Number Systems in Digital Electronics

The digital computer is a digital system that performs various computational tasks. The word digital implies that the information in the computer is represented by variables that take a limited number of discrete values. The count of these limited number is denoted by Base or Radix.

Base – A number system of base or radix “r” is a system that uses r distinct symbols to represent any number

The table below summarizes four number systems, their base and the distinct symbols used by these number systems to represent any number:

Number System	Base / Radix	Distinct Symbols
Binary	2	0, 1
Octal	8	0 – 7
Decimal	10	0 – 9
Hexadecimal	16	0 – 9, A, B, C, D, E, F

Note: In hexadecimal format, A=>10, B=>11, C=>12, D=>13, E=>14, F=>15

Now let’s take an example of an absolute number and a fractional number and see how to convert from one base to another.

Convert from Binary to Decimal

1010 = 1*2³ + 0*2² + 1*2¹ + 0*2⁰ = 8 + 0 + 2 + 0 = 10

1010.01 = 1*2³ + 0*2² + 1*2¹ + 0*2⁰ + 0*2^-1 + 1*2^-2 = 8 + 0 + 2 + 0 + 0 + 0.25 = 10.25

Here 2^-1 means ½ and 2^-2 means 1/2²

(1010)₂ = (10)₁₀

(1010.01)₂ = (10.25)₁₀

Convert from Binary to Octal

Since octal is represented using 3 digits (as 8 = 2³), assemble the digits in a group of 3 starting from right to left to convert from binary to octal. For fractional part, grouping needs to be done from left to right. Extra zeros should be added to form a group of 3 as shown below:

1010 = 1 010 = 001 010 = 12

1010.01 = 1 010.010 = 001 010.010 = 12.2

(1010)₂ = (12)₈

(1010.01)₂ = (12.2)₈

Convert from Binary to Hexadecimal

Since hexadecimal numbers are represented using 4 digits (as 16 = 2⁴), assemble the digits in a group of 4 starting from right to left to convert from binary to hexadecimal. For fractional part, grouping needs to be done from left to right. Extra zeros should be added to form a group of 4 as shown below:

1010 = 1010 = A

1010.01 = 1010.0100 = A.4

(1010)₂ = (A)₁₆

(1010.01)₂ = (A.4)₁₆

Convert from Octal to Binary

12 = 001 010 = 1010

12.2 = 001 010.0100 = 1010.01

(12)₈ = (1010)₂

(12.2)₈ = (1010.01)₂

Convert from Octal to Decimal

12 = 1*8¹ + 2*8⁰ = 8 + 2 * 1 = 10

12.2 = 1*8¹ + 2*8⁰ + 2*8^-1 = 8 + 2 * 1 + 2/8 = 8 + 2 + 0.25 = 10.25

(12)₈ = (10)₁₀

(12.2)₈ = (10.25)₁₀

Convert from Octal to Hexadecimal

Since hexadecimal numbers are represented using 4 digits (as 16 = 2⁴), assemble the digits in a group of 4 starting from right to left to convert from octal to hexadecimal. For fractional part, grouping needs to be done from left to right. Extra zeros should be added to form a group of 4 as shown below:

12 = 001 010 = 00 1010 = 0000 1010 = 0A = A

12.2 = 001 010.010 = 00 1010.0100 = 0000 1010.0100 = 0A.4 = A.4

(12)₈ = (A)₁₆

(12.2)₈ = (A.4)₁₆

Convert from Decimal to Binary

To convert from decimal to binary, given number should be divided by 2, until the quotient can not be divided further.

(10)₁₀ = (1010)₂

0.25 * 2 = 0. 50 | 0 (Most Significant Digit)

0.50 * 2 = 1. 00 | 1 (MSD)

(0.25)₁₀ = (01)₂

(10.25)₁₀ = (1010.01)₂

Convert from Decimal to Octal

To convert from decimal to octal, given number should be divided by 8, until the quotient can not be divided further.

(10)₁₀ = (12)₈

0.25 * 8 = 2. 00 | 2 (Most Significant Digit)

(10.25)₁₀ = (12.2)₈

Convert from Decimal to Hexadecimal

To convert from decimal to hexadecimal, given number should be divided by 16, until the quotient can not be divided further.

(10)₁₀ = (A)₁₆

0.25 * 16 = 4. 00 | 4 (Most Significant Digit)

(10.25)₁₀ = (A.4)₁₆

Convert from Hexadecimal to Binary

A = 1010

(A)₁₆ = (1010)₂

A.4 = 1010.0100 = 1010.01

(A.4)₁₆ = (1010.01)₂

Convert from Hexadecimal to Octal

Since octal is represented using 3 digits (as 8 = 2³), assemble the digits in a group of 3 starting from right to left to convert from hexadecimal to octal. For fractional part, grouping needs to be done from left to right. Extra zeros should be added to form a group of 3 as shown below:

A = 1010 = 1 010 = 001 010 = 12

A.4 = 1010.100 = 1 010.100 = 001 010.100 = 12.4

(A)₁₆ = (12)₈

(A.4)₁₆ = (12.4)₈

Convert from Hexadecimal to Decimal

A = 10*16⁰ = 10 * 1 = 10

A.4 = 10*16⁰ + 4 * 16^-1 = 10 * 1 + 4/16 = 10 + 0.25 = 10.25

(A)₁₆ = (10)₁₀

(A.4)₁₆ = (10.25)₁₀

FAQs about Number System

Top of Form

Q. Why we section off 3 bits in Octal and 4 bits in Hexadecimal?

A.As per definition of base given above – A number system of base or radix “r” is a system that uses r distinct symbols to represent any number. Thus, octal with a base of 8, uses 8 distinct symbols (which can be represented by 3 digits as 8=2^3) and hexadecimal with a base of 16, uses 16 distinct symbols (which can be represented by 4 digits as 16=2^4). So as octal is represented using 3 digits, we section off 3 binary numbers starting from right, and as hexadecimal number is represented using 4 digits, we section off 4 binary numbers starting from right.

Q. How to convert any non-decimal number to a number with base x?

A. To convert a number to some other base say x, the procedure is to first convert the number to decimal and then divide it by x repeatedly till the quotient is greater than or equal to x.

Bottom of Form

Above are the examples of conversions from one base to another and can be utilized to covert any number from any base to any base. Please feel free to leave your footprints in the comments section below for any clarifications to above and anything related to Number System in Digital Electronics

In our daily life we use "decimal number system". Now don't give me that look, decimal number system is just a numbering system which we use to count things, measure quantities, maintain our bills, bank maintains their accounts etc..etc. Yes, we use only 10 symbols to do any kind of counting, these 10 symbols are '0', '1', '2', '3', '4', '5', '6', '7', '8' and '9'. Since there are only 10 symbols this numbering system it is called deci-mal(deci means 10).

There are also other numbering systems namely..

Octa Decimal : Here we use only eight symbols (octa means 8), these are '0', '1', '2', '3', '4', '5', '6' and '7'

Hexa Decimal : This numbering system uses total 16 symbols: 0 to 9, 'A', 'B' , 'C', 'D', 'E' and 'F'

Binary : Here we use only two symbols '0' and '1'

Buy why on the earth will any body will require these weired numbering system ? Yes, you are correct..all these weird numbering systems are computer/digital system friendly. These number systems are widely used in digital systems like microprocessor, logic circuits, computers etc. All System designers prefer to use these numbering systems, as these are more continent from digital systems point of view. For example any digital system understands only two things "no voltage" and "voltage" i.e '0' and '1' at input/output. Hence always use Hexa-decimal and binary numbering system while designing. It is good idea not to combine decimal and hexa-decimal while putting specifications.

To understand how other numbering systems can be used, lets re-analyze how we use decimal numbering system..

In decimal system we first use all the symbols while counting and when we run out of symbols we re-use these symbols on the ten's position in the same sequence. But symbols at ten's position changes only when we run out of symbols at one's position. For example, lets say we are counting bottles. First we will use 0 to 9 to count 9 bottles now the next counted bottle will be counted as 10, observe this carefully, since we ran out of symbol we started using ten's position, again symbol at ten's position will change when we run out of symbol at one's position i.e When our counting moves from 19 to 20. This rule applies to ones, tens, hundred, thousand ... to all decimal places.

The same rule is applied when we count using any other numbering system. Lets say you have 12 bottles (decimal), but you want to count it using octa-decimal numbering system.

Bottle count in :

 

         Decimal  Octa-Decimal  Hexa-Decimal  Binary

            0          0            0            0

            1          1            1            1

            2          2            2           10

            3          3            3           11

            4          4            4          100

            5          5            5          101

            6          6            6          110

            7          7            7          111 

            8         10            8         1000

            9         11            9         1001

           10         12            A         1010

           11         13            B         1011

   

Observe that, while counting in octa-decimal when we run out of symbol after 7, we started using ten's position. In our daily life we start from 1 instead of 0, but for simplicity in above example we started counting from 0 even in decimal. Remember that in hexa decimal and binary we always count from 0. we will see other arithmetic operation on binary in next topic.

Number Systems

There are infinite ways to represent a number. The four commonly associated with modern computers and digital electronics are: decimal, binary, octal, and hexadecimal.

Decimal (base 10) is the way most human beings represent numbers. Decimal is sometimes abbreviated as dec.

Decimal counting goes:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, and so on.

Binary (base 2) is the natural way most digital circuits represent and manipulate numbers. (Common misspellings are “bianary”, “bienary”, or “binery”.) Binary numbers are sometimes represented by preceding the value with '0b', as in 0b1011. Binary is sometimes abbreviated as bin.

Binary counting goes:
0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 10000, 10001, and so on.

Octal (base 8) was previously a popular choice for representing digital circuit numbers in a form that is more compact than binary. Octal is sometimes abbreviated as oct.

Octal counting goes:
0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 20, 21, and so on.

Hexadecimal (base 16) is currently the most popular choice for representing digital circuit numbers in a form that is more compact than binary. (Common misspellings are “hexdecimal”, “hexidecimal”, “hexedecimal”, or “hexodecimal”.) Hexadecimal numbers are sometimes represented by preceding the value with '0x', as in 0x1B84. Hexadecimal is sometimes abbreviated as hex.

Hexadecimal counting goes:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, 10, 11, and so on.

All four number systems are equally capable of representing any number. Furthermore, a number can be perfectly converted between the various number systems without any loss of numeric value.

At first blush, it seems like using any number system other than human-centric decimal is complicated and unnecessary. However, since the job of electrical and software engineers is to work with digital circuits, engineers require number systems that can best transfer information between the human world and the digital circuit world.

It turns out that the way in which a number is represented can make it easier for the engineer to perceive the meaning of the number as it applies to a digital circuit. In other words, the appropriate number system can actually make things less complicated.

Fundamental Information Element of Digital Circuits

Almost all modern digital circuits are based on two-state switches. The switches are either on or off. It doesn’t matter if the switches are actually physical switches, vacuum tubes, relays, or transistors. And, it doesn’t matter if the 'on' state is represented by 1.8 V on a cutting-edge CPU core, -12 V on a RS-232 interface chip, or 5 V on a classic TTL logic chip.

Because the fundamental information element of digital circuits has two states, it is most naturally represented by a number system where each individual digit has two states: binary.

For example, switches that are 'on' are represented by '1' and switches that are 'off' are represented by '0'. It is easy to instantly comprehend the values of 8 switches represented in binary as 10001101. It is also easy to build a circuit to display each switch state in binary, by having an LED (lit or unlit) for each binary digit.

Making Values More Compact

“Binary digit” is a little unwieldy to say over and over, so the term was contracted to “bit”. Not only is the term “binary digit” a little unwieldy, but so is the length of a binary number, since each digit can only represent one switch. As digital circuits grew more complex, a more compact form of representing circuit information became necessary.

An octal number (base 8) can be up to 1/3 the length of a binary number (base 2). 8 is a whole power of 2 (2³=8). That means three binary digits convert neatly into one octal digit.

A hexadecimal number (base 16) can be up to 1/4 the length of a binary number. 16 is a whole power of 2 (2⁴=16). That means four binary digits convert neatly into one hexadecimal digit.

Unfortunately, decimal (base 10) is not a whole power of 2. So, it is not possible to simply chunk groups of binary digits to convert the raw state of a digital circuit into the human-centric format.

Let’s see how to convert a number from binary...

Binary to Octal

An easy way to convert from binary to octal is to group binary digits into sets of three, starting with the least significant (rightmost) digits.

Binary: 11100101 =	11 100 101
	011 100 101	Pad the most significant digits with zeros if necessary to complete a group of three.

Then, look up each group in a table:

Binary:	000	001	010	011	100	101	110	111
Octal:	0	1	2	3	4	5	6	7

Binary =	011	100	101
Octal  =	3	4	5	= 345 oct

Binary to Hexadecimal

An equally easy way to convert from binary to hexadecimal is to group binary digits into sets of four, starting with the least significant (rightmost) digits.

Binary: 11100101 = 1110 0101

Then, look up each group in a table:

Binary:	0000	0001	0010	0011	0100	0101	0110	0111
Hexadecimal:	0	1	2	3	4	5	6	7

Binary:	1000	1001	1010	1011	1100	1101	1110	1111
Hexadecimal:	8	9	A	B	C	D	E	F

Binary =	1110	0101
Hexadecimal =	E	5	= E5 hex

Binary to Decimal

They say there are only 10 people in this world: those that understand binary and those that don’t. Ha ha.

If you don’t get that joke, you'll need a method to convert from binary to decimal. One method involves addition and multiplication.

1. Start the decimal result at 0.
2. Remove the most significant binary digit (leftmost) and add it to the result.
3. If all binary digits have been removed, you’re done. Stop.
4. Otherwise, multiply the result by 2.
5. Go to step 2.

Here is an example of converting 11100000000 binary to decimal:

Binary Digits	Operation	Decimal Result	Operation	Decimal Result
11100000000	+1	1	× 2	2
1100000000	+1	3	× 2	6
100000000	+1	7	× 2	14
00000000	+0	14	× 2	28
0000000	+0	28	× 2	56
000000	+0	56	× 2	112
00000	+0	112	× 2	224
0000	+0	224	× 2	448
000	+0	448	× 2	896
00	+0	896	× 2	1792
0	+0	1792	done.

A repeated division and remainder algorithm can convert decimal to binary, octal, or hexadecimal.

1. Divide the decimal number by the desired target radix (2, 8, or 16).
2. Append the remainder as the next most significant digit.
3. Repeat until the decimal number has reached zero.

Decimal to Binary

Here is an example of using repeated division to convert 1792 decimal to binary:

Decimal Number	Operation	Quotient	Remainder	Binary Result
1792	÷ 2 =	896	0	0
896	÷ 2 =	448	0	00
448	÷ 2 =	224	0	000
224	÷ 2 =	112	0	0000
112	÷ 2 =	56	0	00000
56	÷ 2 =	28	0	000000
28	÷ 2 =	14	0	0000000
14	÷ 2 =	7	0	00000000
7	÷ 2 =	3	1	100000000
3	÷ 2 =	1	1	1100000000
1	÷ 2 =	0	1	11100000000
0	done.

Decimal to Octal

Here is an example of using repeated division to convert 1792 decimal to octal:

Decimal Number	Operation	Quotient	Remainder	Octal Result
1792	÷ 8 =	224	0	0
224	÷ 8 =	28	0	00
28	÷ 8 =	3	4	400
3	÷ 8 =	0	3	3400
0	done.

Decimal to Hexadecimal

Here is an example of using repeated division to convert 1792 decimal to hexadecimal:

Decimal Number	Operation	Quotient	Remainder	Hexadecimal Result
1792	÷ 16 =	112	0	0
112	÷ 16 =	7	0	00
7	÷ 16 =	0	7	700
0	done.

The only addition to the algorithm when converting from decimal to hexadecimal is that a table must be used to obtain the hexadecimal digit if the remainder is greater than decimal 9.

Decimal:	0	1	2	3	4	5	6	7
Hexadecimal:	0	1	2	3	4	5	6	7

Decimal:	8	9	10	11	12	13	14	15
Hexadecimal:	8	9	A	B	C	D	E	F

The addition of letters can make for funny hexadecimal values. For example, 48879 decimal converted to hex is:

Decimal Number	Operation	Quotient	Remainder	Hexadecimal Result
48879	÷ 16 =	3054	15	F
3054	÷ 16 =	190	14	EF
190	÷ 16 =	11	14	EEF
11	÷ 16 =	0	11	BEEF
0	done.

Other fun hexadecimal numbers include: AD, BE, FAD, FADE, ADD, BED, BEE, BEAD, DEAF, FEE, ODD, BOD, DEAD, DEED, BABE, CAFE, C0FFEE, FED, FEED, FACE, BAD, F00D, and my initials DAC.

Octal to Binary

Converting from octal to binary is as easy as converting from binary to octal. Simply look up each octal digit to obtain the equivalent group of three binary digits.

Octal:	0	1	2	3	4	5	6	7
Binary:	000	001	010	011	100	101	110	111

Octal  =	3	4	5
Binary =	011	100	101	= 011100101 binary

Octal to Hexadecimal

When converting from octal to hexadecimal, it is often easier to first convert the octal number into binary and then from binary into hexadecimal. For example, to convert 345 octal into hex:

(from the previous example)

Octal  =	3	4	5
Binary =	011	100	101	= 011100101 binary

Drop any leading zeros or pad with leading zeros to get groups of four binary digits (bits):
Binary 011100101 = 1110 0101

Then, look up the groups in a table to convert to hexadecimal digits.

Binary:	0000	0001	0010	0011	0100	0101	0110	0111
Hexadecimal:	0	1	2	3	4	5	6	7

Binary:	1000	1001	1010	1011	1100	1101	1110	1111
Hexadecimal:	8	9	A	B	C	D	E	F

Binary =	1110	0101
Hexadecimal =	E	5	= E5 hex

Therefore, through a two-step conversion process, octal 345 equals binary 011100101 equals hexadecimal E5.

Octal to Decimal

Converting octal to decimal can be done with repeated division.

1.      Start the decimal result at 0.

2.     Remove the most significant octal digit (leftmost) and add it to the result.

3.     If all octal digits have been removed, you’re done. Stop.

4.     Otherwise, multiply the result by 8.

5.     Go to step 2.

Octal Digits	Operation	Decimal Result	Operation	Decimal Result
345	+3	3	× 8	24
45	+4	28	× 8	224
5	+5	229	done.

The conversion can also be performed in the conventional mathematical way, by showing each digit place as an increasing power of 8.

345 octal = (3 * 8²) + (4 * 8¹) + (5 * 8⁰) = (3 * 64) + (4 * 8) + (5 * 1) = 229 decimal

Hexadecimal to Binary

Converting from hexadecimal to binary is as easy as converting from binary to hexadecimal. Simply look up each hexadecimal digit to obtain the equivalent group of four binary digits.

Hexadecimal:	0	1	2	3	4	5	6	7
Binary:	0000	0001	0010	0011	0100	0101	0110	0111

Hexadecimal:	8	9	A	B	C	D	E	F
Binary:	1000	1001	1010	1011	1100	1101	1110	1111

Hexadecimal =	A	2	D	E
Binary =	1010	0010	1101	1110	= 1010001011011110 binary

Hexadecimal to Octal

When converting from hexadecimal to octal, it is often easier to first convert the hexadecimal number into binary and then from binary into octal. For example, to convert A2DE hex into octal:

(from the previous example)

Hexadecimal =	A	2	D	E
Binary =	1010	0010	1101	1110	= 1010001011011110 binary

Add leading zeros or remove leading zeros to group into sets of three binary digits.

Binary: 1010001011011110 = 001 010 001 011 011 110

Then, look up each group in a table:

Binary:	000	001	010	011	100	101	110	111
Octal:	0	1	2	3	4	5	6	7

Binary =	001	010	001	011	011	110
Octal =	1	2	1	3	3	6	= 121336 octal

Therefore, through a two-step conversion process, hexadecimal A2DE equals binary 1010001011011110 equals octal 121336.

Hexadecimal to Decimal

Converting hexadecimal to decimal can be performed in the conventional mathematical way, by showing each digit place as an increasing power of 16. Of course, hexadecimal letter values need to be converted to decimal values before performing the math.

Hexadecimal:	0	1	2	3	4	5	6	7
Decimal:	0	1	2	3	4	5	6	7
Hexadecimal:	8	9	A	B	C	D	E	F
Decimal:	8	9	10	11	12	13	14	15

A2DE hexadecimal:
= ((A) * 16³) + (2 * 16²) + ((D) * 16¹) + ((E) * 16⁰)
= (10 * 16³) + (2 * 16²) + (13 * 16¹) + (14 * 16⁰)
= (10 * 4096) + (2 * 256) + (13 * 16) + (14 * 1)
= 40960 + 512 + 208 + 14
= 41694 decimal

MEMORY:

SAM (Sequentially Access Memory) is accessed by stepping through each memory location until the desired location is reached. Magnetic tape is an example of SAM.

The second category of memory devices is called RAM (Random Access Memory) where the memory can be randomly accessed at any instant, without having to step through each memory location. It is generally faster to access a RAM compared to SAM. Most of the electronics gadgets memory are of RAM type.

Random Access Memory (RAM) Memory Device

RAM memory is "volatile" which means that the information stored in the RAM will be lost once the power to it is removed. Two common types of RAM are DRAM (Dynamic RAM) and SRAM (Static RAM). DRAM stored a bit as the presence or absence of charge on MOSFET gate substrate capacitance.

As the capacitance has leakage, it must be refreshed every few miliseconds. SRAM is an array of flip flops of which the bit stored in the flip flop will remain until power is removed or another bit replaces it. SRAM does not need to be refreshed. DRAM is usually 1.5 to 4 times as dense as SRAM and hence cheaper. However, SRAM has faster access times than DRAM.

Read Only Memory (ROM) Memory Device

ROM is non volatile in that its contents are not lost when power to it is removed. All ROMs can be programmed at least once. Mask ROMs are programmed by having "1"s and "0"s etched into their semiconductors at the time of manufacturing.

Programmable ROM (PROM) can be written after manufacturing by electrically burning specific transistors or diodes in the memory array. EPROM can be erased and reprogrammed by using ultraviolet light.

EEPROM (electronically erasable PROM) data can be erased electronically but it takes longer time compared to RAM. Read and write time for RAM is almost the same but PROM has slower write times. PROM must be erased before they can be reprogrammed and it often needs a higher programming voltage than its operating voltage.

ROM is usually used to store data or programs that do not change frequently and must still be there when power supply cuts off.

Non Volatile RAM (NVRAM) Memory Device

A NVRAM chip consists of both RAM and ROM. During power on reset, the contents of the ROM are copied to RAM. Before the power turns off, the system will copy the entire contents of the RAM into ROM for non volatile storage. The RAM in an NVRAM is called shadow RAM. NVRAM fills the gap between easily written memory and non volatile memory.

Modem

Introduction:

        A modem is a device or program that enables a computer to transmit data over, for example, telephone or cable lines. Computer information is stored digitally, whereas information transmitted over telephone lines is transmitted in the form of analog waves. A modem converts between these two forms.

        Fortunately, there is one standard interface for connecting external modems to computers called RS-232. Consequently, any external modem can be attached to any computer that has an RS-232 port, which almost all personal computers have. There are also modems that come as an expansion board that you can insert into a vacant expansion slot. These are sometimes called onboard or internal modems. 

 

Modem Operation

Modes of Modem  operation

a). Simplex mode

     In this mode, the data sets are transmitted in only one direction. This type of data set uses only one transmission channel. So that no signaling is available from the receiver to the transmitter. It is an economical method of data transfer.  It is not suited for error correction and requests for retransmission methods.

b). Half duplex

     Some modems provide for data transfer in both directions, but the data will flow in one direction at a time. It requires only one transmission channel, but the channel must be bidirectional. The speed of transmission is low.

c). Full duplex

     Full duplex operation permits transmission in both directions at the same time. It requires two 2 wire circuits one 4-wire circuit. Modems are placed at each end of the circuits to provide modulation and demodulation.

Serial Interface Standard – RS 232

RS 232:

     Short for recommended standard-232C, a standard interface approved by the Electronic Industries Alliance (EIA) for connecting serial devices. In 1987, the EIA released a new version of the standard and changed the name to EIA-232-D. And in 1991, the EIA teamed up with Telecommunications Industry association (TIA) and issued a new version of the standard called EIA/TIA-232-E. Many people, however, still refer to the standard as RS-232C, or just RS-232.

     Almost all modems conform to the EIA-232 standard and most personal computers have an EIA-232 port for connecting a modem or other device. In addition to modems, many display screens, mice, and serial printers are designed to connect to a EIA-232 port. In EIA-232 parlance, the device that connects to the interface is called a Data Communications Equipment (DCE) and the device to which it connects (e.g., the computer) is called a Data Terminal Equipment (DTE).

     The EIA-232 standard supports two types of connectors -- a 25-pin D-type connector (DB-25) and a 9-pin D-type connector (DB-9). The type of serial communications used by PCs requires only 9 pins so either type of connector will work equally well.

     Although EIA-232 is still the most common standard for serial communication, the EIA has recently defined successors to EIA-232 called RS-422 and RS-423. The new standards are backward compatible so that RS-232 devices can connect to an RS-422 port.

The standard 25-pin RS-232 connector

Features:

1.    It is used for serial communication

2.    It is a protocol standard as well as electrical standard.

3.    It is is used for short distances, upto 50ft.

4.    It is maximum data reate is 20,000 bd.

5.    It is not TTL compatible.

Errror Detection and correction

Introduction

Transmission of a signal is always subject to an error which may be caused due to various kinds of noises. These are all grouped into the so called transmission line spikes.

Normally noise interference with data by noise created in the channel itself or from the transmitting equipments. In the receiver this noise may give wrong data. As shown in fig 10, due to positive noise voltage, a 0 bit may be represented as 1 bit. Similarly, due to negative noise voltage a 1bit may be represented as 0 bit. Thus due to noise interference, there is a possibility to receive wrong data at the receiver end.

Error control coding involves systematic addition of extra digits to the transmitted message. These extra check digits convey no information but they are used to detect or correct errors in the regenerated message digits.

ERROR DETECTION CODES

Coding for error detection without correction is simpler than error correction coding. When a two way channel exists between source and destination, the receiver can request transmission of information containing detected errors.

The codes developed for providing automatic error detection are 1. Redundant Codes 2. Parity Check Codes

Cyclic Redundancy check Codes (CRC)

These codes are also one type of parity check codes. These code uses shift registers with feedback to create parity bits based on polynomial representation of the data bits. Basically it involves treating both transmitted and received data with the same polynomial.

The remainder after the receive processing will be zero if no errors have occurred. Cyclic codes provide the highest level of error detection to the same redundancy of any parity check code.

Parity Check Codes

The simplest form of error detection is parity; Single parity is established as follows. First, the information is coded in the normal manner using one of the standard binary codes. Each character is then examined to determine whether it contains an even or an odd number of 1 bits for example

Pulse	Number of one's	Parity
10101	3	Odd
11011	4	Even

If even parity is to be established, a 1 bit is added to each odd character, and a 0 bit is added to each even character. The result is that all the characters now contain an even number of 1 bits.

After transmission each character is examined to see if it still contains an even number of 1 bits. If it does not, the presence of an error is indicated. If it does, the parity bit is removed and the data passed to the user.

This form of parity will detect errors only if an odd number of bits are disturbed. An even number of errors within the same character will compensate for each other and go undetected.

Error Correction:

After having detected the error, it needs correction or minimization of errors. Usually three methods have been adopted for error correcting purposes.

Digital Modulation Techniques

       The techniques used to modulate digital information so that it can be transmitted via microwave, satellite or down a cable pair are different to that ofanalogue transmission. The data transmitted via satellite or microwave is transmitted as an analogue signal. The techniques used to transmit analogue signals are used to transmit digital signals. The problem is to convert the digital signals to a form that can be treated as an analogue signal that is then in the appropriate form to either be transmitted down a twisted cable pair or applied to the RF stage where is modulated to a frequency that can be transmitted via microwave or satellite.

The equipment that is used to convert digital signals into analogue format is a modem. The word modem is made up of the words “modulator” and “demodulator”. A modem accepts a serial data stream and converts it into an analogue format that matches the transmission medium. There are many different modulation techniques that can be utilised in a modem. These techniques are

·         Amplitude shift key modulation(ASK)

·         Frequency shift key modulation(FSK)

·         Binary-phase shift key modulation(BP SK)

·         Quadrature-phase shift key modulation(QP SK)

·         Quadrature amplitude modulation(QAM)

Classification of digital system

According to the modes of Modem operation

a.     Simplex

b.    Half duplex

c.     Full duplex

According to the modem interconnection

1.    Hard wired modem

2.    Acoustically coupled modem,

According to the modem data transmission period

1.    Low speed

2.    Medium speed

3.    High speed

According to the modulation technique

1.    FSK modulation

2.    PSK modulation

3.    Four phase PSK

4.    Eight phase PSK

5.    Quadrature AM

6.    Vestigial sideband AM

Advantages of digital modulation: 

1.    Noise in the system does not have the same detrimental effect (damage) on the received signal.

2.    It is possible to encode messages in special ways so that the receiving system can able to detect an error.

3.    Digital modulation and demodulation circuitry may be easier to implement for certain types of modulation.

4.    Digital signals are easier to combine together so that they can then be modulated as a group using multiplexing process.

Disadvantages:

1.    The cost of the conversion process may be too high.

2.    Requires more bandwidth than amplitude modulation

ASK Modulation

     Amplitude-shift keying (ASK) is a form of modulation that represents digital data as variations in the amplitude of a carrier wave. The amplitude of an analog carrier signal varies in accordance with the bit stream (modulating signal), keeping frequency and phase constant. The level of amplitude can be used to represent binary logic 0s and 1s. We can think of a carrier signal as an ON or OFF switch. In the modulated signal, logic 0 is represented by the absence of a carrier, thus giving OFF/ON keying operation.

     Like AM, ASK is also linear and sensitive to atmospheric noise, distortions, propagation conditions on different routes in PSTN, etc. Both ASK modulation and demodulation processes are relatively inexpensive. The ASK technique is also commonly used to transmit digital data over optical fiber. For LED transmitters, binary 1 is represented by a short pulse of light and binary 0 by the absence of light. Laser transmitters normally have a fixed "bias" current that causes the device to emit a low light level. This low level represents binary 0, while a higher-amplitude lightwave represents binary 1.

FSK Modulation

     Frequency-shift keying (FSK) is a frequency modulation scheme in which digital information is transmitted through discrete frequency changes of a carrier wave. The simplest FSK is binary FSK (BFSK). BFSK literally implies using a pair of discrete frequencies to transmit binary (0s and 1s) information. With this scheme, the "1" is called the mark frequency and the "0" is called the space frequency.

Other forms of FSK:

Minimum-shift keying

     Minimum frequency-shift keying or minimum-shift keying (MSK) is a particularly spectrally efficient form of coherent FSK. In MSK the difference between the higher and lower frequency is identical to half the bit rate. Consequently, the waveforms used to represent a 0 and a 1 bit differ by exactly half a carrier period. This is the smallest FSK modulation index that can be chosen such that the waveforms for 0 and 1 are orthogonal. A variant of MSK called GMSK is used in the GSM mobile phone standard. FSK is commonly used in Caller ID and remote metering applications: see FSK standards for use in Caller ID and remote metering for more details.

Audio FSK

·         Audio frequency-shift keying (AFSK) is a modulation technique by which digital data is represented by changes in the frequency (pitch) of an audio tone, yielding an encoded signal suitable for transmission via radio or telephone. Normally, the transmitted audio alternates between two tones: one, the "mark", represents a binary one; the other, the "space", represents a binary zero.

·         AFSK differs from regular frequency-shift keying in performing the modulation at baseband frequencies. In radio applications, the AFSK-modulated signal normally is being used to modulate an RF carrier (using a conventional technique, such as AM or FM) for transmission.

·         AFSK is not always used for high-speed data communications, since it is far less efficient in both power and bandwidth than most other modulation modes. In addition to its simplicity, however, AFSK has the advantage that encoded signals will pass through AC-coupled links, including most equipment originally designed to carry music or speech.

PSK Modulation

In this method the phase of the carrier signal is varying according to the binary level. When binary level is 0 the carrier will be in the reference phase. When binary level is 1 the carrier phase will deviate 1800 from reference. Thus whenever the signal level changes the carrier phase will vary by 180°. This is called as binary PSK. The wave forms are shown in fig.

However, PSK and FSK systems are widely used in practice because of their robustness with respect to changes in their amplitude that may be caused by transmission over a non linear channel

Digital Codes

Introduction:

Different types of equipments are used in computer systems to send and receive data through the devices like keyboards, video terminals, printers, paper tape punches and readers and magnetic storage devices. Each of these types of equipment generates and receives data in the form of codes.

Some codes are advantageous when used in different applications. Modern computers can easily deal with different codes by simply converting them to the code used by the computer. The common codes used are Baudot code, Binary code, ASCII code, EBCDIC code, Hollerith code etc.

·         Baudot Code

·         ASCII Codes

·         EBCBIC Code

·         BCD Code

Baudot Code

Introduction: 
(Murray codes or International Telegraph Codes) 1845-1903

The Baudot code is a 5-bit code which has been used in telegraphy and paper-tape systems. There are totally 32 (25= 32) different code combination's. This is not sufficient to represent the alphabets (26), numerals (10) and other characters (e.g. +, - , >, This 5 element code uses Letter shift and figure shift symbols to expand the number of combination's it can provide. Line A is used for weather Symbols, line B - for fractions, line C - for communications

The following table explains the Baudot Code Set. The leftmost bit is the Most Significant Bit (MSB), transmitted last. The rightmost bit is the Least Significant Bit (LSB), transmitted first. The associated LETTERS and FIGURES (case) characters are also listed, along with the hexadecimal representation of the character.

BITS     LTRS    FIGS      HEX

-----    ----    ----      ---

00011      A      -        03

11001      B      ?        19

01110      C      :        0E

01001      D      $        09

00001      E      3        01

01101      F      !        0D

11010      G      &        1A

10100      H      STOP     14

00110      I      8        06

01011      J      '        0B

01111      K      (        0F

10010      L      )        12

11100      M      .        1C

01100      N      ,        0C

11000      O      9        18

10110      P      0        16

10111      Q      1        17

01010      R      4        0A

00101      S      BELL     05

10000      T      5        10

00111      U      7        07

11110      V      ;        1E

10011      W      2        13

11101      X      /        1D

10101      Y      6        15

10001      Z      "        11

00000      n/a    n/a      00

01000      CR     CR       08

00010      LF     LF       02

00100      SP     SP       04

11111      LTRS   LTRS     1F

11011      FIGS   FIGS     1B

Uses:
Used in telegraph and paper tape systems.

Drawbacks:
i) It does not provide extra combination of bits to code punctuation and various codes.
ii) All the five bits are used for coding the data. Hence error correction   using parity technique is not possible.

Limitations:

1.    Only 5 bits are available.

2.    The number of combinations are limited only up to 25 = 32 codes.

3.    This code is not sequential.

4.    It is not used for error detection and correction.

ASCII Codes

ASCII stands for the American Standard Code for Information Interchange, As a standard, ASCII was first adopted in 1963 and quickly became widely used throughout the computer world.

ASCII is a way of defining a set of characters which can be displayed by a computer on a screen, as well as some control characters which have special functions. Basic ASCII uses seven bits to define each letter, meaning it can have up to 128 specific identifiers, two to the seventh power. This size was chosen based on the common basic block of computing, the byte, which consists of eight bits. The eighth bit was often set aside for error-checking functions, leaving seven remaining for a character set.

Thirty-three codes in ASCII are used to represent things other than specific characters. The first 32 (0-31) represent things ranging from a chime sound, to a line feed command, to the start of a header. The final code, 127, represents a backspace. Beyond the first 31 bits are the printable characters. Bits 48-57 represent the numeric digits. Bits 65-90 are the capital letters, while bits 97-122 are the lower-case letters. The rest of the bits are symbols of punctuation, mathematical symbols, and other symbols such as the pipe and tilde.

Standard ASCII table:

Extended ASCII Table:

Advantages:
i.) Error detection can be achieved by increasing the total numbers of bits of 8. The parity bit is added as the 8th bit, usually the MSB.
ii.) It can be easily used in a computer. Modern computer uses hexadecimal code for their internal computations. Since ASCII is an 8 bit code with parity bit, it can be easy accommodated in computer as 8-bit data.

Use:
It is widely use in modern computers

EBCBIC Code

       EBCDIC code stands for Extended Binary Coded Decimal Inter Change code. It is also based on the binary coded decimal format. It is an 8 bit code. Here all the 8 bits are used for representing the information. This code also follows a standard binary progression for coding. This code has totally 256 combinations. In this code, the letter ‘A’ is represented as ‘1100 1000’ and ‘J’ is represented as 1101 1000. Here the last two bits are same, but the first 4 bits change progressively from 0000 to 1111. To represent letter A, the code is 1100 1000, and to represent letter B, the code is 1100 0100. Here the first 4 bits are same, but last 4 bits vary. Here also it varies progressively from 0000 to 1111, in reverse direction.

Advantages:

1.    It is similar to ASCII code. It can be readily used in telegraph and  computer.

2.    Total number of combinations is higher.

Disadvantages

Here all the 8-bits are used for data encoding. There is no provision for parity bit. Here error correction is not possible.

BCD Code

     BCD stands for Binary Coded Decimal numbers. If 8 bits are used, then 256 combinations are possible.

    For representing numbers, the binary code was modified so that only the lower 4 bits are needed. The sequence was a second 8 bit word to represent each successive decimal column. As one binary word reaches decimal number 10, it returns to zero and a carry is added to next word. The number 9567 is represented as
    1001 0101 01100111

    An extension of BCD code is alphanumeric code. Two extra bits are needed to represent letters and punctuation marks. Seventh bit is added to provide parity bit for error detection. BCD encoding is used for data representation on magnetic tape. Recording of data is made on several tracks. An ‘1’ results in magnetized spot on the tape: A ‘0’ leaves the spot unmagnetized.

Disadvantages:

Unmodified 8 bit code did not permit any means of error detection. Hence we have to add a parity bit for error detection.

CDMA

 Code division multiple access (CDMA) is a channel access method utilized by various radio communication technologies. It should not be confused with the mobile phone standards called cdmaOne and CDMA2000 (which are often referred to as simply "CDMA"), which use CDMA as an underlying channel access method.

 One of the basic concepts in data communication is the idea of allowing several transmitters to send information simultaneously over a single communication channel. This allows several users to share a bandwidth of different frequencies. This concept is called multiplexing. CDMA employs spread-spectrum technology and a special coding scheme (where each transmitter is assigned a code) to allow multiple users to be multiplexed over the same physical channel. By contrast, time division multiple access (TDMA) divides access by time, while frequency-division multiple access (FDMA) divides it by frequency. CDMA is a form of "spread-spectrum" signaling, since the modulated coded signal has a much higher data bandwidth than the data being communicated.

 An analogy to the problem of multiple access is a room (channel) in which people wish to communicate with each other. To avoid confusion, people could take turns speaking (time division), speak at different pitches (frequency division), or speak in different languages (code division). CDMA is analogous to the last example where people speaking the same language can understand each other, but not other people. Similarly, in radio CDMA, each group of users is given a shared code. Many codes occupy the same channel, but only users associated with a particular code can understand each other.

FDMA

     Frequency Division Multiple Access or FDMA is a channel access method used in multiple-access protocols as a channelization protocol. FDMA gives users an individual allocation of one or several frequency bands, or channels. Multiple Access systems coordinate access between multiple users. The users may also share access via different methods such as TDMA, CDMA, or SDMA. These protocols are utilized differently, at different levels of the theoretical OSI model. Disadvantage:Crosstalk which causes interference on the other frequency and may disrupt the transmission.

Features:

·         FDMA requires high-performing filters in the radio hardware, in contrast to TDMA and CDMA.

·         FDMA is not vulnerable to the timing problems that TDMA has. Since a predetermined frequency band is available for the entire period of communication, stream data (a continuous flow of data that may not be packetized) can easily be used with FDMA.

·         Due to the frequency filtering, FDMA is not sensitive to near-far problem which is pronounced for CDMA.

·         Each user transmits and receives at different frequencies as each user gets a unique frequency slot

     It is important to distinguish between FDMA and frequency-division duplexing (FDD). While FDMA allows multiple users simultaneous access to a certain system, FDD refers to how the radio channel is shared between the uplink and downlink (for instance, the traffic going back and forth between a mobile-phone and a base-station). Furthermore, frequency-division multiplexing (FDM) should not be confused with FDMA. The former is a physical layer technique that combines and transmits low-bandwidth channels through a high-bandwidth channel. FDMA, on the other hand, is an access method in the data link layer.

     FDMA also supports demand assignment in addition to fixed assignment. Demand assignment allows all users apparently continuous access of the radio spectrum by assigning carrier frequencies on a temporary basis using a statistical assignment process. The first FDMA demand-assignment system for satellite was developed by COMSAT for use on the Intelsat series IVA and V satellites.

TDMA

               Time division multiple access (TDMA) is a channel access method for shared medium networks. It allows several users to share the same frequency channel by dividing the signal into different time slots. The users transmit in rapid succession, one after the other, each using his own time slot. This allows multiple stations to share the same transmission medium (e.g. radio frequency channel) while using only a part of its channel capacity. TDMA is used in the digital 2G cellular systems such as Global System for Mobile Communications (GSM), IS-136, Personal Digital Cellular (PDC) and iDEN, and in the Digital Enhanced Cordless Telecommunications (DECT) standard for portable phones. It is also used extensively in satellite systems, and combat-net radio systems. For usage of Dynamic TDMA packet mode communication, see below.