A byte comprises eight bits. The number shown above is a binary number, which is one byte in length. The rightmost 1st bit, i.e., is known as the Least Significant Bit (LSB) and the leftmost 8th bit, i.e., is called the Most Significant Bit

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A byte comprises eight bits. The number shown
above is a binary number, which is one byte in
length. The rightmost 1st bit, i.e., is
known as the Least Significant Bit (LSB) and the
leftmost 8th bit, i.e., is called the Most
Significant Bit (MSB) of the byte.
2
A decimal number has base 10 (ten), whereas a
binary number has base 2 (two). Similar to the
decimal number system, each position of a binary
number comprised of bit streams has a unique
weight. The weight of each position is calculated
in terms of power over two. For instance, the
weight of bit position 5 i.e., is 16.
3
The octal number system uses 8 as its base and
accommodates digits 0, 1, 2, 3, 4, 5, 6 and 7.
The weight of each position is determined similar
to decimal and binary number systems
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Octal to binary conversion
In this figure the given octal number is (4267)8.
Its binary equivalent is (100010110111)2, which
is a 12 bit binary number.
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Binary to Octal conversion
To convert binary to octal number, simply break
the binary number into a group of three bits,
starting from the least significant bit. Then
convert the 3-bit binary number to its octal
equivalent.
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The hexadecimal number system is derived from
four-bit binary numbers. This number system can
be used to represent the same values as the
decimal and binary number systems. Just like the
decimal number system represents a power of 10,
each hexadecimal number represents a power of 16
(b)
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Similarly, to convert binary to hexadecimal,
break the binary number into a group of four
bits, starting from the least significant bit.
Convert the 4-bit binary number to its
hexadecimal equivalent.
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The BCD number system has ten combinations of
nibble (a group of 4-bit is called a nibble)
corresponding to each decimal number. The
conversion of decimal to BCD or BCD to decimal is
similar to the conversion of hexadecimal to
binary and vice versa. To convert from BCD to
decimal, just reverse the process.
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A typical example of a position detection system
using gray code. A change in adjacent location
only affects one bit for a Gray code pattern,
whereas using a binary code pattern up to four
bits could change, giving rise to wildly
incorrect readings.
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The following fundamental rules are employed in
adding binary numbers. (Also refer
Figure-4.10)   0 0 0 (carry
nil) 0 1 1 (carry nil) 1 0 1
(carry nil) 1 1 10
(carry 1) 1 1 1 11 (carry 1)
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1 1 1
0 0
Negative logic Lower voltage represents 1 and
the higher voltage represents 0
Positive logic Higher voltage represents 1
and the lower voltage represents 0
1 1 1
0 0
0 volts
- 5 volts
Theoretically and numerically these two logic
states are expressed as 1 or 0 and
electrically they are realized as V and or
vice versa, where and essentially are
two distinct voltage levels. If V represents
1, then has to represent 0 or vice
versa.
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The basis of logic circuits are logic functions.
Basic logic functions are OR, AND and NOT. These
logic functions are in fact realized by means of
solid state electronics, which in turn are defined
as Logic Gates. In view of that there are three
logic gates such as OR gate, AND gate and NOT
gate. Logic gates, circulating the fundamental
principle of logic functions, constitute the
functional building blocks in designing the
digital circuits or digital system.
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Using the variables, A, B, C, D, and Y, the truth
table of this gate is shown in the figure.
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The simplest gate is an inverter or NOT gate. It
takes a bit or state as input and produces its
opposite as output. If the input is 0, the
output is 1. If the input is 1 then the
output is 0. NOT gates can be realized using a
transistor.
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Figure (a) describes three input NAND gate and
its truth table. Figure (b) describes a NOR gate
and its truth table. From the figure is it
appropriate to say OR-NOT and AND-NOT rather than
NOT-OR and NOT-AND, respectively.
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The XOR gate output is 1 if any one of the
inputs are 1 the output is 0 when all inputs
are 0 or 1. The notation is used to
describe the operation. It narrates that Y is
exclusively A or B.
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An XOR gate is called a half-adder electronic
circuit. One can see from the rows of the truth
table that the gate adds two input bits.
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A simple type of flip-flop that can retain or
store a single bit is an SR flip-flop. S stands
for set and R stands for reset. The truth
table of an SR flip-flop is given. (b)
illustrates the realization of SR flip-flops
using NAND and NOR gates. (c) is its symbol.
Although the SR flip-flop behaves as a single bit
memory cell but they can be used for sequencing
and triggering applications.
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The clock input given to the SR flip-flop is to
sequence the operation in terms of displaying the
next-time status of the circuit.
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Although SR flip-flops are realized either using
NAND gates or NOR gates, in their basic form the
SR circuit is simply two transistors connected
back to back as shown.
As soon as the terminals S and R are made high
one would saturate (go to active reason) faster
than the other. There would be a race between the
two transistors so connected.
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D
S R
Clock
(a)
(b)
The SR flip-flop can retain a bit at the output Y
in response to setting and resetting of input. In
essence, once a bit is retained it is memorized.
This retaining ability of the flip-flop forms the
basis in designing real memory devices. Indeed,
several flip-flops are put together to store
large set of DNS data. Some of the flip-flops are
very good as far as design of real memory devices
are concerned. D flip-flops are good in this
respect.
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A JK flip-flop is one of the most versatile
flip-flop and frequently used in digital control
systems. It has two data inputs like an SR
flip-flop and a clock input. The two data inputs
are called the J and K terminals. It does not
inherit the race problem.
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A large memory block is composed of a set of many
small units called registers.
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Multiplexers are sequential logical devices used
for various applications such as data selector,
data multiplexing, and even for the genera-tion
of Boolean functions.
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The truth table of the expression ABC.D and
that of the expression (AB) B.C A.D C.D
are same
A truth table could have more than one Boolean
expression. Boolean expression contains redundant
terms. So minimisation is necessary.
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(a) Box number of K-map corresponding to the two
variable truth table (b) Box number of K-map
corresponding to the three variable truth table
(c) Box number of K-map corresponding to the
four variable truth table (d) Sequence number of
K-map corresponding to the two variable truth
table (e) Sequence number of K-map
corresponding to the three variable truth table
(f) Sequence number of K-map corresponding to
the four variable truth table.
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Figure illustrates how K-map has been derived
from the given two-, three-, and four-variable
truth tables.
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One of the step in the K-map procedure is to form
groups of adjacent 1s. Groups of 2, groups of
4, and groups of 8 could be formed. These groups
are called pairs, quads, and octets,
respectively, as shown in the figure. The
presence of pairs, quads, and octets purely
depends on the truth table at hand. The K-map
uses some rules as far as grouping of adjacent
boxes containing 1s is concerned.
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An example of K-map based minimisation process.
Consider a K-map, as shown in figure, which has
been derived from a given truth table (also
shown). There are three groups, one singular
group (sequence number 4), a pair, and a quad.
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Realisation of logic expression
32
The NOT gate and SR flip-flop illustrated earlier
uses bipolar transistors. Such gates and logic
circuits can also be designed even using diodes.
Above figure shows the realization of a two-input
AND gates, and a two-input OR gate using diodes.
(c) shows CMOS logic of a NOT gate.
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Semiconductor-based memory cells that are
fabricated on a single piece of material (wafer)
can accommodate a large number of data bytes.
These are called memory ICs. Figure shows a
schematic illustration of a semiconductor memory.
The semiconductor memory has 16 locations,
Location-0 to Location-15. Each location is a
byte long containing, eight bits. The bit
positions of each byte are defined by B0 through
B7. B0 is called the Least Significant Bit (LSB)
and B7 is called the Most Significant Bit (MSB).
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The above figure shows a schematic pin out
diagram of a typical RAM chip. The chip is
selected prior to a reads and writes operation.
If it is a read operation the Read/Write signal
is
low and if it is a write operation the
Read/Write signal is high. The chip must have
an address lines in order to address the location
in which the data is to be stored or from which
the stored data is to be retrieved. The number of
address lines solely depends on the number of
locations the chip has.