into sets where all members of each set have the same output combination 12 (a,d) (b,c,f,g) Applying the procedure again, (a(1,2),d(2,2))(b(1,2),c(1,2),f(1,2),g(1,2))

148. Linear Sequential Machines148 Here a(1,2), means, a goes to a when X = 0 andgoes to b when X = 1 from the table, and a is in group 1 and b is in group2, so, we write as a(1,2) Applying the procedure again 123 ad (b,c,f,g) (b(2,3),c(1,3),f(2,3),g(1,3))

149. Linear Sequential Machines149 Applying the procedure again, 1234 ad b,f c,g (b(2,4),f(2,4)) (c(1,3),g(1,3)) so, there is no need to apply the previous procedure again

150. Linear Sequential Machines150 Finally, f can be replaced by b and g can be replaced by c