# Sequential Logic Examples

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CS 150 - Fall 2000 - Sequential Logic Examples - 1 Sequential Logic Examples Finite State Machine Concept FSMs are the decision making logic of digital designs Partitioning designs into datapath and control elements When inputs are sampled and outputs asserted Basic Design Approach: 4-step Design Process Implementation Examples and Case Studies Finite-string pattern recognizer Complex counter Traffic light controller Door combination lock

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CS 150 - Fall 2000 - Sequential Logic Examples - 2 General FSM Design Procedure (1) Determine inputs and outputs (2) Determine possible states of machine – State minimization (3) Encode states and outputs into a binary code – State assignment or state encoding – Output encoding – Possibly input encoding (if under our control) (4) Realize logic to implement functions for states and outputs – Combinational logic implementation and optimization – Choices in steps 2 and 3 have large effect on resulting logic

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CS 150 - Fall 2000 - Sequential Logic Examples - 3 Finite String Pattern Recognizer (Step 1) Finite String Pattern Recognizer One input (X) and one output (Z) Output is asserted whenever the input sequence …010… has been observed, as long as the sequence 100 has never been seen Step 1: Understanding the Problem Statement Sample input/output behavior: X: 0 0 1 0 1 0 1 0 0 1 0 … Z: 0 0 0 1 0 1 0 1 0 0 0 … X: 1 1 0 1 1 0 1 0 0 1 0 … Z: 0 0 0 0 0 0 0 1 0 0 0 …

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CS 150 - Fall 2000 - Sequential Logic Examples - 4 Finite String Pattern Recognizer (Step 2) Step 2: Draw State Diagram For the strings that must be recognized, i.e., 010 and 100 Moore implementation

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CS 150 - Fall 2000 - Sequential Logic Examples - 5 Finite String Pattern Recognizer (Step 2, cont’d) Exit conditions from state S3: have recognized …010 If next input is 0 then have …0100 = ...100 (state S6) If next input is 1 then have …0101 = …01 (state S2) Exit conditions from S1: recognizes strings of form …0 (no 1 seen); loop back to S1 if input is 0 Exit conditions from S4: recognizes strings of form …1 (no 0 seen); loop back to S4 if input is 1

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CS 150 - Fall 2000 - Sequential Logic Examples - 6 Finite String Pattern Recognizer (Step 2, cont’d) S2 and S5 still have incomplete transitions S2 = …01; If next input is 1, then string could be prefix of (01)1(00) S4 handles just this case S5 = …10; If next input is 1, then string could be prefix of (10)1(0) S2 handles just this case Reuse states as much as possible Look for same meaning State minimization leads to smaller number of bits to represent states Once all states have complete set of transitions we have final state diagram

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CS 150 - Fall 2000 - Sequential Logic Examples - 7 module string (clk, X, rst, Q0, Q1, Q2, Z); input clk, X, rst; output Q0, Q1, Q2, Z; reg state[0:2]; ‘define S0 = [0,0,0]; //reset state ‘define S1 = [0,0,1]; //strings ending in ...0 ‘define S2 = [0,1,0]; //strings ending in ...01 ‘define S3 = [0,1,1]; //strings ending in ...010 ‘define S4 = [1,0,0]; //strings ending in ...1 ‘define S5 = [1,0,1]; //strings ending in ...10 ‘define S6 = [1,1,0]; //strings ending in ...100 assign Q0 = state; assign Q1 = state; assign Q2 = state; assign Z = (state == ‘S3); always @(posedge clk) begin if rst state = ‘S0; else case (state) ‘S0: if (X) state = ‘S4 else state = ‘S1; ‘S1: if (X) state = ‘S2 else state = ‘S1; ‘S2: if (X) state = ‘S4 else state = ‘S3; ‘S3: if (X) state = ‘S2 else state = ‘S6; ‘S4: if (X) state = ‘S4 else state = ‘S5; ‘S5: if (X) state = ‘S2 else state = ‘S6; ‘S6: state = ‘S6; default: begin \$display (“invalid state reached”); state = 3’bxxx; endcase end endmodule Finite String Pattern Recognizer (Step 3) Verilog description including state assignment (or state encoding)

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CS 150 - Fall 2000 - Sequential Logic Examples - 8 Finite String Pattern Recognizer Review of Process Understanding problem Write down sample inputs and outputs to understand specification Derive a state diagram Write down sequences of states and transitions for sequences to be recognized Minimize number of states Add missing transitions; reuse states as much as possible State assignment or encoding Encode states with unique patterns Simulate realization Verify I/O behavior of your state diagram to ensure it matches specification

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CS 150 - Fall 2000 - Sequential Logic Examples - 9 Complex Counter Synchronous 3-bit counter has a mode control M When M = 0, the counter counts up in the binary sequence When M = 1, the counter advances through the Gray code sequence binary: 000, 001, 010, 011, 100, 101, 110, 111 Gray: 000, 001, 011, 010, 110, 111, 101, 100 Valid I/O behavior (partial)

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CS 150 - Fall 2000 - Sequential Logic Examples - 10 Complex Counter (State Diagram) Deriving State Diagram One state for each output combination Add appropriate arcs for the mode control

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CS 150 - Fall 2000 - Sequential Logic Examples - 11 Complex Counter (State Encoding) Verilog description including state encoding module string (clk, M, rst, Z0, Z1, Z2); input clk, X, rst; output Z0, Z1, Z2; reg state[0:2]; ‘define S0 = [0,0,0]; ‘define S1 = [0,0,1]; ‘define S2 = [0,1,0]; ‘define S3 = [0,1,1]; ‘define S4 = [1,0,0]; ‘define S5 = [1,0,1]; ‘define S6 = [1,1,0]; ‘define S7 = [1,1,1]; assign Z0 = state; assign Z1 = state; assign Z2 = state; always @(posedge clk) begin if rst state = ‘S0; else case (state) ‘S0: state = ‘S1; ‘S1: if (M) state = ‘S3 else state = ‘S2; ‘S2: if (M) state = ‘S6 else state = ‘S3; ‘S3: if (M) state = ‘S2 else state = ‘S4; ‘S4: if (M) state = ‘S0 else state = ‘S5; ‘S5: if (M) state = ‘S4 else state = ‘S6; ‘S5: if (M) state = ‘S7 else state = ‘S7; ‘S5: if (M) state = ‘S5 else state = ‘S0; endcase end endmodule

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CS 150 - Fall 2000 - Sequential Logic Examples - 12 traffic light controller timer TL TS ST Traffic Light Controller as Two Communicating FSMs Without Separate Timer S0 would require 7 states S1 would require 3 states S2 would require 7 states S3 would require 3 states S1 and S3 have simple transformation S0 and S2 would require many more arcs C could change in any of seven states By Factoring Out Timer Greatly reduce number of states 4 instead of 20 Counter only requires seven or eight states 12 total instead of 20

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CS 150 - Fall 2000 - Sequential Logic Examples - 13 machines advance in lock step initial inputs/outputs: X = 0, Y = 0 Communicating Finite State Machines One machine's output is another machine's input

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CS 150 - Fall 2000 - Sequential Logic Examples - 14 "puppet" "puppeteer who pulls the strings" control data-path status info and inputs control signal outputs state Datapath and Control Digital hardware systems = data-path + control Datapath: registers, counters, combinational functional units (e.g., ALU), communication (e.g., busses) Control: FSM generating sequences of control signals that instructs datapath what to do next

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CS 150 - Fall 2000 - Sequential Logic Examples - 15 Digital Combinational Lock Door Combination Lock: Punch in 3 values in sequence and the door opens; if there is an error the lock must be reset; once the door opens the lock must be reset Inputs: sequence of input values, reset Outputs: door open/close Memory: must remember combination or always have it available Open questions: how do you set the internal combination? Stored in registers (how loaded?) Hardwired via switches set by user

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CS 150 - Fall 2000 - Sequential Logic Examples - 16 Implementation in Software integer combination_lock ( ) { integer v1, v2, v3; integer error = 0; static integer c = 3, 4, 2; while (!new_value( )); v1 = read_value( ); if (v1 != c) then error = 1; while (!new_value( )); v2 = read_value( ); if (v2 != c) then error = 1; while (!new_value( )); v3 = read_value( ); if (v2 != c) then error = 1; if (error == 1) then return(0); else return (1); }

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CS 150 - Fall 2000 - Sequential Logic Examples - 17 Determining Details of the Specification How many bits per input value? How many values in sequence? How do we know a new input value is entered? What are the states and state transitions of the system?

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CS 150 - Fall 2000 - Sequential Logic Examples - 18 Digital Combination Lock State Diagram States: 5 states Represent point in execution of machine Each state has outputs Transitions: 6 from state to state, 5 self transitions, 1 global Changes of state occur when clock says its ok Based on value of inputs Inputs: reset, new, results of comparisons Output: open/closed closed closed closed C1==value & new C2==value & new C3==value & new C1!=value & new C2!=value & new C3!=value & new closed reset not new not new not new S1 S2 S3 OPEN ERR open

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CS 150 - Fall 2000 - Sequential Logic Examples - 19 Datapath and Control Structure Datapath Storage registers for combination values Multiplexer Comparator Control Finite-state machine controller Control for data-path (which value to compare)

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CS 150 - Fall 2000 - Sequential Logic Examples - 20 State Table for Combination Lock Finite-State Machine Refine state diagram to take internal structure into account State table ready for encoding

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CS 150 - Fall 2000 - Sequential Logic Examples - 21 mux is identical to last 3 bits of state open/closed is identical to first bit of state therefore, we do not even need to implement FFs to hold state, just use outputs Encodings for Combination Lock Encode state table State can be: S1, S2, S3, OPEN, or ERR Needs at least 3 bits to encode: 000, 001, 010, 011, 100 And as many as 5: 00001, 00010, 00100, 01000, 10000 Choose 4 bits: 0001, 0010, 0100, 1000, 0000 Output mux can be: C1, C2, or C3 Needs 2 to 3 bits to encode Choose 3 bits: 001, 010, 100 Output open/closed can be: open or closed Needs 1 or 2 bits to encode Choose 1 bit: 1, 0

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CS 150 - Fall 2000 - Sequential Logic Examples - 22 Datapath Implementation for Combination Lock Multiplexer Easy to implement as combinational logic when few inputs Logic can easily get too big for most PLDs

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CS 150 - Fall 2000 - Sequential Logic Examples - 23 oc open-collector connection (zero whenever one connection is zero, one otherwise – wired AND) tri-state driver (can disconnect from output) Datapath Implementation (cont’d) Tri-State Logic Utilize a third output state: “no connection” or “float” Connect outputs together as long as only one is “enabled” Open-collector gates can only output 0, not 1 Can be used to implement logical AND with only wires

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CS 150 - Fall 2000 - Sequential Logic Examples - 24 non-inverting tri-state buffer 100 In OE Out Tri-State Gates Third value Logic values: “0”, “1” Don't care: “X” (must be 0 or 1 in real circuit!) Third value or state: “Z” — high impedance, infinite R, no connection Tri-state gates Additional input – output enable (OE) Output values are 0, 1, and Z When OE is high, the gate functions normally When OE is low, the gate is disconnected from wire at output Allows more than one gate to be connected to the same output wire As long as only one has its output enabled at any one time (otherwise, sparks could fly) In Out OE

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CS 150 - Fall 2000 - Sequential Logic Examples - 25 when Select is high Input1 is connected to F when Select is low Input0 is connected to F this is essentially a 2:1 mux Tri-State and Multiplexing When Using Tri-State Logic (1) Never more than one "driver" for a wire at any one time (pulling high and low at same time can severely damage circuits) (2) Only use value on wire when its being driven (using a floating value may cause failures) Using Tri-State Gates to Implement an Economical Multiplexer

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CS 150 - Fall 2000 - Sequential Logic Examples - 26 open-collector NAND gates with ouputs wired together using "wired-AND" to form (AB)'(CD)' Open-Collector Gates and Wired-AND Open collector: another way to connect gate outputs to same wire Gate only has the ability to pull its output low Cannot actively drive wire high (default – pulled high through resistor) Wired-AND can be implemented with open collector logic If A and B are "1", output is actively pulled low If C and D are "1", output is actively pulled low If one gate output is low and the other high, then low wins If both outputs are "1", the wire value "floats", pulled high by resistor Low to high transition usually slower than if gate pulling high Hence, the two NAND functions are ANDed together

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CS 150 - Fall 2000 - Sequential Logic Examples - 27 Digital Combination Lock (New Datapath) Decrease number of inputs Remove 3 code digits as inputs Use code registers Make them loadable from value Need 3 load signal inputs (net gain in input (4*3)–3=9) Could be done with 2 signals and decoder (ld1, ld2, ld3, load none)

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CS 150 - Fall 2000 - Sequential Logic Examples - 28 Section Summary FSM Design Understanding the problem Generating state diagram Implementation using synthesis tools Iteration on design/specification to improve qualities of mapping Communicating state machines Four case studies Understand I/O behavior Draw diagrams Enumerate states for the "goal" Expand with error conditions Reuse states whenever possible

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