dlc pre requisite

7/30/2019 DLC Pre Requisite

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UNIT - I

Numerical Presentation

In science, technology, business, and, in fact, most other fields of endeavor, we

are constantly dealing with quantities. Quantities are measured, monitored,recorded, manipulated arithmetically, observed, or in some other way utilized in

most physical systems. It is important when dealing with various quantities that

we be able to represent their values efficiently and accurately. There are basicallytwo ways of representing the numerical value of quantities: analogand digital.

Analog Representation

In analog representation a quantity is represented by a voltage, current, or meter

movement that is proportional to the value of that quantity. Analog quantities

such as those cited above have an important characteristic: they can vary over acontinuous range of values. Below is a diagram ofanalog voltage vs time:

Digital Representation

In digital representation the quantities are represented not by proportionalquantities but by symbols called digits. As an example, consider the digital watch,

which provides the time of day in the form of decimal digits which represent

hours and minutes (and sometimes seconds). As we know, the time of day

changes continuously, but the digital watch reading does not change continuously;rather, it changes in steps of one per minute (or per second). In other words, this

digital representation of the time of day changes in discrete steps, as compared

with the representation of time provided by an analog watch, where the dialreading changes continuously. Below is a diagram ofdigital voltage vs. time:


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The major difference between analog and digital quantities, then, can be simply

stated as follows:

Analog = continuousDigital = discrete (step by step)

Advantages and Limitations of Digital Techniques

Advantages

1. Easier to design. Exact values of voltage or current are not important, only the

range (HIGH or LOW) in which they fall.2. Information storage is easy.

3. Accuracy and precision are greater.

4. Operation can be programmed. Analog systems can also be programmed, but thevariety and complexity of the available operations is severely limited.

5. Digital circuits are less affected by noise. As long as the noise is not large enough

to prevent us from distinguishing a HIGH from a LOW.

6. More digital circuitry can be fabricated on IC chips.

Limitations

There is really only one major drawback when using digital techniques:

The real world is mainly analog.

Most physical quantities are analog in nature, and it is these quantities that areoften the inputs and outputs that are being monitored, operated on, and controlled

by a system.

To take advantage of digital techniques when dealing with analog inputs and

outputs, three steps must be followed:

1. Convert the real-world analog inputs to digital form. (ADC)

2. Process (operate on) the digital information.3. Convert the digital outputs back to real-world analog form. (DAC)


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The following diagram shows a temperature control system that requires

analog/digital conversions in order to allow the use of digital processing

techniques.

Number Systems

A number system specifies how values are represented. Human uses DECIMAL

Number System. There are ten digits in Decimal Number System:0,1,2,3,4,5,6,7,8,9

Digital Computers use Binary Numbers, which have only two digits: 0,1

There are other number systems, including: Octal and Hexadecimal

Number System Terminology

In Number System, a value of an n-digit number a n-1a n-2a1a 0 is:N = a n-1 x r

n-1 + a n-2 x rn-2 ++ a 1 x r

1 + a 0 x r0

where a n-1,a n-2,,a1,a 0 are coefficients

r is called the Base or RadixDecimal is Base-10 system, r = 10

Binary is Base-2 system, r = 2

Octal is Base-8 system, r = 8Hexadecimal is Base-16 system, r = 16

Decimal Number System

The decimal system is composed of 10 numerals or symbols. These 10 symbols

are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9; using these symbols as digits of a number, we can

express any quantity. The decimal system, also called the base-10 system becauseit has 10 digits. Values are represented by the digits and their positions in the

number and the type of number system is called Positional Number System


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103 102 101 100 10-1 10-2 10-3

=1000 =100 =10 =1 . =0.1 =0.01 =0.001

Most

Significant

Digit

Decimal

point

Least

Significant

Digit

8973 is Eight Thousand Nine Hundred and Seventy Three:

8 = 8000 = 8 x 103 (Thousands Place)

9 = 900 = 9 x 102 (Hundreds Place)7 = 70 = 7 x 101 (Tens Place)

3 = 3 = 3 x 100 (Units Place)

Binary Number System

An n-bit binary number a n-1a n-2a1a 0 has a value:

N = a n-1 x 2n-1 + a n-2 x 2

n-2 ++ a 1 x 21 + a 0 x 2

0

This base-2 system can be used to represent any quantity that can be represented

in decimal or other number system.

23 22 21 20 2-1 2-2 2-3

=8 =4 =2 =1 . =1/2 =1/4 =1/8

Most

Significant Bit

Binary

point

Least

Significant Bite.g. A 4-bit binary number 10112 is:

N = 1 x 23 + 0 x 22 + 1 x 21 + 1 x 20

= 8 + 0 + 2 + 1 = 1110e.g. A 6-bit binary number 1100102 is:

N = 1x25 + 1x24 + 0x23 + 0x22 + 1x21 + 0x20

= 32 + 16 + 0 + 0 + 2 = 5010

The Binary counting sequence is shown in the table:


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Representing Binary Quantities

In digital systems the information that is being processed is usually presented in

binary form. Binary quantities can be represented by any device that has only twooperating states or possible conditions. Eg. a switch has only open or closed. We

arbitrarily (as we define them) let an open switch represent binary 0 and a closed

switch represent binary 1. Thus we can represent any binary number by usingseries of switches.

Typical Voltage Assignment

Binary 1: Any voltage between 2V to 5VBinary 0: Any voltage between 0V to 0.8V

Not used: Voltage between 0.8V to 2V, this may cause error in a digital circuit.


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We can see another significant difference between digital and analog systems. In

digital systems, the exact value of a voltage is notimportant; eg, a voltage of 3.6V

means the same as a voltage of 4.3V. In analog systems, the exact value of avoltage is important.

Binary to Decimal Number Conversion

Binary numbers can be converted to Decimal numbers by using:N = a n-1 x 2

n-1 + a n-2 x 2n-2 ++ a 1 x 2

1 + a 0 x 20

1 1 0 1 1 2(binary)

24+23+0+21+20= 16+8+0+2+1

= 2710 (decimal)

and

1 0 1 1 0 1 0 1 2(binary)

27+0+25+24+0+22+0+20= 128+0+32+16+0+4+0+1

= 18110 (decimal)

Decimal to Binary Number Conversion

Decimal numbers can be converted to binary numbers by dividing the decimal

number by 2 successively.

This method uses repeated division by 2. Ex. Convert 2510 to binary

25/ 2 = 12+ remainder of1 1 (Least Significant Bit)

12/ 2 = 6 + remainder of 0 0

6 / 2 = 3 + remainder of 0 0

3 / 2 = 1 + remainder of 1 1

1 / 2 = 0 + remainder of1 1 (Most Significant Bit)

Result 2510 = 1 1 0 0 12

The Flow chart for repeated-division method is as follow:


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e.g. 16810 = 101010002

Octal Number System

Octal Numbers are base 8Octal has 8 digits: 0, 1, 2, 3, 4, 5, 6, 7

An n-bit octal number a n-1a n-2a1a 0 has a decimal value:

a n-1 x 8n-1 + a n-2 x 8

n-2 ++ a 1 x 81 + a 0 x 8

0


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The conversion of octal to decimal can be done with the above equatione.g. 2638 = 2x8

2 + 6x81 + 3x80

= 128 + 48 + 3 = 17910

e.g 24.68 = 2 x (81) + 4 x (80) + 6 x (8-1) = 20.7510

Decimal to Octal Number Conversion

Can be done by successive division of 8

This method uses repeated division by 8.

e.g. Convert 17710 to octal and binary:

177/8 = 22+ remainder of1 1 (Least Significant Bit)

22/ 8 = 2 + remainder of 6 6


Result 17710 = 2618

Convert to Binary = 0101100012

e.g. 93810 = 16528

Hexadecimal Number System

83 82 81 80 8-1 8-2 8-3

=512 =64 =8 =1 . =1/8 =1/64 =1/512

Most

Significant

Digit

Octalpoint

Least

Significant

Digit


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Hexadecimal Numbers are base 16

There are 16 digits: 0 to 9, A, B, C, D, E, F

163 162 161 160 16-1 16-2 16-3

=4096 =256 =16 =1 . =1/16 =1/256 =1/4096

Most

Significant

Digit

Hexadec.

point

Least

Significant

Digit

An n-bit hexadecimal number a decimal value:a n-1 x 16

n-1 + a n-2 x 16n-2 ++ a 1 x 16

1 + a 0 x 160

The conversion of hexadecimal to decimal can be done with the above equation

e.g. B5E16 = 11x162

+ 5x161

+ 14x160

= 2816 + 80 + 14 = 291010

Decimal to Hexadecimal Conversion

This method uses repeated division by 16.

e.g. Convert 37810 to hexadecimal and binary:

378/16 = 23+ remainder of10 A (Least Significant Bit)

23/ 16 = 1 + remainder of 7 7


Result 37810 = 17A16

Convert to Binary = 0001 0111 10102

e.g. 279310 = AE916


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Conversion between Binary and Octal

Conversion between Binary and Octal is convenientEach Octal digit equals to 3 bits

e.g. 3 6 2 8 - 011 110 010 25 4 1 8 - 101 100 001 2

e.g. 010 101 110 2 - 2 5 6 8111 010 001 2 - 7 2 1 8

Conversion between Binary and Hex

Conversion between Binary and Hexadecimal is also convenient

Each Hexadecimal digit equals to 4 bits

E 2 8 0 16 - 1110 0010 1000 0000 2

F B 1 16 - 1111 1011 0001 2

Conversion among number systems

7318 = 111 011 0012


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= 0001 1101 10012= 1 D 9 16= 1x162 + 13x161 + 9x160

= 256 + 208 + 9

= 47310

7318 = 7x82

+ 3x81

+ 1x80

= 448 + 24 + 1

= 47310

Signed Binary Numbers

In ordinary arithmetic, a minus sign - is used to represent negative

numbers,e.g.-38In digital electronic circuits, everything is represented with bits (0, 1).There are

several ways to represent the signed binary numbers with just bits, e.g.:

Signed Magnitude Representation

1s Complement Representation 2s Complement Representation

Signed Magnitude Number System

The most significant bit (MSB) is a sign bitIf the MSB is 0, the number is positive

If the MSB is 1, the number is negative

e.g. 01101 = +1311101 = -13

e.g. 00000 = +0

10000 = -0Disadvantage (a):2 patterns represent 0

(b): Handle sign bit separately

1s Complement Number System

Positive numbers and the corresponding negative numbers complement each other

Complement is inversion (Logic NOT)e.g. 01101 = +13

10010 = -1

e.g. 00000 = +0

11111 = -0Disadvantage (a):2 patterns represent 0

(b): Handle sign bit separately25

2s Complement Number Conversion

To find the 2s complement number for a negative decimal, we can find thebinary of


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the positive decimal and then take its 2scomplement.Twos complement is

obtained by adding one to the ones complement

e.g. -9109 = 01001

Invert bits: 10110

Plus 1: 10111 = -910Check: -16 + 0 + 4 + 2 + 1 = -9

Value Range

For 4-bit binary, the range is:

Value Range

For n-bit binary, the range is:

CODES


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Group of bits assigned to represent, identify or relate to multivalued items of

information. By assigning each item of information a unique combination of bits the

information is transferred from one form to another. The group of bits may be numbers,alphabets, control functions and special characters.

An n-bit binary code is a group of n bits that assume up to 2 n combinations of 1s

and 0s with each combination representing one element of the set being enclosed.

Types

Weighted codes

Non weighted codes

Self complementing codes

Reflective codesAlphanumeric codes

Error detecting and correcting codes

1.Weighted Codes

Each bit has a positional value of 8,4,2 or 1 in binary codes. Examples are8421, 2421, 3321, 4221, 5211, 5311, 5421, 6311,7421, 7421,

8421

All the above codes are used to represent a given decimal digit into four bitbinary word.

S.No. DecimalNumber

8421 2421 3321 4221 5311 5421 6311 7421 7421

8421

1. 0 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000

2. 1 0001 0001 0001 0001 0001 0001 0001 0001 0111 01113. 2 0010 0010 0010 0010 0011 0010 0011 0010 0110 0110

4. 3 0011 0011 0011 0011 0100 0011 0100 0011 0101 0101

5. 4 0100 0100 0101 1000 0101 0100 0101 0100 0100 0100

6. 5 0101 1011 0110 0111 1000 0101 0111 0101 1010 1011

7. 6 0110 1100 0111 1100 1001 0110 1000 0110 1001 1010

8. 7 0111 1101 1101 1101 1010 0111 1001 0111 1000 1001

9. 8 1000 1110 1110 1110 1100 1011 1011 1001 1111 1000

10. 9 1001 1111 1111 1111 1101 1100 1100 1010 1110 1111

2. Non-weighted Codes:

Each bit has no positional value1. Excess-3 code2.Gray code

3.Five bit BCD

3. Self complementing codes (or) Reflective codes

Code for one digit will be the complement of other


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1.2421

2.5211

3. Excess-3

4. Sequential Codes

Succeeding number is one more than the previous one

1. 84212. Excess-3

5. Alphanumeric codes

1. ASCII2. EBCDIC

3. Hollerith

6. Error detecting and correcting codes

For reliable transmission and storage of digital data, error detection and correction

is required. Below are a few examples of codes which permit error detection and

error correction after detection

Error Detecting Codes

When data is transmitted from one point to another, like in wireless transmission,

or it is just stored, like in hard disks and memories, there are chances that data

may get corrupted. To detect these data errors, we use special codes, which areerror detection codes.

Parity

In parity codes, every data byte, or nibble (according to how user wants to use it)

is checked if they have even number of ones or even number of zeros. Based on

this information an additional bit is appended to the original data. Thus if weconsider 8-bit data, adding the parity bit will make it 9 bit long.

At the receiver side, once again parity is calculated and matched with the received

parity (bit 9), and if they match, data is ok, otherwise data is corrupt.

There are two types of parity:

Even parity: Checks if there is an even number of ones; if so, parity bit is

zero. When the number of ones is odd then parity bit is set to 1.

Odd Parity: Checks if there is an odd number of ones; if so, parity bit is

zero. When number of ones is even then parity bit is set to 1.

Error-Correcting Codes

Error-correcting codes not only detect errors, but also correct them. This is used

normally in Satellite communication, where turn-around delay is very high as is

the probability of data getting corrupt.


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ECC (Error correcting codes) are used also in memories, networking, Hard disk,

CDROM, DVD etc. Normally in networking chips (ASIC), we have 2 Error

detection bits and 1 Error correction bit.

Hamming Code

Hamming code adds a minimum number of bits to the data transmitted in a noisy

channel, to be able to correct every possible one-bit error. It can detect (not

correct) two-bits errors and cannot distinguish between 1-bit and 2-bitsinconsistencies. It can't - in general - detect 3(or more)-bits errors.

The idea is that the failed bit position in an n-bit string (which we'll call X) can be

represented in binary with log2(n) bits, hence we'll try to get it adding just log 2(n)

bits.

First, we set m = n + log2(n) to the encoded string length and we number each bit

position starting from 1 through m. Then we place these additional bits at power-of-two positions, that is 1, 2, 4, 8..., while remaining ones (3, 5, 6, 7...) hold the

bit string in the original order.

Now we set each added bit to the parity of a group of bits. We group bits this

way: we form a group for every parity bit, where the following relation holds:

position(bit) AND position(parity) = position(parity)

(Note that: AND is the bit-wise boolean AND; parity bits are included in the

groups; each bit can belong to one or more groups.)

So bit 1 groups bits 1, 3, 5, 7... while bit 2 groups bits 2, 3, 6, 7, 10... , bit 4

groups bits 4, 5, 6, 7, 12, 13... and so on.

Thus, by definition, X (the failed bit position defined above) is the sum of the

incorrect parity bits positions (0 for no errors).

To understand why it is so, let's call Xn the nth bit of X in binary representation.

Now consider that each parity bit is tied to a bit of X: parity1 -> X 1, parity2 -> X2,

parity4 -> X3, parity8 -> X4 and so on - for programmers: they are the respective

AND masks. By construction, the failed bit makes fail only the parity bits which

correspond to the 1s in X, so each bit of X is 1 if the corresponding parity iswrong and 0 if it is correct.

Note that the longer the string, the higher the throughput n/m and the lower the

probability that no more than one bit fails. So the string to be sent should bebroken into blocks whose length depends on the transmission channel quality (the

cleaner the channel, the bigger the block). Also, unless it's guaranteed that at most


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one bit per block fails, a checksum or some other form of data integrity check

should be added.

Alphanumeric Codes

The binary codes that can be used to represent all the letters of the alphabet,numbers and mathematical symbols, punctuation marks, are known as

alphanumeric codes or character codes. These codes enable us to interface the

input-output devices like the keyboard, printers, video displays with the computer.

ASCII Code

ASCII stands for American Standard Code for Information Interchange. It has

become a world standard alphanumeric code for microcomputers and computers.

It is a 7-bit code representing 27 = 128 different characters. These characters

represent 26 upper case letters (A to Z), 26 lowercase letters (a to z), 10 numbers

(0 to 9), 33 special characters and symbols and 33 control characters.

The 7-bit code is divided into two portions, The leftmost 3 bits portion is called

zone bits and the 4-bit portion on the right is called numeric bits.

An 8-bit version of ASCII code is known as USACC-II 8 or ASCII-8. The 8-bitversion can represent a maximum of 256 characters.

EBCDIC Code

EBCDIC stands for Extended Binary Coded Decimal Interchange. It is mainly

used with large computer systems like mainframes. EBCDIC is an 8-bit code andthus accomodates up to 256 characters. An EBCDIC code is divided into two

portions: 4 zone bits (on the left) and 4 numeric bits (on the right).

Hollerith code

Hollerith developed a way of feeding information into digital computers using

punched cards. The code used in this system to represent alphanumericinformation is known as Hollerith code. Punch card has 80 columns and 12 rows.

Each column represents an alphanumeric character with holes in appropriate

rows. A hole is sensed as 1 and absence of hole is sensed as 0 by the circuit in

card reader. The 12 rows are marked starting from top as12,11,0,1,2,3,4,5,6,7,8,9,. Each row is 1-bit information. So, Hollerith code is

a12-bit code. The first 3 rows are zone punch rows and the remaining 9 arenumeric punch rows. The numbers are represented in the column by single punch

whereas alphabets are represented using 2 punches.

Binary Coded Decimal (BCD)


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Binary numbers are used by computers and human beings are familiar with

decimals.

To facilitate the easy conversion between binary and decimal, BCD is used.In BCD number system, each decimal digit is represented by 4 bits

Group of 4 binary bits is a nibble. A nibble representing a number greaterthan 9 is invalid BCD

Decimal to BCD Conversion

Binary 1010 to 1111 are not used in BCD

Example 1. Decimal 56 in BCD is 0101 0110

in binary is 00111000

BCD addition Add the given BCD numbers using binary addition

If the sum nibble (group of 4 bits) is equal to or less than 9 (without

carry) then it is valid BCD.

If the sum nibble (group of 4 bits) is greater than 9 (or) if a carry is

generated then it is invalid BCD.

To make the invalid BCD valid add 01102 to the nibble and if a carry is

generated add it to the next higher order BCD digit.

Examples

1. Add 4 and 5

4-> 0100

5-> 0101

---------1001 -> 9 -> valid BCD

---------

2. Add 4 and 8


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4-> 0100

8-> 1000

---------1100

---------

1100 is invalid BCD since it is greater than 1001 so add 0110 to 1100and

the result is 1 0010 which is equal to 12.

3. Add 8 and 9

8-> 1000

9-> 1001---------

10001

---------

In the addition carry is generated so the result is invalid BCD. To make it

valid add 0110 to 10001 and the result is 1 0111 which is equal to 17.

Excess-3 (XS3) code

1. Non weighted BCD code

2. Add three to each nibble of BCD code to get the XS3 code.

3. This code helps in performing subtraction operations in the earlier

computers.4. The table below shows the BCD and XS-3 codes for decimal digits 0 to

9

Decimal BCD Excess-3

0 0000 00111 0001 0100

2 0010 0101

3 0011 0110

4 0100 01115 0101 1000

6 0110 1001

7 0111 10108 1000 1011

9 1001 1100

Decimal to XS-3 conversion

(Ex).Convert 436 into XS-3 code

BCD equivalent of 436 is 0100 0011 0110


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Add 3 (i.e) 0011 to each group of 4 bits

0100 0011 0110

0011 0011 0011----------------------

0111 0110 1001

----------------------7 6 9 - Xs-3 in decimal

Gray code

It is a minimum change code where only one bit in the code group changes

while going from one step to the next. Binary numbers have more than 1 bits

changing when increasing or decreasing continuouslye.g. 0011 to 0100 (3 to4): 3 bits change.In digital electronic circuits, sometimes it is desired to have

only 1 bit changed In this case, Gray Code can be used In Gray Code, the

adjacent numbers have only 1 bit different

It is also called as unit distance code.

In this code no weight can be attached with each bit position therefore it is

unsuitable for arithmetic operations.

Gray codes are useful in input and output devices and in analog and digital

converters.

Gray to Binary conversion

MSB of the gray code is same as binary number. So write the MSB as it is.

Add the MSB of the output with bit immediately on right in input and record

the sum. If a carry is generated it is ignored.

Continue adding the bits in the output to immediate input bit in right till the

LSB is reached.

The binary equivalent has same number of bits as the gray code.

This can be achieved by using X-OR gate.

ExampleConvert 1011 to binary


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Binary to Gray conversion

MSB of the binary is same as MSB of gray code.

Add the MSB of binary with immediate bit to right in binary and that is

the next gray bit. If a carry is generated it is ignored.

Repeat the procedure till the LSB is reached.

Example:

Convert 1011 to gray code


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