Download The Digital Logic (level-0)

CE 454
Computer Architecture
Lecture 5
Ahmed Ezzat
The Digital Logic,
Ch-3.1, 3.2
1
Outline
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Transistors and Logic Gates
Boolean Algebra
Implementation of Boolean Algebra
Circuit Equivalence
Integrated Circuits
Combinational Circuits
Arithmetic Circuits
Clocks
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Transistors and Logic Gates
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Border of Computer Science and Electrical Engineering
Digital circuit has 2 logical values: (0 – 2.5) volts and (3.5
– 5) volts. Values outside these ranges are not allowed.
Gates are the devices that can compute various functions
for these two-values signals. Gates are the basis of
digital computers.
The heart of the Gate logic is the transistor and its use as
a binary switch!
Details of how to build gates (device level) is not in our
scope – pure Electrical Engineering.
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Transistors and Logic Gates
Transistor Types:
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Bipolar Transistor:
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
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MOSFET (Metal Oxide Semiconductor Field Effect Transistors):
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TTL (Transistor-Transistor Logic)
ECL (Emitter-Coupled Logic)
PMOS: p-channel (switch is ON when input is low)
NMOS: n-channel (switch is ON when input is high)
CMOS: Complementary Metal Oxide Semiconductor
Bipolar is faster than CMOS. CMOS is much more dense and uses
less power than Bipolar. Functionality is the same. Most modern
CPUs (logic functions) and Memories (storage elements) use CMOS
technology
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Transistors and Logic Gates
Drain
Gate
Source
Switch
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Transistors and Logic Gates
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Transistors and Logic Gates
XOR Logic Gate:
Input A
Output
Input B
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Input
A
Input
B
Output
0
0
0
0
1
1
1
0
1
1
1
0
Logical function “Exclusive OR”
AB + AB alternatively A’B + AB’
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Transistors and Logic Gates
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Transistors can be connected together with wires, and create
logic functions – these functions are called logic gates
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In modern computer chips, the transistors and wires are very,
very, very small...
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Boolean Algebra – George Boole
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New type of algebra is needed to describe these binary
devices/Gates – Boolean Algebra! Boolean algebra can take on
variables and functions that can have only binary values.
Functions in boolean algebra can have one or more input variables
and yield a result that only depends on these inputs!
F(), can be defined as f(A) = 1 if A = 0, and f(A) = 0 if A = 1 –- this
is the “NOT” function
A boolean function of n variables has only 2n possible
combinations on inputs, as a result it can be described by a table
with 2n rows, each row stating the outcome for specific combination
of the inputs –- called a truth table.
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Boolean Algebra
M = f(A, B, C):
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M = ABC + ABC + ABC + ABC
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Implementation of Boolean Algebra
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A boolean function can be implemented by logic gates in more
than one way.
A Generalized Procedure for Generating a Digital Logic Design
from a Truth Table:
1.
2.
3.
4.
5.
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Write down the truth table for the function
Provide inverters to generate the complement of each input
Draw an AND gate for each term with a 1 in the result column
Wire the AND gates to the appropriate inputs
Feed the output of all the AND gates into one OR logic gate.
The above model uses AND, OR and NOT logic gates, however,
we can also use NANDs and NORs logic gates as well
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Implementation of Boolean Algebra
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Implementation of Boolean Algebra
Switching Expression  Digital Logic Circuit:
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Example: M = xy + z’
x
y
M
z
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Circuit Equivalence
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Typically start with
Boolean function,
apply laws of
Boolean algebra to
find simpler but
equivalent one.
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Circuit Equivalence
Identity Table for Boolean Algebra:
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Each law has 2 forms
that are dual of each
other: by interchanging
AND and OR, and 0
and 1, either form can
be produced from the
other.
Except DeMorgan’s
law, the absorption
law, and the AND form
of the distributive law,
the result is intuitive!
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Circuit Equivalence
Alternative Boolean Algebra Symbols for the same Gates:
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Circuit Equivalence
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Circuit Equivalence
Same Gate can Compute Different Functions (conventions used):
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+ve logic is an AND function, and –ve logic is an OR function.
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Circuit Equivalence
Working with Switching Algebra
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F(A,B,C) = A’ + B’ + A’BC = A’ + B’ = (AB)’
F(X,Y,Z,W) = XY + W’XYZ + X’Y = XY + X’Y = Y
Input A
Output
Input B
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Logic function = ((A + B’)’ B’)’ … using DeMorgan’s law:
= (A + B’) + B = A + 1 = 1
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Integrated Circuits
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Gates are not manufactured or sold individually, but in units called
ICs or chips.
IC classes:
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SSI: Small Scale Integrated Circuit – 1 to 10 gates
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MSI: Medium Scale Integrated Circuit – 10 - 100 gates
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LSI: Large Scale Integrated Circuit – 100 – 100,000 gates
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VLSI: Very Large Scale Integrated Circuit – > 100,000 gates
ICs introduce delay (1 – 10 nsec) due to gate delay and switching
time delay.
Current state-of-the art is close to 100,000,000 transistor on a chip.
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Combinational Circuits
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Digital logic that is designed so that the outputs are dependent only
upon the current set of inputs is called COMBINATIONAL LOGIC –
this is in contrast to SEQUENTIONAL LOGIC where output is a
function of input and the current state (i.e., circuits containing
memory elements).
In the following we will cover few frequently-used combinational
circuits:
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Multiplexers
Decoders
Comparators
Programmable Logic Arrays (PLA)
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Combinational Circuits
Multiplexers:
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Eight data inputs, and
three control inputs.
The control lines
(A,B,C) select which of
the 8 input lines to be
“gated” out.
Each AND is enabled
by a different
combination of the
control lines.
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Combinational Circuits
Decoders:
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The set of n-inputs
are decoded to
select exactly one
of the 2n output
lines.
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Combinational Circuits
Comparators:
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Compares 2 input words,
and produces 1 iff they
are equal, otherwise it
produces 0.
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Combinational Circuits
PLA:
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This PLA consists of: 12 inputlines and their inverse (24inputs), 50 AND gates, each can
have any subset of the 24 inputs
(supplied by the user), and 6 OR
gates as output lines.
Total # pins = 12 + 6 + power,
ground = 20 pins.
With appropriate internal
connections, you can use PLA
to implement multiple separate
functions.
Custom PLA are more common
and cheaper.
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Arithmetic Circuits
Shifters:
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8-input, 8-output, 1control line (C) to
determine the direction
of shift (left/right).
Don’t deal with
overflow.
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Arithmetic Circuits
Half Adders (Integers):
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1-bit integer addition
logic, with 2-bits input
and 2-bits as out (sum,
carry).
Not suitable by-itself as
it does not propagate
carry as input to the
next bit!
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Arithmetic Circuits
Full Adders (Integers):
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1-bit integer addition logic,
with 3-bits input (including
carry-bit from previous
stage) and 2-bits as out
(sum, carry).
Full adder is built out of 2
half adders.
This type of adder is called
ripple carry adder.
Addition do not complete
until the carry has rippled
to the rightmost bit!
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Arithmetic Circuits
Full Adders (Integers):
A3
Carry
Full
Adder
Sum3
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A2
B3
Carry
B2
Full
Adder
Sum2
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A1
Carry
A0
B1
Full
Adder
Sum1
Carry
B0
Full
Adder
Sum0
Carry
“0”
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Arithmetic Circuits
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Full Adders (Integers):
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Consider breaking a 32-bit adder up into a 16-bit lower half and
two upper 16-bit halves.
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The circuit consists of three 16-bit adders.
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Let the lower half start as usual, feed 0 in one of the two upper
halves, and feed 1 in the other.
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After 16-bit addition times, it will be known what the carry is into
the upper half is, so select the correct upper half immediately.
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This reduces the addition time by factor of 2!
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Such an adder is called carry select adder.
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This trick can then be repeated to build each 16-bit adder out of
replicated 8-bit adder, etc.
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Arithmetic Circuits
Arithmetic Logic Unit (ALU):
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ALU is the math engine for
modern digital computers.
Performs 4 simple functions:
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A AND B // bitwise AND
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A OR B // bitwise OR
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NOT B // bitwise NOT

A + B // addition operator
Operates on single bits “A”
and “B”
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Arithmetic Circuits
ALU Block Diagram:
Inv A
A
Enable A
B
Enable B
Carry In
A
B
Logic Unit
(AND, OR,
NOT)
Output
A
B
Cntl 1
Cntl 0
32
2 to 4
Decoder
En
En
En
En
OR
AND
NOT
ADD
Sum
Full Adder
Carry Out
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Arithmetic Circuits
Bit Slice ALU:
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Four Functions (AND, OR, NOT B, ADDITION) for single bits Ai and Bi
Inputs
–
–
–
–
–
–
–
–
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Inverse of Ai
The single bit Ai
Enable the A input
The single bit Bi
Enable the B input
Control signal 0
Control signal 1
Carry-in signal (usually the carry out signal from the previous state)
Outputs
–
–
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INV A
A
ENA
B
ENB
FO
F1
Carry In
Output
Result of ALU computation
Carry Out Carry out signal from full adder circuit
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Arithmetic Circuits
Control Signals F0 and F1:
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What do the control signals
F0 and F1 do?
Both signals go into the 2 to
4 decoder
Between the two signals,
one of the four ALU
functions is selected to be
the ALU’s output
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F0
F1
Output
Function
0
0
A AND B
0
1
A OR B
1
0
NOT B
1
1
A + B
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Arithmetic Circuits
How would the ALU subtract two integers?
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A–B
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Think about the two’s complement procedure
1.
2.
3.
4.
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NOT B
(take the complement of B)
NOT B + 1 (add one to the result of step 1) 2’s complement of B
A + (NOT B + 1) (add A to the result of step 2)
Ignore any carry out signals
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Arithmetic Circuits
How would the ALU multiply or divide two integers?
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How about a shifter?
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Multiplication and division can be thought of as “shift and add” or
“shift and subtract” procedures
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Clocks
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Just about like everything else in this world, many digital
logic designs “run on the clock”
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Digital circuit designs that use a clock are called
synchronous circuits
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The clock synchronizes when things happen in the digital
circuit
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Clocks
Clock Signals:
 Clocks send out a series of pulses, where the signal
alternates between high and low
 Ideally, a clock signal will look like a square wave
Rising Clock Edge
Falling Clock Edge
High
Low
One Clock Cycle
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Clocks
Synchronous Digital Circuits:
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Synchronous digital circuits operation can be thought of as a series
of discrete events
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The clock is used to trigger these discrete events
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Clock “triggers” can be on:
– Rising clock edge
– Falling clock edge
– Clock low
– Clock high
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Clocks
Asymmetric Clock:
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Introducing delay can
generate asymmetric clock
In (b) events can happen
when C1 or when C2 is high
for example
If more intervals are
needed, more clocks can be
provided, or the state of the
clocks can overlap partially
which produces 4 distinct
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 
intervals: (C1 AND C2), (C1
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AND C2), (C1 AND C2),
and (C1 AND C2).
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Clocks
How Might You Use a Clock with ALU to Subtract 2 Integers:
1.
Cycle 1:
•
•
•
–
2.
Cycle 2:
•
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Get the value of B from a register and onto the input lines of the ALU
Enable B
Select the NOT B function
(store the intermediate results back into a register)
–
Get the value of B from the intermediate results register and place on
the input lines of the ALU
Enable B
Select the increment function (add with first stage’s carry in = 1)
(store the intermediate results back into a register)
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Clocks
How Might You Use a Clock with ALU to Subtract 2 Integers:
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Cycle 3:
•
•
•
•
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Get the values of A and (NOT B + 1) from the appropriate registers
and onto the input lines of the ALU
Enable both A and B
Select the ADD A + B function
Store the result in the appropriate register
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Clocks
How else Might You Use the ALU to Subtract 2 Integers:
1.
Cycle 1:
•
•
•
•
•
•
•
–
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Get the value of “B” from a register and place it onto the input lines
of the ALU (make it the “A” value)
Enable A
Enable INV A
Note: You now have “the complement of A” going into the ALU
Get the value of “A” from a register and place it onto the input lines
of the ALU (make it the “B” value)
Configure the first stage of the ALU to have “Carry In” signal set to
“1” (equivalent of adding 1 or incrementing)
Add A and B inputs
(store the results into a register)
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