CSE140: Components and Design Techniques
for Digital Systems
Tajana Simunic Rosing
1
Sources: TSR, Katz, Boriello & Vahid
Where we are now…
• What we covered thus far:
–
–
–
–
Number representations
Logic gates
Boolean algebra
Introduction to CMOS
• HW#2 due, HW#3 assigned
• Where we are going:
–
–
–
–
Logic representations
SOP and POS
K-maps
Algorithm for simplification
2
Sources: TSR, Katz, Boriello & Vahid
Combinational Circuit Introduction
Digital circuit
• We’ll start with a simple form of circuit:
– Combinational circuit
• A digital circuit whose outputs depend solely on
the present combination of the circuit inputs’
values
• Built out of simple components: switches and
gates
a
b
a
b
1
Combinational
0
digital circuit
1
0
1
Sequential
?
digital circuit
F
F
3
Sources: TSR, Katz, Boriello & Vahid
Combinational Logic Design Process
Step
2.7
Description
Step 1 Capture the
function
Create a truth table or equations to describe the
desired behavior of the combinational logic.
Step 2 Convert to
equations
This step is only necessary if you captured the
function using a truth table instead of equations.
Simplify the equations if desired.
Step 3 Implement
as a gatebased
circuit
For each output, create a circuit corresponding
to the output’s equation.
4
Sources: TSR, Katz, Boriello & Vahid
Example: Three 1s Detector
• Problem: Detect 3 consecutive
1s in 8-bit input: abcdefgh
• 00011101 
• 01111000 
• 10101011 
a
– Step 1: Capture the function
• Truth table or equation?
a
b
c
abc
bcd
d
cde
– Step 2: Convert to equation
e
y
def
f
– Step 3: Implement with gates
efg
g
fgh
h
5
Sources: TSR, Katz, Boriello & Vahid
Example: Seat Belt Warning Light System
•
•
Design circuit for warning light
Sensors
– s=1: seat belt fastened
– k=1: key inserted
– p=1: person in seat
•
Capture logic equation
– What are conditions for warning
light to go on?
•
Convert equation to circuit
6
Sources: TSR, Katz, Boriello & Vahid
Example: Number of 1s Count
• Problem: Output in binary on
two outputs yz the number of 1s
on three inputs
• 010 
• 101 
• 000 
– Step 1: Capture the function
• Truth table or equation?
– Step 2: Convert to equation
– Step 3: Implement as gates
a
b
c
a
b
c
a
b
c
a
b
y
a
b
c
z
a
b
c
a
b
c
7
Sources: TSR, Katz, Boriello & Vahid
Design example: 1-bit binary adder
Cout Cin
• Inputs: A, B, Carry-in
• Outputs: Sum, Carry-out
A
0
0
0
0
1
1
1
1
B
0
0
1
1
0
0
1
1
Cin Cout S
0
1
0
1
0
1
0
1
A
B
Cin
A
B
A
B
A
B
A
B
A
B
S
S
S
S
S
S
Cout
8
Sources: TSR, Katz, Boriello & Vahid
CSE140: Components and Design Techniques
for Digital Systems
Representation of logic functions
Tajana Simunic Rosing
9
Sources: TSR, Katz, Boriello & Vahid
Canonical Form -- Sum of Minterms
• Truth tables are too big for numerous inputs
• Use standard form of equation instead
– Known as canonical form
– Regular algebra: group terms of polynomial by power
• ax2 + bx + c
(3x2 + 4x + 2x2 + 3 + 1 --> 5x2 + 4x + 4)
– Boolean algebra: create a sum of minterms
• Minterm: product term with every literal (e.g. a or a’) appearing
exactly once
Determine if F(a,b)=ab+a’ is same function as F(a,b)=a’b’+a’b+ab, by
converting the first equation to the canonical form
10
Sources: TSR, Katz, Boriello & Vahid
Sum-of-products canonical forms
• Also known as disjunctive normal form
• Minterm expansion:
F = 001
011
101
110
111
F = A’B’C + A’BC + AB’C + ABC’ + ABC
A
0
0
0
0
1
1
1
1
B
0
0
1
1
0
0
1
1
C
0
1
0
1
0
1
0
1
F
0
1
0
1
0
1
1
1
F’
1
0
1
0
1
0
0
0
F’ = A’B’C’ + A’BC’ + AB’C’
11
Sources: TSR, Katz, Boriello & Vahid
Sum-of-products canonical form (cont’d)
• Product minterm
– ANDed product of literals – input combination for which output is 1
– each variable appears exactly once, true or inverted (but not both)
A
0
0
0
0
1
1
1
1
B
0
0
1
1
0
0
1
1
C
0
1
0
1
0
1
0
1
minterms
A’B’C’ m0
A’B’C m1
A’BC’ m2
A’BC m3
AB’C’ m4
AB’C m5
ABC’ m6
ABC
m7
short-hand notation for
minterms of 3 variables
F in canonical form:
F(A, B, C) = m(1,3,5,6,7)
= m1 + m3 + m5 + m6 + m7
= A’B’C + A’BC + AB’C + ABC’ + ABC
canonical form  minimal form
F(A, B, C) = A’B’C + A’BC + AB’C + ABC + ABC’
= (A’B’ + A’B + AB’ + AB)C + ABC’
= ((A’ + A)(B’ + B))C + ABC’
= C + ABC’
= ABC’ + C
= AB + C
12
Sources: TSR, Katz, Boriello & Vahid
Product-of-sums canonical form
• Also known as conjunctive normal form
• Also known as maxterm expansion
• Implements “zeros” of a function
F=
000
010
100
F = (A + B + C) (A + B’ + C) (A’ + B + C)
A
0
0
0
0
1
1
1
1
B
0
0
1
1
0
0
1
1
C
0
1
0
1
0
1
0
1
F
0
1
0
1
0
1
1
1
F’
1
0
1
0
1
0
0
0
F’ = (A + B + C’) (A + B’ + C’) (A’ + B + C’) (A’ + B’ + C) (A’ + B’ + C’)
13
Sources: TSR, Katz, Boriello & Vahid
Product-of-sums canonical form (cont’d)
• Sum term (or maxterm)
– ORed sum of literals – input combination for which output is false
– each variable appears exactly once, true or inverted (but not both)
A
0
0
0
0
1
1
1
1
B
0
0
1
1
0
0
1
1
C
0
1
0
1
0
1
0
1
maxterms
A+B+C
A+B+C’
A+B’+C
A+B’+C’
A’+B+C
A’+B+C’
A’+B’+C
A’+B’+C’
short-hand notation for
maxterms of 3 variables
M0
M1
M2
M3
M4
M5
M6
M7
F in canonical form:
F(A, B, C) = M(0,2,4)
= M0 • M2 • M4
= (A + B + C) (A + B’ + C) (A’ + B + C)
canonical form  minimal form
F(A, B, C) = (A + B + C) (A + B’ + C) (A’ + B + C)
= (A + B + C) (A + B’ + C)
(A + B + C) (A’ + B + C)
= (A + C) (B + C)
14
Sources: TSR, Katz, Boriello & Vahid
Mapping between canonical forms
• Minterm to maxterm conversion
– use maxterms whose indices do not appear in minterm expansion
– e.g., F(A,B,C) = m(1,3,5,6,7) = M(0,2,4)
• Maxterm to minterm conversion
– use minterms whose indices do not appear in maxterm expansion
– e.g., F(A,B,C) = M(0,2,4) = m(1,3,5,6,7)
• Minterm expansion of F to minterm expansion of F’
– use minterms whose indices do not appear
– e.g., F(A,B,C) = m(1,3,5,6,7)
F’(A,B,C) = m(0,2,4)
• Maxterm expansion of F to maxterm expansion of F’
– use maxterms whose indices do not appear
– e.g., F(A,B,C) = M(0,2,4)
F’(A,B,C) = M(1,3,5,6,7)
15
Sources: TSR, Katz, Boriello & Vahid
Incompletely specified functions
• Example: binary coded decimal increment by 1
– BCD digits encode the decimal digits 0 – 9
•
A
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
B
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
C
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
D
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
W
0
0
0
0
0
0
0
1
1
0
X
X
X
X
X
X
X
0
0
0
1
1
1
1
0
0
0
X
X
X
X
X
X
Y
0
1
1
0
0
1
1
0
0
0
X
X
X
X
X
X
Z
1
0
1
0
1
0
1
0
1
0
X
X
X
X
X
X
Don’t cares and canonical forms
–
–
–
so far, only represented on-set
also represent don’t-care-set
need two of the three sets (on-set, off-set, dc-set)
off-set of W
on-set of W
don’t care (DC) set of W
these inputs patterns should
never be encountered in practice
– "don’t care" about associated
output values, can be exploited
in minimization
16
Sources: TSR, Katz, Boriello & Vahid
Alternative two-level implementations of
F = AB + C
A
B
F1
canonical sum-of-products
C
minimized sum-of-products
F2
canonical product-of-sums
F3
minimized product-of-sums
F4
17
Sources: TSR, Katz, Boriello & Vahid
CSE140: Components and Design Techniques
for Digital Systems
Logic simplification
Tajana Simunic Rosing
18
Sources:
Sources:
TSR, Katz,
Katz, Boriello
Boriello &
& Vahid
Vahid
Key to simplification: the uniting theorem
• Uniting theorem: A (B’ + B) = A
• Essence of simplification of two-level logic
– find two element subsets of the ON-set where only one variable
changes its value – this single varying variable can be eliminated
and a single product term used to represent both elements
F = A’B’+AB’ = (A’+A)B’ = B’
A
B
F
0
0
1
0
1
0
1
0
1
1
1
0
B has the same value in both on-set rows
– B remains
A has a different value in the two rows
– A is eliminated
19
Sources: TSR, Katz, Boriello & Vahid
Boolean cubes
• Visual technique for applying the uniting theorem
• n input variables = n-dimensional "cube"
11
01
0
1-cube
1
Y
X
00
X
111
3-cube
Y Z
000
101
2-cube
10
0111
1111
4-cube
Y
X
0000
Z
W
X
1000
20
Sources: TSR, Katz, Boriello & Vahid
Mapping truth tables onto Boolean cubes
• Uniting theorem combines two “faces" of a cube
into a larger “face"
• Example:
A
B
F
0
0
1
0
1
0
1
0
1
1
1
0
F
11
01
two faces of size 0 (nodes)
combine into a face of size 1(line)
B
00
A
10
A varies within face, B does not
this face represents the literal B'
ON-set = solid nodes
OFF-set = empty nodes
DC-set = 'd nodes
21
Sources: TSR, Katz, Boriello & Vahid
Three variable example
• Binary full-adder carry-out logic
(A'+A)BCin
A
0
0
0
0
1
1
1
1
B
0
0
1
1
0
0
1
1
Cin
0
1
0
1
0
1
0
1
Cout
0
0
0
1
0
1
1
1
111
B C
000
AB(Cin'+Cin)
101
A
A(B+B')Cin
the on-set is completely covered by
the combination (OR) of the subcubes
of lower dimensionality - note that “111”
is covered three times
Cout = BCin+AB+ACin
22
Sources: TSR, Katz, Boriello & Vahid
Higher dimensional cubes
• Sub-cubes of higher dimension than 2
011
111
110
010
000
on-set forms a square - a cube of dimension 2
001
B C
A
F(A,B,C) = m(4,5,6,7)
101
100
This subcube represents the
literal A
•
In a 3-cube (three variables):
–
–
–
–
•
a 0-cube, i.e., a single node, yields a term in 3 literals
a 1-cube, i.e., a line of two nodes, yields a term in 2 literals
a 2-cube, i.e., a plane of four nodes, yields a term in 1 literal
a 3-cube, i.e., a cube of eight nodes, yields a constant term "1"
In general,
– an m-subcube within an n-cube (m < n) yields a term with n – m literals
23
Sources: TSR, Katz, Boriello & Vahid
Karnaugh maps
• Flat map of Boolean cube
– wrap–around at edges
– hard to draw and visualize for more than 4 dimensions
– virtually impossible for more than 6 dimensions
• Alternative to truth-tables to help visualize adjacencies
– guide to applying the uniting theorem
– on-set elements with only one variable changing value are
adjacent unlike the situation in a linear truth-table
B
A
0
1
0
0
1
1
0
2
3
A
B
F
1
0
0
1
1
0
1
0
0
1
0
1
1
1
0
24
Sources: TSR, Katz, Boriello & Vahid
Karnaugh maps (cont’d)
• Numbering scheme based on Gray–code
– e.g., 00, 01, 11, 10
– only a single bit changes in code for adjacent map cells
C
AB
0
C 1
0
1
00
11
01
2
3
6
7
A
5
011
111
110
010
001
B C
A
0
2
6
4
1
3
7
5
B
A
4
B
C
10
000
A
101
100
C
0
4
12
8
1
5
13
9
3
7
15
11
2
6
14
10
B
D
25
Sources: TSR, Katz, Boriello & Vahid
Karnaugh map examples
A
• F=
B
1
1
0
0
• Cout =
A
0
0
1
0
Cin 0
1
1
1
B
• f(A,B,C) = m(0,4,5,7)
C
AB
0
C 1
00
11
01
A
0
2
6
4
1
3
7
5
B
A
10
C
1
0
0
1
0
0
1
1
B
26
Sources: TSR, Katz, Boriello & Vahid
CSE140: Components and Design Techniques
for Digital Systems
Logic simplification cont.
Tajana Simunic Rosing
27
Sources:
Sources:
TSR, Katz,
Katz, Boriello
Boriello &
& Vahid
Vahid
CSE140a HW2 Stats
Students (%)
30.0%
HW2 Grade Distribution
20.0%
10.0%
0~5
5 ~ 10
10 ~ 15
15 ~ 20
20 ~ 25
25 ~ 30
30 ~ 35
35 ~ 40
40 ~ 45
45 ~ 50
50 ~ 55
55 ~ 60
60 ~ 65
65 ~ 70
70 ~ 75
75 ~ 80
80 ~ 85
85 ~ 90
90 ~ 95
95 ~ 100
0.0%
Bucket of Points
•
•
•
•
•
•
buckets # students
0~5
14
5 ~ 10
0
10 ~ 15
0
15 ~ 20
0
20 ~ 25
0
25 ~ 30
1
30 ~ 35
0
35 ~ 40
0
40 ~ 45
0
45 ~ 50
1
50 ~ 55
3
55 ~ 60
0
60 ~ 65
4
65 ~ 70
6
70 ~ 75
11
75 ~ 80
18
80 ~ 85
11
85 ~ 90
29
90 ~ 95
11
95 ~ 100
42
students (%)
9.27%
0.00%
0.00%
0.00%
0.00%
0.66%
0.00%
0.00%
0.00%
0.66%
1.99%
0.00%
2.65%
3.97%
7.28%
11.92%
7.28%
19.21%
7.28%
27.81%
Total students enrolled : 151
# Solutions received : 137
Max score : 100.00
Min score (w/o 0s) : 30.00
Mean score : 86.03
Median score : 89.00
Sources: TSR, Katz, Boriello & Vahid
Where we are now…
• What we covered thus far:
–
–
–
–
Chap 1
Logic representations
SOP and POS
K-maps
• HW#3 due, HW#4 assigned
• Midterm #1 next week on Th
• Where we are going:
– K-maps – more examples
– Mux and Demux
29
Sources: TSR, Katz, Boriello & Vahid
Karnaugh map: 4-variable example
• F(A,B,C,D) = m(0,2,3,5,6,7,8,10,11,14,15)
F=
A
C
1
0
0
1
0
1
0
0
1
1
1
1
1
1
1
1
0111
D
C
0000
D
A
B
1111
1000
B
find the smallest number of the largest possible
subcubes to cover the ON-set
(fewer terms with fewer inputs per term)
30
Sources: TSR, Katz, Boriello & Vahid
Karnaugh maps: don’t cares
• f(A,B,C,D) = m(1,3,5,7,9) + d(6,12,13)
without don't cares
with don’t cares
f=
f=
A
C
A
0
0
X
0
1
1
X
1
1
1
0
0
0
X
0
0
B
D
0
0
X
0
1
1
X
1
1
1
0
0
0
X
0
0
D
C
B
don't cares can be treated as 1s or 0s
depending on which is more advantageous
31
Sources: TSR, Katz, Boriello & Vahid
Another Example
• F = m(0, 2, 7, 8, 14, 15) + d(3, 6, 9, 12, 13)
A
C
1
0
X
1
0
0
X
X
X
1
1
0
1
X
1
0
D
B
32
Sources: TSR, Katz, Boriello & Vahid
Design example: two-bit comparator
N1
N2
A
B
C
D
LT
EQ
GT
AB<CD
AB=CD
AB>CD
block diagram
and
truth table
A
0
B
0
0
1
1
0
1
1
C
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
D
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
LT
0
1
1
1
0
0
1
1
0
0
0
1
0
0
0
0
EQ
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
GT
0
0
0
0
1
0
0
0
1
1
0
0
1
1
1
0
we'll need a 4-variable Karnaugh map
for each of the 3 output functions
33
Sources: TSR, Katz, Boriello & Vahid
Design example: two-bit comparator (cont’d)
A
C
A
0
0
0
0
1
0
0
0
1
1
0
1
1
1
0
0
D
C
A
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
D
C
0
1
1
1
0
0
1
1
0
0
0
0
0
0
1
0
B
B
B
K-map for LT
K-map for EQ
K-map for GT
D
LT = A' B' D + A' C + B' C D
EQ = A' B' C' D' + A' B C' D + A B C D + A B' C D’ = (A xnor C) • (B xnor D)
GT = B C' D' + A C' + A B D'
LT and GT are similar (flip A/C and B/D)
34
Sources: TSR, Katz, Boriello & Vahid
Design example: 2x2-bit multiplier
A1
A2
B1
B2
P1
P2
P4
P8
block diagram
and
truth table
A2 A1 B2
0 0 0
0
1
1
0 1 0
0
1
1
1 0 0
0
1
1
1 1 0
0
1
1
B1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
P8
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
P4
0
0
0
0
0
0
0
0
0
0
1
1
0
0
1
0
P2
0
0
0
0
0
0
1
1
0
1
0
1
0
1
1
0
P1
0
0
0
0
0
1
0
1
0
0
0
0
0
1
0
1
4-variable K-map
for each of the 4
output functions
35
Sources: TSR, Katz, Boriello & Vahid
Design example: 2x2-bit multiplier (cont’d)
A2
B2
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
K-map for P8
K-map for P4
B1
B2
A2
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
1
A1
A1
A2
B2
0
0
0
0
0
0
1
1
0
1
0
1
0
1
1
0
A1
B1
K-map for P2
K-map for P1
B1
B2
A2
0
0
0
0
0
1
1
0
0
1
1
0
0
0
0
0
B1
A1
36
Sources: TSR, Katz, Boriello & Vahid
Design example: BCD + 1
I1
I2
I4
I8
O1
O2
O4
O8
block diagram
and
truth table
I8
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
I4
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
I2
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
I1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
O8
0
0
0
0
0
0
0
1
1
0
X
X
X
X
X
X
O4
0
0
0
1
1
1
1
0
0
0
X
X
X
X
X
X
O2
0
1
1
0
0
1
1
0
0
0
X
X
X
X
X
X
O1
1
0
1
0
1
0
1
0
1
0
X
X
X
X
X
X
4-variable K-map for each of
the 4 output functions
37
Sources: TSR, Katz, Boriello & Vahid
Design example: BCD + 1 (cont’d)
I8
I2
0
0
X
1
0
0
X
0
0
1
X
X
0
0
X
X
O8
O4
I1
I2
I8
0
1
X
0
0
1
X
0
1
0
X
X
0
1
X
X
I4
I4
I8
I2
0
0
X
0
1
1
X
0
0
0
X
X
1
1
X
X
I4
I1
O2
O1
I1
I2
I8
1
1
X
1
0
0
X
0
0
0
X
X
1
1
X
X
I1
I4
38
Sources: TSR, Katz, Boriello & Vahid
CSE140: Components and Design Techniques
for Digital Systems
Muxes and demuxes
Tajana Simunic Rosing
39
Sources: TSR, Katz, Boriello & Vahid
Contention: X
• Contention: circuit tries to drive output to 1 and 0
–
–
–
–
Actual value somewhere in between
Could be 0, 1, or in forbidden zone
Might change with voltage, temperature, time, noise
Often causes excessive power dissipation
A=1
Y=X
B=0
• Warnings:
– Contention usually indicates a bug.
– X is used for “don’t care” and contention - look at the
context to tell them apart
Sources: TSR, Katz, Boriello & Vahid
Transmission Gate:
Mux/Tristate building block
• nMOS pass 1’s poorly
• pMOS pass 0’s poorly
• Transmission gate is a better switch
– passes both 0 and 1 well
EN
A
B
• When EN = 1, the switch is ON:
– EN = 0 and A is connected to B
• When EN = 0, the switch is OFF:
EN
– A is not connected to B
Sources: TSR, Katz, Boriello & Vahid
Floating: Z
• Floating, high impedance, open, high Z
• Floating output might be 0, 1, or somewhere in
between
– A voltmeter won’t indicate whether a node is
floating
Tristate Buffer
E
Y
A
E
0
0
1
1
A
0
1
0
1
Y
Z
Z
0
1
Sources: TSR, Katz, Boriello & Vahid
Tristate Busses
• Floating nodes are used in tristate busses
– many different drivers, but only one is active at once
processor
en1
to bus
from bus
video
en2
to bus
Ethernet
en3
sharedbus
from bus
to bus
from bus
memory
en4
to bus
from bus
Sources: TSR, Katz, Boriello & Vahid
2:1 Multiplexer or Mux
• Selects between one of N inputs to connect to output
• log2N-bit select input – control input
• Example:
2:1 Mux
Y
S
S
0
0
0
0
1
1
1
1
D1
0
0
1
1
0
0
1
1
D0
0
D1
1
D0
0
1
0
1
0
1
0
1
S
Y
Logic gates
D0 D1
00
01
11
10
0
0
1
1
S
0
1
Pass gates
S
D0
1
Y
0
1
0
1
0
0
1
1
0
Tristates
Y
D0
D1
0
1
1
0
Y = D 0S + D1S
Y
D1
D0
S
D1
Y
Sources: TSR, Katz, Boriello & Vahid
Multiplexers
• 2:1 mux:
• 4:1 mux:
• 8:1 mux:
Z = A'I0 + AI1
Z = A'B'I0 + A'BI1 + AB'I2 + ABI3
Z = A'B'C'I0 + A'B'CI1 + A'BC'I2 + A'BCI3 +
AB'C'I4 + AB'CI5 + ABC'I6 + ABCI7
• In general: Z = 
2 n -1(m
k=0
kIk)
– in minterm shorthand form for a 2n:1 Mux
I0
I1
2:1
mux
A
Z
I0
I1
I2
I3
4:1
mux
A B
Z
I0
I1
I2
I3
I4
I5
I6
I7
8:1
mux
Z
A B C
45
Sources: TSR, Katz, Boriello & Vahid
Logic using Multiplexers
• Using the mux as a lookup table
A
0
0
1
1
B
0
1
0
1
Y
0
0
0
1
Y = AB
AB
00
01
10
Y
11
Sources: TSR, Katz, Boriello & Vahid
Logic using Multiplexers
• Reducing the size of the mux
Y = AB
A
0
0
1
1
B
0
1
0
1
Y
0
0
0
1
A
Y
0
0
1
B
A
0
B
1
Y
Sources: TSR, Katz, Boriello & Vahid
Mux example: Logical function unit
C0
0
0
0
0
1
1
1
1
C1
0
0
1
1
0
0
1
1
C2
0
1
0
1
0
1
0
1
Function
1
A+B
(A • B)'
A xor B
A xnor B
A•B
(A + B)'
0
Comments
always 1
logical OR
logical NAND
logical xor
logical xnor
logical AND
logical NOR
always 0
0
1
2
3 8:1 MUX
4
5
6
7
S2 S1 S0
C0
C1
F
C2
48
Sources: TSR, Katz, Boriello & Vahid
Mux as general-purpose logic
•
A 2n-1:1 multiplexer can implement any function of n variables
– with n-1 variables used as control inputs and
– the data inputs tied to the last variable or its complement
•
Example: F(A,B,C) = AC + BC' + A'B‘C
49
Sources: TSR, Katz, Boriello & Vahid
Demux or Decoder
• N inputs, 2N outputs
• One-hot outputs: only one output HIGH at once
2:4
Decoder
A1
A0
A1
0
0
1
1
A0
0
1
0
1
Y3
0
0
0
1
11
10
01
00
Y3
Y2
Y1
Y0
Y2
0
0
1
0
Y1
0
1
0
0
Y0
1
0
0
0
Sources: TSR, Katz, Boriello & Vahid
Decoder: logic equations
• Decoders/demultiplexers
– control inputs (called “selects” (S)) represent binary index of
output to which the input is connected
– data input usually called “enable” (G)
1:2 Decoder:
O0 = G  S’
O1 = G  S
2:4 Decoder:
O0 = G  S1’ 
O1 = G  S1’ 
O2 = G  S1 
O3 = G  S1 
S0’
S0
S0’
S0
O0
O1
O2
O3
O4
O5
O6
O7
3:8 Decoder:
= G  S2’  S1’  S0’
= G  S2’  S1’  S0
= G  S2’  S1  S0’
= G  S2’  S1  S0
= G  S2  S1’  S0’
= G  S2  S1’  S0
= G  S2  S1  S0’
= G  S2  S1  S0
51
Sources: TSR, Katz, Boriello & Vahid
Decoder Implementation
A1
A0
Y3
Y2
Y1
Y0
Sources: TSR, Katz, Boriello & Vahid
Logic Using Decoders
• OR minterms
A
B
2:4
Decoder
11
10
01
00
Y = AB + AB
= A  B
Minterm
AB
AB
AB
AB
“1”
0
1
2
3
3:8 DEC 4
5
6
7
S2 S1 S0
A
B
A'B'C'
A'B'C
A'BC'
A'BC
AB'C'
AB'C
ABC'
ABC
C
Y
Sources: TSR, Katz, Boriello & Vahid
Example of demux as general-purpose logic
F1 = A'BC'D + A'B'CD + ABCD
F2 = ABC'D' + ABC
F3 = (A' + B' + C' + D')
Enable
0
1
2
3
4
5
6
4:16 7
DEC 8
9
10
11
12
13
14
15
A'B'C'D'
A'B'C'D
A'B'CD'
A'B'CD
A'BC'D'
A'BC'D
A'BCD'
A'BCD
AB'C'D'
AB'C'D
AB'CD'
AB'CD
ABC'D'
ABC'D
ABCD'
ABCD
A B C D
Sources: TSR, Katz, Boriello & Vahid
Another example
•
F(A,B,C) = M(0,2,4)
55
Sources: TSR, Katz, Boriello & Vahid
CSE140: Components and Design Techniques
for Digital Systems
Timing and hazards
Tajana Simunic Rosing
56
Sources: TSR, Katz, Boriello & Vahid
Timing
• Delay between input change and output changing
• How to build fast circuits?
A
Y
delay
A
Y
Time
Sources: TSR, Katz, Boriello & Vahid
Propagation & Contamination Delay
• Propagation delay: tpd = max delay from input to output
• Contamination delay: tcd = min delay from input to output
A
Y
tpd
A
Y
tcd
Time
Sources: TSR, Katz, Boriello & Vahid
Propagation & Contamination Delay
• Delay is caused by
– Capacitance and resistance in a circuit
– Speed of light limitation
• Reasons why tpd and tcd may be different:
– Different rising and falling delays
– Multiple inputs and outputs, some of which are faster
than others
– Circuits slow down when hot and speed up when cold
Sources: TSR, Katz, Boriello & Vahid
Critical (Long) & Short Paths
Critical (Long) Path: tpd = 2tpd_AND + tpd_OR
Short Path: tcd = tcd_AND
Critical Path
A
B
n1
n2
C
Y
D
Short Path
Sources: TSR, Katz, Boriello & Vahid
Glitches or Hazards
• Glitch occurs when an input change causes multiple output changes;
circuit with a potential for a glitch has a hazard
• There are 3 types of hazards:
•
•
•
Static-0 : output should be 0 but has a 1 glitch
Static-1 : output should be 1 but has a 0 glitch
Dynamic: transition 0->1 or 1->0 with a glitch
• Example: B = A1 = 00
0
Critical Path
1
n1
Y=1
0
1
What happens if
A = 0, C = 1, & Y
B falls?
A
B
n2
C=1
1
0
C
Short Path
Y
B
AB
00
01
11
10
0
1
0
0
0
1
1
1
1
0
C
n2
n1
Y = AB + BC
Y
Time
glitch
Sources: TSR, Katz, Boriello & Vahid
Fixing a hazard
Y
AB
00
01
11
10
0
1
0
0
0
1
1
1
1
0
C
AC
Y = AB + BC + AC
A=0
B=1 0
Y=1
C=1
Sources: TSR, Katz, Boriello & Vahid
Another example
A
• F(A,B,C,D)=m(1,3,5,7,8,9,12,13)
• Test two single bit input transitions:
– 1100 -> 1101
– 1100 -> 0101
A
C
C’
0
0
1
1
1
1
1
1
1
1
0
0
0
0
0
0
D
B
Z
D
63
Sources: TSR, Katz, Boriello & Vahid
CSE140: Components and Design Techniques
for Digital Systems
Two and Multilevel logic implementation
Tajana Simunic Rosing
64
Sources: TSR, Katz, Boriello & Vahid
Multiple-Output Circuits: Priority Circuit
•
Output asserted corresponds to the most significant TRUE input
A3
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
A2
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
A1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
A0
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
Y3
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
Y2
0
0
0
0
1
1
1
1
0
0
0
0
0
0
0
0
Y1
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
Y0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
A3
0
0
0
0
1
A2
0
0
0
1
X
A1
0
0
1
X
X
A0
0
1
X
X
X
Y3
0
0
0
0
1
Y2
0
0
0
1
0
A3 A 2 A1 A0
A3
Y3
A2
Y2
A1
Y1
A0
Y0
PRIORITY
CiIRCUIT
Y1
0
0
1
0
0
Y0
0
1
0
0
0
Y3
Y2
Y1
Y0
Sources: TSR, Katz, Boriello & Vahid
Multiple-Output Circuits
• Many circuits have more than one output
• Can give each a separate circuit, or can share gates
• Ex: F = ab + c’, G = ab + bc
Option 1: Separate circuits
Option 2: Shared gates
66
Sources: TSR, Katz, Boriello & Vahid
Multi-level logic
•
x=ADF + AEF + BDF + BEF + CDF + CEF + G
– reduced sum-of-products form – already simplified
– 6 x 3-input AND gates + 1 x 7-input OR gate (that may not even exist!)
– 25 wires (19 literals plus 6 internal wires)
•
x = (A + B + C) (D + E) F + G
– factored form – not written as two-level S-o-P
– 1 x 3-input OR gate, 2 x 2-input OR gates, 1 x 3-input AND gate
– 10 wires (7 literals plus 3 internal wires)
A
B
C
X
D
E
F
G
Sources: TSR, Katz, Boriello & Vahid
Multiple-Output Example:
BCD to 7-Segment Converter
a
f
b
g
e
c
d
abcdefg =
(a)
1111110
0110000
1101101
(b)
a = w’x’y’z’ + w’x’yz’ + w’x’yz + w’xy’z +
w’xyz’ + w’xyz + wx’y’z’ + wx’y’z
b = w’x’y’z’ + w’x’y’z + w’x’yz’ + w’x’yz +
w’xy’z’ + w’xyz + wx’y’z’ + wx’y’z
68
Sources: TSR, Katz, Boriello & Vahid