Digital design chap 1

DIGITAL ELECTRONICS
CHAPTER 1
DEE 204

DIGITAL ELECTRONICS
• Information representation
Sign and magnitude.
Complement number systems, addition, subtraction,
division, multiplication of binary numbers.
Character coding ASCII, hexadecimal number systems.
Hexadecimal.
• Digital fundamentals
Difference between analog and digital systems.
Logic levels and families.
Logic values, truth tables and logical operations.
Boolean algebra.
De Morgan’s Theorem

DIGITAL ELECTRONICS
• Function of combination logic
Conversion of BCD to 7 segment decoder.
Multiplexer, tri-state output, fan out, address, half
adder, full adder, comparator.
Logic minimisation and Karnaugh maps.
• Function of sequential logic
Sequential logic.
Latches gated and edge-triggered S-R latch.
Edge-trigged D-type latches.
Toggle flip-flop.
Asynchronous counters, registers.
State machines.

DIGITAL ELECTRONICS
• Memory organisation
Principles of storage, RAM, ROM, PROM, EPROM.
Dynamic theory, logic diagram of a single bit within a
RAM and its operation.
Coincident address selection.
Block diagram of a single bit within a RAM and its
operation.
• Conversation between analoque and systems
function of A-D and D-A Converters.
Principle of operation of a D-A converter.
Principle of operation of a single ramp.
A continuously balanced and successive
approximation A-D converter.

INFORMATION REPRESENTATION
Information representation
- Number systems actually represent data
before they are processed by any digital
system
- There are various number systems but all
number systems are related to a
common parameter

- Different characteristics define different
number systems including
 number of independent digits
 place values of different digits
 maximum number able to be written in the
number system
• The most fundamental characteristic is the
number of independent digits or symbols used
in the number system known as radix or base

• the most familiar number system is the decimal
system which has 10 independent digits or a
radix of 10
 the decimal number consists of 0,1,2,3,4,5,6,7,8,9
• the binary number system has only 2
independent digits, i.e. 0,1
• the octal and hexadecimal number system has
8 and 16 independent digits respectively
 octal: 0,1,2,3,4,5,6,7
 hexadecimal: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F

• the place value of different digits in the
integer are given as rn
, where r is the radix of
the system and represents the maximum
number that can be written with n digits:
24
23
22
21
20
=16 =8 =4 =2 =1
1 0 1 1 0
= 101102 = 2210

Sign/magnitude notation
• is the simplest and one of the most obvious
methods of encoding positive and negative
numbers.
– Assign the leftmost (most significant) bit to be
the sign bit. If the sign bit is 0, this means the
number is positive. If the sign bit is 1, then the
number is negative. The remaining m-1 bits are
used to represent the magnitude of the binary
number in the unsigned binary notation.

Example:
Add 00101 + 00110 (+5 and +6)
00101
+ 00110
-------
01011 = +11 (decimal)

Example:
Add +7 and –2 : +7 = 00111
+2 = 00010, so flip the bits: 11101 and add 1: 11110
00111
+ 11110
-------
100101 Discard the extra carry to give 00101 = +5

Example:
Add –5 + -4
5 = 00101. Flip the bits to get 11010, and add 1 to get 11011
4 = 00100. Flip the bits to get 11011, and add 1 to get 11100
11011
+ 11100
----------
110111 = 10111 when we discard the carry
10111 is negative, as indicated by the leading 1.
Flip the bits to get 01000. Add 1 to get 01001. The result is
9. Since it is negative, we really have –9.

• 1's Complement Arithmetic
The Formula
N *= (2n
-1) - N
where: n is the number of bits per word
• N is a positive integer
• N* is -N in 1's complement notation
• For example with an 8-bit word and N = 6, we
have:
N* = (28
-1) - 6 = 255 - 6 = 249 = 111110012

• In Binary
An alternate way to find the 1's complement is to simply
take the bit by bit complement of the binary number.
For example: N = +6 = 000001102
N *= -6 = 111110012
Conversely, given the 1's complement we can find the
magnitude of the number by taking it's 1's complement.
The largest number that can be represented in 8-bit 1's
complement is 011111112 = 127 = $7F. The smallest is
100000002 = -127. Note that the values 000000002 and
111111112 both represent zero.

• Addition
End-around Carry.
When the addition of two values results in a
carry, the carry bit is added to the sum in the
rightmost position. There is no overflow as
long as the magnitude of the result is not
greater than 2n
-1.

• 2's Complement Arithmetic
The Formula
N ** = 2n
- N
where: n is the number of bits per word
• N is a positive integer
• N** is -N in 2's complement notation
• For example with an 8-bit word and N = 6, we
have:
N** = 28
- 6 = 256 - 6 = 250 = 111110102

• In Binary
An alternate way to find the 2's complement is to
start at the right and complement each bit to the left
of the first "1".
For example: N = +6 = 000001102
N** = -6 = 111110102
Conversely, given the 2's complement we can find the
magnitude of the number by taking it's 2's complement.
The largest number that can be represented in 8-bit 2s
complement is 011111112 = 127. The smallest is
100000002 = -128.

• Addition
When the addition of two values results in a
carry, the carry bit is ignored. There is no
overflow as long as the magnitude is not
greater than 2n
-1 nor less than -2n
.

Addition
• Adding 1+1 = 1, with carry of ‘1’ to the left bit

Addition

Subtraction
• Subtracting 0 – 1 = 1, with borrow of ‘1’ from
the left bit

Subtraction

Addition/Subtraction
1101
+ 101
______
1001
+ 111
______
11110
- 10110
________
1111101
- 1101011
_________

Division/Multiplication
11
X 11
11
11
1001
111
X 101
111
000
+ 111
100011

10
000
11
11011
11
000
010
010
10
11010
2
0
6
63
3
0
6
62

* Division or multiplication of binary numbers
could also be done by converting the value
into decimal, divided or multiplied, and then
converted back into binary

• Binary/octal/hexadecimal – decimal conversion
Example:
(1101)2 in decimal
1X23
+ 1X22
+ 0X21
+ 1X20
= 8 + 4 + 0 + 1 = 13
(234)8 in decimal
2X82
+ 3X81
+ 4X80
= 2X64 + 3X8 + 4X1
= 128 + 24 + 4 = 156

• Binary/octal/hexadecimal – decimal
conversion
Example:
(5B1)16 in decimal
5X162
+ 11X161
+ 1X160
= 5X256 + 11X16 + 1X1
= 1280 + 176 + 1 = 1457

Conversion binary-octal-hexadecimal-decimal
Example:
Decimal – binary conversion of (13)10
Solution:
In this case integer = 13

Solution:
For the integer part = 13
Thus, binary equivalent of 13 = 1101

• Example:
Decimal – octal conversion of (73)10
Solution:
integer = 73

• Solution
For the integer part = 73
Thus, octal equivalent of 73 = 111

• Hexadecimal-octal-binary conversion
Example:
(1101111001101001)2 in octal
(1101111001101001)2 = 1571518
1 101 111 001 101 0012
001 101 111 001 101 001
1 5 7 1 5 1

• Hexadecimal-octal-binary conversion
Example:
(1101111001101001)2 in hexadecimal
(1101111001101001)2 = DE6916
1101 1110 0110 10012
1101 1110 0110 1001
13 14 6 9
D E 6 9

• ASCII code
- stands for American Standard Code for
Information Interchange
- a seven bit code based on the English
alphabet used to represent alphanumeric data
in computers or communication equipment
- able to represent 128 characters: A to Z, a to
z, 0 to 9, a few mathematical symbols,
punctuation marks and space character

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Digital design chap 1