in this lecture: lecture 2: data...

E1.2 Digital Electronics 1 2.1 16 October 2008

Lecture 2: Data Representation

Dr Pete Sedcole

Department of E&E Engineering

Imperial College London

http://cas.ee.ic.ac.uk/~nps/

(Floyd 2.1 – 2.4, 2.8, 2.10 – 2.11)

(Tocci 2.1 – 2.8)


In this lecture:

• What do we mean by data?

• How can data be represented electronically?

• What number systems are usually used, and why?

• How do number systems in different bases work?

• How do you convert a number from decimal to binary and binary to decimal?


What do we mean by data?

• It depends on context, but we will say:

data = physical representation of information

• Data can be stored

– e.g.: computer disk, DVD, SIM card

• Data can be transmitted

– e.g.: fax, text message

• Data can be processed

– e.g.: cash till


Electronic representation of data

• Information can be complex

– e.g.: numbers, music, pictures

• What can we do using electronics?

– Set up voltages and currents

– Change voltages and currents

• A useful device is a switch

– closed: V = 0 Volts

– open: V = 5 Volts

5 Volts

Switch

R

V


• We can represent information using voltage levels

• The simplest information is TRUE/FALSE

– This can be represented by two voltage levels:

• 5 Volts = TRUE

• 0 Volts = FALSE

• A unit of information that has only two possible values is also called a BIT (short for Binary Digit)

• Why do we use binary data in electronics?

– Simple to implement in hardware: we only need

switches

– Good tolerance to noise


Number systems

Decimal

(base 10)

Octal

(base 8)

Binary

(base 2)

Hexadecimal

(base 16)


Decimal numbers

• Deci = ten

• The decimal number system has ten digits:0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

• The decimal number system has a base of 10, with each digit position weighted by a power of 10:

decimal point

10-410-310-210-1∏100101102103

ones

thou

sand

ths

tent

hs

hund

redt

hs

ten-

thou

sand

ths

thou

sand

s

tens

hund

reds

LSDMSD

MSD = most

significant digit

LSD = least

significant digit


Binary numbers

• Bi = two

• The binary number system has two digits:

0 and 1.

• The binary number system has a base of 2 with each

digit position weighted by a power of 2:

2-42-32-22-1∏20212223

binary point

MSB LSB

MSB = most

significant bit

LSB = least

significant bit


Binary number system

• A binary number uses the digits 0 and 1

• Subscript 2 indicates the number is in binary notation

(base 2)

• Example: 100112 is:

11001

2021222324

1 x 24 + 0 x 23 + 0 x 22 + 1 x 21 + 1 x 20 = 19


Integers and fractions in binary

• Binary numbers can represent fractional values as well

as integers

– e.g.: 10011.0112

• 10011.0112 is an 8-bit number

• It is in Q3 format – it has 3 bits after the binary point

• What if we want to represent (19.376)10 with an 8-bit

binary number?

0

2-1

1

2-2

1

20

∏ 11001

2-321222324

= (19.375)10


Conversion: decimal to binary (method 1)

• Express the decimal number as the sum of powers of 2

• 1s and 0s written in the corresponding bit positions

Example 1:

5010 = 32 + 18

= 32 + 16 + 2

= 1x25 + 1x24 + 1x21

5010 = 1100102

Example 2:

338.510 = 256 + 82.5

= 256 + 64 + 18.5

= 256 + 64 + 16 + 2.5

= 256 + 64 + 16 + 2 + 0.5

= 1x28 + 1x26 + 1x24 + 1x21 + 1x2-1

338.510 = 101010010.12


Conversion: decimal to binary (method 2)

• Repeated division

quotient remainder

50/2 = 25 0 LSB

25/2 = 12 1

12/2 = 6 0

6/2 = 3 0

3/2 = 1 1

1/2 = 0 1 MSB

5010 = 1100102


Conversion: binary to decimal

• Determine the power of 2 for the position of each 1, and then sum

• Example:

(10101)2 = 24 + 22 + 20 = 16 + 4 + 1 = (21)10

• Not the only way to do it…


Binary addition

• Firstly, recall decimal addition:

6302Sum

287+B

4521A

11

• Binary addition follows the same pattern, but

0 + 0 = 0 carry 0

0 + 1 = 1 + 0 = 1 carry 0

1 + 1 = 0 carry 1

1 + 1 + 1 = 1 carry 11011Sum

0110+B

1110A

11


Binary addition

• Note that to calculate each bit sn of the sum, we need to

consider the values of 3 input bits:

– The corresponding bits of A and B, (an and bn)

– The carry-out from the previous addition

• Each bit or column of the binary addition generates two outputs

– sum sn and carry-out


Hexadecimal numbers

• Hexadeci = sixteen

• Base 16 representation

• Easy to convert to and from binary numbers

• More compact to write, easier for us to read than binary


Hexadecimal number conversions

• Binary-to-hexadecimal

1. Separate the binary number into 4-bit groups

2. Replace each group with the hexadecimal equivalent

• Hexadecimal-to-decimal

1. Re-write each hexadecimal digit as the 4-bit binary equivalent

2. Convert the binary number to decimal

• Decimal-to-hexadecimal

– Repeated division by 16


Binary Coded Decimal (BCD)

• Use 4-bit binary to represent one decimal digit

• Easy to convert between decimal ↔ binary

• Wastes bits (4 bits can represent 16 values, but only 10 values are used)

• Used extensively in financial calculations


Binary Coded Decimal (BCD)

• Convert 0110100100010111 (BCD) to its decimal equivalent:

0110 1001 0001 0111

6 9 1 7

• Convert the BCD number 011111000001 to decimal

0111 1100 0001

7 ? 1

The forbidden code group indicates an error has occurred


Putting it all together


Gray Codes

• Only one bit in the code

changes in the sequence

• Useful for industrial control


• Binary code results in glitches

• Gray code avoids glitches


ASCII code

• Codes representing letters of the alphabet, punctuation marks and other special characters are called alphanumeric codes

• The most widely used alphanumeric code is the American Standard Code for Information Interchange, or ASCII

• ASCII is pronounced “askee”

• It is a 7-bit code

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