boolean expressions
DESCRIPTION
Boolean Expressions. Lecture 3 Digital Design and Computer Architecture Harris & Harris Morgan Kaufmann / Elsevier, 2007. Logistics. Do the reading Handout Lecture Notes On web: Lab 1 solutions, Lab 2. Overview. Boolean Algebra K-maps X’s and Z’s. Boolean Expressions. - PowerPoint PPT PresentationTRANSCRIPT
Boolean ExpressionsBoolean Expressions
Lecture 3Digital Design and Computer Architecture
Harris & HarrisMorgan Kaufmann / Elsevier, 2007
LogisticsLogistics
• Do the reading
• Handout– Lecture Notes– On web: Lab 1 solutions, Lab 2
OverviewOverview
• Boolean Algebra
• K-maps
• X’s and Z’s
Boolean ExpressionsBoolean Expressions
A B Y0 00 11 01 1
minterm
A BA BA B
A B
2.2.2 Sum-of-Products Form2.2.2 Sum-of-Products Form
Boolean ExpressionsBoolean Expressions
A + BA B Y
0 00 11 01 1
maxterm
A + BA + BA + B
2.2.3 Product-of-Sums Form2.2.3 Product-of-Sums Form
Boolean EquationsBoolean Equations
• You are going to the Hoch for lunch– You won’t eat lunch (E) if it’s not open (O) or– If they only serve corndogs (C)
• Write a truth table for determining if you will eat lunch (E).
SOP & POS FormSOP & POS Form
• SOP – sum-of-products
• POS – product-of-sums
O C E0 00 11 01 1
mintermO CO CO CO C
O + CO C Y
0 00 11 01 1
maxterm
O + CO + CO + C
SOP & POS FormSOP & POS Form
• SOP – sum-of-products
• POS – product-of-sums
A B Y0 00 11 01 1
minterm
A BA BA B
A B
A + BA B Y
0 00 11 01 1
maxterm
A + BA + BA + B
2.4 From Logic to Gates2.4 From Logic to Gates
• Fig. 2.23 shows Schematic of • It is an example of Gate Array
CBACBACBA
2.4 From Logic to Gates2.4 From Logic to Gates
• Two-level logic: 8 forms into 2 groups
Multiple Output CircuitsMultiple Output Circuits
A0
A1
PRIORITYENCODER
A2
A3
PriorityEncoder
Y0
Y1
Y2
Y3
A1 A0
0 00 11 01 1
0000
Y3 Y2 Y1 Y0
0000
0011
0100
A3 A2
0 00 00 00 0
0 0 0 1 0 00 10 11 01 10 0
0 10 10 11 0
0 11 01 01 10 00 1
1 01 01 11 1
1 01 11 11 1
0001
1110
0000
0000
1 0 0 01111
0000
0000
0000
1 0 0 01 0 0 0
Priority Encoder HardwarePriority Encoder Hardware
A1 A00 00 11 01 1
0000
Y3 Y2 Y1 Y0
0000
0011
0100
A3 A20 00 00 00 0
0 0 0 1 0 00 10 11 01 10 0
0 10 10 11 0
0 11 01 01 10 00 1
1 01 01 11 1
1 01 11 11 1
0001
1110
0000
0000
1 0 0 01111
0000
0000
0000
1 0 0 01 0 0 0
A3A2A1A0Y3
Y2
Y1
Y0
2.6 Don’t Cares2.6 Don’t Cares
A1 A00 00 11 01 1
0000
Y3 Y2 Y1 Y0
0000
0011
0100
A3 A20 00 00 00 0
0 0 0 1 0 00 10 11 01 10 0
0 10 10 11 0
0 11 01 01 10 00 1
1 01 01 11 1
1 01 11 11 1
0001
1110
0000
0000
1 0 0 01111
0000
0000
0000
1 0 0 01 0 0 0
A1 A0
0 00 11 XX X
0000
Y3 Y2 Y1 Y0
0001
0010
0100
A3 A2
0 00 00 00 1
X X 1 0 0 01 X
2.3.3 Boolean Axioms & Theorems2.3.3 Boolean Axioms & Theorems
Boolean Axioms & TheoremsBoolean Axioms & Theorems
無法靠直覺
• What is the Boolean expression for this circuit?
Bubble PushingBubble Pushing
AB
YCD
2.7 Karnaugh Maps (K-Maps)2.7 Karnaugh Maps (K-Maps)
• Sum-of-products (SOP) form can be tedious to simplify using Boolean algebra
• K-maps allow us to do the same thing graphically
A B Y0 0 00 1 01 0 11 1 1
Truth TableA
B0 1
0
1
0 1
0 1
YK-Map
m0
m3
3-input K-map3-input K-map
C 00 01
0
1
Y
11 10AB
ABC
ABC
ABC
ABC
ABC
ABC
ABC
ABC
1 0
B C Y0 0 00 1 01 01 1 1
Truth Table
C 00 01
0
1
Y
11 10ABA
0000
0 0 00 1 11 0 01 1 0
1111
0
1
1
0
0
0
1
K-Map
3-input K-map3-input K-map
C 00 01
0
1
Y
11 10AB
ABC
ABC
ABC
ABC
ABC
ABC
ABC
ABC
1 0
B C Y0 0 00 1 01 01 1 1
Truth Table
C 00 01
0
1
Y
11 10ABA
0000
0 0 00 1 01 0 01 1 1
1111
0
1
1
1
0
0
0
K-Map
A’B
BC’
K-map DefinitionsK-map Definitions
• Complement: variable with a bar over it
• Literal: variable or its complement
• Implicant: product of literals
• Prime implicant: implicant corresponding to the larges circle in the K-map
K-map RulesK-map Rules
• Each circle must span a power of 2 (i.e. 1, 2, 4) squares in each direction
• Each circle must be as large as possible• A circle may wrap around the edges of the K-
map• A one in a K-map may be circled multiple
times• A “don't care” (X) is circled only if it helps
minimize the equation
4-input K-map4-input K-map
01 11
1
0
0
1
0
0
1
101
1
1
1
1
0
0
0
1
11
10
AB00
00
10CD
Y
(m2) (m10)
(m15)
7-segment display7-segment display
• See example 2.10 for it. It is useful in implementing clock
2.6 Don’t Cares2.6 Don’t Cares
01 11
1
0
0
X
X
X
1
101
1
1
1
1
X
X
X
X
11
10
CD
AB00
00
10
Y
Contention: XContention: X
• Not don’t care• Not just 1’s and 0’s• Contention: X
A = 1
Y = X
B = 0
Floating: ZFloating: Z
Tri-state Buffer
E A Y0 0 Z0 1 Z1 0 01 1 1
A
E
Y
2.8 Building Blocks2.8 Building Blocks
• Multiplexer vs. Demultiplexer• Decoders vs. Encoder• Priority Encoder
Next TimeNext Time
• Timing• Hazards• Sequential Circuits