# L D OGIC ESIGN

```LOGIC DESIGN
By
Mustafa Saied
INTRODUCTION

digital has become part of our everyday
vocabulary

computers, automation, robots, medical science
and technology, transportation, entertainment,
space exploration, and on and on

exciting educational journey
NUMERICAL REPRESENTATIONS

Quantities

There are basically two ways of representing the
numerical value of quantities: analog and
digital.
ANALOG REPRESENTATIONS

represented by a voltage, current, or meter movement
that is proportional to the value of that quantity


Speedometer

mercury thermometer

audio microphone
they can vary over a continuous range of values
DIGITAL REPRESENTATIONS

the quantities are represented not by proportional
quantities but by symbols called digits
digital watch
 time of day changes continuously, but the digital watch





discrete steps
analog = continuous
digital = discrete (step by step)
Because of the discrete nature of digital
representations, there is no ambiguity when reading
the value of a digital quantity, whereas the value of
an analog quantity is often open to interpretation.
DIGITAL AND ANALOG SYSTEMS

digital system is a combination of devices designed
to manipulate logical information or physical
quantities that are represented in digital form
digital computers
 Calculators
 digital audio and video equipment
 telephone system


analog system contains devices that manipulate
physical quantities that are represented in analog
form.
audio amplifiers
 magnetic tape recording and playback equipment
 light dimmer switch

Digital systems are generally easier to design
 Information storage is easy
 Accuracy and precision are greater
 Operation can be programmed
 Digital circuits are less affected by noise
 More digital circuitry can be fabricated on IC
chips


There is really only one major drawback when
using digital techniques:
The real world is mainly analog

We are in the habit of expressing these quantities
digitally

as when we say that the temperature is 64&deg;
(63.8&deg; when we want to be more precise); but we
are really making a digital approximation to an
inherently analog quantity.


Process (operate on) the digital information.

Convert the digital outputs back to real-world analog
form.
EXAMPLES








sounds from instruments and human voices produce
an analog voltage signal in a microphone
this analog signal is converted to a digital format
using an analog-to-digital conversion process
the digital information is stored on the CD's surface
during playback, the CD player takes the digital
information from the CD surface and converts it into
an analog signal which is then amplified and fed to a
speaker where it can be picked up by the human ear
Complexity
Expense
extra time
electrical power
NUMBER SYSTEMS AND CODES

decimal

Binary

Octal

DECIMAL SYSTEM
composed of 10 numerals or symbols
 base-10 system
 using these symbols as digits
 positional-value system
 453
 digit 4 actually represents 4 hundreds, the 5
represents 5 tens, and the 3 represents 3 units
 most significant digit (MSD)
 least significant digit (LSD)
 2745.214 , (2 * 10+3) + (7 * 10+2) + (4 * 101) + (5 *
100)+ (2 * 10-1) + (1 * 10-2) + (4 * 10-3)

DECIMAL COUNTING
N places or digits we can count through -------- different numbers,
starting with and including zero. The largest number will always be
---------.
BINARY SYSTEM
It is very easy to design simple, accurate
electronic circuits that operate with only two
voltage levels
 almost every digital system uses the binary
(base-2) number system as the basic number
system of its operations
 only two symbols or possible digit values
 base-2 system
 positional-value system
 (1011.101)2, = (1 * 23) + (0 * 22) + (1 * 21) + (1 X
2&deg;) + (1 * 2-1) + (0 * 2-2) + (1 * 2-3) = 8 + 0 + 2+1 +
0.5 + 0 + 0.125 = (11.625)10


The most significant bit (MSB)

The least significant bit (LSB)
BINARY COUNTING

16 possible digit symbols

base 16

represents a group of four binary digits

positional-value system
BINARY-TO-DECIMAL
CONVERSIONS


procedure is to find the weights (i.e., powers of 2)
for each bit position that contains a 1, and then
1
0
1
1
0
1
0
1
DECIMAL-TO-BINARY
CONVERSIONS
OCTAL-TO-DECIMAL CONVERSION
DECIMAL-TO-OCTAL CONVERSION
OCTAL-TO-BINARY CONVERSION


performed by converting each octal digit to its
three-bit binary equivalent
Octal Digit
Binary Equivalent
0
000
1
001
2
010
3
011
4
100
5
101
6
110
7
111
54318
BINARY-TO-OCTAL CONVERSION

The bits of the binary number are grouped into
groups of three bits starting at the LSB

1 0 0 1 1 1 0 1 0

0 1 1 0 1 0 1 1 0
HEX-TO-DECIMAL CONVERSION



A hex number can be converted to its decimal
equivalent by using the fact that each hex digit
position has a weight that is a power of 16
35616 = 3 * l62 + 5 * 161 + 6 * 16&deg; = 768 + 80 + 6 =
85410
2AF16 = 2 * 162 + 10 * 161 + 15 * 16&deg; = 512 + 160
+ 15 = 68710
DECIMAL-TO-HEX CONVERSION

Recall that we did decimal-to-binary conversion
using repeated division by 2, and decimal-to-octal
using repeated division by 8. Likewise, decimalto-hex conversion can be done using repeated
division by 16
HEX-TO-BINARY CONVERSION

Each hex digit is converted to its four-bit binary
equivalent

9F2l6

BA6l6
BINARY-TO-HEX CONVERSION


The binary number is grouped into groups of four
bits, and each group is converted to its equivalent
hex digit
1010111112
SUMMARY OF CONVERSIONS






Right now your head is probably spinning as you try to keep straight all of
these different conversions from one number system to another. You probably
realize that many of these conversions can be done automatically on your
calculator just by pressing a key, but it is important for you to master these
conversions so that you understand the process. Besides, what happens if your
calculator battery dies at a crucial time and you have no handy replacement?
practice!
When converting from binary [or octal or hex] to decimal, use the method of
taking the weighted sum of each digit position.
When converting from decimal to binary [or octal or hex], use the method of repeatedly dividing by 2 [or 8 or 16] and collecting remainders.
3- When converting from binary to octal [or hex], group the bits in groups of
three [or four], and convert each group into the correct octal [or hex] digit.
4. When converting from octal [or hex] to binary, convert each digit into its
three-bit [or four-bit] equivalent.
5. When converting from octal to hex [or vice versa], first convert to binary;
then convert the binary into the desired number system
BCD
numbers, letters, or words special group of
symbols encoded code
 straight binary coding.
 Binary-Coded-Decimal Code
 each digit of a decimal number is represented by
its binary equivalent
 four bits are required
8
7
4 (decimal)
 1000
0111
0100 (BCD)
9
4
3 (decimal)
 1001
0100 0011 (BCD)

BCD
each digit of the decimal number by a four-bit
binary number
 1010, 1011, 1100, 1101, 1110, and 1111 does not
 If it happen???
 BCD is not another number system like binary,
 137
 10001001 8bits
 0001 0011 0111 12 bits
 Inefficient
 relative ease of converting to and from decimal

Decimal
Binary
Octal
BCD
0
0
0
0
0000
1
1
1
1
0001
2
10
2
2
0010
3
11
3
3
0011
4
100
4
4
0100
5
101
5
5
0101
6
110
6
6
0110
7
111
7
7
0111
8
1000
10
8
1000
9
1001
11
9
1001
10
1010
12
A
0001 0000
11
1011
13
B
0001 0001
12
1100
14
C
0001 0010
13
1101
15
D
0001 0011
14
1110
16
E
0001 0100
15
1111
17
F
0001 0101
LOGIC GATES
 OR
Gate
 AND
Gate
 NOT
Gate
 NOR
Gate
 NAND
Gate
OR GATE
x = A + B , x = A + B + C is read as (x equals A
OR B OR C) (Truth table)
 x is a logic 1 for every combination of input levels
where one or more inputs are 1
 OR operation produces 1 + 1 = 1, not 1 + 1 = 2

A
B
x=A+B
0
0
0
0
1
1
1
0
1
1
1
1
EXAMPLE
AND GATE
x=A.B
 x is a logic 1 only when both A and B are at the
logic 1 level. For any case where one of the inputs
is 0, the output is 0


A
B
x=A•B
0
0
0
0
1
0
1
0
0
1
1
1
we have x = A . B . C = ABC. This is read as &quot;x
equals A AND B AND C,&quot; (Truth Table)
NOT GATE
Single input variable
x=A
 &quot;x equals NOT A&quot; or &quot;x equals the inverse of A' or
&quot;x equals the complement of A“

A
x=A
NOT
A
0
1

Inverter
1
0
♦x=A
Presence of small circle always
denotes inversion
NOR GATE
A
B
A+B
A+B
0
0
0
1
0
1
1
0
1
0
1
0
1
1
1
0
NAND GATE
AB
AB
AB
0
0
0
1
0
1
0
1
1
0
0
1
1
1
1
0
TIMING DIAGRAMS
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