ECE 368: CAD-Based Logic Design

1. ECE 368: CAD-Based Logic Design Lecture Notes # 5 Sequential Circuit (Finite-State Machine) Design SHANTANU DUTT Department of ECE University of Illinois, Chicago Phone: (312) 355-1314: e-mail: dutt@ece.uic.edu URL: http://www.ece.uic.edu/~dutt

2. Finite State Machine (FSM) Design Sampling instances x O/p y Tlogic + Tsetup FSM CLK CLK x # of 1s even (0) odd (1) even (2) odd (3) odd (3) • FSMs are different from counters in the sense that they have external I/Ps, and state transitions are dependent on these I/Ps and the current state. • Example : Problem Statement There is a bit-serial I/P line. Design an FSM that outputs a ‘0’ if an even # of 1’s have been received on the I/P line and the outputs a ‘1’ otherwise. Note : If a synchronous sequential circuit is being designed, the counting of the # of 1s occur every clock cycle.

3. Approach to determining states of an FSM: • First determine the min # of useful information classes about past i/ps required to solve the problem (requires analytical thinking about the problem) • Each info class  a potential state • From this 1st cut at possible states, determine if there are well-defined transitions from each state for all possible i/p values. • If so then these states can be the final states; otherwise some states may need to be refined into multiple states to achieve well-defined transitions (see FSM word prob. 1). • • In this problem, only 2 classes of information are reqd: whether an even # of 1s have been received so far, or an odd # of 1s have been received so far & there are well-defined transitions between them. Thus these 2 classes become 2 states. Solution 2: (Moore) 0 Solution 1: (Mealy) 0/0 Reset Even Reset Output [0] Even Input 1 1 1/1 O/P is dependent on current state and input in Mealy 1/0 Input Odd [1] Output Output is dependent only on current state Odd Transition Arc 0 0/1 Moore Machine: Output is associated with the state and hence appears after the state transition take place. Mealy Machine: Output is associated with the state transition, and appears before the state transition is completed (by the next clock pulse).

4. Determining a Reset State: • A reset state is a state the the FSM (seq ckt) should be in when it is just powered on. • In other words, a reset state is a state the FSM should be in, when it has recvd no i/ps • Based on the above definition, decide if any of the states determined so far can be a reset state. E.g., in the parity detector problem, the even state qualifies to be the reset state, as in the reset state no i/ps recvd  zero 1’s recvd  even # of 1’s recvd  it can be the even state • If not, then need to have a separate reset state, and have the correct transitions from this state to the other states (depending on the problem solved by the FSM). Solution 1: (Mealy) Solution 2: (Moore) 0 0/0 Reset Reset Even Even Input Output [0] 1/1 O/P is dependent on current state and input in Mealy 1/0 1 1 Input Output Odd Odd [1] Transition Arc Output is dependent only on current state 0 0/1 Mealy Machine: Output is associated with the state transition, and appears before the state transition is completed (by the next clock pulse). Moore Machine: Output is associated with the state and hence appears after the state transition take place.

5. External I/Ps Next State Comb. Logic External I/Ps External O/Ps m1 m1 Comb. Logic m2 FFs n n FFs Output Logic n n CLK External Outputs m2 CLK even  odd Moore Machine Model Mealy Machine Model Time t : Even I/P  = propagation delay of logic of Mealy M/C 2 = propagation delay of O/P logic unit of Moore M/C t+TCLK+2 t+ t+TCLK t Even x=1 O/P=0 Odd O/P=1 (Moore) O/P=1 (Mealy)

6. State Transition Table y2 x x N.S. Logic N.S. & O/P Logic Q FF Q FFs DA A DA A O/P Logic y1 CLK (Even-Parity Checker) Even State: 0 ; Odd State: 1; State Variable A Next State Present State Mealy O/P D-FF Excit. Moore O/P T-FF Excit. Input A x A+ y1 y2 DA TA 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 0 1 0 0 1 Output functions Input variables to comb. logic DA= Ax (same for Moore and Mealy); TA= x (same for Moore and Mealy); y1 = A for Moore y2 = Ax for Mealy Or

7. Reset 0 0/0 State=0 Even Reset State=0 Even [0] 1/1 1/0 1 1 State=1 Odd N.S. Logic x State=1 Odd [1] Q 0/1 0 D- FF Q D Mealy Moore CLK S.T. is complete. Assume single bit state information stored in a D-FF State Transition is occurring State Transition is occurring S.T. is complete. CLK x D even even even odd Q (state) odd odd y2 (Mealy O/P) y1 Moore O/P)

8. Reset Reset Reset Reset DA= Ax ; TA= x; y1 = A for Moore; y2 = Ax for Mealy Moore M/C Implementation a) D-FF b) T-FF 0 x A y2 A y2 T Q D Q x=1 CLK Q R Q R CLK Moore O/P is synchronized with clock. Mealy M/C Implementation y1 x y1 0 T Q 1 A D Q x=1 CLK Q R Q R CLK b) T-FF a) D-FF Mealy O/P is not synchronized with clock. Note: Here Moore and Mealy state transition function is the same. Will not always be the case.

9. Difference Between Mealy and Moore Machine Mealy Moore (1) O/Ps depend on the present O/Ps depend only on the state and present I/Ps present state (2) The O/P change asyn Since the O/Ps change -chronously with the when the state changes, enabling clock edge and the state change is synchronous with the enabling clock edge, O/Ps change synchronously with this clock edge (3) A counter is not a Mealy A counter is a Moore machine machine (4) A Mealy machine will have the same # or fewer states than a Moore machine

10. Moore machine 0 Reset Even [0] 1 1 x N.S. Logic Odd [1] C.S. Q N.S. 0 FFs O/P Logic y entity fsm1_1 is port (x, reset:in bit; y: out bit:=`0’) end entity fsm1_1 architecture behav of fsm1_1 is type states is (even, odd); begin state_mc: process (x, reset) is variable curr_state, next_state: states := even; -- initial state is reset state begin if reset = ‘1’ then curr_state := even; y <= ‘0’; else case curr_state is when even => y <=`0’; -- assert output -- next determine transition based on input if x=`1’ then next_state := odd; else next_state := even; end if; when odd => y <= `1’; -- assert output if x=`1’ then next_state := even; else next_state := odd; end if; end case; curr_state := next_state; end if; end process state_mc; end architecture behav; Behavioral FSM Descriptions in VHDL – 1st Cut • Simple very high-level behavioral description w/o timing (no clock, no delays) • Checks only the high-level FSM logic • No parallelism • If timing and/or parallelism of the underlying hardware is desired we need to introduce clocks, delays and multiple processes

11. Moore machine 0 Reset Even [0] 1 1 x N.S. Logic Odd [1] C.S. Q N.S. 0 FFs O/P Logic y architecture behav of fsm1_2 is type states is (even, odd); begin state_mc: process is variable curr_state, next_state: states := even; -- reset state begin wait until (clk’event and clk=`1’) or (reset=`1’); if reset = ‘1’ then curr_state := even; y <= `0’ after 2 ns; else -- note need of curr_state to be variable for immediate effect case curr_state is when even => y <=`0’ after 2 ns; -- assert outputs -- next determine transition based on input if x=`1’ then wait for 2 ns; next_state := odd; else wait for 2 ns; next_state := even; end if; when odd => y <= `1’ after 2 ns; -- assert outputs if x=`1’ then wait for 2 ns; next_state := even; else wait for 2 ns; next_state := odd; end if; end case; curr_state := next_state; end if; end process state_mc; end architecture behav; Behavioral FSM Descriptions in VHDL – 2nd Cut entity fsm1_2 is port (x, clk, reset:in bit; y: out bit:=`0’) end entity fsm1_2 Problem w/ the description: The N.S. logic is simulated only on the clock edge. Actually, it should be simulated anytime x or C.S. changes. The latter happens in the code but not the former.

12. x N.S. Logic C.S. Q N.S. FFs O/P Logic y Process 2 clock Process 1 Process3 General solution to behavioral description of a system (described in an arch) – when to have multiple processes: -- If a design has different components, each operating or triggered w/ different timing events, then separate them into individual processes. -- In other words a single process should not describe multiple components triggered by different timing events -- Furthermore if signal assignments to, say, signal X, in a process should affect the rest of the logic in it in the current iteration and X cannot be made a variable (if it is also an o/p of the process, e.g., when it is an i/p signal of another process), separate out the X-dependent logic as a separate process -- In a Moore machine, we have 3 modules, FFs, N.S. logic and O/P logic. FF is triggered by the clock +ve edge. Its o/p C.S. triggers the O/P logic; so it is also ultimately triggered by the clock. The N.S. logic is triggered by C.S. and x, where x has no relation to the clock. -- So a possible breakup is Process 1: FFs + O/P Logic and Process 2: N.S. Logic. But not a good idea as: -- C.S. needs to be a signal i/p to the N.S. Logic and FF processes to communicate -- Since the O/P Logic depends on C.S., if it is in the same process as the FF, there will not be any effect from the FF to the O/P Logic modules in the current iteration (in other words, combined FF+O/P process will need to be simulated more times (i.e., it has to be activated by more events, which may or may not happen) to propagate final values to o/p of O/P module). So separate out the O/P Logic as a 3rd process -- Large processes resulting from the above partitioning methodology may be further broken up functionally (i.e., by functional modules, e.g., ALU and Register File in a Datapath process of a CPU) and/or if some parallelism of the underlying hardware is desired to be simulated and/or communication between these modules need to be simulated Behavioral FSM Descriptions in VHDL – Final Cut Process A X <= ….. Process B Y <= X nand Z;

13. Moore machine 0 Even Reset Process 2 [0] 1 1 x N.S. Logic Odd [1] clock C.S. Process 1 Q N.S. 0 FFs O/P Logic Process3 y architecture behav of fsm1_3 is type states is (even, odd); signal next_state, curr_state: states := even; begin ns_proc: process (curr_state, x) is begin case curr_state is when even => if x=`1’ then next_state <= odd after 2 ns; else next_state <= even after 2 ns; end if; when odd => if x=`1’ then next_state <= even after 2 ns; else next_state <= odd after 2 ns; end if; end case; end process ns_proc; ff_proc: process is begin wait until (clk’event and clk=`1’) or (reset=`1’); if reset = ‘1’ then curr_state <= even; else curr_state <= next_state; end if; end process ff_proc; op_proc: process (curr_state) is begin case curr_state is when even => y <=`0’ after 2 ns; -- assert outputs when odd => y <= `1’ after 2 ns; end case; end process op_proc; end architecture behav; Behavioral FSM Descriptions in VHDL – Final Cut Note: When all processes block, sim. time is advanced by the fictitious delta time or min. delay across signals, resulting in assigned signal values appearing and causing corresponding processes to unblock (those w/ wait statements dependent on these signals). entity fsm1_3 is port (x, clk, reset:in bit; y: out bit:=`0’) end entity fsm1_3

14. Mealy Machine 0/0 Reset Even 1/1 1/0 Odd Transition Arc 0/1 architecture behav of fsm2_1 is type states is (even, odd); begin state_mc: process is variable next_state, curr_state: states := even; -- reset state begin wait until (clk’event and clk=`1’) or (reset=`1’); if reset = ‘1’ then curr_state := even; else curr_state := next_state; end if; case curr_state is when even => if x=`1’ then y <= `1’ after 2 ns; wait for 2 ns; next_state := odd; else y <= `0’ after 2 ns; wait for 2 ns; next_state := even after 2 ns; end if; -- note: above times are arbitrary; just for e.g. when odd => if x=`1’ then y <=`0’ after 2 ns; wait for 2 ns; next_state := even after 2 ns; else y <= `1’ after 2 ns; wait for 2 ns; next_state := odd after 2 ns; end if; end case; end process state_mc; end architecture behav; Behavioral FSM Descriptions in VHDL– First Cut y2 x N.S. & O/P Logic Q FF N.S. C.S. CLK entity fsm2_1 is port (x, clk, reset:in bit; y: out bit:=‘0’) end entity fsm2_1 Problem w/ the description: The N.S. & O/P logic is simulated only on the clock edge. Actually, it should be simulated anytime x or C.S. changes. The latter happens but not the former.

15. Mealy Machine 0/0 Reset Even 1/1 1/0 Odd Transition Arc 0/1 architecture behav of fsm2_2 is type states is (even, odd); signal next_state, curr_state: states := even; begin ns_op_proc: process (x, curr_state) is begin case curr_state is when even => if x=`1’ then y <= `1’ after 2 ns; next_state <= odd after 2 ns; else y <= `0’ after 2 ns; next_state <= even after 2 ns; end if; -- note: above times are arbitrary; just for e.g. when odd => if x=`1’ then y <=`0’ after 2 ns; next_state <= even after 2 ns; else y <= `1’ after 2 ns; next_state <= odd after 2 ns; end if; end case; end process ns_op_proc; ff_proc: process is begin wait until (clk’event and clk=`1’) or (reset=`1’); if reset = ‘1’ then curr_state <= even; else curr_state <= next_state; end if; end process ff_proc; end architecture behav; Behavioral FSM Descriptions in VHDL– Final Cut y2 x N.S. & O/P Logic Q FF N.S. C.S. CLK entity fsm2_2 is port (x, clk, reset:in bit; y: out bit:=‘0’) end entity fsm2_2

16. Another example: A simple vending machine Timer 0C 5C 10C 15C Here is how the control is supposed to work. The vending machine delivers a package of gum after it has received 15 cents in coins. The machine has a single coin slot that accepts nickels and dimes, one coin at a time. A mechanical sensor indicates to the control whether a dime or a nickel has been inserted into the coin slot. The controller’s output causes a single package of gum to be released down a chute to the customer. One further specification: We will design our machine so it does not give change. A customer who pays with two dimes is out 5 cents! Coin Sensor Gum Release Mechanism Vending Machine FSM Open CLK Reset Reset (ext.) OR time_out Vending Machine block diagram States:

17. Reset )/0 Reset 0 cent [0] Reset N 5 cent [0] D N D 10 cent [0] N+D 15 cent [1] Reset / 1 Reset Moore machine — The figure below show the Moore and Mealy machine state transition diagrams. Reset / 0 )/0 Reset 0 cent Reset / 0 N / 0 D / 0 5 cent D/1 N / 0 10 cent N+D/1 15 cent Mealy machine Moore and Mealy machine state diagrams for the vending machine FSM

18. Behavioral VHDl description of Moore vending machine controller Reset Reset )/0 0 cent [0] N Reset 5 cent [0] D N D 10 cent [0] Moore machine N+D 15 cent [1] entity vend_moore N Reset NS logic process next_state D reset <= ext_ reset or time_out after 2 ns; ext_reset time_out FF proc o/p logic proc Timer proc curr_state open1 open1 Arch. behav clk reset ff_proc: process is begin wait until (clk’event and clk=‘1’) or ( reset=`1’); if reset = ‘1’ then curr_state <= S0 after 1 ns; else curr_state <= next_state after 1 ns; end if; end process ff_proc; op_proc: process (curr_state) is begin case curr_state is when S0 => open1 <=` 0’ after 2 ns; when S5 => open1 <= `0’ after 2 ns; when S10 => open1 <=` 0’ after 2 ns; when S15 => open1 <= `1’ after 2 ns; end case; end process op_proc; entity vend_moore is port(N, D, ext_reset, clk: in bit; open1:inout bit); end entity vend_moore; architecture behav of vend_moore is type states is (S0, S5, S10, S15); signal next_state, curr_state: states := even; signal reset, time_out : bit := `0’; begin reset <= ext_ reset or time_out after 2 ns; -- concurrent signal assignment statement timer: process is wait until open1 = `1’; time_out = `1’ after 20 sec; wait for 20 sec; time_out = `0’ after 10 ns; -- timeout signal is high for 10 ns; if open1 = `1’ then wait until open1 = `0’; end process timer;

19. Behavioral VHDl description of Moore vending machine controller Reset )/0 Reset 0 cent [0] N 5 cent [0] D N D 10 cent [0] N+D 15 cent [1] Reset ns_proc: process (N, D, curr_state) is case curr_state is when S0 => if N=1 then next_state <= S5 after 2 ns; elsif D=1 next_state <= 10 after 2 ns; else next_state <= S0 after 2 ns; end if; when S5 => if N=1 then next_state <= S10 after 2 ns; elsif D=1 next_state <= S15 after 2 ns; else next_state <= S5 after 2 ns; end if; when S10 => if N=1 or D=1 then next_state <= S15 after 2 ns; else next_state <= S10 after 2 ns; end if; when S15 => next_state <= S15 after 2 ns; end case; end process ns_proc; end architecture behav; Reset Moore machine

20. —State transition table for Moore and Mealy M/C.(Next state also gives D-FF excitation). Present State Inputs Next State Moore Output Mealy Output Q1 Q2 D N Q1+ Q2+ Open Open 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 0 1 1 x x x x 0 1 0 0 0 1 0 0 0 1 1 0 0 0 1 0 1 1 0 1 1 1 x x x x 1 0 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 0 1 1 1 x x x x 1 1 0 0 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 x x x x Q+ = D Q Q+ D 0 0 0 0 1 1 1 0 0 1 1 1 Encoded vending machine state transition table. Note: Do not have to design for the reset input if FFs have a direct reset inputs. Make sure though that reset state is encoded as all 0’s if possible; otherwise need FFS w/ asynch. reset as well as set inputs

21. Q1Q0 Q1Q0 Q1Q0 Q1Q0 00 01 11 10 00 01 11 10 00 01 11 10 00 01 11 10 DN DN DN DN 00 00 00 00 01 01 01 01 11 11 11 11 10 10 10 10 Implementation using D-FFs 0 0 1 1 0 1 1 0 0 0 1 0 0 1 1 1 1 0 1 1 0 0 1 0 x x x x x x x x x x x x 1 1 1 1 0 1 1 1 0 0 1 0 K-map for Open (Moore) K-map for D1 K-map for D0 D1 = Q1 + D + Q0·N 0 0 1 0 0 0 1 1 OPEN = Q1·Q0 OPEN = Q1·Q0 + D·Q0 + D·Q1 + N·Q1 Moore x x x x Mealy 0 1 1 1 K-map for Open (Mealy)

22. Reset Reset Q1 D1 Q1 D D Q Q0 Similarly, a Mealy implementation; only the OPEN function changes. CLK Q R N OPEN N Q0 D0 Q0 D Q CLK Q1 Q R N Q1 D Vending machine FSM implementation based on D flip-flops(Moore).

23. Implementation using J-K FFS J-K Excitation Q1 Q2 D N Q1+ Q2+ J1 K1 J0 K0 0 0 0 0 0 0 0 x 0 x 0 1 0 1 0 x 1 x 1 0 1 0 1 x 0 x 1 1 x x x x x x 0 1 0 0 0 1 0 x x 0 0 1 1 0 1 x x 1 1 0 1 1 1 x x 0 1 1 x x x x x x 1 0 0 0 1 0 x 0 0 x 0 1 1 1 x 0 1 x 1 0 1 1 x 0 1 x 1 1 x x x x x x 1 1 0 0 1 1 x 0 x 0 0 1 1 1 x 0 x 0 1 0 1 1 x 0 x 0 1 1 x x x x x x Q Q+ J K 0 0 0 x 0 1 1 x 1 0 x 1 1 1 x 0 Remapped next-state functions for the vending machine example.

24. Q1Q0 Q1Q0 Q1Q0 Q1Q0 00 01 11 10 00 01 11 10 00 01 11 10 00 01 11 10 DN DN DN DN 00 00 00 00 01 01 01 01 11 11 11 11 10 10 10 10 0 0 x x x x 0 0 0 1 x x x x 0 0 x x x x x x x x 1 1 x x x x 0 0 K-map for J1 K-map for K1 0 x x 0 x 0 0 x 1 x x 1 x 1 0 x x x x x x x x x 0 x x 1 x 0 0 x K-map for J0 K-map for K0 K-maps for J-K flip-flop implementation of vending machine. J1 = D + Q0·N K1 = 0

25. Reset N Q0 Q1 J Q CLK D Q R K OPEN N Q1 Q0 D J Q CLK Q R K N J-K flip-flop implementation for the vending machine example (Moore). Similarly, a Mealy implementation; only the OPEN function changes.

26. Basic Steps in the FSM Design Procedure 1. Understand the problem and the different information classes (minimal number) required to solve it. Also make sure there are well-defined transitions possible between these classes; otherwise break down some or all classes into smaller classes to get well-defined transitions. 2a. Convert these information classes into distinct states, and determine the state transition diagram of the FSM. Determine reset state. 2b Optional: Perform state minimization 3a. Encode states in binary, and obtain state transition table and FF excitation (for desired FF type). 3b Optional: Perform state assignment for minimizing logic. 4. Minimize the FF excitation functions and output functions (using K-Maps, for example) and implement the FSM using these FFs and logic gates (or MUXes, PLAs, etc.) that implement the output and FF excitation functions.

27. FSM Word Problem 1: • Design a system that outputs a ‘1’ whenever it receives a multiple of 3 # of 1’s (i.e., 0, 3, 6, 9, etc. # of 1’s) on a serial input line x. — Relevant information classes needed to solve the problem: (A) A multiple of 3 # is received. (B) A non-multiple of 3 # is received. Questions to consider: (1) How do we go from (A)(B) Ans.: If a ‘1’ is received (2) How do we go from (B)(A) Ans.: Not clear. Need to split up (B) further into (B1): 3y+1 # of 1’s received. (B2): 3y+2 # of 1’s received. Where y is an integer  0.

28. Note: (A): is 3y+0 = 3y # of 1’s received. • Now the transitions between the3 classes of information is clear: (A)  (B1)  (B2)  (A) 1 received 1 received 1 received • Hence these classes of information can be considered states of the required as states of the required FSM: These 3 states can be represented by 3y+I, i = 0,1,2 0 Output 00 0/1 Reset Reset i=0 i=0 [1] Input 0/0 1 0 1/0 i=1 1/1 01 1 i=1 [0] i=2 1/0 i=2 [0] 10 1 0/0 0 Mealy Machine Moore Machine

29. FSM Word Problem 2: • Design a system that outputs a ‘1’ whenever it receives: (a) A multiple of 3 # of 1’s AND (b) A non-zero even # of 0’s E.g., ((0,2) , (3,2) , (3,4) , (6,2) ,···) — Relevant classes of information: - For # of 1’s: 3y+i, i = 0,1,2 [3 classes] - For # of 0’s: 2z+j, j = 0,1 For j = 0, we need to distinguish between zero (z = 0) and non-zero (z > 0) # of 0’s - Thus we have 3 classes: 2z+0, z = 0 ( 0 ) 2z+0, z > 0 ( non-zero even ) 2z+1 ( odd ) # of 1’s # of 0’s

30. The relevant # of 1’s can be represented by i = { 0, 1, 2 } ( # of 1’s = 3y+i ) — The relevant # of 0’s can be represented by j= { 00 , 0>0 , 1 } ( # of 0’s = 2z+j ) where the subscript of the 0 indicates whether z=0 or z>0. — Since at any point time, a certain # of 1’s and # of 0’s will have been received, the state of the system will be given by a combination of relevant # of 1’s and # of 0’s. — There are 9 combinations: { 0, 1, 2, } X { 00, 0>0, 1 } = (0,00), (0,0>0), (0,1), (1,00), (1,0>0), (1,1), (2,00), (2,0>0), (2,1)  # of 1’s # of 0’s Cartesian Product

31. (0,00) (0,0>0) (1,00) (0,1) (2,00) (1,1) (2,1) (1,0>0) (2,0>0)

32. Note: 0>0 2z+j, j = 0 z > 0 Reset (0,00) 0/0 1/0 1/0 (0,0>0) (1,00) 0/0 1/0 0/0 0/1 1/0 (0,1) (2,00) (1,1) 1/0 0/0 0/0 1/0 0/0 (2,1) (1,0>0) 1/1 0/0 1/0 0/0 (2,0>0) 1/0