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# Class Note for ECE 380 at UA-Digital Logic(5)

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This 2 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Alabama - Tuscaloosa taught by a professor in Fall. Since its upload, it has received 12 views.

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Date Created: 02/06/15

ECE380 Digital Logic State Minimization amine AcumvmerEnginSQring Dr in J Jackson LedurelZl State minimization o For simple FSMs it is easy to see from the state diagram that the number of states used is the minimum possible 7 FSMs for counters are good examples 0 For more complex FSMs it is likely that an initial state diagram may have more states than are necessaw to perform a required function 0 Minimizing states is of interest because fewer states implies fewer flipflops to implement the circuit 7 Complexity of combinational logic may also be reduced 0 Instead of trying to show which states are equivalent it is often easier to show which states are definitely not equivalent 7 This can be exploited to define a minimizau on procedure amine ABDMDmHEnginSQring Dr in J Jaclson LedurelZZ State equivalence 0 Definition Two states 5 and S are equivalent if and only if for every possible input sequence the same output sequence will be produced regardless of whether 5 or S is the initial state 0 Ifan input w0 is applied to an FSM in state S and the FSM transitions to state SH then SM is termed a 0successor of 5 Similarly ifw1 and the FSM transitions to Sy then 5V is a 1successor of S o The successors of S are its ksuccessors 0 With one input k can only be 0 or 1 but if there are multiple inputs then k represents all the valuations of the inputs amine AcumvmerEnginSQring Dr in J Jackson LedureClZCl Partitioning minimization 0 From the equivalence definition if S and SJ are equivalent then their corresponding ksuccessors are also equivalent Using this we can construct a minimization procedure that involves considering the states of the machine as a set and then breaking that set into partitions that comprise subsets that are definitely not equivalent Definition A partition consists ofone or more blocks where each block comprises a subset of states that may be equivalent but the states in one block are definitely not equivalent to the states in other blocks amine ABDMDmHEnginSQring Dr in J Jaclson LedureClZA Partition minimization example 0 Consider the following state table Presen Next state Output sham W 0 W 1 z A B c 1 B D F 1 c F E 0 D B G 1 E F c 0 F E D 0 G F G 0 An initial partition contains all the states in a single block P1ABCDEFG amine AcumvmerEnginSQring Dr in J Jackson LectureClZ Partition minimization example o The next partition separates the states that have different outputs P2ABDCEFG 0 Now consider all 0 and lsuccessors of the states in each block For ABD the Osuccessors are BDB I Since these are all in the same block we must still consider A B and D equivalent The 1successors of ABD are CFG We must still cons der A B and D equivalent Now consider the CEFG block amine ABDMDmHEnginSQring Dr in J Jaclson Lecturele imization example o P2ABDCEFG o For CEFG the 0successors are FEFF which are all in the same block in P2 7 c E F and G must still be considered equivalent 0 The 1successors of CEFG are ECDG 7 Since these are not in the same block in l2 then at least one of the slates in CEFG is not equivalent to the others state F must be different from c E and G because ifs 1s successorD is in a different block than E c and G 0 Therefore P3ABDCEGF 0 At this point we know that state F is unique since it is in a block by itself acclnca AEoNDmerEngmasrlng Dr D J Jackson LectureClZT Partition minimization example 0 P3ABDCEGF o The process repeats yielding the following 0successors of ABD are BDB A B and D are still considered equivalent 1successors of ABD are CFG which are not in the same block I B Cannot be equivalent to A and D Since F is in a different block than C and G The 0 and 1successors of CEG are FFF and ECG c E and G must still be considered equivalent Thus P4ADBCEGF acclnca AEoNDmaEngmssring Dr D J Jacison Lecturele imization example o If we repeat the process to check the 0 and lsuccessors of the blocks AD and CEG we find that P5ADBCEGF 0 Since P5P4 and no new blocks are generated it follows that states in each block are equivalent A and D are equivalent C E and G are equivalent acclnca AEoNDmerEngmasrlng Dr D J Jackson Lecturele Partition minimization example o The state table can be rewritten removing the rows for D E and G and replacing all occurrences of D with A and all occurrences of E or G with C o The resulting state table is as follows Presen Emma Output sham w0 w1 z A B c F nmgtw gt010 OOHH acclnca AEoNDmaEngmssring Dr D J Jacison La urelzl Design Example 0 Determine which states if any are equivalent in the following Moore state diagram acclnca AEoNDmerEngmasrlng Dr D J Jackson Lecturele

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