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# sol9 - Chapter 9 9.1 The next-state and output expressions...

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Chapter 9 9.1. The next-state and output expressions for the circuit in Figure P9.1 are Y 1 = w 1 + y 1 y 2 Y 2 = w 2 + y 1 + w 1 y 2 z 1 = y 1 z 2 = y 2 This gives the excitation table Present Next state state w 2 w 1 = 00 01 10 11 z 2 z 1 y 2 y 1 Y 2 Y 1 A 00 11 10 11 10 11 B 01 11 11 0 1 0 1 10 C 10 11 1 0 11 1 0 01 D 11 1 1 10 01 10 00 The resulting flow table is Present Next state state w 2 w 1 = 00 01 10 11 z 2 z 1 A D C D C 11 B D D B B 10 C D C D C 01 D D C B C 00 The behavior is the same as described in the flow table in Figure 9.21 a , if the state interchanges A D and B C are made. 9-1

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9.2. The waveforms are 1 2 3 2 1 2 3 2 c z 1 z 2 The flow table is Present Next state Outputs, z 2 z 1 state C = 0 1 0 1 0 1 0 00 10 1 1 0 01 00 The circuit generates a non-overlapping 2-phase clock. 9.3. The partitioning procedure gives P 1 = ( ADGJMPT )( BEHR )( CF )( ILOSV )( KNU ) P 2 = ( AD )( GP )( JMT )( B )( E )( HR )( C )( F )( ILOSV )( KNU ) P 3 = ( A )( D )( GP )( JMT )( B )( E )( HR )( C )( F )( ILOSV )( KNU ) P 4 = P 3 This gives the flow table Present Next state state w 2 w 1 = 00 01 10 11 z A A B C - 0 B D B - - 0 C G - C - 0 D D E F - 0 E G E - - 0 F J - F - 0 G G H I - 0 H J H - - 0 I A - I - 1 J J K I - 0 K A K - - 1 9-2
The corresponding merger diagram is E D A B C G J I H F K This leads to the reduced flow table Present Next state state w 2 w 1 = 00 01 10 11 z A A B C - 0 B D B - - 0 C G C C - 0 D D C F - 0 F J F F - 0 G G F I - 0 I A I I - 1 J J I I - 0 9.4. The partitioning procedure gives P 1 = ( AF )( BEGL )( CJ )( DK )( HM ) P 2 = ( AF )( BG )( EL )( CJ )( DK )( HM ) P 3 = ( A )( F )( BG )( EL )( CJ )( DK )( HM ) P 4 = P 3 9-3

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Replacing states B and G , E and L , C and J , D and K , and H and M with new states B , E , C , D , and H , respectively, produces the following flow table: Present Next state Output state w 2 w 1 = 00 01 10 11 z A A B C - 0 B D B - H 0 C F - C H 0 D D E C - 1 E A E - H 0 F F E C - 0 H - B C H 1 The merger diagram is A H B C F D E Only C and F can be merged if the Moore model is to be preserved. Therefore, the reduced flow table is Present Next state state w 2 w 1 = 00 01 10 11 z A A B C - 0 B D B - H 0 C C E C H 0 D D E C - 1 E A E - H 0 H - B C H 1 9-4
9.5. Relabel the flow table as Present Next state Output z state w 2 w 1 = 00 01 10 11 00 01 10 11 A 1 3 7 2 0 - 1 1 B 5 3 4 8 0 1 0 0 C 5 6 4 2 0 1 0 1 D 1 6 7 8 - - 1 0 The transition diagram is D A B 3 1 7 , 4 5 , C 2 8 6 The diagonal transitions cannot be avoided without introducing additional states. A possible modification is D A B 3 1 7 , 4 5 , C 2 8 6 4 5 , G 8 F 6 E This transition diagram can be embedded onto a 3-cube as follows: D A B F G C E 8 6 3 4 5 , 1 7 , 8 y 2 y 3 y 1 2 6 4 5 , 9-5

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Then the modified flow table is Present Next state Output z state w 2 w 1 = 00 01 10 11 00 01 10 11 A A B D A 0 - 1 1 B G B B F 0 1 0 0 C C C G A 0 1 0 1 D A E D D - - 1 0 E - C - - - 1 - - F - - - D - - - 0 G C - B - 0 - 0 - The excitation table is Present Next state Output state w 2
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