Course contents

• Chapter 1 - section 1.6• Chapter 2 - all sections• Chapter 4 - 4.1 – 4.7, and 4.12• Chapter 5 - 5.1-5.3, 5.6-5.7• Chapter 6 - all sections• Chapter 7 - all sections• Chapter 8 - 8.1-8.9

1

1.6 Binary numbers

• An electronic signal in logic circuits carries one digit of information. – Each digit is allowed to take on only two possible

values, usually denoted as 0 and 1.– -> Information in logic circuits is represented as

combinations of 0 and 1 digits.

• Q: How to represent numbers (E.g., positive integers) using only binary digits 0 and 1?

2

Decimal (base-10) number system

• A decimal integer is expressed by an n-tuple comprising n decimal digits

D = dn-1dn-2 ∙ ∙ ∙ d1d0

which represents the value

V(D) = dn-1×10n-1 + dn-2×10n-2 + ∙ ∙ ∙ + d1×101 + d0×100

• This is referred to as the positional number representation.

3

Binary (base-2) number system

• Logic circuits use the binary system whose positional number representation is

B = bn-1bn-2 ∙ ∙ ∙ b1b0

bn-1 is the most significant bit (MSB),

b0 is the least significant bit (LSB),

Every bit bi can only have two values: 0 or 1.

4

Numbers in decimal and binary

Decimal representation

Binary representation

00 0000

01 0001

02 0010

03 0011

04 0100

05 0101

06 0110

07 0111

08 1000

Decimal representation

Binary representation

09 1001

10 1010

11 1011

12 1100

13 1101

14 1110

15 1111

5

Conversion from binary to decimal

• Compute a weighted sum of every binary digit contained in the binary number

B = bn-1bn-2 ∙ ∙ ∙ b1b0

V(B) = bn-1×2n-1 + bn-2×2n-2 + ∙ ∙ ∙ + b1×21 + b0×20

E.g., (1101)2 = 1×23 + 1×22 + 0×21+1×20=(13)10

6

Conversion from decimal to binary

• Perform the successive division by 2 until the quotient becomes 0. Remainder

857 / 2 = 428 1 LSB

428 / 2 = 214 0

214 / 2 = 107 0

107 / 2 = 53 1

53 / 2 = 26 1

26 / 2 = 13 0

13 / 2 = 6 1

6 / 2 = 3 0

3 / 2 = 1 1

1 / 2 = 0 1 MSB7

Chapter 2Introduction to Logic Circuits

Outline

2.1 Variables and Functions

2.2 Inversion

2.3 Truth tables

2.4 Logic gates and networks

2.5 Boolean algebra

2.6 Synthesis using AND, OR and NOT gates

2.7 NAND and NOR logic networks

2.8 Design examples

9

Figure 2.1. A binary switch.

x 1 = x 0 =

(a) Two states of a switch

S

x

(b) Symbol for a switch

2.1 Variables

10

Figure 2.2. A light controlled by a switch.

(a) Simple connection to a battery

S

(b) Using a ground connection as the return path

Battery Light

Power supply

S

Light

x

x

An application

11

Figure 2.3. Two basic functions.

(a) The logical AND function (series connection)

S

Power supply

S

S

Power supply S

(b) The logical OR function (parallel connection)

Light

Light x1 x2

x1

x2

Functions

12

S

Power supply S

Light

S

X1

X2

X3

L(x1, x2, x3) = (x1 + x2) x3

A series-parallel connection

13

Figure 2.5. An inverting circuit.

S Light Power supply

R

x

2.2 Inversion (complement, not)

 

14