WEBVTT
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In this lecture you'll learn various techniques
for creating combinational logic circuits
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that implement a particular functional specification.
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A functional specification is part of the
static discipline we use to create the combinational
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logic abstraction of a circuit.
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One approach is to use natural language to
describe the operation of a device.
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This approach has its pros and cons.
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In its favor, natural language can convey
complicated concepts in surprisingly compact
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form and it is a notation that most of us
know how to read and understand.
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But, unless the words are very carefully crafted,
there may be ambiguities introduced by words
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with multiple interpretations or by lack of
completeness since it's not always obvious
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whether all eventualities have been dealt
with.
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There are good alternatives that address the
shortcomings mentioned above.
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Truth tables are a straightforward tabular
representation that specifies the values of
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the outputs for each possible combination
of the digital inputs.
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If a device has N digital inputs, its truth
table will have 2^N rows.
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In the example shown here, the device has
3 inputs, each of which can have the value
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0 or the value 1.
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There are 2*2*2 = 2^3 = 8 combinations of
the three input values, so there are 8 rows
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in the truth table.
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It's straightforward to systematically enumerate
the 8 combinations, which makes it easy to
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ensure that no combination is omitted when
building the specification.
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And since the output values are specified
explicitly, there isn't much room for misinterpreting
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the desired functionality!
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Truth tables are an excellent choice for devices
with small numbers of inputs and outputs.
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Sadly, they aren't really practical when the
devices have many inputs.
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If, for example, we were describing the functionality
of a circuit to add two 32-bit numbers, there
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would be 64 inputs altogether and the truth
table would need 2^64 rows.
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Hmm, not sure how practical that is!
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If we entered the correct output value for
a row once per second, it would take 584 billion
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years to fill in the table!
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Another alternative specification is to use
Boolean equations to describe how to compute
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the output values from the input values using
Boolean algebra.
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The operations we use are the logical operations
AND, OR, and XOR, each of which takes two
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Boolean operands, and NOT which takes a single
Boolean operand.
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Using the truth tables that describe these
logical operations, it's straightforward to
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compute an output value from a particular
combination of input values using the sequence
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of operations laid out in the equation.
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Let me say a quick word about the notation
used for Boolean equations.
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Input values are represented by the name of
the input, in this example one of "A", "B",
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or "C".
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The digital input value 0 is equivalent to
the Boolean value FALSE and the digital input
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value 1 is equivalent to the Boolean value
TRUE.
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The Boolean operation NOT is indicated by
a horizontal line drawn above a Boolean expression.
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In this example, the first symbol following
the equal sign is a "C" with line above it,
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indicating that the value of C should be inverted
before it's used in evaluating the rest of
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the expression.
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The Boolean operation AND is represented by
the multiplication operation using standard
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mathematical notation.
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Sometimes we'll use an explicit multiplication
operator - usually written as a dot between
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two Boolean expressions - as shown in the
first term of the example equation.
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Sometimes the AND operator is implicit as
shown in the remaining three terms of the
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example equation.
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The Boolean operation OR is represented by
the addition operation, always shown as a
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"+" sign.
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Boolean equations are useful when the device
has many inputs.
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And, as we'll see, it's easy to convert a
Boolean equation into a circuit schematic.
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Truth tables and Boolean equations are interchangeable.
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If we have a Boolean equation for each output,
we can fill in the output columns for a row
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of the truth table by evaluating the Boolean
equations using the particular combination
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of input values for that row.
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For example, to determine the value for Y
in the first row of the truth table, we'd
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substitute the Boolean value FALSE for the
symbols A, B, and C in the equation and then
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use Boolean algebra to compute the result.
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We can go the other way too.
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We can always convert a truth table into a
particular form of Boolean equation called
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a sum-of-products.
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Let's see how…
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Start by looking at the truth table and answering
the question "When does Y have the value 1?"
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Or, in the language of Boolean algebra: "When
is Y TRUE?".
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Well, Y is TRUE when the inputs correspond
to row 2 of the truth table, OR to row 4,
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OR to rows 7 OR 8.
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Altogether there are 4 combinations of inputs
for which Y is TRUE.
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The corresponding Boolean equation thus is
the OR for four terms, where each term is
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a boolean expression which evaluates to TRUE
for a particular combination of inputs.
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Row 2 of the truth table corresponds to C=0,
B=0, and A=1.
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The corresponding Boolean expression is (NOT
C) AND (NOT B)
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AND A, an expression that evaluates to TRUE
if and only if C is 0, B is 0, and A is 1.
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The boolean expression corresponding to row
4 is (NOT C) AND B AND A.
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And so on for rows 7 and 8.
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The approach will always give us an expression
in the form of a sum-of-products.
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"Sum" refers to the OR operations and "products"
refers to the groups of AND operations.
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In this example we have the sum of four product
terms.
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Our next step is to use the Boolean expression
as a recipe for constructing a circuit implementation
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using combinational logic gates.
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As circuit designers we'll be working with
a library of combinational logic gates, which
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either is given to us by the integrated circuit
manufacturer, or which we've designed ourselves
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as CMOS gates using NFET and PFET switches.
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One of the simplest gates is the inverter,
which has the schematic symbol shown here.
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The small circle on the output wire indicates
an inversion, a common convention used in
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schematics.
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We can see from its truth table that the inverter
implements the Boolean NOT function.
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The AND gate outputs 1 if and only if the
A input is 1 *and* the B input is 1, hence
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the name "AND".
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The library will usually include AND gates
with 3 inputs, 4 inputs, etc., which produce
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a 1 output if and only if all of their inputs
are 1.
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The OR gate outputs 1 if the A input is 1
*or* if the B input is 1, hence the name "OR".
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Again, the library will usually include OR
gates with 3 inputs, 4 inputs, etc., which
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produce a 1 output when at least one of their
inputs is 1.
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These are the standard schematic symbols for
AND and OR gates.
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Note that the AND symbol is straight on the
input side, while the OR symbol is curved.
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With a little practice, you'll find it easy
to remember which schematic symbols are which.
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Now let's use these building blocks to build
a circuit that implements a sum-of-products
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equation.
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The structure of the circuit exactly follows
the structure of the Boolean equation.
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We use inverters to perform the necessary
Boolean NOT operations.
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In a sum-of-products equation the inverters
are operating on particular input values,
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in this case A, B and C.
To keep the schematic easy to read we've used
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a separate inverter for each of the four NOT
operations in the Boolean equation, but in
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real life we might invert the C input once
to produce a NOT-C signal, then use that signal
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whenever a NOT-C value is needed.
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Each of the four product terms is built using
a 3-input AND gate.
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And the product terms are ORed together using
a 4-input OR gate.
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The final circuit has a layer of inverters,
a layer of AND gates and final OR gate.
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In the next section we'll talk about how to
build AND or OR gates with many inputs from
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library components with fewer inputs.
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The propagation delay for a sum-of-products
circuit looks pretty short: the longest path
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from inputs to outputs includes an inverter,
an AND gate and an OR gate.
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Can we really implement any Boolean equation
in a circuit with a tPD of three gate delays?
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Actually not, since building ANDs and ORs
with many inputs will require additional layers
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of components, which will increase the propagation
delay.
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We'll learn about this in the next section.
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The good news is that we now have straightforward
techniques for converting a truth table to
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its corresponding sum-of-products Boolean
equation, and for building a circuit that
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implements that equation.