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ECEN 248: INTRODUCTION TO
DIGITAL SYSTEMS DESIGN
Lecture 3
Dr. Shi
Dept. of Electrical and Computer Engineering
Understanding Decimal Numbers


Decimal numbers are made of decimal digits:
(0,1,2,3,4,5,6,7,8,9)
Number representation:


8653 = 8x103 + 6x102 + 5x101 + 3x100
What about fractions?


97654.35 = 9x104 + 7x103 + 6x102 + 5x101 + 4x100 +
3x10-1 + 5x10-2
Informal notation  (97654.35)10
Understanding Binary Numbers

Binary numbers are made of binary digits (bits):


Number representation:


(1011)2 = 1x23 + 0x22 + 1x21 + 1x20 = (11)10
What about fractions?


0 and 1
(110.10)2 = 1x22 + 1x21 + 0x20 + 1x2-1 + 0x2-2
Groups of eight bits are called a byte (B)

(11001001) 2
Convert an Integer from Decimal to Another
Base
For each digit position:
1. Divide decimal number by the base (e.g. 2)
2. The remainder is the lowest-order digit
3. Repeat first two steps until no divisor
remains.
Example for (13)10:
Integer Remainder Coefficient
Quotient
13/2 =
6/2 =
3/2 =
1/2 =
6
3
1
0
1
0
1
1
a0 = 1
a1 = 0
a2 = 1
a3 = 1
Answer (13)10 = (a3 a2 a1 a0)2 = (1101)2
Convert a Fraction from Decimal to Another Base
For each digit position:
1. Multiply decimal number by the base (e.g. 2)
2. The integer is the highest-order digit
3. Repeat first two steps until fraction
becomes zero.
Example for (0.625)10:
Integer
0.625 x 2 =
0.250 x 2 =
0.500 x 2 =
1
0
1
Fraction
+
+
+
0.25
0.50
0
Coefficient
a-1 = 1
a-2 = 0
a-3 = 1
Answer (0.625)10 = (0.a-1 a-2 a-3 )2 = (0.101)2
The Growth of Binary Numbers
n
2n
n
2n
0
20=1
8
28=256
1
21=2
9
29=512
2
22=4
10
210=1024
3
23=8
11
211=2048
4
24=16
12
212=4096
5
25=32
20
220=1M
6
26=64
30
230=1G
Giga
7
27=128
40
240=1T
Tera
Kilo
Mega
250=Peta
Binary Addition



Binary addition is very simple.
This is best shown in an example of adding two binary
numbers…
Procedure easy to implement by circuits
1 1 1 1
1 1 1 0 1
+
1 0 1 1 1
--------------------1 0 1 0 1 0 0
1
1
1
carries
Binary Subtraction
° We can also perform subtraction (with borrows in place
of carries).
° Let’s subtract (10111)2 from (1001101)2…
° Procedure hard to implement by circuit
1
0 10 10
1
0
10
0 0 10
0 1 1 0 1
1 0 1 1 1
-----------------------1 1 0 1 1 0
borrows
Binary Multiplication

Binary multiplication is much the same as decimal multiplication,
except that the multiplication operations are much simpler…
1
0 1 1 1
X
1 0 1 0
----------------------0 0 0 0 0
1 0 1 1 1
0 0 0 0 0
1 0 1 1 1
----------------------1 1 1 0 0 1 1 0
Binary Data Storage & Representation
•
Binary cells store individual bits of data
•
Multiple cells form a register.
•
Data in registers can indicate different values
•
Hexadecimal: (41)16=(65)10
•
BCD (Binary Coded Decimal): (41)10
•
ASCII (American Standard Code for
Information Interchange): letter “A”
0
Binary Cell
1
0
0
0
0 0
1
Register Transfer



Data can move from register to register.
Digital logic used to process data.
We will learn to design this logic.
Register A
Register B
Digital Logic
Circuits
Register C
Digital Systems

Analysis problem:
Inputs


·
·
Logic
Circuit
·
·
Outputs
Determine binary outputs for each combination of inputs
Design problem: given a task, develop a circuit that
accomplishes the task


Many possible implementations.
Try to develop “best” circuit based on some criterion (size, power,
performance, etc.)
Boolean Algebra

George Boole 1815-1864
 English
mathematician
 Boolean algebra or Boolean logic
 Sound



 In
and complete system
Everything is either true or false, and can proven
Nothing is unknown
Case closed
contrast, mathematics about integers are incomplete



Many things can NOT be proven true nor false
Godel’s incomplete theorem
Endless effort …
Boolean Algebra

A Boolean algebra is defined as a closed
algebraic system containing a set of elements
 Every
variable can take only two values 0 or 1
 Only two operators, * and +

Identity elements
a
+0=a
a * 1 = a


0 is the identity element for the + operation·
1 is the identity element for the * operation·
Commutativity and Associativity of
the Operators

The Commutative Property: For every a and b
a
+b=b+a
a * b = b * a

The Associative Property: For every a, b, and c
a
+ (b + c) = (a + b) + c
 a * (b * c) = (a * b) * c
The Distributive Property

The Distributive Property:
For every a, b, and c,
a+(b*c)=(a+b)*(a+c)
a*(b+c)=(a*b)+(a*c)

Distributivity of the Operators
and Complements

The Existence of the Complement:
For every a there exists a unique element called
a’ (complement of a) such that,
a + a’ = 1
 a * a’ = 0


To simplify notation, the * operator is frequently
omitted. When two elements are written next to
each other, the AND (*) operator is implied…
a+b*c=(a+b)*(a+c)
 a + bc = ( a + b )( a + c )

Claude Shannon 1916-2001


American mathematician
 Father of information theory and
cryptography
In 1937, at age 21, he was a master
student at MIT
 Noticed engineers were messing
around with logic circuits without a
sound foundation
 Wrote the “most important master’s
thesis of all time” to demonstrate
Boolean algebra could be used to
resolve any logical relationship
 Since then, Boolean algebra becomes
widely used
Describing Circuit Functionality:
Inverter
Truth Table
A
Y
A
Y
0
1
1
0
Symbol
Input


Basic logic functions have symbols.
The same functionality can be represented with truth
tables·


Output
Truth table completely specifies outputs for all input
combinations.
The above circuit is an inverter.


An input of 0 is inverted to a 1.
An input of 1 is inverted to a 0.
The AND Gate
A
B



This is an AND gate
Logic operation Y= A* B
So, if the two inputs signals
are asserted (high) the
output will also be asserted.
Otherwise, the output will
be deasserted (low).
Y
Truth Table
A
B
Y
0
0
0
0
1
0
1
0
0
1
1
1
The OR Gate
A
B
This is an OR gate.

Logic operation Y = A+B

So, if either of the two
input signals are
asserted, or both of
them are, the output
will be asserted.
Note: If there are n inputs, then the truth
table will have 2n rows. Since we
can fill any of the entries on output
with 0 or 1, there are 2^2^n ways to
fill a truth table of n input.
Therefore, there are 2^2^n gates of
n input

Y
A
B
Y
0
0
0
0
1
1
1
0
1
1
1
1
The NAND Gate

This is a NAND gate. It
is a combination of an
AND gate followed by
an inverter. Its truth
table shows this.
A
B
Y
A
B
Y
0
0
1
0
1
1
1
0
1
1
1
0
The NOR Gate
A
B

Y
This is a NOR gate. It is a combination of an OR
gate followed by an inverter. It’s truth table
A
B
shows this
Y
0
0
1
0
1
0
1
0
0
1
1
0
The XOR Gate (Exclusive-OR)
A
B



This is a XOR gate.
XOR gates assert their output
when exactly one of the inputs
is asserted, hence the name.
The switching algebra symbol
for this operation is , i.e.
1  1 = 0 and 1  0 = 1.
Y
A
B
Y
0
0
0
0
1
1
1
0
1
1
1
0
Describing Circuit Functionality:
Waveforms
AND Gate



A
B
Y
0
0
0
0
1
0
1
0
0
1
1
1
Waveforms provide another approach for representing
functionality.
Values are either high (logic 1) or low (logic 0).
Can you create a truth table from the waveforms?
Consider three-input gates
3 Input OR Gate
Ordering Boolean
Functions

How to interpret A · B+C?




Is it A · B ORed with C ?
Is it A ANDed with B+C ?
Order of precedence for Boolean algebra: AND before
OR.
Note that parentheses are needed here :
Toll Booth Controller



Consider the design of a toll booth controller.
Inputs: quarter, car sensor.
Outputs: gate lift signal, gate close signal
$0.25?
Car?




Logic
Circuit
Raise gate
Close gate
If driver pitches in quarter, raise gate.
When car has cleared gate, close gate.
Call quarter detector signal Q, car detector C
Call raise gate signal R=Q*C, and lower gate signal L=C’
Boolean Algebra
Boolean Algebra

A Boolean algebra is defined with:
A
set of elements S={0,1}
 A set of operators
 A number of unproved axioms or postulates
Boolean Algebra (cont.)

A binary operator:
 Defined
on a set S of elements
 A rule that assigns to each pair of elements from S a
unique element from S
 Example a*b=c
 aS and bS it must hold that a*bS

A set S is closed with respect to a binary operator
if for every two elements of S, the binary
operator specifies a unique element of S
Definition of Boolean Algebra

Boolean algebra is an algebraic structure,
defined by a set of elements S and two binary
operators, +, and ·, with the following postulates:
The structure is closed with respect to + and *
Element 0 is an identity element for +
Element 1 is an identify element for *
1.
2.


x+0=0+x=x
x · 1=1 · x=x
Definition of Boolean Algebra (cont.)
3.
The structure is commutative with respect to + and · :
a
+b=b+a
a·b=b·a
4.
The operator · is distributive over +
The operation + is distributive over ·
a
·(b+c)=(a·b)+(a·c)
a+(b·c)=(a+b)·(a+c)