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Section 1.3 Asynchronous Circuits Exercises

Section 1.3 Asynchronous Circuits Exercises. Alfredo Benso Politecnico di Torino, Italy Alfredo.benso@polito.it. #1. Find a minimal race free state coding for the following transition table. Also, find the boolean functions representing the future state. c=0 c=1. ab.

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Section 1.3 Asynchronous Circuits Exercises

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  1. Section 1.3Asynchronous CircuitsExercises Alfredo Benso Politecnico di Torino, Italy Alfredo.benso@polito.it

  2. #1 • Find a minimal race free state coding for the following transition table. • Also, find the boolean functions representing the future state. c=0 c=1 ab 00 01 11 10 00 01 11 10 A A A B C D A A B B A - C B A D C B C C C C C B B D C D A C - A D D A D

  3. #1: stable states c=0 c=1 00 01 11 10 00 01 11 10 A A A B C D A A B B A - C B A D C B C C C C C B B D C D A C - A D D A D

  4. #1: Vicinity graph D C c=0 c=1 00 01 11 10 00 01 11 10 A A A B C D A A B B A - C B A D C B C C C C C B B D C D A C - A D D A D A B

  5. #1: Minimize the vicinity graph C D c=0 c=1 00 01 11 10 00 01 11 10 A A A B C D A A B B A - C B A D C B C C C C C B B D C D A C - A D D A D A B

  6. #1: Minimize the vicinity graph C D 10 11 00 01 c=0 c=1 00 01 11 10 00 01 11 10 A A A B C B A A B B A - C B D D A B C C C C C D B A C D B C - C D D B D A B

  7. #2 • Design the reduced transition table of an asynchronous circuit functioning in fundamental mode. The circuit has three inputs a, b, and c, and one output U, which is 1 only if all inputs are equal to ‘1’, and they went to ‘1’ in the order a, b, and c. U returns to 0 when all inputs returned to ‘0’. • Also, find the minimum number of state variables required to have a race free state assignment, and find a possible state assignment.

  8. #2: Transition table • We have to recognize the following sequence: 000  100  110  111 c=0 c=1 00 01 11 10 00 01 11 10 A Waiting for 100 B Waiting for 110 C Waiting for 111 D OK! Now waiting for all 0 E Wrong sequence.

  9. #2: Transition table • We have to recognize the following sequence: 000  100  110  111 c=0 c=1 00 01 11 10 00 01 11 10 A Waiting for 100 B Waiting for 110 C Waiting for 111 D OK! Now waiting for all 0 E Wrong sequence.

  10. #2: Minimize the transition table c=0 c=1 00 01 11 10 00 01 11 10 A Waiting for 100 B Waiting for 110 C Waiting for 111 D OK! Now waiting for all 0 E Wrong sequence. States A and B are compatible, so I can collapse them in one state

  11. #2: Minimize the transition table c=0 c=1 00 01 11 10 00 01 11 10 A Waiting for 100 & 110 C Waiting for 111 D OK! Now waiting for all 0 E Wrong sequence. States A and B are compatible, so I can collapse them in one state

  12. #2: Vicinity graph E D c=0 c=1 00 01 11 10 00 01 11 10 A C A C D E A-C and C-E are essential, so we can try to eliminate A-E and C-D

  13. #2: Vicinity graph E D 00 10 11 01 c=0 c=1 A C 00 01 11 10 00 01 11 10 A C D E

  14. #3 • Find the Future State and Output functions of an asynchronous circuit functioning in fundamental mode. The circuit has three inputs X, Y, and Z and one output U, which goes to ‘1’ only when: • There is a rising edge of X AND • While the value of X was ‘0’, Y and Z followed the sequence 00 - 10 – 11 (even if BEFORE or AFTER this sequence Y and Z changed to different values). • U returns to ‘0’ as soon as X returns to ‘0’. • The circuit must have a race free state assignment.

  15. #3: Transition Table • We have to recognize the following sequence on Y and Z with X=0: 00  10  11 X=0 X=1 00 01 11 10 00 01 11 10 A Wait for 00 B Wait for 10 C Wait for 11 D Now wait for rising edge X E Wait for X to return to 0

  16. #3: Transition Table Minimization • B and C are compatible so they can be collapsed in one state X=0 X=1 00 01 11 10 00 01 11 10 A Wait for 00 B Wait for 10 & 11 D Now wait for rising edge X E Wait for X to return to 0

  17. #3: Transition Table Minimization D E X=0 X=1 00 01 11 10 00 01 11 10 A B D E A B I can remove E-B going trough E - A - B

  18. #3: Transition Table Minimization D E 11 10 X=0 X=1 00 01 11 10 00 01 11 10 A B D E A B 00 01

  19. #3: Output function X=0 X=1 00 01 11 10 00 01 11 10 00 01 11 10

  20. #3: Output function X=0 X=1 00 01 11 10 00 01 11 10 00 01 11 10 U = X S0

  21. #3: Future state function X=0 X=1 00 01 11 10 00 01 11 10 00 01 11 10

  22. #3: Future state function X=0 X=1 00 01 11 10 00 01 11 10 00 01 11 10 S0 = S0S1 + XS0 + YZS1

  23. #3: Future state function X=0 X=1 00 01 11 10 00 01 11 10 00 01 11 10 S1 = S0S1X + XYZS1 + XYZS0 + XYZS1

  24. #5 • Design the reduced transition table of an asynchronous circuit functioning in fundamental mode. The circuit has two inputs X and Y, and one output U, which is 1 only if X=1 and, during the last two rising-edges of X, Y maintained the same value. • Also, find out the minimum number of state variables required to have a race free state assignment.

  25. #5: Primitive Transition table XY 00 01 11 10 A A,0 A,0 E,0 B,0 wait for the first transition of X B C,0 C, 0 B,0 B,0 1st trans. & Y=0 – wait for X to go low C C,0 C,0 E,0 D,- wait for another transition of X D H,- H,- D,1 D,1 2nd trans. & Y=0 – wait for X to go low E F,0 F,0 E,0 E,0 1st trans. & Y=1 –wait for X to go low F F,0 F,0 G,- B,0 wait for another transition G I,- I,- G,1 G,1 2nd trans. & Y=1 – wait for X to go low H H,0 H,0 E,0 D,- last 2 trans Y=0 – wait for a new transition I I,0 I,0 G,- B,0 last 2 trans Y=1 – wait for a new transition

  26. #5: Reduced Transition table XW 00 01 11 10 A A,0 A,0 E,0 B,0 Equivalent states: C & H  C B C,0 C, 0 B,0 B,0 F & I  F C C,0 C,0 E,0 D,- No compatible states D H,- H,- D,1 D,1 E F,0 F,0 E,0 E,0 F F,0 F,0 G,- B,0 G I,- I,- G,1 G,1 H H,0 H,0 E,0 D,- I I,0 I,0 G,- B,0

  27. #5: Reduced Transition table XW 00 01 11 10 A A,0 A,0 E,0 B,0 Equivalent states: C & H  C B C,0 C, 0 B,0 B,0 F & I  F C C,0 C,0 E,0 D,- No compatible states D H,- H,- D,1 D,1 E F,0 F,0 E,0 E,0 F F,0 F,0 G,- B,0 G I,- I,- G,1 G,1 H H,0 H,0 E,0 D,- I I,0 I,0 G,- B,0

  28. #5: Reduced Transition table XW 00 01 11 10 A A,0 A,0 E,0 B,0 Equivalent states: C & H  C B C,0 C, 0 B,0 B,0 F & I  F C C,0 C,0 E,0 D,- No compatible states D H,- H,- D,1 D,1 E F,0 F,0 E,0 E,0 F F,0 F,0 G,- B,0 G I,- I,- G,1 G,1 H H,0 H,0 E,0 D,- I I,0 I,0 G,- B,0

  29. #5: Reduced Transition table XW 00 01 11 10 A A,0 A,0 E,0 B,0 Equivalent states: C & H  C B C,0 C, 0 B,0 B,0 F & I  F C C,0 C,0 E,0 D,- No compatible states D C,- C,- D,1 D,1 E F,0 F,0 E,0 E,0 F F,0 F,0 G,- B,0 G F,- F,- G,1 G,1

  30. #5: Vicinity Graph • Max outdegree is 3. I need at least three state variables A B C D E F G

  31. #5: Vicinity Graph B C D E F G A ? ? 011 111 ? ? 001 101 ? ? 110 010 ? ? 000 100 • It seems that there is no solution with three variables. • Either we need 4 state variables

  32. #6 • Design the gate level model of an asynchronous circuit functioning in fundamental mode. The circuit has two inputs P (pulse) and R (reset) and one output Z, which is normally at ‘0’. Z follows the following rules: • It goes to ‘1’ when a rising edge is detected on P and R=0. • It goes to ‘0’ when R=1.

  33. #6: Transition table P R Let’s start with the correct sequence. R=0 and P=0 and we wait for a transition on P. 00 01 11 10 A 0 - - B - A C - C - B 1 B 1 B We can stay in B until R changes back to 1. C If R goes to 1, Z goes to 0 but we have to wait for R to go to 0 again before checking for another rising edge on P

  34. #6: Transition table P R We can exit C only when both R and P are ‘0’. 00 01 11 10 A 0 C 0 - - B - A C - C - B 1 B 1 If R goes to 1 while we are waiting for a rising edge on P, we go out-of-sequence. B A 0 C 0 C 0 C 0 C

  35. #6: Vicinity graph 00 01 P R 00 01 11 10 A 2 C A 0 C 0 - - B - A C - C - B 1 B 1 B 1 2 A 0 C 0 C 0 C 0 C B

  36. #6: Vicinity graph P R 00 01 11 10 A 2 C A 0 C 0 C 0 B - A A 0 A 0 B 1 B 1 B 1 2 A 0 C 0 C 0 C 0 C B

  37. #6: Race free state assignment P R 00 10 00 01 11 10 A C A 0 C 0 C 0 B - A A 0 A 0 B 1 B 1 B A 0 C 0 C 0 C 0 C 01 B

  38. #6: Output function P R P R 00 01 11 10 00 01 11 10 S0 S1 A 0 C 0 C 0 B - 00 00 A 0 A 0 B 1 B 1 01 01 A 0 C 0 C 0 C 0 10 11 10 Z = S0S1R

  39. #6: State function P R P R 00 01 11 10 00 01 11 10 S0 S1 A 0 C 0 C 0 B - 00 00 A 0 A 0 B 1 B 1 01 01 A 0 C 0 C 0 C 0 10 11 10

  40. #6: S0 variable S0 = S0 P + S1 R P R P R 00 01 11 10 00 01 11 10 S0 S1 A 0 C 0 C 0 B - 00 00 A 0 A 0 B 1 B 1 01 01 A 0 C 0 C 0 C 0 10 11 10

  41. #6: S1 variable P R P R 00 01 11 10 00 01 11 10 S0 S1 A 0 C 0 C 0 B - 00 00 A 0 A 0 B 1 B 1 01 01 A 0 C 0 C 0 C 0 10 11 10 S1 = S1 R + S0 P R

  42. #6: Final circuit Z P R S0 S1

  43. #10 • Design an asynchronous circuit functioning in fundamental mode. The circuit has two inputs A and B, and one output Z, which is normally at ‘0’. Z changes value when it recognize the sequence 00 – 01 – 00 on the inputs. 00 can be considered concurrently the end of a sequence and the beginning of a new one. • Show: • A Minimized Transition table • A race free state assignment • The Boolean function for Z

  44. #10: Transition Table Out of sequence 00 01 11 10 A Wait for 00 Wait for 01; Sequence OK B Wait for 00 C Sequence OK; wait for 01 D Wait for 00 E F

  45. #10: Transition Table 00 01 11 10 A Out of sequence; Wait for 00 U=0 Wait for 01 Sequence OK B Wait for 00 C Sequence OK; wait for 01 D U=1 Wait for 00 E F Out of sequence; wait for 00

  46. #10: Vicinity Graph • There are no equivalent or compatible states. 00 01 11 10 A C A B B C D E E F F D

  47. #10: Vicinity Graph • We can eliminate A – C and F - E 00 01 11 10 A C A B B C D E E F F D

  48. #10: Vicinity Graph • We can eliminate A – C and F - E 00 01 11 10 A C A B B C D E E F F D

  49. #10: Vicinity Graph • A race free state assignment is: 00 01 11 10 111 010 A B 110 C D 100 E F 001 000

  50. #10: Output Function • Be careful! To cover the table, you have to order the rows correctly: it has to be a K-map! 00 01 11 10 111 110 010 000 100 001

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