lecture 5 circuits simplification cont
DESCRIPTION
Lecture 5 Circuits Simplification Cont. CSCE 211 Digital Design. Topics Karnaugh Maps Sums-of-Products Form Products-of-Sums Form Readings 4.2-4.3. September 4, 2003. Overview. Last Time Basic Gates Adders: Half-adder, full-adder, ripple-carry adder Boolean Algebra: Axioms, Theorems, - PowerPoint PPT PresentationTRANSCRIPT
Lecture 5Circuits Simplification Cont.
Lecture 5Circuits Simplification Cont.
TopicsTopics Karnaugh Maps Sums-of-Products Form Products-of-Sums Form
Readings 4.2-4.3Readings 4.2-4.3
September 4, 2003
CSCE 211 Digital Design
– 2 – CSCE 211H Fall 2003
OverviewOverview
Last TimeLast Time Basic Gates Adders: Half-adder, full-adder, ripple-carry adder Boolean Algebra: Axioms, Theorems,
On last Time’s Slides(what we didn’t get to)On last Time’s Slides(what we didn’t get to) Principle of Duality N-variable Theorems
NewNew Combinational Circuit Analysis Algebraic analysis, Truth tables, Logic Diagrams Sums-of-Products and Products-of-Sums Circuit Simplification: Karnuagh Maps
VHDL – Half-AdderVHDL – Half-Adder
– 3 – CSCE 211H Fall 2003
Karnaugh MapsKarnaugh MapsTabular technique for simplifying circuitsTabular technique for simplifying circuits
two variable maps three variable maptwo variable maps three variable map
0000 1010
0101 1111
0
0
1
1
000000 010010 110110 100100
001001 011011 111111 101101
00 01 11 10
0
1
00 01 11 10
00 22 66 44
11 33 77 55
0
1
00 22
11 33
0 1
0
1
XYZ
XY
Z
XY
X
Y
– 4 – CSCE 211H Fall 2003
Karnaugh Map SimplificationKarnaugh Map Simplification
F(X,Y,Z) = F(X,Y,Z) =
11 11 11 11
11 11
ZXY 00 01 11 10
0
1 Z
Sum of minterms formSum of minterms form
F(X,Y,Z)=F(X,Y,Z)=
Minimize ? Fewer gates, fewer inputsMinimize ? Fewer gates, fewer inputs
F(X,Y,Z)=F(X,Y,Z)=
F(X,Y,Z)=F(X,Y,Z)=
– 5 – CSCE 211H Fall 2003
Karnaugh Map TerminologyKarnaugh Map Terminology
F(X,Y,Z) = F(X,Y,Z) =
11 11
11 11
ZXY 00 01 11 10
0
1 Z
Implicant set - rectangular group of size 2Implicant set - rectangular group of size 2ii of adjacent of adjacent containing onescontaining ones
Each implicant set of size 2Each implicant set of size 2ii of corresponds to a of corresponds to a product term in which i variables are true and the product term in which i variables are true and the rest falserest false
Implicant Sets:Implicant Sets:
– 6 – CSCE 211H Fall 2003
Karnaugh Map TerminologyKarnaugh Map Terminology
F(X,Y,Z) = F(X,Y,Z) =
ZXY 00 01 11 10
0
1 Z
Prime implicant – an implicant set that is as large as Prime implicant – an implicant set that is as large as possiblepossible
Implies – We say P implies F if everytime P(XImplies – We say P implies F if everytime P(X11, X, X22, … X, … Xnn) ) is true then F (Xis true then F (X11, X, X22, … X, … Xnn) is true also.) is true also.
If P(XIf P(X11, X, X22, … X, … Xnn) is a prime implicant then P implies F) is a prime implicant then P implies F
– 7 – CSCE 211H Fall 2003
Karnaugh Map TerminologyKarnaugh Map Terminology
F(X,Y,Z) = F(X,Y,Z) =
11 11 11
11 11
ZXY 00 01 11 10
0
1 Z
Prime implicants – Prime implicants –
If P(XIf P(X11, X, X22, … X, … Xnn) is a prime implicant then P implies F ) is a prime implicant then P implies F and if we delete any variable from P this does not and if we delete any variable from P this does not imply F. imply F.
– 8 – CSCE 211H Fall 2003
Karnaugh Map SimplificationKarnaugh Map Simplification
F(X,Y,Z) = F(X,Y,Z) =
ZXY 00 01 10 11
0
1 Z
F(X,Y,Z) = F(X,Y,Z) =
– 9 – CSCE 211H Fall 2003
Karnaugh Map SimplificationKarnaugh Map Simplification
F(X,Y,Z) = F(X,Y,Z) =
ZXY 00 01 10 11
0
1 Z
F(X,Y,Z) = F(X,Y,Z) =
– 10 – CSCE 211H Fall 2003
4 Variable Map Simplification4 Variable Map Simplification
F(W,X,Y,Z) = F(W,X,Y,Z) =
YZWX 00 01 11 10
00
01
11
10
Z
00000000 01000100 11001100 10001000
00010001 01010101 11011101 10011001
00110011 01110111 11111111 10111011
00100010 01100110 11101110 10101010
W
X
Y
– 11 – CSCE 211H Fall 2003
4 Variable Map Simplification4 Variable Map Simplification
F(W,X,Y,Z) = F(W,X,Y,Z) =
YZWX 00 01 11 10
00
01
11
10
Z
00 44 1212 88
11 55 1313 99
33 77 1515 1111
22 66 1414 1010
W
X
Y
– 12 – CSCE 211H Fall 2003
Karnaugh Map SimplificationKarnaugh Map Simplification
F(W,X,Y,Z) = F(W,X,Y,Z) =
YZWX 00 01 11 10
00
01
11
10
Z
W
X
Y
– 13 – CSCE 211H Fall 2003
Karnaugh Map SimplificationKarnaugh Map Simplification
F(W,X,Y,Z) = F(W,X,Y,Z) =
YZWX 00 01 11 10
00
01
11
10
Z
W
X
Y
– 14 – CSCE 211H Fall 2003
Karnaugh Map SimplificationKarnaugh Map Simplification
F(W,X,Y,Z) = F(W,X,Y,Z) =
YZWX 00 01 11 10
00
01
11
10
Z
W
X
Y
– 15 – CSCE 211H Fall 2003
Larger MapsLarger Maps
Five variable maps - Figure X4.72 page 307Five variable maps - Figure X4.72 page 307
Six variable maps - Figure X4.74 page 308Six variable maps - Figure X4.74 page 308
But who cares, use Quine-McKluskey section 4.5But who cares, use Quine-McKluskey section 4.5
– 16 – CSCE 211H Fall 2003
Don’t Care ConditionsDon’t Care Conditions
F(W,X,Y,Z) = F(W,X,Y,Z) =
YZWX 00 01 11 10
00
01
11
10
Z
W
X
Y
– 17 – CSCE 211H Fall 2003
Don’t Care ConditionsDon’t Care Conditions
F(W,X,Y,Z) = F(W,X,Y,Z) =
YZWX 00 01 11 10
00
01
11
10
Z
W
X
Y
– 18 – CSCE 211H Fall 2003
Don’t Care ConditionsDon’t Care Conditions
F(W,X,Y,Z) = F(W,X,Y,Z) =
YZWX 00 01 11 10
00
01
11
10
Z
W
X
Y
– 19 – CSCE 211H Fall 2003
Don’t Care ConditionsDon’t Care Conditions
F(W,X,Y,Z) = F(W,X,Y,Z) =
YZWX 00 01 11 10
00
01
11
10
Z
W
X
Y
– 20 – CSCE 211H Fall 2003
SummarySummary
HomeworkHomework
1.1. 4.6b4.6b
2.2. 4.314.31
3.3. 4.39b4.39b
4.4. 4.404.40
5.5. 4.554.55
6.6. 4.13d4.13d
7.7. 4.19c,e4.19c,e