Digital Logic
http://www.pds.ewi.tudelft.nl/~iosup/Courses/2011_ti1400_1.ppt
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Outline
1.
2.
3.
4.
5.
Basics of Boolean algebra and digital implementation
Sum of products form and digital implementation
Functional Units
Repeated Operations
Other Building Blocks
Unit of Information
• Computers consist of digital (binary) circuits
• Unit of information: bit (Binary digIT), e.g. 0 and 1
• There are two interpretations of 0 and 1:
• as data values
• as truth values (true and false)
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Bit Strings
• By grouping bits together we obtain bit strings
• e.g <10001>
which can be given a specific meaning
• For instance, we can represent non-negative numbers by
bitstrings:
0
1
2
3
<00>
<01>
<10>
<11>
<10>
<00>
<01>
<11>
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Boolean Logic
• We want a computer that can calculate, i.e transform
strings into other strings:
1 +2 = 3  <01>  <10> = <11>
• To calculate we need an algebra being able to use only two
values
• George Boole (1854) showed that logic (or symbolic
reasoning) can be reduced to a simple algebraic system
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
Boolean algebra
• Rules are the same as school algebra:
xyyx
x y  y x
Commutative Law
x(y  z)  x  y  x  z
Distributive Law
(x  y)  z  x  (y  z)
Associative Law
• There is, however, one exception:
!
x x  x
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Boolean algebra
• To see this we have to find out what the operations “+”
and “.” mean in logic
• First the “.” operation: x.y (or x  y )
• Suppose x means “black” and y means “cows”.
Then, x.y means “black cows”
• Hence “.” implies the class of objects that has both
properties. Also called AND function.
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Boolean algebra
• The “+” operation merges independent objects:
x + y (or x  y)
• Hence, if x means “man” and y means “woman”
• Then x+y means “man or woman”
• Also called OR function
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Boolean algebra
• Now suppose both objects are identical, for example x
means “cows”
• Then x.x comprises no additional information
• Hence
x x  x  x
xxx
2
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Boolean algebra
• Next, we select “0” and “1” as the symbols in the algebra
• This choice is not arbitrary, since these are the only
number symbols for which holds x2 = x
• What do these symbols mean in logic?
• “0” : Nothing
• “1” : Universe
• So 0.y = 0 and 1.y = y
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Boolean algebra
• Also, if x is a class of objects, then 1-x is the complement
of that class
• It holds that x(1-x) = x -x2 = x-x =0
• Hence, a class and its complement have nothing in
common
• We denote 1-x as x
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

Boolean algebra
• A nice property of this system that we write any function
f(x) as
f (x)  a x  b(1 x)
• We can show this by observing that virtually every
mathematical function can be written in polynomial form,
i.e
f (x)  a0 x  a1 x  a2 x 2  ...
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Boolean algebra
• Now
• Hence,
f (x)  a0 a1x
• Let b = a0 and a = a0 + a1
• Then we have f (x)  a
• From this it follows that
x  b(1 x)
f (1)  a
f (0)  b

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Boolean algebra
• So
f (x)  f (1) x  f (0) x
• More dimensional functions can be derived in an identical
way:

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f (x, y)  f (1,1)  x  y  f (1,0)  x  y 
f (0,1)  x  y  f (0,0)  x  y
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Binary addition
• We apply this on the modulo-2 addition
x
y

0
1
0
1
0
0
1
1
0
1
1
0
x  y  x y  y x
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Binary multiplication
• Same for modulo-2 multiplication
x  y  x y
x
y

0
1
0
1
0
0
1
1
0
0
0
1
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Functions
• Let X denote bitstring, e.g., <x4 x3 x2 x1 >
• Any polynomial function Y=f(X) can be constructed using
Boolean logic
• Also holds for functions with more arguments
• Functions can be put in table form or in formula form
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Gates
• We use basic components to represent primary logic
operations (called gates)
• Components are made from transistors
x
y
x+y
OR
x
x
y
AND
x
INVERT
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x.y
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Networks of gates
• We can make networks of gates
x
x y  x y
y

xy
EXOR
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
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Outline
1.
2.
3.
4.
5.
Basics of Boolean algebra and digital implementation
Sum of products form and digital implementation
Functional Units
Repeated Operations
Other Building Blocks
TI1400/11-PDS
TU-Delft
Sum of product form
x
y
f
0
1
0
1
0
0
1
1
0
1
1

1
f  x y  x y  x y
simplify
f
x
y
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Minimization of expressions
• Logic expressions can often be minimized
• Saves components
• Example:
f  x yz  x yz x yz x yz
f  x  y (z  z)  y  z(x  x)
f  x  y 1 y  z 1
f  x y  yz
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Karnaugh maps (1)
• Alternative geometrical method
xy
00
01
11
10
1
1
0
1
01
1
1
0
1
11
0
0
0
0
10
1
1
1
0
vw
00
f v  x v w x v w y v  x y
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Karnaugh maps (2)
Different drawing
y
w
v
x
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Don’t Cares
• Some outputs are indifferent
• Can be used for minimization
x
y
f
0
1
0
1
0
0
1
1
0
1
d
1
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NAND and NOR gates
• NAND and NOR gates are universal
• They are easy to realize
x y  x y
x y  x y
de Morgan’s Laws

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
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Outline
1.
2.
3.
4.
5.
Basics of Boolean algebra and digital implementation
Sum of products form and digital implementation
Functional Units
Repeated Operations
Other Building Blocks
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Delay
• Every network of gates has delays
transition time
1
input
0
1
output
propagation delay
time
0
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Packaging
Vcc
Gnd
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Making functions
nand gates
A
Y
ADD
B
A,B
Y

delay
time
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Functional Units
• It would be very uneconomical to construct separate
combinatorial circuits for every function needed
• Hence, functional units are parameterized
• A specific function is activated by a special control string F
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Arithmetic and Logic Unit
A
B
F
add
subtract
compare
or
f1 f0
0
0
1
1
Y
F
0
1
0
1
F
A
B
F
F
Y
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Outline
1.
2.
3.
4.
5.
Basics of Boolean algebra and digital implementation
Sum of products form and digital implementation
Functional Units
Repeated Operations
Other Building Blocks
TI1400/11-PDS
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Repeated operations
• Y : = Y + Bi, i=1..n
• Repeated addition requires feedback
• Cannot be done without intermediate storage of results
B
F
Y
F
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Registers
B
Y
F
F
= storage element
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SR flip flop
• Storage elements are not transient and are able to hold a
logic value for a certain period of time
R
S
Qa
Qb
S
R Qa Qb
0
0
0
1
0
1
1
0
1
0
1
1
0
0
0/1 1/0
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Clocks
• In many circuits it is very convenient to have the state
changed only at regular points in time
• This makes design of systems with memory elements easier
• Also, reasoning about the behavior of the system is easier
• This is done by a clock signal
clock period
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D flip flop
• D flip flop samples at clock is high and stores if clock is low
Qn
C
D
D
Qn+1
0
0
1
1
Qn
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Edge triggered flip flops
• In reality most systems are built such that the state only
changes at rising edge of the clock pulse
• We also need a control signal to enable a change
state change
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Basic storage element
enables a state change
I
R/W
C
C
I
C
D
Q
O
R/W
O
time
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Outline
1.
2.
3.
4.
5.
Basics of Boolean algebra and digital implementation
Sum of products form and digital implementation
Functional Units
Repeated Operations
Other Building Blocks
TI1400/11-PDS
TU-Delft
4-bit register
I
R/W
I
I
I
C
C
D
C
D
C
D
C
D
Q
Q
Q
Q
O
O
O
O
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Some basic circuits
A
m
B
Y = A if m=1
Y = B if m=0
MPLEX
Y
Y
Decoder
Only output yA= 1, rest is 0
A
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Decoder
Y
Decoder
Only output yA= 1, rest is 0
A
a1
3
2
1
a2
a1 a2 #y
0 0
0
0 1
1
1 0
2
1 1 3
0
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Multiplexer
A
m
B
Y = A if m=1
Y = B if m=0
MPLEX
Y
b
y
a
m
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End of Lecture
• Comments?
• Questions?
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Memory
Din
REG1
decoder
Address
R/W
mplex
Dout
REG2
REG3
REG4
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Counter
preset
R/W
MPLEX
REG
INC
0001
output
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Sequential circuits
• The counter example shows that systems have state
• The state of such systems depend on the current inputs
and the sequence of previous inputs
• The state of a system is the union of the values of the
memory elements of that system
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State diagrams
• We call the change from one state to another a state transition
• Can be represented as a state diagram
code
S0
S1
S2
0
0
1
0
0
1
0
0
state
S0
S1
S2
S0
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Conditional Change
x=0
S0
x=1
Present
state
Next
x=0
State
x=1
S0
S1
S2
S1
S2
S2
S2
S0
S0
S1
S2
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Coding of State
Present
state
yz
00
Next
x=0
YZ
01
State
x=1
YZ
10
01
10
10
10
00
00
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Put in Karnaugh map
y
x
0
1
0
0
Y
d
d
Y  x y z
1
1

y
z
Z  x  y  z  (x  z)  y
x
1
0
0
0
Z
d
d
0
0
z
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Scheme
x y
x
xz
y

Z

 z
x y z
Q D
Q D

(x  z)  y
Y
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General scheme
Outputs
Inputs
Combinatorial Logic
Delay elements
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Procedure FST
1.
2.
3.
4.
5.
Make State Diagram
Make State Table
Give States binary code
Put state update functions in Karnaugh Map
Make combinatorial circuit to realize functions
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