Combinational Networks

1
Combinational Networks

• Chapter 10
• CIS-2261

2
HOMEWORK CHAPTER 10
Exercises 1a, 5a, 7a, 13, 14a-f, 20a-c, 25, 28b,
30a-c, 32a-b, 35a-b, 41, 49
Hope you have a VERRRY NICE day !!
3
OVERVIEW

• Boolean Algebra and Logic Gates
• Combinational Analysis
• Combinational Design
• Combinational Devices

4
Introduction

• Time to think about how we actually turn logic
  into hardware that can be used to form the heart
  of the modern day computer
• There are two basic types of logic systems found
  in computers
• Combinational Logic Systems (retains no memory of
  prior events)
• Sequential Logic Systems (retains memory of
  events which have already occurred)
• Both use Logic Gates as basic components

5
Introduction

• The building block of all logic gates is the
  circuit
• A collection of transistors and wires
• Performs a Boolean operation on one or more
  inputs and produces an output
• A combination of logic gates taken together form
  combinational system (or network)
• Can describe the action of a combinational system
  in terms of a truth table
• Can use Boolean Algebra to reduce a combinational
  system to smallest size

6
Combinational Networks

• Think of in terms of inputs and outputs
• The inputs determine the outputs
• The actual output is determined by the set of
  logic gates found in the combinational network

A
B
X f(A,B,C,D)
C
D
7
Combinational Networks

• Three common methods for describing the behavior
  of a combinational network or system
• Truth Tables
• Boolean Algebraic Expressions
• Logic Diagrams

8
Combinational Networks
A
X f(A,B,C)
B
C
Truth Table
A
B
C
X
0
0
1
0
0
0
1
1
0
1
1
0
0
1
0
1
Truth Table describes operation of the
Combinational network
1
0
1
0
1
0
1
1
1
1
0
0
1
1
0
1
9
Boolean Algebra

• Three basic logical operations associated with
  binary variables
• AND Ex. X Y Z or XY Z. Interpreted as Z 1
  if and only if X 1 AND Y1 else Z0
• OR Ex XY Z. Interpreted as Z1 if X1 OR Y1
  OR if both X1 and Y1. Z0 if and only if both
  X0 AND Y0.
• NOT X Z. Interpreted as Z is what X is NOT. If
  X 1, then Z 0 and if X0, then Z1. Also
  referred to as the complement operation.

10
Truth Tables of Boolean Operators
NOT
X
X
0
1
1
0
11
Boolean Algebra

• Algebra that deals with binary variables and
  logic operations
• All complex logic functions are built on the
  three basic logic operations AND, OR, and NOT
  which operate on binary variables and the
  constants 1 and 0
• Think about DUALITY - used extensively in design
  of logic systems

12
Basic Identities for Boolean Algebra
1. X 0 X
3. X 1 1
5. X X X
10. X Y Y X
11. XY YX
Comm
12. X (Y Z) (X Y) Z
13. X(YZ) (XY)Z
Assoc
14. X(Y Z) XY XZ
15. X YZ (XY)(XZ)
Dist
DeMorg
13
DeMorgans Truth Tables
X
Y
XY
(XY)
X
Y
XY
XY
X Y
XY
0
0
0
1
1
1
0
1
1
1
0
1
1
0
1
0
0
1
0
1
1
0
1
0
0
1
0
1
0
1
1
1
1
0
0
0
1
0
0
0
14
Boolean Manipulation

• Why worry about manipulating Boolean equations?
• Simplification of digital circuits
• Fewer gates required
• Potential for faster operation
• Smaller is always better

15
Boolean Manipulation

• Duality principle revisited
• Used in simplifying logic equations
• Obtain the dual by interchanging OR and AND
  operations and replacing 1s with 0s and 0s
  with 1s
• The dual of
• XYZW is (X Y)(ZW)

16
Boolean Manipulation

• Complements
• Can derive the complement of a function
  algebraically by applying DeMorgans theorem
• or can obtain via truth table by interchanging
  1s and 0s for the function F
• or can obtain by interchanging AND and OR
  operations and complementing each variable (which
  is a straightforward application of DeMorgans
  theorem)

17
Boolean ManipulationComplement Functions
Function
Complement Function
F (AB)(AC)(BC)
18
Boolean ManipulationComplement Functions
Find the complement of F
F ABC ABC
Step 1 Find the Dual
F (ABC)(ABC)
Step 2 Complement each variable
F (ABC)(ABC)
Results in Complement Function
19
Boolean ManipulationConsensus Theorem

• Useful in simplifying Boolean expressions
• Eliminate unnecessary terms

1
Show
Truth Table
F
XY XZ YZ XY XZ YZ(XX)
X
Y
Z
F
G
0
0
0
0
0
XY XZ XYZ XYZ
0
0
1
1
1
XY XYZ XZ XYZ
0
1
0
0
0
0
1
1
1
1
XY(1 Z) XZ(1 Y)
1
0
0
0
0
XY XZ
G
1
0
0
1
0
1
1
1
1
0
1
1
1
1
1
1
?YZ is an unnecessary term
20
Logic Diagrams

• The third way of representing the behavior of a
  combinational network
• An interconnection of Logic Gates
• Gives a hardware perspective of the behavior
• Each of the three basic Boolean operators (AND,
  OR, NOT) has a logic gate associated with it
• There are more logic gates besides these three
  (XOR, NAND, XNOR, NOR)

21
Basic Digital Logic Gates
Truth Tables
Logic Gate
Operator
Boolean Function
AND
ZXY
OR
ZXY
NOT
22
Basic Digital Logic Gates
Truth Tables
Logic Gate
Operator
Boolean Function
NAND
ZXY
Invert the output
NOR
ZXY
XOR
23
OVERVIEW

• Boolean Algebra and Logic Gates
• Combinational Analysis
• Combinational Design
• Combinational Devices

24
Boolean Representation
The connection between the three representations
of Boolean behavior
25
Logic Diagram Example
Draw the Logic Diagram for F((ABBC)A)
A
B
F
C
26
Two Level Networks

• Logic can have any number of levels of gates to
  go through
• Fewer levels means faster operation in real life
• All combinational networks can be reduced to two
  levels
• A change in the input requires only two gate
  delays to propagate to the output
• A good thing to do when constructing digital
  logic circuits!

27
Two-Level Networks
Level 1
Level 2
28
Ubiquitous NAND
X
Y
XYZ
Z
Three Input NAND
EQUIVALENT TO
X
Y
X Y Z
Z
Three INVERTED Input OR
Because of De Morgans Theorem!!
29
Ubiquitous NOR
X
Y
XYZ
Z
Three Input NOR
EQUIVALENT TO
X
Y
X Y Z
Z
Three INVERTED Input AND
30
An And-OR ImplementationUsing NAND Gates
31
OVERVIEW

• Boolean Algebra and Logic Gates
• Combinational Analysis
• Combinational Design
• Combinational Devices

32
Boolean Functions in Standard Form(Canonical
Expressions)

• Boolean functions are in standard form when
  expressed in terms of product terms or sum terms
  - enhances simplification procedures
• A product term where all the input variables
  appear exactly once either complemented or
  uncomplemented is called a minterm
• A sum term where all the input variables appear
  exactly once either complemented or
  uncomplemented is called a maxterm

33
Standard FormMinterms and Maxterms for 3
Variables
Minterms
Maxterms
Product Term
Symbol
Sum Term
Symbol
X
Y
Z
0
0
XYZ
0
m0
XYZ
M0
0
0
1
m1
XYZ
M1
XYZ
0
1
0
m2
XYZ
M2
XYZ
0
1
1
m3
XYZ
M3
XYZ
1
0
0
m4
XYZ
M4
XYZ
1
0
1
m5
XYZ
M5
XYZ
1
1
0
m6
XYZ
M6
XYZ
1
1
1
m7
XYZ
M7
XYZ
Note A Minterm and a Maxterm with the same
subscript number are compliments of each other.
M0 m0
34
Putting a Function in Standard Form

• A Boolean Function can be expressed
  algebraically from a given
• truth table by forming the logical sum of all
  minterms which
• produce a 1 in the function

F 1 for each of the combinations of variables
XYZ, XYZ, XYZ, XYZ,
So F
XYZ XYZ XYZ XYZ
F m0 m2 m5 m7 ?m(0,2,5,7)
And
F m1 m3 m4 m6 ?m(1,3,4,6)
F m1 m3 m4 m6
F m1 m3 m4 m6
F M1 M3 M4 M6 ?M(1,3,4,6)
35
Focus on Minterms

• We will only focus on using Minterms in this class

36
Important Properties of Minterms

• There are 2n minterms for n Boolean variables
• Any Boolean function can be expressed as a
  logical sum of minterms
• The compliment of a function contains those
  minterms not included in the original function
• A function that includes all the 2n minterms is
  equal to logic 1

37
Sum of Products
A Boolean Function in Sum of Minterms form can be
simplified by algebraic manipulation or map
simplification
F 1 for each of the combinations of variables
XYZ, XYZ, XYZ, XYZ, XYZ
Truth Table
X
Y
Z
F
0
0
1
0
m0
So
0
0
1
1
m1
0
1
1
0
m2
F XYZ XYZ XYZ XYZ XYZ
0
1
0
1
m3
F ?m(0,1,2,4,5)
1
0
1
0
m4
F can be simplified to by algebraic or map
simplification techniques to Sum of Products
1
0
1
1
m5
1
1
0
0
m6
1
1
0
1
m7
F Y XZ
Which equivalent equation is easier to build??
38

• Map Simplification

39
Karnaugh Map Simplification

• Working with algebraic equations often tedious
  and error-prone
• Use a mapping technique to reduce an algebraic
  equation to its simplest form
• Can simplify equations of up to 5 variables
  readily by hand but beyond 5 becomes unweildy

40
Two-Variable Map

• There will be 4 minterms for a 2 variable
  equation, therefore use a Karnaugh map with 4
  squares
• 1s and 0s on the left side and top of map
  designate the values of the variable
• As shown, Variable X is complemented in row 0 and
  uncomplemented in row 1
• Variable Y appears complemented in column 0 and
  uncomplemented in column 1

41
Two-Variable Map
Equates

• The concept is to put a one in each square that
  has a
• corresponding minterm in the Boolean Equation
• Only four terms possible

42
Two-Variable Map Example
Simplify

• Put 1s in boxes of corresponding terms
• Put 0s (or leave blank) in all other boxes

• Simplified expression for F is obtained by
  grouping adjacent 1s
• and eliminating unneeded variables.
• In this problem there are two ways of grouping
  1s, 1 way is to
• circle the row where X 1 and the other is to
  circle the column
• where Y 1
• Y is eliminated from the two product terms
  where X 1 and X is
• eliminated from the two product terms where Y 1

ANSWER F X Y
43
Three-Variable Map

• There will be 8 minterms for a 3 variable
  equation, therefore use a Karnaugh map with 8
  squares
• The minterm numbers do not follow the normal
  binary sequence
• Only one bit changes from one adjacent column to
  the next

44
Three -Variable Map
Equates to
45
Three -Variable Map Example
Simplify
F(X, Y, Z) ?m(1, 2, 5, 6)
m1 m5 X Y Z X Y Z
(X X)(Y Z)
YZ
00
01
11
10
? X is redundant
X
0
1
0
1
0
m2 m6 X Y Z X Y Z
(X X)(Y Z)
1
0
1
0
1
? X is redundant
ANSWER F YZ YZ
46
Three -Variable Map Example
Simplify
F(X, Y, Z) ?m(1, 3, 5, 7)
m1 m3 m5 m7 X Y Z X Y Z X Y Z
X Y Z
((X(Y Y) X(Y Y))Z
Y
(X X)Z
YZ
00
01
11
10
? X and Y are redundant
X
0
1
1
0
0
ANSWER F Z
X
1
0
1
1
0
Z
47
Three -Variable Map Example
Simplify
F(X, Y, Z) ?m(3, 4, 6, 7)
Y
YZ
00
01
11
10
X
0
0
1
0
0
X
1
1
0
1
1
Z
Y is redundant
X is redundant
ANSWER F YZ XZ
48
Four -Variable Map
Y
YZ
00
01
11
10
WX
m0
m1
m3
m2
00
01
m4
m5
m7
m6
X
m12
m13
m15
m14
11
W
10
m8
m9
m11
m10
Z
49
Four -Variable Example
Truth Table
Y
X
Y
Z
F
W
YZ
0
0
0
0
0
00
01
11
10
WX
0
0
1
1
0
0
1
1
0
0
0
1
0
1
00
0
1
0
1
0
1
0
1
0
0
01
1
0
1
1
0
1
1
0
1
1
1
0
0
0
X
1
1
1
1
0
1
1
1
1
11
0
0
0
1
0
W
0
0
1
1
1
0
1
0
1
1
0
1
0
1
10
0
1
1
1
0
1
0
0
1
1
1
0
1
1
1
Z
1
1
0
1
1
1
1
1
1
1
ANSWER F XY YZ YZ WX
50
Dont Care Conditions

• Often times you will generate a truth table that
  has several combinations that you as the designer
  dont care about
• Use DONT CARES to simplify your expression
• Sometimes you can eliminate essential prime
  implicants
• Example Consider the following incompletely
  specified function
• F(W,X,Y,Z) ?m(1,3,7,11,15)
• d(W,X,Y,Z) ?m(0,2,5) - These are dont cares

51
Dont Care Example
Truth Table
Y
X
Y
Z
F
W
YZ
0
0
x
0
0
00
01
11
10
WX
0
0
1
1
0
0
1
x
0
0
X
1
1
X
00
0
1
1
1
0
1
0
0
0
0
01
1
0
x
1
0
0
X
1
0
1
1
0
0
0
X
1
1
1
1
0
0
0
1
0
11
0
0
0
1
0
W
0
0
1
1
0
0
1
0
1
0
0
0
1
0
10
0
1
1
1
1
1
0
0
1
0
1
0
1
1
0
Z
1
1
0
1
0
1
1
1
1
0
ANSWER F YZ WX
52
Dont Care Example
Truth Table
Y
X
Y
Z
F
W
YZ
0
0
x
0
0
00
01
11
10
WX
0
0
1
1
0
0
1
x
0
0
X
1
1
X
00
0
1
1
1
0
1
0
0
0
0
01
1
0
x
1
0
0
X
1
0
1
1
0
0
0
X
1
1
1
1
0
0
0
1
0
11
0
0
0
1
0
W
0
0
1
1
0
0
1
0
1
0
0
0
1
0
10
0
1
1
1
1
1
0
0
1
0
1
0
1
1
0
Z
1
1
0
1
0
1
1
1
1
0
ANSWER F YZ WZ
Second solution
53
OVERVIEW

• Boolean Algebra and Logic Gates
• Combinational Analysis
• Combinational Design
• Combinational Devices

54
Combinational Devices

• Several combinational devices are widely used in
  computer design
• Half and Full Adders
• Multiplexers
• Decoders
• Arithmetic Logic Units
• Demultiplexers

55
Multiplexers

• A multiplexer selects binary information from one
  of many inputs and directs that input to a single
  output
• Use selector variables to determine which input
  line is selected
• For 2n input lines there are n selection lines
  needed
• Used extensively in computer systems to move data
  through system

56
Multiplexer
D0
Truth Table
D1
S1
S0
F
F
0
0
D0
D2
0
1
D1
1
0
D2
D3
1
1
D3
S0
S1
57
4 to 1 Line Multiplexer

S0

S1


D0

D1
Y

D2
D3
58
Decoders

• Binary code of n bits is capable of representing
  up to 2n distinct elements
• A decoder can convert n coded inputs to a maximum
  of 2n unique ouputs
• How might you use a decoder in building a
  computer?

59
3 to 8 Line Decoder Truth Table
D5
D6
A2
A1
A0
D7
D4
D3
D2
D1
D0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
0
0
1
0
0
1
0
0
0
0
0
0
1
0
0
0
1
0
1
0
0
0
1
0
0
0
1
0
0
0
0
0
1
0
0
0
0
1
0
0
1
0
1
0
0
0
0
0
1
1
0
0
1
0
0
0
0
0
0
1
1
1
1
0
0
0
0
0
0
0
60
3 to 8 Line Decoder
D0 A2A1A0


D1 A2A1A0


A0

D2 A2A1A0



A1

D3 A2A1A0



D4 A2A1A0
A2


D5 A2A1A0


D6 A2A1A0

D7 A2A1A0
61
Demultiplexers

• Does the opposite of the multiplexer
• Selects 1 of 2n outputs to direct the input to
• Use selector variables to determine which output
  line is selected
• For 2n output lines there are n selection lines
  needed
• Used extensively in computer systems to move data
  through system

62
Demultiplexer
D0
Truth Table
D1
S1
S0
D0
D1
D2
D3
F
0
0
F
0
0
0
D2
0
1
0
F
0
0
1
0
0
0
F
0
D3
1
1
0
0
0
F
S0
S1
63
1 to 4 Line Demultiplexer

S0

S1


D0

D1


D2

D3
Input

64
Arithmetic Circuits

• A combinational circuit that performs arithmetic
  operations with binary numbers or decimal numbers
  in a binary code
• Most basic operation is the addition of binary
  digits
• Most basic circuit is the half-adder
• Two outputs Sum and Carry
• Two inputs A and B

65
Basic Half-Adder
Inputs
Outputs
A
Sum
A
B
Sum
Carry
0
0
0
0
B
1
0
1
0
1
0
1
0
Carry
1
1
0
1
66
Basic Full-Adder Construction
Odd function therefore implement using XOR gates
Sum
BCin
00
01
11
10
A
0
1
0
1
0
1
1
0
1
0
Cout
BCin
00
01
11
10
A
0
0
1
0
0
1
0
1
1
1
Cout AB ACin BCin Sum A xor B xor Cin
67
Basic Full-Adder
68
End Chapter 10