1 digital systems and binary numbers ee 208 – logic design chapter 1 sohaib majzoub

1

Digital Systems and Binary Digital Systems and Binary NumbersNumbers

EE 208 – Logic Design

Chapter 1

Sohaib Majzoub

2

DigitalDigital Electronics …. EverywhereElectronics …. Everywhere

3

ObjectiveObjectiveLEARN BASICS BEHIND COMPUTER SYSTEMS• Hardware architecture and organization• Digital logic design

– Switching and logic gate design

• Computer architecture building blocks– Adders/subtractors/counters

– Shift/rotate registers

– Multiplexers/demultiplexers and encoders/decoders

– Controllers and sequencers

4

Generic Computer SystemGeneric Computer System

5

Hierarchy in a SystemHierarchy in a System

6

NAND Gate ExampleNAND Gate Example

7

Analog and Digital SignalAnalog and Digital Signal

Analog signals vary continuously over a specified range with infinite values.

t

X(t)

Analog signal

Digital signals can assume only finite values.

7

Sampled

Quantized

Digital

8

Binary Logic LevelsBinary Logic Levels

9

Number SystemsNumber Systems• Number Systems?• What are the different forms to represent the number

systems? – Decimal

– Hexadecimal representation

– Binary representation

– Arithmetic Operations

10

Decimal Number SystemDecimal Number System

• Radix or base is equal to 10, e.g. N = 1256.932

N = 1x103+2x102+5x101+6x100+9x10-1+3x10-2+2x10-3

11

Polynomial Representation of Numbers Polynomial Representation of Numbers

an is the most significant digit or MSD and

a-m is the least significant digit or LSD

• Radix or base can be any integer, most known systems are - r = 2 (binary)- r = 3 (Ternary)- r = 4 (Quaternary)- r = 8 (Octal)- r = 16 (Hexadecimal)

12

Commonly Used Number SystemsCommonly Used Number Systems

Name Decimal Binary Octal Hexadecimal

Radix 10 2 8 16

Possible

digit

values

0 to 9 0 and 1 0 to 7 0 to 9

A to F

13

Number SystemsNumber Systems

14

Conversion to DecimalConversion to Decimal

• Binary to Decimal: N = (11010.1101)2

• N = 1x24 +1x23 +0x22 +1x21 +0x20 +1x2-1 +1x2-2 +0x2-3 +1x2-4 = 26.8125

15

Conversion to DecimalConversion to Decimal

• Octal: N = (4 5 2 6 . 2 3)8

N = 4x83 +5x82 +2x81 +6x80 +2x8-1 +3x8-2 = 2390 + 19/64

• Hexadecimal: N = (A 1 F .1 C)16

N = Ax162 +1x161 +Fx160 +1x16-1 +Cx16-2 =2592+28/256

16

Conversion from DecimalConversion from Decimal

• Radix Divide Technique: Divide the given integer successively by the target radix, noting the reminder in each step, until the quotient is zero. Collect the from each step starting from last to first.

• Convert (245)10 = (?)2

17


• Convert: (245)10 = (?)8

• Convert: (245)10 = (?)16

18


• Radix Multiply Technique: Successively multiply the given fraction by the required base, noting the integer portion of the product at each step. Collect the integer digits starting from first to last.

• Convert: (0.345)10 = (?)2

19


• Covert: (242.45)10= (?)2

20

Conversion Between SystemsConversion Between Systems

• From Binary to:

– Octal: arrange bits in groups of 3 and substitute

(10111011001)2 = (010 111 011 001)2 = (2731)8

– Hex: arrange bits in groups of 4 and substitute (10111011001)2 = (0101 1101 1001)2 = (5D9)16 = (5D9)H

Each single hex digit (4 bits) is called a nibble

– Decimal: use general positional summation method

(10111011001)2 = 1x210 + 0x29 + 1x28 + 1x27 +

1x26 + 0x25 + 1x24 + 1x23 + 0x22 + 0x21 + 1x20 = (1497)10

21

Conversion Between SystemsConversion Between Systems• From Octal to:

– Binary: Substitute each Oct digit with 3 binary bits

(1234)8 = (001 010 011 100)2

– Hex: Convert to binary, then to Hex

(1234)8 = (001 010 011 100)2

= (0010 1001 1100)2 = (29C)16


(1234)8 = 1x83 + 2x82 + 3x81 + 4x80

= 1x512 + 2x64 + 3x8 + 4x1 = (668)10

22

Conversion Between SystemsConversion Between Systems

• From Hex to:

– Binary: Substitute each Hex digit with 4 binary bits

(C0DE)16 = (1100 0000 1101 1110)2

– Octal: Convert to binary, then to Hex

(C0DE)16 = (1100 0000 1101 1110)2

= (001 100 000 011 011 110)2 = (140336)8


(C0DE)16 = 12x163 + 0x162 + 13x161 + 14x160

= 12x4096 + 0x256 + 13x16 + 14x1 = (49374)10

23

ArithmeticArithmetic

• Arithmetic rules in all Number Systems are the same (-, +, *, / ), just remember the base

• Binary Arithmetic:

Binary Addition

24

Binary ArithmeticBinary Arithmetic

• Binary Subtraction:

• Binary Multiplication:

25

Binary ArithmeticBinary Arithmetic

• Binary Division

26

Representation of Negative NumbersRepresentation of Negative Numbers• Why do we need a negative representation?• Computers do not understand the negative sign.

• Example: (178)10 is equivalent to (10110010)2

• Sign-Magnitude system: bn-1bn-2 …… b1b0

• Example:

-178 - 10110010 is equivalent to 1 10110010

+178 +10110010 is equivalent to 0 10110010• If the computer has 9 bits to represent numbers use the

MSB for the sign

Signbit

N-1 magnitude bits

27

The Complement Number SystemThe Complement Number System

• A complement is a negative representation of a positive number (and vice versa)

• The complement of a number (N)r defined by:

• Twos Complement when r = 2 (Binary System)• Examples:

- The twos complement of (01010)2 is

25 – (01010) = 100000 – 01010 = 10110• Complement numbers simplify subtractions

– Subtraction is converted to addition– Simplifies hardware implementation

28

Twos ComplementTwos Complement

• A shortcut to find the Twos complement:

• Similar to regular binary addition• Ignore carry out of MSB

29

Ones ComplementOnes Complement

• Another method to represent negative numbers, same as Two complement but do not add one.

• Example: the ones complement of (01010)2 is 10101.

30

Negative Representation in ComputersNegative Representation in Computers

31

Subtractions with Twos ComplementSubtractions with Twos Complement

• Twos complement to represent negative numbers or subtractions.

• Example: A – B means A + (-B), i.e. calculate the Twos complement of B, and then add both.

• Consider A =5, B =4, then A + B is

• Consider A – B, or A + (-B), then

32

Subtractions with Twos ComplementSubtractions with Twos Complement

• B – A or B + (-A) then:

33

Subtraction with Ones ComplementSubtraction with Ones Complement

• A=5, and B = 4, then A + (– B) is equal to:

• Consider B + (– A):

34

OverflowOverflow

• What is overflow?– Result of arithmetic exceeds the range

• i.e. No enough bits to represent number

Example: using 4 bit two’s complement binary:

6 + 5 = 11 0110 + 0101 = 1011 = -5 WRONG!

The result 11 cannot be represented with 4 bits using two’s complement notation

35

When Will Overflow Occur?When Will Overflow Occur?• In unsigned arithmetic:

– Adding two positive numbers, if we have a carry out from the MSB, then overflow occurred (overflow = MSB carry out)

– Example: using 4 bit unsigned binary:9 + 10 = 19 1001 + 1010 = 1 0011 = 3 Overflow!

• In signed two’s complement arithmetic:– Adding a positive and a negative number: overflow can never

happen. Why? The result is always between the two numbers can be represented with same number of bits

– Adding two positive or two negative numbers, overflow can happen.

How to detect? If sign of result is different than sign of two numbers being added Overflow

Remember: Carry out from MSB is always ignored!

36

Overflow ExamplesOverflow Examples

• Add the following 8-bit two’s complement numbers. Find if there is an overflow or not.

1001011001110100------------

1101101010010010------------

0110111001011101------------

37

Binary Coded DecimalBinary Coded Decimal

• Computers operate on binary numbers, but when communicate with humans, they use decimal.

• Conversion from binary to decimal and decimal to binary is necessary.

• The digits 0 to 9 are coded into binary bits (this is not decimal binary conversion, it is digit coding)

• In the BCD code, each decimal digit is represented by 4 bits. The BCD code is the 8,4,2,1 code.

• BCD is a weighted code (8, 4, 2, and 1 are weights)• This code is the simplest, most intuitive binary code for

decimal digits and uses the same powers of 2 as a binary number, but only encodes the first ten values from 0 to 9.

38

BCD CodeBCD Code

• Binary Coded Decimal (BCD) – Decimal number 86 is

represented in BCD with 8 bits as 1000 0110, with each group of 4 bits is an input to a seven segment display.

– A decimal number in BCD is the same as its equivalent binary number only when the number is between 0 and 9.

– The binary combinations 1010 through 1111 are not used.

39

BCD CodeBCD Code• Example:

– Consider decimal number 185 and its corresponding value in BCD and binary:

• BCD addition (hint: Add +6 if the sum >9)

40

BCD CodeBCD Code• Example:

– Consider the addition of 184 + 576 = 760 in BCD:

41

Gray CodeGray Code

– Advantage: only one bit is changed between any two successive codes

42

ASCII CodeASCII Code• American Standard Code for Information Interchange (ASCII)

42

43

ASCII PropertiesASCII Properties

– Digits 0 to 9 span Hexadecimal values 3016 to 3916

– Upper case A-Z span 4116 to 5A16

– Lower case a-z span 6116 to 7A16

– Lower to upper case translation (and vice versa) occurs by flipping bit 6.

44

Other CodesOther Codes

45

Error DetectionError Detection

• Error-Detecting Code – To detect errors in data communication and processing, an

eighth bit is sometimes added to the ASCII character to indicate its parity.

– A parity bit is an extra bit included with a message to make the total number of 1's either even or odd.

• Example: characters with even and odd parity

46

Warning: Conversion or Coding?Warning: Conversion or Coding?

• Do NOT mix up conversion of a decimal number to a binary number with coding a decimal number with a BINARY CODE.

• 1310 = 11012 (This is conversion)

• 13 0001|0011 (This is coding)

47

Chapter 1 47

UNICODEUNICODE

• UNICODE extends ASCII to 65,536 universal characters codes

– For encoding characters in world languages

– Available in many modern applications

– 2 byte (16-bit) code words

1 digital systems and binary numbers ee 208 – logic design chapter 1 sohaib majzoub

Documents