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This Lecture
Review of last lecture: Analysis Chapter 5: State Reduction, Design Procedure
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Analysis: Obtaining a table/diagram for the time sequence of inputs/outputs/internal states.
Examples: State Equations, State Table, State Diagram
Analysis of Clocked Sequential Circuits
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Analysis of Clocked Sequential Circuits
Example of state equation:
A(t+1) = A(t)x(t) + B(t)x(t)B(t+1) = A’(t)x(t)
A(t+1)=Ax+BxB(t+1)=A’x
y(t)=(A(t)+B(t)).x’(t) = (A+B)x’
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Example of state tables
Present state input Next State Output A B x A B y 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 1 1 1 1 0 1 0 0 0 0 1 1 0 1 1 0 0 1 1 0 0 0 1 1 1 1 1 0 0
State equation:
A(t+1) = A(t)x(t) + B(t)x(t)B(t+1) = A’(t)x(t)
y(t)=(A(t)+B(t)).x’(t)
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Example of state tables-2nd form
Present state Next State Output x=0 x=1 x=0 x=1 AB AB AB y y 00 00 01 0 0 01 00 11 1 0 10 00 10 1 0 11 00 10 1 0
State equation:
A(t+1) = A(t)x(t) + B(t)x(t)B(t+1) = A’(t)x(t)
y(t)=(A(t)+B(t)).x’(t)
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Example of state diagram
Present state Next State Output x=0 x=1 x=0 x=1AB AB AB y y 00 00 01 0 0 01 00 11 1 010 00 10 1 0 11 00 10 1 0
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Mealy & Moore
Mealy machine: Output depends on both input & present state Moore machine: Output only depends on present state.
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Example of Mealy Machine
Present state Next State Output x=0 x=1 x=0 x=1AB AB AB y y 00 00 01 0 0 01 00 11 1 010 00 10 1 0 11 00 10 1 0
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Example of Moore Machine
Present state input Next State A B x A B 0 0 0 0 1 0 0 1 0 0 0 1 0 1 1 0 1 1 1 0 1 0 0 1 1 1 0 1 1 0 1 1 0 0 0 1 1 1 1 1
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State Reduction and Assignment
Goal: Reduce the number of states while keeping the external input-output requirements.
2m states need m flip-flops, so reducing the states may reduce flip-flops.
If two states are equal, one can be removed but what are equal states?
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State Reduction Example
As an example consider the input sequence below:
010101110100 applied and start from state a.
State a a b c d e f f g f g ainput 0 1 0 1 0 1 1 0 1 0 0 output 0 0 0 0 0 1 1 0 1 0 0
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State Reduction Example
Present State Next State Output x=0 x=1 x=0 x=1
a a b 0 0 b c d 0 0 c a d 0 0 d e f 0 1 e a f 0 1 f g f 0 1 g a f 0 1
States e and g are equal since for each member of the set of inputs, they give the same output and send the circuit either to the same state or an equivalent state.
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State Reduction Example
Present State Next State Output x=0 x=1 x=0 x=1
a a b 0 0 b c d 0 0 c a d 0 0 d e f 0 1 e a f 0 1 f e f 0 1
Table and state diagram after the first reduction: g is removed and replaced by state e.
NEW equal states: d and f
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State Reduction Example
Present State Next State Output x=0 x=1 x=0 x=1
a a b 0 0 b c d 0 0 c a d 0 0 d e d 0 1 e a d 0 1
Table and state diagram after the second reduction: f is removed and replaced by state d.
If we apply the same sequence
State a a b c d e d d e d e ainput 0 1 0 1 0 1 1 0 1 0 0 output 0 0 0 0 0 1 1 0 1 0 0
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Design Procedure
First Step: From the word description of the problem derive a state diagram
example:design a circuit to detect three or more consecutive 1’s in a string of bits coming through an input line.
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Design steps
1.From word description, derive state diagram 2.Reduce the number of states 3.Assign binary values to states 4.Obtain the binary coded state table 5.Choose the type of flip-flop used 6.Derive the simplified flip-flop input and output equations 7.Draw the logic diagram
steps 4 to 7can be implemented by exact algorithms and can be automated.
The part of the design that is well-defined is referred to as synthesis.
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State Table for Sequence Decoder
Present State Input Next State Output A B x A B y 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 1 1 1 0 0 1 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 0 1 1 1 1 1 1 1
A(t+1)= Σ(3,5,7)
B(t+1)= Σ(1,5,7)
Y(A,B,x)= Σ(6,7)
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Synthesis Using D Flip-Flops
A(t+1)=DA(A,B,x)= Σ(3,5,7)
B(t+1)=DB(A,B,x)= Σ(1,5,7)
Y(A,B,x)= Σ(6,7)
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Logic Diagram for a Sequence Detector
DA = Ax + BxDB= Ax + B’xy=AB
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Excitation Tables
Using flip-flops other than D can be complicated.
Why? Input equations for the circuit must be derived indirectly from the state table
Excitation tables can help.
Excitation tables give us the flip-flop input for every state transition.
Example : JK- Recall Q(t+1) = JQ’(t) + K’Q(t) Q(t) Q(t+1) J K 0 0 0 X 0 1 1 X 1 0 X 1 1 1 X 0
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Excitation Tables- T flip-flop
Example : JK- Recall Q(t+1) = TQ’(t) + T’Q(t) = T XOR Q Q(t) Q(t+1) T 0 0 0 0 1 1 1 0 1 1 1 0
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Synthesis Using JK Flip-Flops
Present State Input Next State Flip-Flop Inputs A B x A B JA KA JB KB 0 0 0 0 0 0 x 0 x 0 0 1 0 1 0 x 1 x 0 1 0 1 0 1 x x 1 0 1 1 0 0 0 x x 0 1 0 0 0 0 x 0 0 x 1 0 1 1 1 x 0 1 x 1 1 0 0 0 x 0 x 0 1 1 1 1 1 x 1 x 1
We also include J, K input conditions, derived from the excitation table.
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Synthesis Using JK Flip-Flops
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Synthesis Using JK Flip-Flops
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Synthesis Using T Flip-Flops
Example: 3-bit Binary Counter
The counter counts the clock.
Clock does not appear explicitly in the state diagram.
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Synthesis Using T Flip-Flops
Present State Next State Flip-Flop InputsA2 A1 A0 A2 A1 A0 TA2 TA1 TA00 0 0 0 0 1 0 0 10 0 1 0 1 0 0 1 10 1 0 0 1 1 0 0 10 1 1 1 0 0 1 1 11 0 0 1 0 1 0 0 1 1 0 1 1 1 0 0 1 11 1 0 1 1 1 0 0 11 1 1 0 0 0 1 1 1
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Synthesis Using T Flip-Flops
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Synthesis Using T Flip-Flops
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Summary
State Reduction, Synthesis
