review
TRANSCRIPT
Sequential Circuits
All state changes (transitions from present state Qn to next state Qn+1) in memory elements are controlled by a system clock
Review of Flipflops
D flipflop
T flipflop
SR flipflop
JK flipflop
Circuit Analysis Example
Draw the equivalent state diagram of the following synchronous sequential circuit which has a 1-bit input S and 1-bit output Z
Step 1: Determine Flipflop Input and System Output Equations
All equations should be in terms of the system input and flipflop outputs only
D = S’Q + SQ’
Z = Q
Step 2: Create the State Transition Table
Input Current State Excitation Next State Output
S Qn D Qn+1 Z
0 0 0 0 0
0 1 1 1 1
1 0 1 1 0
1 1 0 0 1
• Write all possible combinations of system inputs and current state (flipflop outputs)
Step 2: Create the State Transition Table
Input Current State Excitation Next State Output
S Qn D Qn+1 Z
0 0 0 0 0
0 1 1 1 1
1 0 1 1 0
1 1 0 0 1
• Using the equation for the flipflop input, complete the excitation column
• D = S’Q + SQ’
Step 2: Create the State Transition Table
Input Current State Excitation Next State Output
S Qn D Qn+1 Z
0 0 0 0 0
0 1 1 1 1
1 0 1 1 0
1 1 0 0 1
• Knowing the flipflop behavior given the flipflop input, determine the next state (next ff output)
Step 2: Create the State Transition Table
Input Current State Excitation Next State Output
S Qn D Qn+1 Z
0 0 0 0 0
0 1 1 1 1
1 0 1 1 0
1 1 0 0 1
• Using the equation for the system output, complete the table.
• Z = Q
• Note: Q dependent
Step 3: Draw the State Diagram
Create circles (your current states)
Create arrows pointing to the next states for every input combination
Follow the convention on the arrow: input/output
Input Current State Excitation Next State Output
S Qn D Qn+1 Z
0 0 0 0 0
0 1 1 1 1
1 0 1 1 0
1 1 0 0 1
Step 3: Draw the State Diagram
0/0 0/1
1/0
1/1
1 0
Input Current State Excitation Next State Output
S Qn D Qn+1 Z
0 0 0 0 0
0 1 1 1 1
1 0 1 1 0
1 1 0 0 1
State Transition Diagrams
Usual convention: state-> bubbles, transitions->arrows
Input and output values are indicated beside the transition it is associated with
Mealy or Moore Machines
Mealy machines Output depends on the state and the input
Output values are indicated beside the transition (arrow)
There is a combinational path between the output and the input
Moore machines Output depends on the state alone
Output values can be indicated in the state (bubble)
No direct combinational path between the output and the input
Mealy or Moore Machine
0 0
1
0
0
0
1
1
1
A B 0/0
0/0
1/0
1/1
A/0 B/0
C/1
Recall: State Minimization
Two states are equivalent if it satisfies the following: Corresponding outputs for all input combinations are the same
Corresponding next states for all input combinations are equivalent
Three techniques were taught under EEE 21 Reduction by Inspection (Very tedious, result may still be reducible)
Reduction by Partitioning (Usually involves many steps, gets the job done)
Reduction by Implication (Can be done in one table, gets the job done)
Adel’s personal favorite!
Reduction by Inspection
Reduction by Inspection
Reduction by Inspection
Reduction by Inspection
Reduction by Implication
Reduction by Implication
X
X
X
X
X
X
X
X
X
X
X
X
X X
X
X
X X X X
BD CG
AD CF
CD AC
EH AD
EH AD
EG AH
BD CG
AD CF
CD AC
GH DH
AB FG
BC AG
GH DH
AC AF
/
/
Reduction by Implication
X
X
X
X
X
X
X
X
X
X
X
X
X X
X
X
X X X X
BD CG
AD CF
CD AC
EH AD
EH AD
EG AH
BD CG
AD CF
CD AC
GH DH
AB FG
BC AG
GH DH
AC AF
/
/
X
X X
X X
X
X X
X
/
/
/
/
/
Reduction by Implication
X
X
X
X
X
X
X
X
X
X
X
X
X X
X
X
X X X X
BD CG
AD CF
CD AC
EH AD
EH AD
EG AH
BD CG
AD CF
CD AC
GH DH
AB FG
BC AG
GH DH
AC AF
/
/
X
X X
X X
X
X X
X
/
/
/
/
/
A = D = G
B = C = F
E = H
I
Reduction by Implication
A = D = G
B = C = F
E = H
I
Comparison
Reduction by Inspection:
6 states
Reduction by Implication:
4 states
Reduction by Partitioning
Group states according to their output pattern (class).
Once grouped, check next states and determine what class the next states belong to.
If states belonging on the same class has the same next class patterns, they are equivalent states.
Else, create a new class for the state/s violating the previous rule (step 3).
Since a new class has been created, redo the process (starting at step 2).
Once step 3 is not violated anymore, stop.
Reduction by Partitioning
Reduction by Partitioning
Reduction by Partitioning
Reduction by Partitioning