Chapter 2 Combinatorial Logic Circuits

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Chapter 2Combinatorial Logic Circuits
In this chapter we will discuss logic gates which
is the most primitive logic elements in digital
circuits. In a combinatorial logic circuit the
output at time t depends only on the input at
time t.
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Digital Design

• The first step in the digital design process is
  to create a formal description of the
  requirements of the problem. This formal
  description is called a behavioral description
  because it defines precisely what the eventual
  circuit is supposed to do. The objective of a
  behavioral description is to provide a formal
  specification that satisfies the following
• complete All possible outputs are accounted
  for. That is, there is no input sequence for
  which we have not also identified the
  corresponding output.
• unambiguous Each input sequence has a unique
  corresponding output sequence.

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Behavioral Description

• A behavioral description specifies what a circuit
  does, but not how. Behavioral descriptions ALWAYS
  consist of two parts
• An entity definition that identifies the signal
  lines that make up the inputs and outputs and the
  ports that interface the device with those signal
  lines. A signal line is any connection that can
  transmit one bit at a time. This will be
  discussed later in Chapter 3.
• A functional specification that specifies how
  the outputs are determined by the inputs. This
  information can be conveyed by a table,
  algebraically, or algorithmically. Initially the
  functional specification will be examined using
  function tables (truth-tables are a special case)
  and algebraic expressions (including Boolean
  expressions).

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Function Tables and Logic Operations

• A function table or truth table is a
  listing of the output expected for every possible
  assignment of values to the inputs that is, for
  every possible binary input sequence.
• The three basic logical operations in the
  order of precedence are
• NOT,
• AND,
• OR.
• The truth tables of these logical
  operations are discussed in the next slide.

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Truth Tables for the basic logical operations
The truth tables and the logic diagrams for the
binary operations are
AND OR NOT
A B A.B A B A
0 0 0 0 1
0 1 0 1 1
1 0 0 1 0
1 1 1 1 0
See Figure 2-1 on page 32.
Note NOT gates are often referred to as
inverters.
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Equivalency
When are two boolean functions equal? When are
two logic circuit diagrams the same? Two boolean
functions are equal when to every instance of the
input sequence they give the same output.
Likewise two logic circuit diagrams are the same
if to every instance of the same input, they give
the same output.
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Boolean Algebra
This is an algebra dealing with binary variables
and logic operations. The alphabet on which a
Boolean algebra is defined can have two or more
elements. The three basic operations of this
algebra are NOT, AND, OR. This algebra is useful
in simplifying statement forms of complex logic
circuits by using boolean identities given in
Table 2-3 on page 35 of the text. As a result of
which the complex expressions/forms are
transformed into simpler ones which evidently
makes our design more parsimonious. The
statement forms are referred to as multiple
output boolean functions. A multiple output
boolean function is a mapping from from each of
the possible combinations of values 0 and 1 on
the function variables to combinations of 0 and 1
on the function outputs.
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Truth Tables of functions with n-input variables
A function using two input variables (binary
functions) has 22 possible input sequences as
shown in Slide 5 of this chapter. A function
using n input variables has 2n possible input
sequences. The table below shows the input
sequence of a function using three input
variables.
a b c F(a, b, c)
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
Note There are 23 possible input sequences
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Duality Principle

• By the dual of a boolean expression/form, we mean
  expression obtained by substituting AND for OR,
  OR for AND, 0 for 1, and 1 for 0.
• The dual of A B is A . B.
• The dual of A . B is A B.
• The dual of A is A.
• The dual of A 0 is A . 1.
• Duality is often used in finding the complement
  of a complex boolean expressions.

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Complementation using Duality
The following steps are used in finding the
complement of a boolean expression, F. Step 1
Find the dual of F. Step 2 Complement each
literal after finding the literal. See examples
on page 41 of the text.
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Sum terms and Product terms
Product terms for 3-input boolean functions are
of the form F1(a, b, c) a . b . c, F2 (a, b,
c) a . b . c, . . . 23 such functions.
Sum terms for 3-input boolean functions are of
the form G1 (a, b, c) a b c, G2 (a, b, c)
a b c, . . . 23 such functions.
Note H(a, b, c) ab bc c is not a sum term
example.
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Minterms and Maxterms
Minterms and Maxterms are special product and
sum terms, respectively. In a minterm term of
the type f(a, b, c) a . b . c, a 1 appears
exactly once in the truth-values of the function.
In a maxterm term of the type g(a, b, c) a b
c, a 0 appears exactly once in the
truth-values of the function. See Table 2-6, 2-7
on page 42-43.
a e F(a,e) a e a e a e
0 0 1
0 1 1
1 0 1
1 1 0

The boolean expression in the table, F(a,e), is
not a maxterm even though there is only one 0
in the truth values because the terms contain
more than one literal.
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Sum of products
From the values in a table we can obtain a
boolean expression by obtaining the minterms and
then summing them up. This process is called
Sum of Products. This process can, also, be
used identifying the rows containing a 1 with
the corresponding minterm. For example, in the
table shown below rows 1, 3, and 4 contain a 1.
These are the only rows one needs to consider in
order to obtain the Boolean expression. Using
these rows we obtain the minterms and then sum
them up.
a e U(a,e) minterm
0 0 1 m0
0 1 0 m1
1 0 1 m2
1 1 1 m3
U(a,e) m0 m2 m3 a e a e a e
U(a,e) m1 a e
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Products of sums
From the values in a table we can obtain a
boolean expression by obtaining the maxterms and
then multiplying them up. This process is called
Product of Sums. This process can, also, be
used identifying the rows containing a 1 with
the corresponding maxterm, and then taking the
products.
Product of Sums can, also, be obtained for a
Sum of Products by taking the dual of each term,
and then taking the complement of each literal.
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Implicants
Implicants are product terms with special
properties as they relate to a function F that we
are trying to implement An implicant P of a
function F is a product term P such that F 1
whenever P 1. An implicant P covers an
implicant Q if P 1 whenever Q 1. That is, Q
is an implicant of P. By definition, an implicant
can cover itself. However a useful covering
implicant has fewer literals than any implicant
it covers. A prime implicant P of F is any
implicant that is not covered by any other
implicant of F. An essential prime implicant of a
function F is one that covers a minterm of F not
covered by any other prime implicant of F. Two
implicants are adjacent if all but one of their
literals are identical, and the differing
literals consist of a variable in one implicant
and its complement in the other.
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Cost Evaluation

• Cost evaluation for a given statement form or a
  logic circuit diagram is calculated as follows
• Find the sum of
• All the literals appearances,
• The number of terms excluding terms that consist
  only of a single literal, and, optionally
• The number of distinct complemented literals.

The function G ab ca cda has 7 literals, 3
terms, and 1 distinct complemented literal. So
cost value is 11.
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Karnaugh Maps (K-Maps)
This is a pictorial method for obtaining minimal
sum-of-products. Algebraic determination of the
prime implicants of a function is tedious for
simple functions and impractical for more complex
ones. Therefore alternate strategies are
available that simplify the process of
identifying prime implicants, such as the
Tabulation or Karnaugh Map method The Karnaugh
Map (K-Map) representation is a variation on
function table representation where minterms are
arranged so that adjacent minterms are next to
each other, either vertically or horizontally in
the K-Map table.
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Two Variable K-Map
This consists of a 2 by 2 table containing the
minterms, m0, m2 , m3 , m4.
A B m0 m1 m2 m3
0 0 1 0 0 0
0 1 0 1 0 0
1 0 0 0 1 0
1 1 0 0 0 1
B A 0 1
0 m0 m1
1 m2 m3
B A 0 1
0 1 1
1
F(a,b) a
Two adjacent squares differing in one statement
letter represents one literal
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Three Variable K-Map
This consists of a 3 by 3 table containing the
minterms, m0 , m2 , m3 , m4 , m5 , m6, m7 ,
m8.
B C A 00 01 11 10
0 1 1
1 1 1
F(a,b,c) m2 m3 m4 m5
F(a,b,c) a b
Two adjacent squares differing in one statement
letter represents one literal
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FINDING PRIME IMPLICANTS FROM A K-MAP
1.) Enter a 1 in the appropriate position of the
K-Map Table corresponding to each minterm that is
present in the function to be minimized. 2.)
Group into pairs any two 1s that are
side-by-side to produce 1x2 or 2x1 arrays of
1s. 3.) Group into fours any two pairs that are
side-by-side to produce 1x4, 4x1 or 2x2 arrays of
1s. 4.) Group into eights any two fours that
are side-by-side to produce 2x4 or 4x2 arrays of
1s. While every possible "group-of-one",
pairing, group of four, and group of eight
defines an implicant, the prime implicants are
those groups that do not belong to any larger
group.
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Optimization using Prime Implicants
For example to find the prime implicants of
f(x,y,z) Sm(1,3,6,7), the Karnaugh map of f is
given by
Three prime implicants are identified, along with
four additional implicants corresponding to the
minterm 1s. To identify an algebraic
representation for each implicant, construct a
product term that includes a literal for each
variable, if that variable does not change value
for any minterm within the group representing the
implicant.
yz x 00 01 11 10
0 1 1
1 1 1
In the example above, the prime implicants are
xz, yz, and xy. Of these, xz and xy are
essential prime implicants.
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To construct a minimal sum-of-products
representation for a function
1.) Choose all the essential prime implicants.
These have to be included since by definition
they are the only prime implicants that cover
certain minterms. 2.) If such a choice does not
cover all the minterms of the function, then
choose the smallest number of remaining prime
implicants possible so as to cover all the
remaining minterms of the function. 3.) The
required sum-of-products is obtained by "summing"
all selected prime implicants.
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Optimization using Dont Care Conditions
In many design problems there exists certain
conditions which are impossible or for which no
requirement is made. These are referred to as
Dont Care Conditions. The boolean functions of
these circuits have unspecified outputs and are
called incomplete specified functions. The Dont
Care Conditions are the unspecified minterms of
the functions.
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Multilevel Optimization Transformations

• The following transformations can be used in cost
  optimization.
• Factoring,
• Decomposition,
• Extraction,
• Substitution,
• Elimination.

See description and details on page 67 of text.
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Entity Diagrams
Entity Diagram are of the type
a
f
f
A
b
g
B
g
c
The input/output ports correspond to external
signal lines of the logic diagram that provide
the input and output values. There are
combinations of logic circuits connected within
these rectangles.
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Entity diagrams in digital systems
We determine the behavioral descriptions of all
components. We begin with components whose
inputs are only connected to external signal
lines which determine the output from each
component for each row of the function table
according to the behavioral description of the
component. This may determine additional columns
in the function table corresponding to values on
internal signal lines for each input assignment.
We construct a column in the function table for
each internal signal line or external output
signal line from each component as soon as the
values of all inputs to the component have been
defined. When all external output signals have
been defined, the function table is complete.
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Example of an entity diagram
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Behavioral description of the entity diagram
u1 c u2 m n
x y s(x,y) t(x,y) z(x,y)
0 0 0 0 0
0 1 0 1 1
1 0 0 1 1
1 1 1 0 1
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OTHER TYPES OF GATES
The laws of Boolean Algebra provide an effective
means for transforming algebraic representations
of circuits into other representations that
define different but equivalent circuits.
However, the transformation rules are applicable
only to expressions that employ the AND and OR
operations. Four other important binary
operations, called NAND, NOR, XOR, and XNOR, have
useful applications even though do not define
boolean algebras and therefore may not satisfy a
given tranformation rule. To manipulate
expressions that employ these operators, it is
necessary to reduce Boolean expressions that
include these functions to ones that use only
AND, OR, and NOT. Reduction Rules for achieving
this can be obtained by obtaining a minimal
sum-of-products rerepsentation from the function
tables for each operation. See Diagrams on Pages
72-73 of text.
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Odd Functions
A function, f(x,y,z) is odd, if f(x,y,z) 1
whenever x 1, y 1, or z 1, exclusively. In
a more generalized form we can say that for any
n-ary function, the number of 1s in any input
sequence must be odd.
B C A 00 01 11 10
0 1 1
1 1 1
f(x,y,z) x ? y ? z
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Buffers
A buffer produces the logical function Z X,
since the binary value of the output is equal to
the binary value of the input. This circuit is
used primarily to amplify an electrical signal to
permit more gates to be attached to the output or
to decrease the time it takes for signals to
propagate through the circuit.
X F
0 0
1 1
F X
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The 3-State Buffers
The 3-state (buffer) logic output exhibits three
distinct states. Two of the states are the
logic 1 and logic 0 of conventional logic. The
third state is the high impedance, Hi-Z value,
that, for three-state logic, is referred to as
Hi-Z state.
EN IN OUT
0 X Hi-Z
1 0 0
1 1 1
OUT
IN
Enable Input, EN
If EN 1, OUT is equal to IN, behaving like a
normal buffer. The only time we get high
impedance output is when EN 0.