CPSC 359
Assignment 2 Prep
PhD Student
Fall 2018
Today
Preparatory material for assignment 2:
- Stateful problem example
- 5 variable k-map explanation and example
Pedestrian crosswalk
Problem
Design a crosswalk system in which a user presses a button at the crosswalk, singling a traffic light to turn yellow for a brief moment, then eventually red for at least two seconds.
It's not a great crosswalk, but it's easy to illustrate the design process
Pedestrian crosswalk
Buttons?
A button isn't great for these designs.
We want behaviour similar to an elevator button.
This is just an SR latch...
Problem
Inputs:
- Button press -- we'll name the input variable b
Outputs:
- Red light: r
- Yellow light: y
- Green light: g
- Button reset: reset
How many flip flops do we need?
\(G\)
Assume state transitions happen every 2 seconds
No button press
Button press
\(Y\)
\(R_0\)
\(R_1\)
reset button state
State Diagram
State Diagram
We have \(2^2=4\) states so we need two flip flops to represent this state. We'll name them A and B
00
Assume state transitions happen every two seconds
01
11
10
0/1000
1/0100
X/0010
X/0010
X/1001
b/g,y,r,reset
inputs/outputs
| A | B | b | A | B | Ja | Ka | Jb | Kb |
|---|---|---|---|---|---|---|---|---|
| 0 | 0 | 0 | ||||||
| 0 | 0 | 1 | ||||||
| 0 | 1 | 0 | ||||||
| 0 | 1 | 1 | ||||||
| 1 | 0 | 0 | ||||||
| 1 | 0 | 1 | ||||||
| 1 | 1 | 0 | ||||||
| 1 | 1 | 1 |
State
Next State
J K flip-flop inputs for next state
Excitation Table
\(G\)
\(Y\)
\(R_1\)
\(R_0\)
| A | B | b | A | B | Ja | Ka | Jb | Kb |
|---|---|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 | ||||
| 0 | 0 | 1 | 0 | 1 | ||||
| 0 | 1 | 0 | 1 | 1 | ||||
| 0 | 1 | 1 | 1 | 1 | ||||
| 1 | 0 | 0 | 0 | 0 | ||||
| 1 | 0 | 1 | 0 | 0 | ||||
| 1 | 1 | 0 | 1 | 0 | ||||
| 1 | 1 | 1 | 1 | 0 |
State
Next State
J K flip-flop inputs for next state
Excitation Table
\(G\)
\(Y\)
\(R_1\)
\(R_0\)
| A | B | b | A | B | Ja | Ka | Jb | Kb |
|---|---|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 | 0 | X | 0 | X |
| 0 | 0 | 1 | 0 | 1 | ||||
| 0 | 1 | 0 | 1 | 1 | ||||
| 0 | 1 | 1 | 1 | 1 | ||||
| 1 | 0 | 0 | 0 | 0 | ||||
| 1 | 0 | 1 | 0 | 0 | ||||
| 1 | 1 | 0 | 1 | 0 | ||||
| 1 | 1 | 1 | 1 | 0 |
State
Next State
J K flip-flop inputs for next state
Excitation Table
\(G\)
\(Y\)
\(R_1\)
\(R_0\)
| A | B | b | 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 | 1 | ||||
| 0 | 1 | 1 | 1 | 1 | ||||
| 1 | 0 | 0 | 0 | 0 | ||||
| 1 | 0 | 1 | 0 | 0 | ||||
| 1 | 1 | 0 | 1 | 0 | ||||
| 1 | 1 | 1 | 1 | 0 |
State
Next State
J K flip-flop inputs for next state
Excitation Table
\(G\)
\(Y\)
\(R_1\)
\(R_0\)
| A | B | b | 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 | 1 | 1 | X | X | 0 |
| 0 | 1 | 1 | 1 | 1 | 1 | X | X | 0 |
| 1 | 0 | 0 | 0 | 0 | X | 1 | 0 | X |
| 1 | 0 | 1 | 0 | 0 | X | 1 | 0 | X |
| 1 | 1 | 0 | 1 | 0 | X | 0 | X | 1 |
| 1 | 1 | 1 | 1 | 0 | X | 0 | X | 1 |
State
Next State
J K flip-flop inputs for next state
Excitation Table
\(G\)
\(Y\)
\(R_1\)
\(R_0\)
| A | B | b | A | B | g | y | r | reset |
|---|---|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 |
| 0 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 |
| 1 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 |
| 1 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 0 |
| 1 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 0 |
State
Next State
Combinational Outputs
\(G\)
\(Y\)
\(R_1\)
\(R_0\)
Excitation Table
00
01
11
10
0/1000
1/0100
X/0010
X/0010
X/1001
b/g,y,r,reset
inputs/outputs
| A | B | b | A | B | g | y | r | reset |
|---|---|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 |
| 0 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 |
| 1 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 |
| 1 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 0 |
| 1 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 0 |
State
Next State
Combinational Outputs
\(G\)
\(Y\)
\(R_1\)
\(R_0\)
Excitation Table
k-Maps
| 0 | 0 | 1 | 1 |
| X | X | X | X |
00
01
11
10
0
1
Ja
A
B,b
| 1 | 1 | 0 | 0 |
| X | X | X | X |
00
01
11
10
0
1
A
B,b
Ka
Ja=B
Ka=B'
k-Maps
| 0 | 1 | X | X |
| 0 | 0 | X | X |
00
01
11
10
0
1
Jb
A
B,b
| X | X | 0 | 0 |
| X | X | 1 | 1 |
00
01
11
10
0
1
A
B,b
Kb
Jb=A'b
Kb=A
k-Maps
| 1 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 |
00
01
11
10
0
1
g
A
B,b
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 |
00
01
11
10
0
1
A
B,b
y
g=B'b' + AB'
y=A'B'b
k-Maps
| 0 | 0 | 1 | 1 |
| 0 | 0 | 1 | 1 |
00
01
11
10
0
1
r
A
B,b
| 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 |
00
01
11
10
0
1
A
B,b
reset
r=B
reset=AB
k-Map Recap
Ja=B
Ka=B'
Jb=A'b
Kb=A
g=B'b' + AB'
y=A'B'b
r=B
reset=AB
The minimized functions are:
Logisim Demo
00
01
11
10
0/1000
1/0100
X/0010
X/0010
X/1001
State Diagram Notes
00
01
10
11
0/1000
1/0100
X/0010
X/0010
X/1001
State Diagram Notes
| A | B | b | 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 | 1 | X |
| 0 | 1 | 1 | 1 | 0 | 1 | X | 1 | X |
| 1 | 0 | 0 | 1 | 1 | X | 0 | 1 | X |
| 1 | 0 | 1 | 1 | 1 | X | 0 | 1 | X |
| 1 | 1 | 0 | 0 | 0 | X | 1 | X | 1 |
| 1 | 1 | 1 | 0 | 0 | X | 1 | X | 1 |
State
Next State
J K flip-flop inputs for next state
Excitation Table
\(G\)
\(Y\)
\(R_1\)
\(R_0\)
00
01
10
11
/01
/11
/10
/00
Two Bit Gray Code
Without direction or enable
00
01
11
10
/01
/11
/10
/00
Two Bit Gray Code
Without direction or enable
5-variable k-Maps
| a | b | c | d | e | f(a,b,c,d,e) |
|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 1 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 1 | 1 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 1 | 1 | 1 |
| 0 | 1 | 1 | 0 | 0 | 1 |
| 0 | 1 | 1 | 0 | 1 | 0 |
| 0 | 1 | 1 | 1 | 0 | 0 |
| 0 | 1 | 1 | 1 | 1 | 0 |
5-variable k-Maps
| a | b | c | d | e | f(a,b,c,d,e) |
|---|---|---|---|---|---|
| 1 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 1 | 1 | 1 |
| 1 | 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 1 | 0 | 1 | 0 |
| 1 | 0 | 1 | 1 | 0 | 0 |
| 1 | 0 | 1 | 1 | 1 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 1 | 1 |
| 1 | 1 | 0 | 1 | 0 | 0 |
| 1 | 1 | 0 | 1 | 1 | 1 |
| 1 | 1 | 1 | 0 | 0 | 1 |
| 1 | 1 | 1 | 0 | 1 | 0 |
| 1 | 1 | 1 | 1 | 0 | 0 |
| 1 | 1 | 1 | 1 | 1 | 1 |
5-variable k-Maps
a=0
a=1
bc
bc
de
de
00
01
11
10
00
01
11
10
00
01
11
10
General Strategy: Use two 4 variable maps
00
01
11
10
5-variable k-Maps
| a | b | c | d | e | f(a,b,c,d,e) |
|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 1 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 1 | 1 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 1 | 1 | 1 |
| 0 | 1 | 1 | 0 | 0 | 1 |
| 0 | 1 | 1 | 0 | 1 | 0 |
| 0 | 1 | 1 | 1 | 0 | 0 |
| 0 | 1 | 1 | 1 | 1 | 0 |
5-variable k-Maps
| 0 | 1 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 |
a=0
a=1
bc
bc
de
de
00
01
11
10
00
01
11
10
00
01
11
10
General Strategy: Use two 4 variable maps
00
01
11
10
5-variable k-Maps
| a | b | c | d | e | f(a,b,c,d,e) |
|---|---|---|---|---|---|
| 1 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 1 | 1 | 1 |
| 1 | 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 1 | 0 | 1 | 0 |
| 1 | 0 | 1 | 1 | 0 | 0 |
| 1 | 0 | 1 | 1 | 1 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 1 | 1 |
| 1 | 1 | 0 | 1 | 0 | 0 |
| 1 | 1 | 0 | 1 | 1 | 1 |
| 1 | 1 | 1 | 0 | 0 | 1 |
| 1 | 1 | 1 | 0 | 1 | 0 |
| 1 | 1 | 1 | 1 | 0 | 0 |
| 1 | 1 | 1 | 1 | 1 | 1 |
5-variable k-Maps
| 0 | 1 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 |
| 1 | 1 | 1 | 0 |
| 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 0 |
| 0 | 1 | 1 | 0 |
a=0
a=1
bc
bc
de
de
00
01
11
10
00
01
11
10
00
01
11
10
General Strategy: Use two 4 variable maps
00
01
11
10
5-variable k-Maps
a=0
a=1
5-variable k-Maps
| 0 | 1 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 |
| 1 | 1 | 1 | 0 |
| 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 0 |
| 0 | 1 | 1 | 0 |
a=0
a=1
bc
bc
de
de
00
01
11
10
00
01
11
10
00
01
11
10
Colour indicates the same group
00
01
11
10
bcd'e'
+ c'e
+ a'b'd'e
+ abde
+ ab'c'd'
f(a,b,c,d,e) =
CPSC 359: Assignment 2 Prep
By Joshua Horacsek
