Lecture Digital logic design - Lecture 12: More about combinational analysis and design procedures
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Lecture 12
More about Combinational Analysis and
Design Procedures
1
Logic Circuit Analysis
Analysis:
Determining the behavior of a system given its
description.
The description of the system is often provided
in the form of a circuit diagram.
2
o
o
Logic Circuit Analysis
For two-level circuits, the analysis process is simple.
The Boolean expression representing the circuit can
often be written by inspection.
For multilevel circuits, the analysis process is much
more complicated.
Cannot write a Boolean expression by inspection.
Must follow a procedure to implement the analysis.
3
Logic Circuit Analysis
1. Identify inputs and outputs
2. Track circuit behavior from input to output
3. Determine Boolean expression for output(s)
4. Determine Truth Table
5. Examine circuit timing, power dissipation, etc.
4
Logic Circuit Analysis
x 1
P 1
x 2
P 9
x 5
x 3
P 4
f
P 7
P 2
P 3
x 4
P 6
P 8
P 10
P 5
5
Veri
ficat
ion
° Circuit
analysis involves figuring out what some circuit
does
Circ
uit
• Every circuit computes some function, which can be described with
Ana
Boolean expressions or truth tables
lysi
• So, the goal is to find an expression or truth table for the circuit
s
° The first thing to do is to figure out what the inputs and
outputs of the overall circuit are
Inputs: x, y,z
Output: f
6
Sy
mb
olicstart with the circuit diagram
° We
Ana
• We determine gate output expressions
lysi
• Intermediate expressions are combined in following gates to form
s complex expressions
-
It might help to do some algebraic simplification along the way
• We repeat until we have the output function and expression
° Symbolic analysis gives both the truth table and logic
expression
7
Literal Analysis
° Literal analysis is process of manually assigning a set
of values to the inputs, tracing the results, and
recording the output values
• For ‘n’ inputs there are 2n possible input combinations
• From input values, gate outputs are evaluated to form next set of
gate inputs
• Evaluation continues until gate outputs are circuit outputs
° Literal analysis only gives us the truth table
° Once you know the number of inputs and outputs, list
all the possible input combinations in your truth table
• A circuit with n inputs should have a truth table with 2n rows
x
0
0
0
0
1
1
1
1
y
0
0
1
1
0
0
1
1
z
0
1
0
1
0
1
0
1
f
8
Lite
ral
Ana
° You
lysi can simulate the circuit by hand to find the output
for
s each possible combination of inputs
x
y
z
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
f
1
9
Lite
ral
Ana the same thing for all the other input combinations
° Doing
lysi the complete truth table
yields
s
° This is simple, but tedious
x
y
z
f
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
0
1
1
0
0
1
0
1
10
Circ
uit
Ana
° Remember
that if you already have a Boolean
lysi
expression, you can use that to easily make a truth table
s
° For example, since we already found that the circuit
computes the function
f(x,y,z) = xz + y’z + x’yz’, we can use that to fill in a table:
x
y
z
xz
y ’z
x ’y z’
f
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
0
0
0
0
0
1
0
1
0
1
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
1
1
0
0
1
0
1
11
Circ
uitopposite is also true: it’s easy to come up with an
° The
Ana
expression if you already have a truth table
lysi
s
° Convert a truth table into a sum of minterms expression
for example
° You can then simplify this sum of minterms if desired
using a K-map,
x
y
z
f
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
0
1
1
0
0
1
0
1
f(x,y,z)
= x’y’z + x’yz’ + xy’z + xyz
= m1 + m 2 + m 5 + m7
12
Truth Table → Equation
° Analyze the logic circuit shown below to determine
the circuit’s truth-table. Using the truth table, derive
the logic expression for the output F1.
Did you analyze the circuit
BEFORE you turned the
power on?
13
Truth Table → Equation
a) Add test-points at the output of every gate.
Steps (a)
TP1
TP2
TP3
he Process
a) Add test-points at the output of every
gate.
b) Add a column to the truth table for every
test-point.
c) Working from the inputs to the output,
complete the truth table for each testpoint, ultimately ending at the circuit’s
output.
d) From the completed truth table, identify
the Minterms from the truth table
anywhere the output is one.
e) Using the extracted Minterms, write the
Sum-Of-Products logic expression.
TP4
TP5
14
Truth Table → Equation
Steps (b) & (c)
b) Add a column to the truth table for every
test-point.
c) Working from the inputs to the output,
complete the truth table for each test-point,
ultimately ending at the circuit’s output.
he Process
a) Add test-points at the output of
every gate.
b) Add a column to the truth table for
every test-point.
c) Working from the inputs to the
output, complete the truth table for
each test-point, ultimately ending at
the circuit’s output.
d) From the completed truth table,
identify the Minterms from the truth
table anywhere the output is one.
e) Using the extracted Minterms, write
the Sum-Of-Products logic
X
Y
Z
F1
TP1
TP2
TP3
TP4
TP5
0
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1
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1
1
0
0
0
0
0
0
15
Truth Table → Equation
Steps (d) & (e)
d) From the completed truth table, identify the
Minterms from the truth table anywhere the output is
one.
XYZ
XYZ
XYZ
X
Y
Z
F1
TP1
TP2
TP3
TP4
TP5
0
0
0
0
1
0
0
1
0
0
0
1
0
1
0
0
1
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1
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1
0
0
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1
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1
1
0
0
0
0
0
0
e) Using the extracted Minterms, write the Sum-OfProducts logic expression.
F1
XYZ
XYZ
XYZ
16
Exa
mpl
e
#1:
Example
Circ
Analyze the logic circuit shown below to
uit
Ana
determine the circuit’s truth table. Using the
lysi
truth table, derive the logic expression for the
soutput F .
2
17
Exa
mpl
e
#1:
Solution
Circ
a)
uit
Ana
lysi
s
TP1
TP2
TP3
TP4
TP5
b) & c)
d)
XYZ
XYZ
XYZ
e) TP4
A
B
C
F2
TP1
TP2
TP3
0
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0
e) F2 X Y Z X Y Z X Y Z
TP5
18
Equation → Truth Table Technique
The Process
a) Working from the inputs to the output, write the
cumulating logic expression at the output of
each gate concluding with the expression for
the circuit’s output.
b) Using the circuit’s output logic expression,
derive the circuit’s truth table.
19
Equation → Truth Table Technique
Analyze the logic circuit shown below to determine the
logic expression for the output F1. Using the logic
expression, derive the circuit’s truth table.
20
Equation → Truth Table Technique
a) Working from the inputs to the output, write the Step (a)
cumulating logic expression at the output of
each gate concluding with the expression for
the circuit’s output.
X
XY
XYZ
Y
XYZ
XY
he Process
a) Add test-points at the output of every gate.
b) Add a column to the truth table for every testpoint.
c) Working from the inputs to the output,
complete the truth table for each test-point,
ultimately ending at the circuit’s output.
d) From the completed truth table, identify the
Minterms from the truth table anywhere the
output is one.
e) Using the extracted Minterms, write the Sum-
21
XY
Equation → Truth Table Technique
b) Using the circuit’s output logic expression,
derive the circuit’s truth table.
F1
XYZ
Step (b)
XY
X
Y
Z
F1
0
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0
0
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1
0
0
1
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0
1
1
1
1
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0
1
1
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1
1
1
1
0
0
1
1
1
0
XYZ
XY
he Process
a) Add test-points at the output of every gate.
b) Add a column to the truth table for every testpoint.
c) Working from the inputs to the output,
complete the truth table for each test-point,
ultimately ending at the circuit’s output.
d) From the completed truth table, identify the
Minterms from the truth table anywhere the
output is one.
e) Using the extracted Minterms, write the SumOf-Products logic expression.
22
Exa
mpl
Example
e
#2:
Analyze
the logic circuit shown below to
Circ
determine
the logic expression for the output F2.
uit
Using the logic expression, derive the circuit’s
Ana
truth
lysi table.
s
23
Example #2: Circuit Analysis
Solution
A
a)
AB
ABC
ABC BC
F2
ABC BC
BC
b)
C
A
B
C
F2
0
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1
1
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1
0
1
1
1
1
0
ABC
BC
24
Circ
uit finding the circuit inputs and outputs, you can
° After
Ana up with either an expression or a truth table to
come
lysi
describe
what the circuit does
s
Su can easily convert between expressions and truth
° You
mm
tables
ary
° The analysis and synthesis tools presented are
sometimes based on the fundamental concepts of
Boolean algebra
Find the circuit’s
inputs and outputs
Find a Boolean
expression
for the circuit
Find a truth table
for the circuit
25
