Boolean Algebra and Digital Circuits
Part 3: Logic Gates and
Combinatorial Circuits
Logic Gates
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a gate implements a simple boolean function
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such as AND, OR or NOT
constructed using a few transistors
basic building block of digital design
x
y
x
AND
NOT
x.y
x̅
x
y
OR
x+y
x
y
XOR
x⊕y
(the circle in the NOT gate represents complement)
Universal Gates
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recall that all other gates can be constructed using
NAND gates alone, or NOR gates alone
•
x
y
so these are our universal gates
NAND
x̅.̅y̅
x
y
NOR
x̅+̅y̅
Multiple Inputs and Outputs
x
y
z
OR
x
y
z
AND
x
y
AND
x+y+z
x.y.z
Q = x.y
Q̅ = x̅.̅y̅
three input OR gate
three input AND gate
two output AND gate
adapted from figs 3.6-3.8, Computer Organization & Architecture
Digital Circuits
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we can build boolean functions using gates:
F(x, y, z) = y + x̅.z
y
x
z
and sum of products can make construction simple:
F = x̅yz + xyz + xyz̅
x
y
•
F
z
…and for efficiency we could always simplify further by using kmaps before
construction
F
Subtle Uses of Logic Gates
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the functionality we see in high-level languages is built upon lower level circuits
and sometimes this can be exploited to improve efficiency and performance:
using XOR to swap 2 variables' values:
x = x XOR y
y = x XOR y
x = x XOR y
•
using AND to mask bits:
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for a byte z, to find if its 4th bit is set use z AND 000010002
in general, set the bits you want to keep to 1, set others to 0, then
AND with your byte (/word)
Integrated Circuits
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a modular approach
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however gates aren't usually manufactured individually
•
we use one or many of these ICs to create a boolean
function
we've seen that individual gates can be combined to
perform more complex functions
but instead a collection of them are etched onto an
integrated circuit
IC Small Scale Integration (SSI) Example
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here's an integrated chip with just four NAND gates
(note that modern chips have VLSI or ULSI)
1
2
3
4
5
6
7
figure 3.10, Computer Organization & Architecture
Implementing a Function on the Simple IC
F
F(x,y) = x+y
converted to just NANDs
= (̅x̿.̿x̿)̅.̅(̅y̿.̿y̿)̅
1
x
2
3
4
y
5
6
7
Combinatorial Circuits
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the examples we've seen so far are combinatorial circuits
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so all input values must arrive at the same time for them to work
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the output is based entirely upon the input values
they produce an output at the (notional) instant when the input
values are specified
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they can't 'remember' the inputs if some arrive later than
others
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we'll see later what to do if the inputs arrive sequentially
for now we'll look at some useful combinatorial circuits
Half-Adder
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we want to be able to add binary numbers in the Arithmetic Logic Unit
(ALU)
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let's start with a half-adder which can add two bits together:
•
so could we simply combine a set of half-adders to perform addition on
words of any size we choose?
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no, because we need to account for the carry in
table 3.9, figure 3.11, Computer Organization & Architecture
Full-Adder
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a full-adder takes account of the carry in...
...and we see how the half-adder got it name
figure 3.12, Computer Organization & Architecture
Ripple-Carry Adder
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the full-adder can be combined to create a ripple-carryadder which can add words together
figure 3.13, Computer Organization & Architecture
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note that in reality we now have far more efficient adders
(40%-90% faster!) through the use of optimisations based
upon parallelisation and handling of the carry bit
Decoder
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uses the combination of the input values to select one specific output line
so, with two inputs x and y, there are 4 possible combinations:
x.y, x.y̅, x̅.y, x̅.y̅
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and we create an output line for each:
figure 3.14, Computer Organization & Architecture
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this is useful for building circuits which can determine which memory
address to access, or which function to perform based upon the input
Multiplexer
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selects a single output from several inputs
uses control lines to determine which input to route
like the inputs & output of a home theatre receiver
figure 3.14, Computer Organization & Architecture
Simple 2-bit ALU
control codes:
00 = A + B
01 = NOT A
10 = A OR B
11 = A AND B
figure 3.15, Computer Organization & Architecture
Simple 2-bit ALU
01
10
1
0
OR function enabled
(code 10)
with inputs of 01 & 10
and an output of 11
figure 3.15, Computer Organization & Architecture
To Conclude
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we can build complex circuits in a modular fashion
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either by combining individual gates
or by combining integrated circuits
but so far these circuits require that the inputs arrive and
are handled at the same time
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they are memoryless
so next lecture we'll see how to account for memory
and timing needs
References & Further Learning
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outline primarily based upon:
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Chapter 3, Computer Organization & Architecture (3rd Edition),
Null & Lobur
other material used for reference:
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Chapter 3, Fundamentals of Computer Architecture, Burrell
suggested learning activities:
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make sure that you understand how the full adder works
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perhaps trace the flow for a 4-bit adder
try to follow instructions through the simple 2-bit ALU