9..7 hapter 6 Introduction to state machines Dr.-Ing. Stefan Werner Table of content hapter : Switching Algebra hapter : Logical Levels, Timing & Delays hapter 3: Karnaugh-Veitch-Maps hapter 4: ombinational ircuit Design hapter 5: Latches and Flip Flops hapter 6: Finite State Machines hapter 7: Basic Sequential ircuits hapter 9: Number Systems State Tables hapter 9: Binary Arithmetic hapter : Binary odes Introduction Mealy and Moore Machines State Graphs Transitions in State Graphs Machines and Flip-Flops Synthesizing a Finite State Machine Introduction to State Machine Minimization of 4
9..7 Example: RS-FF, as a FSM S R n n Functionality new State depends on actual state store X X reset set forbidden State Z with corresponding Output Y = State Z with corresponding Output Y = X States Z Output Y 3of 4 Example: RS-FF, as a FSM S R n n Functionality store reset transition function: g(x t, Z t ) if (r,s)=(i,) set g((s,r), ) = if (r,s)=(,) I if (r,s)=(,i) X forbidden Set X=(s,r)={, X I, I} Output Set Y=()={,I} State Set Z=(z)={,I} g((s,r), I) = if (r,s)=(i,) I if (r,s)=(,) I if (r,s)=(,i) 4of 4
9..7 State tables The state table (also called transition table or state transition table) contains the same information as the State Diagram hld holds information if i about Present State and Next State plus input lists all possible state transitions for all possible states for a table using one column for Present state, Next State and consits of n s. n i rows with -n i := no. of input variables -n s := no. of states. Present State S S S3 S3 Next State S3 S 5of 4 State tables A different form of a state table uses one column for the actual state and one column per input variable Such a table consists of ( n i ) columns and n s rows Present State S S S3 S3 Next State S3 S Present State I S S3 S3 S Next State 6of 4 3
9..7 State tables Output Values Moore Machine To assign the outputs there are differences, depending on the type of Machine: for a Moore Machine the outputs are written to an extra table. state table and output table can easily be combined to one Present State Output S S3 Output table of a Moore machine Present State I S S3 S3 S Output 7of 4 State tables Output Values Mealy Machine To assign the outputs there are differences, depending on the type of Machine for a Mealy Machine the output can be provided in an extra table or they can be included in the state table. 8of 4 4
9..7 State tables Output Values Mealy Machine Present Next Output Present State State State I S S / / S / S3 / S3 / S / S3 S3 Output included in S3 S the state table Output provided in an extra table (best choice depends on the actual complexity of the machine) 9of 4 Transition function and transition table Present State x x.. x j.. x m S S S 3 : s n ij =g(xi,zj) S i : S n of 4 5
9..7 Output table Present State x x.. x j.. x m S S S 3 : y ij =f(x i,s j ) S i : S n of 4 State table of a Mealy-Machine Present State x x.. x j.. x m S S S 3 : s ij n / y ij S i : S n of 4 6
9..7 State table of a Moore-Machine Present Output State x x.. x j.. x m S S S 3 : S i : S n s ij n =h(xi,zj) y y y 3... y k... y l 3 of 4 Machines and Flip Flops the simplest form of a Finite State Machine is the flip-flop. flip-flops are the memory of sequential circuits the part that provides the ability of having a state. Example: JK-Flip Flop l J K n Functionality X X n Disabled Preserve n s Preserve Set Reset n Toggle haracteristic equation: n n n = J K 4 of 4 7
9..7 Machines and Flip Flops l J K n Functionality X X n Disabled Preserve n s Preserve Set Reset n Toggle JK-FF has two input lines J and K JK-FF has one (two) Output lines (not ) JK-FF can be in one of two possible states S = Set =I = Reset = uncoded coded Output depends only on the actual state => JK-FF is a Moore Machine 5 of 4 Machines and Flip Flops - State transition table of JK-FF - l J K n JK-FF remains in its actual state Functionality X X n Disabled Preserve if J= and K= n Store s Preserve S = Set with output =I Set = Reset with Output = Reset n Toggle Being in S, JK-FF will transit to state regardless to the K input General structure of table with K= it transits to with k= it remains in S Out Present JK put State I I II S S S S S I Being in, JK-FF will transit to state S regardless to the J input with J= it transits to S with J= it remains in 6 of 4 8
9..7 Machines and Flip Flops - State Graph of JK-FF - State tranistion table of JK-FF X Present State S JK Out put I I II S S S S I S X J = don t care K= J = K= Note, that an input that does not have any influence on a certain transition is marked with an X at the according arrow (don t care). 7 of 4 Machines and Flip Flops Most machines contain more than only one FF Each of the machine s state is then assigned to exactly one possible combination of the flip-flop s states. Therefore the number of possible states grows by the power of with every new flip-flop: n flip-flops k = n possible states k states wanted n = (ld k) flip-flops needed The assignment flip-flop s states machine s state leads to socalled coded states and is basically free. 8 of 4 9
9..7 oding states Example Present State I Z Z Z Z Z Z3 Z3 Z Z Present I Output Output Three States => ld (3) = Flip Flops/ Variables needed (A,B) Even if coding is free of choice, there are usually smart assignments and less smart assignments The first intention is usually to represent the machine s states in such a way that the machine s output can be derived from the flip-flop s output with minimum effort. 9 of 4 Synthesizing a Finite State Machine The development of a Finite State Machine usually includes the steps translating ti the problem into a state t graph putting up state transition tables and output tables finding the simplified functions for the state transitions and outputs deducing the functions for the flip-flop s feedback circuit from the state transitions draw an appropriate circuit of 4
9..7 Synthesizing a Finite State Machine Example: traffic light Development of the control circuit for a traffic light. A traffic light basically runs through a sequence of several different states. As an extra; a control input can switch the traffic light into night mode => the yellow lamp blinks. Red Two different modes: day mode night mode Mode Traffic Light ontroller Yellow lock Green of 4 Synthesizing a Finite State Machine Example: traffic light Two different sequences depending on the mode Day mode Night mode Red S : Red Mode S : Red Yellow S4 : Yellow Traffic Light Yellow ontroller : Green S5 : All off S3 : Yellow S4 : Yellow S : Red lock Green the control line (mode) is used to switch between the modes. If control is, the traffic light shall remain in the present mode, it should switch modes. In day mode: light should start with yellow and continue with red. by power-on or in an undefined state: light should remain dark and switch to the day cycle s yellow with the next clock pulse. of 4
9..7 State Graph for traffic light control 3 of 4 Uncoded state table for traffic light control Present State (ontrol) I Output S S S4 R S S4 R,Y S3 S4 G Table is uncoded - 6 states: => ld(6) = 3 Variables - Output: => 3 Variables S3 S S4 Y S4 S5 S3 Y S5 S4 S3 Dark 4 of 4
9..7 oding the state table for traffic light control The states are to be expressed oded output table with the states of the flip-flops flops. State Output ( ) (RYG) State oded State ( ) S S S3 S4 S5 5 of 4 oded state transition table for traffic light control Present State (,, ) (ontrol) I Output (R,Y,G) I I I I I I II I II I I II I I I II II I II I II i,=f(s i,x i ) Remember:,, 6 of 4 3
9..7 How to read the transition functions from the state transition table Present State (,, ) (ontrol) I Output (R,Y,G) I I I I I I II I II I I II I I I II II I i,=f(s i,x i ) = II I II,, 7 of 4 Transition and output functions for traffic light control = = = R = Y = G = Now: Simplify with KV-Map Keep in mind: Not all combinations are used during the coding of the states 8 of 4 4
9..7 How to read the equations = X X X X = X X X X = = ( ) 9 of 4 All Functions for traffic light control = = ( ) = R = Y = G = 3 of 4 5
9..7 Flip Flop Feedback Functions Next: Design decision required for type of FF. Here: JK-Flip Flop n n = J K n J = K = n n = J K = J = K = n clk J clk K 3 of 4 Flip Flop Feedback Functions Next: Draw the circuit R = J = J = = = K J = ( ) = ( ) Y = K = K = = G = 3 of 4 6
9..7 hapter : Switching Algebra hapter : Logical Levels, Timing & Delays hapter 3: Karnaugh-Veitch-Maps hapter 4: ombinational ircuit Design hapter 5: Latches and Flip Flops hapter 6: Finite State Machines hapter 7: State Machine Minimization hapter 8: Basic Sequential ircuits hapter 9: Number Systems hapter : Binary Arithmetic hapter : Binary odes 33 of 4 Introduction to state machine miminization The minimization of a state machine means to reduce the number of states (if possible). The number of states can be reduced either when states can be eliminated or when they can be summarized with other states. 34 of 4 7
9..7 Elimination of states States can be eliminated if they are not reachable by any other state AND if they are not the initial starting state if they are isolated AND if they are not the initial starting state Sub-graphs can be eliminated if they are not reachable and if they do NOT contain the initial starting state Entire frame contains the state machine R: Reset or Starting Point 35 of 4 State Machine Minimization according to Huffmann/Mealy two states can be summarized/combined to one state, if they are equivalent. requirement for equivalency is that they have for identical input values the same next state with identical output vectors. 36 of 4 8
9..7 Example: State Machine Minimazing using state graphs State 5: X= Z n = Y n =I X=I Z n = Y n = State 6: same states can be combined state 5 remains state 6 can be deleted 37 of 4 Minimized State Graph 38 of 4 9
9..7 State Machine Minimazing using state tables Minimization is also possible with state table. Z n X= X=I / / 3/ 7/ 6/ / 3 / 4/ 4 5/ / 5 / / 6 / / 7 5/ / 39 of 4 State Machine Minimazing using state tables Minimization with state tables means, searching for identical rows Z n is a more systematic process X= X=I / / 3/ 7/ Delete state 6 and state 7 6/ / Rename 3 / 4/ all next states 6 to next state 5 4 5/ / all next states 7 to next state 4 5 / / Repeat searching for identical rows 6 / / 7 5/ / 4 of 4
9..7 State Machine Minimazing using state tables Z n X= X=I / / 3/ 4/ 5/ / 3 / 4/ 4 5/ / 5 / / Z n X= X=I / / 3/ / 5/ / 3 / / 5 / / More systematic and computer based approaches will be discussed in the lecture Logical Design of Digital Systems 4 of 4