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ECE 15A
Fundamentals of Logic Design
Lecture 14
Malgorzata MarekSadowska
Electrical and Computer Engineering Department
UCSB
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Today: Test #2 Material review
QM and Petrick’s methods for functions with
don’t cares.
PI chart reduction rules
Mux and decoder; expansion
MUX as circuit element
Modular design
Propagation delays/circuit hazards
Generating test patterns
2level circuits with NOR and NAND gates
QuineMcCluskey method for
functions with don’t cares.
Find all prime implicants for the function with all
don’t cares temporarily set to 1.
Build the implicant chart (table) for the
minterms in care set.
Find essential PIs and reduce the table.
Apply Petrick’s algorithm on the reduced table.
The selected solution determines don’t care
assignment. Some don’t cares may be set to 0.
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Implicant chart simplification
The fewer rows and columns in the chart, the
simpler the Petrick’s expression is.
Simplification #1 – remove columns and rows
related to essential PIs – we know this!
Can we simplify more? Yes. But 2 cases must be
considered:
Case 1: Only 1 minimal solution is sought
Case 2: All minimal solutions are to be determined
What are minimal solutions?
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Implicant’s Cost
Solution in a SOP form
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a
b
d
c
a’
f
e
F=e + ab + a’cd
Implementation with 3 gates and 8 gate inputs.
We will seek the solution with the fewest # of gates;
if tie – minimum
sum of inputs.
Cost(PI)=#literals +1 if #literals > 1, otherwise Cost(PI) = 1
P1
P2
P3
Cost(P1) = 4
Cost(P2) = 3
Cost(P3) = 1
Column dominance
A column i in a prime implicant table dominates a column
j if i has x’s in all the rows j does, and I has at least 1 x in a
row that j does not.
Two columns
of a prime implicant table are equal if they
have x’s in exactly the same rows.
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m
a
m
b
m
c
m
d
m
e
m
f
P1
x
x
P2
x
x
x
P3
x
x
x
P4
x
x
x
x
x
m
c
dominates m
b
m
d
and m
e
are equal
m
c
can be removed
and one of m
d
or m
e
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2
Row dominance
A row i in a prime implicant table dominates a row j if i
has x’s in all the columns j does, and i has at least 1 x in a
column that j does not.
Two rows
of a prime implicant table are equal if they
have x’s in exactly the same columns.
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m
a
m
b
m
c
m
d
m
e
m
f
P1
x
x
P2
x
x
x
P3
x
x
x
P4
x
x
x
x
x
P4 dominates P3
P3 can be eliminated
If Cost(P4)<Cost(P3)
Example
F(v,w,x,y,z) =
m(1,9,10,11) +
d(0,3,14,25,27)
Pis: A=v’x’z; B=wx’z; C=v’w’x’y’; D=v’wx’y;
E=v’wyz’
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m1
m9
m10
m11
cost
A
x
x
x
4
B
x
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 Spring '08
 M
 PIS, prime implicant table, Mux

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