functions of combinational logic · •an encoder is a combinational logic ckt that essentially...
TRANSCRIPT
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FUNCTIONS OF
COMBINATIONAL LOGIC
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Agenda
•Adders
•Comparators
•Decoders
•Encoders
•Multiplexers
•Demultiplexers
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Adders
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Basic Adders
•Adders are important in
•computers
•other types of digital systems in
which numerical data are processed
•We must know about adders.
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The Half-Adder
• Basic rule for binary addition.
• The operations are performed by a logic
called a half-adder.
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The Half-Adder
• The half-adder accepts two
binary digits on its inputs and
produces two binary digits on its
outputs, a sum bit and a carry bit.
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The Full-Adder • The full-adder accepts two
input bits and an input carry
and generates a sum output
and an output carry.
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Full-Adder Logic
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The Full-Adder
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Parallel Binary Adders • Two or more full adders are connected to form
parallel binary adders.
• To add two binary numbers, a full-adder is required
for each bit in the numbers.
• So, for 2-bit numbers, two adders are needed.
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Parallel Binary Adders • The carry output of each adder is
connected to the carry input of the next
higher-order adder.
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Four-Bit Parallel Adders • A group of 4 bits is called a nibble. A
basic 4-bit parallel adder is implemented
with four full-adder stages as shown.
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Four-Bit Parallel Adders
The carry output of each adder
is connected to the carry input
of the next higher-order adder
as indicated. These are called
internal carries.
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Ripple Carry Adders • Ripple carry adder
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Look-Ahead Carry Adders
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Look-Ahead Carry Adders
• Carry generation occurs
when an output carry is
produced (generated)
internally by the full-adder.
• A carry is generated only
when both input bits are
1s.
• The generated carry, Cg,
is expressed as the AND
function of the two input
bits, A and B.
Cg=AB
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Look-Ahead Carry Adders • Carry propagation occurs when the input carry is rippled to become the output carry.
• An input carry may be propagated by the full-adder when either or both of the input bits are 1s.
• The propagated carry, Cp, is expressed as the OR function of the two input bits.
Cp=A+B
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Confuse? Check this out…
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Look-Ahead Carry Adders • The output carry (Cout) of a full-adder can be expressed in
terms of both:
• the generated carry (Cg), and
• the propagated carry (Cp).
• The relationship of the carries is expressed as:
Cout = Cg + Cp Cin
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Look-Ahead Carry Adders
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Look-Ahead Carry Adders
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Look-Ahead Carry Adders
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Comparators
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Comparators
•To compare the magnitude of two
binary quantities to determine the
relationship of those quantities.
• The simplest form a comparator
determines whether two numbers are
equal.
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• XOR gate can be used as a 2-bit comparator.
• To compare binary numbers containing two bits each:
Equality
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Inequality • Many IC comparators provide additional
outputs that indicate which of the two
binary numbers being compared is the
larger.
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Inequality
• To determine an inequality
of binary numbers A and B,
you first examine the
highest-order bit in each
number:
• If A3=1 and B3=0 number A is
greater than number B
• If A3=0 and B3=1 number A is
less than number B
• If A3=B3 you must examine
the next lower bit position for
an equality
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Decoders
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Decoders • A decoder detects the presence of a
specified combination of bits (code) on
its inputs and indicates the presence of
that code by a specified output level.
• In its general form, a decoder has n input
lines to handle n bits and forms one to 2n
output lines to indicate the presence of one
or more n-bit combinations.
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The Basic Binary Decoder • Suppose we need to determine when a
binary 1001 occurs on the inputs of a digital.
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The 4-Bit Decoder • In order to decode all possible combinations of four bits, 16 decoding gates are required (24=16).
• This type of decoder is commonly called either: • A 4-line-to-16-line decoder, or
• A 1-of-16 decoder
• Decoding functions and truth table for a 4-line-to-16-line decoder with active-LOW outputs see the next slide.
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The 4-Bit Decoder
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The 4-Bit Decoder
74HC154: 1-of-16 decoder
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The BCD-to-Decimal Decoder
• The BCD-to-decimal converts each BCD code into one of ten possible decimal digit indications.
• Called 4-line-to-10-line decoder or 1-of-10 decoder
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The BCD-to-Decimal Decoder
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The BCD-to-7-Segment Decoder
• The BCD-to-7-segment
decoder accepts the BCD
code on its inputs and
provides outputs to drive
7-segment display devices
to produce a decimal
readout.
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The BCD-to-7-Segment Decoder (The Application)
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Encoders
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Encoders • An encoder is a combinational logic ckt that
essentially performs a “reverse” decoder
function.
• An encoder accepts an active level on one
of its inputs representing a digit, such as a
decimal or octal digit, and converts it to a
coded output such as BCD or binary.
• Encoders can also be devised to encode
various symbols and alphabetic characters.
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The Decimal-to-BCD Encoder
• It has 10 inputs and 4
outputs
corresponding to the
BCD code.
• A3 = 8+9
• A2 = 4+5+6+7
• A1 = 2+3+6+7
• A0 = 1+3+5+7+9
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The Decimal-to-BCD Encoder
NOTE: A 0-digit input is not needed because
the BCD outputs are all LOW when there are
no HIGH input.
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The Decimal-to-BCD Encoder (The Application)
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Code Converters
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Code Converters • Binary-to-gray & gray-to-binary conversion
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Multiplexers
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Multiplexers (Data Selectors)
• A MUX is a device that allows digital information
from several sources to be routed onto a single
line for data transmission over that line to a
common destination.
• The basic MUX has several data-input lines and a
single output line.
• It also has data-select inputs, which permit digital
data on any one of the inputs to be switched to
the output line.
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Multiplexers (Data Selectors)
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Multiplexers (Data Selectors)
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Multiplexers (Data Selectors)
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Demultiplexers
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Demultiplexers • A DEMUX basically
reverses the MUX
function.
• It takes digital
information from one line
and distributes it to a
given number of output
lines.
• It also known as data
distributor.
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Demultiplexers