Digital Logic

Digital Logic

CS2052 Computer Architecture

Computer Science & Engineering

University of Moratuwa

Dilum BandaraDilum.Bandara@uom.lk

Outline

Logic gates

Boolean Algebra

Kanaugh Maps

2

Logic Gates

Every digital device is based on a set of chips

designed to store & process information

Basic building blocks of these chips are logic

gates

Implementation of gates can be different

Different materials & fabrication technologies

Different operating voltages

e.g., 5v vs. 0v, 3+v vs. 0.5v

Today, they go as low as 1.9 - 2.1v

But their logical behavior is consistent across all

computers3

Common Logic Gates

NOT/inverter

AND

4

Truth table

Algebraic function

Common Logic Gates (Cont.)

OR

NAND

5

Common Logic Gates (Cont.)

NOR

XOR/EXOR (Exclusive OR)

6

Common Logic Gates (Cont.)

NXOR/EXNOR (Exclusive NOR)

7

Exercise

Implement 3-input AND using 2-inputs ANDs

Implement a NOT gate using a NAND

8

Fundamental Logic Gate

2-input NAND gate can be used to build any

other gate

9

Exercise

Implement AND using NANDs

Homework

Implement following gates using NANDs

OR

XOR

10

Boolean Algebra

Boolean variable

Takes only 2 values – either TRUE (1) or FALSE (0)

Boolean function

Mapping from Boolean variables to a Boolean value

Application

Can represent complex relationships in a digital circuit

Boolean algebra

Deals with binary variables & logic operations

operating on those variables

Application

Can simplify a relationship among multiple inputs in a digital

circuit 11

Basic Identities of Boolean Algebra

Basic operations

AND (.), OR (+), NOT (– or /)

0 – FALSE

1 – TRUE

x + 0 = x x + x = x

x · 0 = 0 x . x = x

x + 1 = 1 x + x/ = 1

x · 1 = x x · x / = 0

12

Basic Identities (Cont.)

Commutativity

x + y = y + x

xy = yx

Associativity

x + ( y + z ) = ( x + y ) + z

x (yz) = (xy) z

Distributivity

x ( y + z ) = xy + xz

x + yz = ( x + y )( x + z)

13

Textbook has a typo

Basic Identities (Cont.)

DeMorgan’s Theorem

( x + y )/ = x/ y/

( xy )/ = x/ + y/

Generalized DeMorgan's Theorem

(a + b + … z) / = a / b / … z /

(a.b … z) / = a / + b / + … z /

Involution

(x /) / = x

14

Exercise

Simplify following functions using Boolean Algebra

a + ab

a(a + b)

a(a / + b)

a + a/ b

(a + b.c) /

a + ab = a(1+b)=a

a(a + b) = a.a +ab=a+ab=a(1+b)=a

a + a'b = (a + a')(a + b)=1(a + b) =a+b

a(a' + b) = a.a' +ab=0+ab=ab

(a + b.c)' = a'.b' + a'.c' 15

Homework

Show steps for following simplifications

(a + b)(a + b') = a

ab + ab'c = ab + ac

(a + b)(a + b' + c) = a + bc

(a(b + z(x + a')))' = a' + b' (z' + x')

(a(b + c) + a'b)'=b'(a' + c')

(a + b)(a' + c)(b + c) = (a + b)(a' + c)

No need to submit answers

16

Minterms

Minterm

Each combination of variables in a truth table

As no of digital inputs (Boolean variables)

increase no of minterms increase

2n minterms for n inputs

As n increases

Large truth tables

Long Boolean functions

Easier to represent using decimal equivalent of

sum of minterms

F(a, b, c) = Σ (2, 4, 5, 7) 17

Exercise

Represent all odd numbers between 0 – 7 as a sum of

minterms

F(a, b, c) = Σ (1, 3, 5, 7)

Use Boolean Algebra to simplify above function

Σ (1, 3, 5, 7) = a‘b‘c + a‘bc + ab‘c + abc

= a‘c(b‘ + b) + ac (b‘ + b)

= a‘c + ac = c (a‘+ a) = c

18

Sum of Products

Karnaugh Map (K-Map)

Pictorial representation of a truth table

Map of squares – each square for each minterm

Adjacent squares change in minterm only by one

variable

19

bc

a

00 01 11 10

0 0 1 3 2

1 4 5 7 6

Decimal equivalent

of minterm

Example – 3 Variable K-Map

Simplify represent of all odd numbers between 0 – 7

F(a, b, c) = Σ (1, 3, 5, 7)

20

ab

c

00

01

11

10

0 1

1

1

1

1

0

0

0

0

= c

Exercise – 3 Variable K-Map

Simplify following expression using a K-map

21

ab

c

00

01

11

10

0 1

1

0

0

0

1

1

0

0

= a’c’ + a’b’

Kanaugh Map – Simplification

Squares containing 1’s are grouped to get sum-

of-product expression

Group size must be 2n

Group must be in rectangular shape

Each group represents an algebraic term

Its OK for groups to overlap

OR of all those terms is the final expression

Maximum group size better simplification

A don’t care (X) can be interpreted as either 0 or 1

when needed, only if it contributes to simplification

Don’t group Xs together!

22

4 Variable K-Map

23

Example – 4 Variable K-Map

24

Exercise – 4 Variable K-Map

What would be a Boolean expression of a circuit

that detects all even numbers between 0 – 15?

Present as a sum of minterms

25

5 Variable K-Map – Option 1

26

BC

DE

A1/0

00 01 11 10

00

01

11

10

0

5

3

1

8124

2

20

11

106

7

14

16

13 9

15

19

18

17

28 24

31

2921

23

3022 26

25

27

Example – 5 Variable K-Map

What would be a Boolean expression of a circuit

to detect all prime numbers between 0 & 31?

27

BC

DE

A1/0

00 01 11 10

00

01

11

10

0

5

3

1

8124

2

20

11

106

7

14

16

13 9

15

19

18

17

28 24

31

2921

23

3022 26

25

27

5 Variable K-Map – Option 2

28

Example – 5 Variable K-Map – Option 2

29

6 Variable K-Map

30

Simplification as Product of Sum

Some times it is useful to obtain the algebraic

expression as a product of sums

Can be obtained by finding sum of products for

F/ & converting it to F

To find F/ Group 0s in a K-map

F(a, b, c) = Σ (1, 3, 5, 7) = Π (1, 3, 5, 7)

31

Exercise – Product of Sum

Find sum-of-product & product-of-sum

representation of following K-map

32

ab

c

00

01

11

10

0 1

1

x

1

1

0

0

0

1

F(a, b, c) = ab’ + c

F/(a, b, c) = a’c’ + bc’

(F/(a, b, c)) / = (a + c)(b’ + c)

= F(a, b, c)