FUNDAMENTALS OF DIGITAL

COMPUTER

KARNAUGH MAP

• It is a visual display of the fundamental products

needed for a sum of products solutions.

THREE VARIABLES MAP

• Logic equation

Y=F(A,B,C)=∑m(2,6,7)

FOUR VARIABLE MAPS

PAIRS AND QUADS AND OCTETS

• The map contain a pair of 1s that are

horizontally adjacent.(next to each other).

• The sum of product equation:

Y=ABCD+ABCD̅

which factor into

Y=ABC(D+D̅)

QUAD

• It is a group of four 1s that are horizontally orvertically adjacent.

• This is may be end to end.

• The Boolean equations of those two pairs:

Y=ABC̅ + ABC

Factor into

Y= AB(C̅ + C)

Which is reduce into

Y=AB

OCTET

• Besides pairs and quads, there is one more

group to adjacent 1s to look .

Y=AC̅ + AC

After factoring

Y=A(C̅ + C)

Reduce to

Y=A

KARNAUGH

SIMPLIFICATION

• Encircle the octet first

• Quad second

• And pairs last

In this way the greatest simplification results.

EQUATION: Y=A̅B̅C+A C̅+CD̅

OVERLAPPING GROUPS

• You are allowed to use the same 1 more than

once.

• The 1 responding the fundamental product

ABC̅D is part of the pair and part of the octet.

• Simplified equation is Y=A+BC̅D

• It is valid to encircle the 1s shown in figure,

but then the isolated one results in a more

complicated

EQUATION :Y=A+A̅BC̅D

ROLLING THE MAP

• The pair result in this equation:

Y=BC̅D̅+BCD̅

Visualize Picking up the karnaugh map and rolling it so

that the left side touches the right side. if you are

visualizing correctly,you will realize the pairs actually

from a quad.to indicate this draw half circles around

each pair shown in fig., from this visualpoint the quad

has the equation:Y=BD̅

DON’T CARE CONDITIONS

• In some digital systems, certain never occur

during normal operations therefore, the

corresponding output never appear. since the

output never appears.it is indicated by an X in

the truth table

• Shows a truth table where the output is low for all the

entries from 0000 to 1000

• The high for input entry 1001 and X for 1010 through

1111

• The X is called a Don’t care conditions.

• Whenever you can see an X in a truth table,you can let

it equal either 0 or 1.

• Shown In the truth table with 1010 to 1111 with don’t

care for all inputs.

• Those don’t care are like wild cards in poker because

you can let them stand for whatever u like.

• Shown in fig2. the most efficient way to encirlce the 1.

• Notice two crucial ideas.

First the 1 is included in a quad,the largest group you

can find if u visualize all X’s as 1’S.

Second, after the 1 has been encircled, all X’s outside

the quad are visualized as 0’S.

In this way the X are used to the best possible

advantage.

As already mentioned, you are free to do this because

don’t care correspond to input condition that never

appear.

The quad boolean equation =Y=AD

• Logical circuit:

• The show in logical circit is AND gate with

inputs of A and D.

• You can check this logic circuit by examining

truth tacble.

• The possible inputs are from 0000 to 1001 in

this range a high A and D produce a high Y

• In this range a high A and D produce high Y

only for input condition 1001

Conditions

• Given the Truth table draw a Karnaugh Map

with 0’S and1s and don’t cares.

• Encircle the actual 1s on the Karnaugh map in

the largest groups you can find by treating the

don’t cares as 1s.

• After the actual 1s have been included in

groups, disregarded the remaining don’t cares

by visualizing them as 0s.

Product of sums method

• With the sum of product s method the design starts

with a truth table that summarizes the desired input –

output conditions.

• Next step to convert the truth table into an equivalent

sum of product equations.

• Final step is to draw the AND –OR network or its

NAND-NAND equivalent.

• The product of sums method is similar .

• Given a truth table u identify the fundamental sums

needed for a logical design.

Converting a truth table to an equation:

What you have to do is locate each output 0 in the

truth table and write down its fundamental sum.

Logical circuit:

• After you have s product of sum equation you can get

the logical circuit by drawing an OR-AND network. or

if you prefer NOR-NOR network.

• Each sum represent the output of a 3 inputs OR gate

• The logical product of y is the output of a 3input AND

gate.

A 3 input OR gate is not available as a TTL chip. shown in

below fig.,is not practical with De Morgan’s first

theorem.

However you can replace the OR-AND circuit

• NOR-NOR circuit:

Conversion between SOP and POS:

We have seen that SOP representation is obtained by

considering ones in a truth table. While POS comes

considering zeros.

PRODUCT OF SUM

SIMPLIFICATION

SUM OF PRODUCT EQUATION:

Y=A̅B̅+AB+AC

DATA PROCESSING CIRCUITS

MULTIPLEXER:

Multiplexer mean many into one.

A multiplexer is a circuit with many inputs but only one output.

16 to 1 multiplexer:

The input bits are labeled D0 to D15,only one of

those is transmitted to the output. Which depends on

the value of ABCD, the control input.

DEMULTIPLEXER

• Demultiplexer means one to many

• A demultiplexer is a logic circuit with one input and

many output

.

• 1 to 16 Demultiplexer

• The input bit is labeled D.

• This data bit is transmitted to the data bit of the

output lines.

• This depends on control input ABCD.

• When ABCD=0 the upper AND gate is enabled and

other and gate are disabled.

• The d is transmitted only to the y0 output.y0=D

• IF D is low y0 is low,

• If D is high Y0 is High

• Y0 is depends on the value of D

• 1 to 16 DECODER

a decoder is similar to a demultiplexer.

With only one exception there is no data input.

The only inpputs are the control bit ABCD.

The 74154 is the decoder de multiplexer.

The ABCD possibilities 0000 to 1111 ,you will find

that the subscript of the high output always equal the

decimal equivalent. that is called binary to decimal

decoder.

BCD TO DECIMAL DECODER

• BCD is an abbreviaton for binary coded decimal.

• The BCD code expresses the each digit in a decimal

number by its nibble equivalent.

• Decial no 429 to its BCD form as follows

4 2 9

0100 0010 1001

• Bcd to Decimal Decoder:

the above shown figure is called 1 to 10 decoder.

because only 1 of these 10 output lines is high for

instance, when ABCD is 0011,only theY3 AND gate

has all High inputs.

Therefore only y3 output is high.

• If ABCD changes to 1000 only the Y8 NAD gate has all high inputs,so the y8 output is high

• PIN DIAGRAM:

• SEVEN SEGMENT DECODER

ENCODER

• AN encoder converts an active input signal into a

coded output signals.

• The n input lines, only one of which is active.

• Internal logic with in the encoder converts this

active input to a coded binary output with m bits.

• DECIMAL TO BCD ENCODER:

The switches are push button switches like those of a

pocket calculator.

When 3 is pressed the C and D,OR gate have high

inputs. therefore the output is ABCD=0011

5 pressed output=ABCD=0101

9 pressed output ABCD=1001

• Pin out diagram:

the 74147 is a Decimal to BCD Encoder

PARITY GENERATOR AND

CHECKERS

• EVEN PARITY:

it means an n bit input has an even number of 1s

ODD PARITY:

It means a n bit input has an ODD number of 1s.

1111 0000 1111 0011 Even parity

1111 0000 1111 0111 Odd parity

• PARITY CHECKER:

PARITY GENERATOR:

• APPLICATION:

BINARY NUMBER SYSTEM

1. It is a system that uses only the digit 0 and 1

as code.

2. All the other digits thrown away(0 to 9).

3. To represent decimal numbers and letters of

the alphabet with the binary code.

4. Where string of dots and dashes are used to

code all numbers and letters.

DECIMAL ODOMETER:

to understand how to count with binary numbers.

it help to review how an odometer counts with decimal

numbers.(miles indicator of a car or bike).

when a new car odometer start 00000

after 1 km 00001 and so on

At the end of the 9th km the number may be changed to

00010.

RESET AND CARRY

The unit wheels has reset to 0 and sent carry to the tens

wheel.

The other wheels of an odometer also reset carry.

For instance after 999 km the odometer shows

00999

What does the next km do?

The unit wheel reset and carries the ten wheel

reset and carries the hundred wheel reset and carries

and the thousand wheel advance by one to get

01000

BINARY ODOMETER-0000 ,0001,0010

the word bit is the abbreviation for binary digit.

When binary number 4 digit called nibble.

The above shown table shows 16 nibble(0000 to

1111)

A binary number 8 bit known as byte.

Bit-x, Nibble-XXXX, Byte-XXXXXXXX

BINARY TO DECIMAL CONVESION:

• The list of binary numbers from 0000 to 1111.

• This session shows how to convert the binary no

quickly and easily into this decimal equivalent.

Positional notation and weights:

we can express any decimal integer(whole number) in

units, tens, hundred, thousand, and so on.

Decimal number 2945 may be written as

2000+900+40+5

in power of 10 becomes

2945=2(103)+9(102)+4(101) +5(100)

Decimal number is an example of positional notation

ASCII CODE: we need to use some kind of

alphanumeric codes.(one for letter,number and other

symbols).

ASCII stand for American Standarard code for

information interchange.

Pronounced as (ask –ee)

Parity bit:

The ASCII code is used for sending digital data over telephone line.as mentioned in the preceding 1 bit error may occur in transmitted data.

A parity bit is usually transmitted along with the original bits.

The parity checker at the receiving end can test for even or odd parity.

The ASCII code uses the 7 bits, the addition of bits is format is parity bit.

X7X6X5X4 X3X2X1X0

X7 IS PARITY BIT

this is an illegal length of digital equaling

EXCESS -3 CODE

the excess 3 code is an important 4 bit code used with

binary coded decimal number.

To convert an any decimal number into its excess-3 form

,add 3 to each decimal digit and then convert the sum

to BCD number.

For example :12

1+3=4 2+3=5

40100 5 0101

So 0100,0101 in the excess 3 code stands for decimal 12

9’s Complement :-

Calculate 9’s complement by subtracting every digit from

9, is defined as (rn - 1) – N

Calculate 10’s complement by adding 1 to the 9’s

Complement, is defined as [(rn - 1) - N] + 1.

Example :- 728.54

9 9 9 9 9 - 7 2 8 . 5 4 =2 7 1 . 4 5

(9’s Complement of (728.54)

Test your Knowledge

546700 9’s Complement – 999999 - 546700 = 453299

012398 9’s Complement – 999999 - 012398 = 987601

GRAY CODE:

A Gray Code represents numbers using a binary encoding

scheme that groups a sequence of bits so that only one

bit in the group changes from the number before and

after.

It is named for Bell Labs researcher Frank Gray, who

described it in his 1947 patent submittal on Pulse Code

Communication. He did not call it a Gray Code, but

noted there was no name associated with the novel code

and referred to it as a Binary Reflected Code for the

way he determined the groupings and number

representations. When the patent was granted in 1953

others began to refer to the encoding scheme as the

Gray Code.

BCD code is some time referred as 8421 code

Decimal (base 10) Binary (base 2) Binary-Reflected (no base)

0 0000 0000

1 0001 0001

2 0101 0011

3 0011 0010

4 0100 0110

5 0101 0111

6 0110 0101

7 0111 0100

8 1000 1100

9 1001 1101

10 1010 1111

From this table we can obtain the

equivalent gray code of the decimal numbers. There

are several steps which will make you understand

how the codes are formed. (1) In case of gray code

one bit will change from its previous in each steps.

One thing must be kept in mind that the change of

bit always occurs from the right side i.e from L.S.B

towards the M.S.B. At first the first three bits are

constant I,e 000 and the fourth bit changes from 0 to

1. We know that for binary digit possible

combination is 0 and 1, so keeping first three bit

constant the possible combination of 4th bit is over

for decimal 0 and 1 respectively.

For n = 1 bit For n = 2 bit For n = 3bit

Binary Gray Binary Gray Binary Gray

0 0 00 00 000 000

1 1 01 01 001 001

10 11 010 011

11 10 011 010

100 110

101 111

110 101

111 100

ARITHMATIC CIRCUITS

Circuits that can perform binary addition and

subtraction are constructed by combining logic

gates.

These circuits are used in the design of the Arithmetic

and logic unit(ALU).

Binary Addition:

numbers represent physical quantities.

FOUR CASES TO REMEMBER:

computer circuits do not process decimal

numbers, they process binary numbers.

0 + 0 = 0

0 + 1 =1

1 + 0 =1

1 + 1 =10

Subscripts:

The foregoing discussion bring up the idea of

subscripts.

Since we already have discussed I four kinds of

numbers.

Decimal,Binary,Octal,and HExaDecimal.

2 -> Binary

8-> octal

10 -> decimal

16-> HexaDecimal

112 represent Binary

238 stands for octal

4510 for decimal

F416 for Hexadecimal

Larger Binary numbers:

11100 + 11010=110110

BINARY SUBTRACTION:

Let us begin with 4 basic cases of binary sbtraction

0-0=0

1-0=1

1-1=0

10-1=1

Larger numbers:

1101-1010=0011

UNSIGNED BINARY NUMBERS:

All data is either positive or negative.

Forget about the + and – sign ,and consentrate on the

magnitude(absolute value) of numbers.

The smallest 8 bit number is 0000 0000 (00H).

The largest 8 bit number is 1111 1111 (FFH).

So the total range of 8 bit number is 0000 0000 to

11111111 this is equivalent to 0 to 255

With 16 bit numbers

0000 0000 0000 0000 (000H)

1111 1111 1111 1111 (FFFFH)

Which represent the magnitude of all decimal no from 0

to 65535

LIMITS:

1ST GENERATIO OF computer can process

only 8 bit at a time.

Therefore each number being added or

subtracted must be between 0 to 255.

If any magnitude are greater than 255,you

should use 16 bit arithmetic

Which means lower 8 bit first

Higher 8 bit is next.

Lower bytes

1010 1100 + 0111 1111= 1 0010 1011

Upper bytes

0000 1111 + 0011 1000 + 1= 0100 1000

Answer is

0100 1000 0010 1011

Overflow:

In 8 bit arithmetic, addition of two

unsigned numbers whose sum is greater than 255

causes an overflow. A carry into the ninth column.

Most microprocessor have a logic circuit called a

carry flag. this circuits detects a carry into the ninth

column and warm you that the 8 bit answer is

invalid.

Example:175+118=293 answer is greater 255

175= AFH=1010 1111

118=76H=0111 0110

10101 1111 + 0111 0110 =1 0010 0101

SIGN MAGNITUDE NUMBERS:

0 is for + sign

1 is for – sign

The foregoing numbers contain a sign bit followed by

magnitude numbers.

The MSB always represent the sign and the remaining

bits always stand for the magnitude.

Some example:

+7= 0000 0111

-16=1001 0000

+25= 0000 0000 0001 1001

-128= 1000 0000 1000 0000

2’s complement representation:

There is a rather unusual number system that

leads to the simple logic circuits for performing arithmetic

known as 2’s complement representation.

1’s complement:

1010=0101

1110 1100=0001 0011

0011 1111 0000 0110 = 1100 0000 1111 1001

2’s complement:

2’s complement= 1’s complement + 1

Example:

1011=0100(1’s complement)

0100 +1=0101( 2’s complement)

2’s complement Arithmetic

Addition and Subtraction can be visualized in

terms of binary odometer.

When you add a positive number, this is equiva

lent to advancing the odometer

When you add a negative number ,this has the

effect of turning the odometer backward.

Addition:

There are 4 possible cases: both numbers positive

number.

Case 1:

+83 and +16 both numbers are converted as follows

+83= 0101 0011 +16= 0001 0000

How addition appears 0101 0011

0001 0000

-----------------

0110 0011 this is converted into

63H99

Case 2:

Positive and smaller negative.

+125 and -68

+1250111 1101

-68 1011 1100

---------------------------

571 0011 100139H57

Case 3:

Small positive larger negative

+37 and -115 =-78

0010 0101

1000 1101

---------------

1011 0010=B2H78

Case 4:

both negative numbers.

-43 and -78=-121

-43 1101 0101

-78 1011 0010

----------------------

-121 1 1000 01111000 0111=83H-121

Subtraction:

There are 4 possible chance to work .

Case 1:

both number is positive.

Case 2:

larger positive and negative no.

Case 3: positive and larger negative no.

Case 4: both negative no.-43 and -78 -43 1101 0101 -781011 0010

-781011 0010 2’s complement is 0100 1110

-431101 0101

+780100 1110

---------------------------

35 1 0010 0011

Overflow:

Sum to be outside the range -128 to +127.

Case 1:

100 +50= 150

0110 0100

0011 0010

---------------

1001 0110150

Case 2:

-85 and -97 add this numbers

1010 1011

1001 1111

----------------

1 0100 1010 0100 1010

ARITHMATIC BUILDING BLOCKS:

A logic circuit that performs 8 bit

arithmetic on positive and negative numbers.

First we need to cover three basic circuits

that will be used as building blocks.

These building blocks are Half adder ,Full

adder and Controlled Inverter.

Only these work it is only a short step to

step come together. that is how computer is able to

add and subtract binary numbers at any length.

Half Adder:

when we add two binary numbers we start with

the least significant column. This means that we

have to add two bits with the possibility of carry. the

circuit used for this is called half adder.

The output of the ex-or gate is called the SUM.

Output of the AND gate is Carry.

The AND gate produces high output only. when the

both inputs are high the Ex-OR output is high

Full Adder:

We have to use a full adder, a logic circuit

that can be add 3 bit at a time that is caller full

adder.

ABC input arte high this agrees with carry

the output is high. the odd number of high inputs

the Ex-OR gate output is high .

Controlled inverter:

It transmits the 8 bit input to the output.

When invert is low it transmit the 8bit input to the

output.

When invert is high it transmit the 1’s complement .

For example:

a low invert y7 …. Y0 = 0110 1110

But a high invert result in y7 …. Y0 = 1001 0001.

FLIP FLOPS:

Any device or circuit that Hs two stable is said to be bistable.

A toggle switch has two table states. it is either up or down,

depending on the position of the switch.

It is a bistable electronic circuit that has two stable states that

is its output is either 0 or +5 Vdc.

It has memory since its output will remain as set until

something is done to change it.

Basic idea:

Nor gate latch:

NAND GATE:

EDGE TRIGGERED RS FLIPFLOPS:

EDGE TRIGGERED D FLIPFLOP:

D latch is used for temporary storage in electronic

instruments. An even more popular kind of D flip

flop is used in digital computer and systems.

Edge triggered JK flip flop:

This is versatile circuit.

Counters can be used to count the number of PTs and

NTs of clock.

JK MASTER SLAVE FLIP FLOP:

master is positive edge triggerd

Salve is negative edge triggered.

REGISTERS:

• A register is a very important building blocks.

• Store a binary information appearing at an output of

an encoding matrix.

• Register must be accept the input an alphanumeric

keyboard.

• Then present in this data at the input of the

microprocessor.

• The asynchronous receiver transmitter(UART)is

chio used to exchange the data in a microprocessor

system.

• UART is constructed a register some control logic.

TYPES OF REGISTERS:

A register simply a group of flip flop that can be used to

store a binary number.

A register to store a 8 bit binary numbers must have the 8

flip flops.

Binary number can be entered(shifted) in to the register

and possibly shifted out.

A group of flip flop connected to provide either or both of

those function is called shift register.

The bits of the binary number moves from one place to

another in either of two ways .

Methods of shift:

1. The first method of shifting the data 1bit at a time in a serial

fashion beginning with either MSB or LSB. this technique is

called serial shifting.

2. The second method involves shifting all the data bits

simultaneously and is referred to as parallel shifting.

There are two ways to shift a data into a register

Serial and parallel.

Types :

1. Serial in serial out-8 bits

2. Serial in parallel out- 8 bits

3. Parallel in serial out- 8 bit

4. Parallel in parallel out- 4 bits

5. Parallel in parallel out-8 bits

SERIAL IN SERIAL OUT:

SERIAL IN PARALLEL OUT:

PARALLEL IN SERIAL OUT:

PARALLEL IN PARALLE OUT:

MEMORY:

Semiconductor memory

1.Bipolar transistor

2.MOS transistor(IC)

Real memory

1.ROM

2.RAM

Large amount of data are generally stored using

magnetic memory techniques.

Magnetic memory includes recording of digital

information on magnetic tape,HDD,FD.

SEMICONDUCTOR MEMORY:

• It consist of rectangular array of memory cells.

• Fabricated on silicon wafer.

• Stored and charge the 1 bit information.

• Memories are classified as bipolar or metal

oxide semiconductor.

• Complementary MOS-used to construct the

individual memory cells.

• It is greater packing and reduced size and cost.

• Lower power requirements.

Characteristics:

Two categories of memory RAM and ROM

RAM

1. SRAM

Smaller and rapid access type of cached memory.

2.DRAM

Bulk of memory and high speed.

PROM-data stored in Permanently.

EPROM- Erased programmed read only only.

Control:

Chip selected

Chip enable

ROM :

NONvolatile data storage.

MAGNETIC MEMORY:

Magnetic tape is produced by deposition of a

film of magnetic material on a large strip of plastic,

which is then wound on reel.

Magnetic recording:

Magnetic tape:

HDD

FD

OPTICAL MEMORY:

CD ROM

CD R

CDRW

DVD