This preview shows pages 1–5. Sign up to view the full content.
1
1
Sources: TSR, Katz, Boriello & Vahid
CSE140:
Components and Design Techniques
for Digital Systems
Logic simplification
Tajana Simunic Rosing
2
Sources: TSR, Katz, Boriello & Vahid
Overview
• What we covered thus far:
– Number representations
– Switches
–MOS
t
r
a
n
s
i
s
t
o
r
s
– Logic gates
– Boolean algebra
– Minterm and maxterm representation of combinational logic
• What is next:
– Combinatorial logic:
• Minimization
• Implementation
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document 2
3
Sources: TSR, Katz, Boriello & Vahid
canonical sumofproducts
minimized sumofproducts
canonical productofsums
minimized productofsums
F1
F2
F3
B
A
C
F4
Alternative twolevel implementations of
F = AB + C
4
Sources: TSR, Katz, Boriello & Vahid
1cube
X
01
Boolean cubes
• Key tool for simplification is uniting theorem: A (B’ + B) = A
• B. Cubes: visual technique for applying the uniting theorem
• n input variables = ndimensional "cube"
2cube
X
Y
11
00
01
10
3cube
X
Y
Z
000
111
101
4cube
W
X
Y
Z
0000
1111
1000
0111
3
5
Sources: TSR, Katz, Boriello & Vahid
ABF
001
010
101
110
ONset = solid nodes
OFFset = empty nodes
DCset =
×
'd nodes
two faces of size 0 (nodes)
combine into a face of size 1(line)
A varies within face, B does not
this face represents the literal B'
Mapping truth tables onto Boolean cubes
• Uniting theorem combines two “faces" of a cube
into a larger “face"
• Example:
A
B
11
00
01
10
F
6
Sources: TSR, Katz, Boriello & Vahid
ABC
i
n
C
o
u
t
000
0
0
0
011
1
100
0
1
1
111
1
Three variable example
• Binary fulladder carryout logic
A
B
C
000
111
101
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document 4
7
Sources: TSR, Katz, Boriello & Vahid
F(A,B,C) =
Σ
m(4,5,6,7)
onset forms a square  a cube of dimension 2
This subcube represents the
literal A
Higher dimensional cubes
• Subcubes of higher dimension than 2
A
B
C
000
111
101
100
001
010
011
110
•
In a 3cube (three variables):
– a 0cube, i.e., a single node, yields a term in 3 literals
– a 1cube, i.e., a line of two nodes, yields a term in 2 literals
– a 2cube, i.e., a plane of four nodes, yields a term in 1 literal
– a 3cube, i.e., a cube of eight nodes, yields a constant term "1"
•
In general,
– an msubcube within an ncube (m < n) yields a term with n – m literals
8
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 02/14/2008 for the course CSE 140 taught by Professor Rosing during the Fall '06 term at UCSD.
 Fall '06
 Rosing

Click to edit the document details