CE1111 :Digital Logic Design
lecture 01
Introduction
Dr. Atef Ali Ibrahim
Course Administration
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Instructor:
Dr. Atef Ali Ibrahim
E_mail:
attif_ali2002@yahoo.com
Office Place: (Room CS-25 )
Office Hrs:
TA:
Tutorial:
Text Book: M. Morris Mano; “Digital Design”, 4thEdition, prentice
hall.
Prerequisite:
Recommended Text:
Notes: Lecture slides and Assignments
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Course Objectives
Design of Digital logic systems.
 Building large systems from
small components (logic gates)
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EEG 481 Lec. 1 Introduction
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Chapter 1
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Course Outlines
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Number Systems (conversion and coding)
Boolean Algebra
Gate Level minimization
Combinational Logic Design
Synchronous sequential logic
Registers and counters
Memories and Programmable logic
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What is the Meaning of Digital
Logic Design?
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Digital means:
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Why Digital?
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Digital systems are easier to design and
implement than analog systems.
EEG 481 Lec. 1 Introduction
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Digital Logic circuit
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11
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NUMBER SYSTEMS
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– Representation
A number with radix r is represented by a string
of digits:
An - 1An - 2 … A1A0 . A- 1 A- 2 … A- m + 1 A- m
in which 0 Ai < r and . is the radix point.
The string of digits represents the power series:
(Number)r
=
(
i=n-1
i=0
Ai
r )+( 
(Integer Portion)
j=-1
i
j=-m
+
Aj r
)
j
(Fraction Portion)
Number Systems – Examples
Radix (Base)
Digits
0
1
2
3
Powers of 4
Radix
5
-1
-2
-3
-4
General
Decimal
Binary
Octal
Hex.
r
10
2
8
16
0 => r - 1
0 => 9
0 => 1
0 => 7
0 => F
r0
r1
r2
r3
r4
r5
r -1
r -2
r -3
r -4
1
10
100
1000
10,000
100,000
0.1
0.01
0.001
0.0001
1
2
4
8
16
32
0.5
0.25
0.125
0.0625
1
8
64
512
4096
32768
0.125
0.015625
….
…..
1
16
256
4096
65536
1048576
0.0625
0.00390625
….
….
Decimal Number System
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Binary Number system
EEG 481 Lec. 1 Introduction
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Octal Number system
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Hexadecimal Number System
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Numbers with different bases
The first 17 numbers
in decimal, Binary,
octal , and
Hexadecimal
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Data Types
CONVERSION OF BASES
Base R to Decimal Conversion
(736.4)8 = 7 x 82 + 3 x 81 + 6 x 80 + 4 x 8-1
= 7 x 64 + 3 x 8 + 6 x 1 + 4/8 = (478.5)10
(110110)2 = ... = (54)10
(110.111)2 = ... = (6.785)10
(F3)16
= ... = (243)10
(0.325)6 = ... = (0.578703703 .................)10
Decimal to Base R number
- Separate the number into its integer and fraction parts and convert
each part separately.
- Convert integer part into the base R number
→ successive divisions by R and accumulation of the remainders.
- Convert fraction part into the base R number
→ successive multiplications by R and accumulation of integer digits
Data Types
EXAMPLE
Convert 41.687510 to base 2.
Integer = 41
41
20 1
10 0
5 0
2 1
1 0
0 1
(41)10
= (101001)2
Fraction = 0.6875
0.6875
x
2
1.3750
x
2
0.7500
x
2
1.5000
x
2
1.0000
(0.6875)10 = (0.1011)2
(41.6875)10 = (101001.1011)2
Exercise
Convert (63)10 to base 5:
Convert (1863)10 to base 8:
Convert (0.63671875)10 to hexadecimal:
(223)5
(3507)8
(0.A3)16
Decimal to Octal Conversion
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Decimal to Hex. Conversion
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Binary to Hex. Conversion
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Binary to Octal Conversion
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Signed Numbers
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Complements
COMPLEMENT OF NUMBERS
Two types of complements for base R number system:
- R's complement and (R-1)'s complement
The (R-1)'s Complement
Subtract each digit of a number from (R-1)
Example
- 9's complement of 83510 is 16410
- 1's complement of 10102 is 01012(bit by bit complement operation)
The R's Complement
Add 1 to the low-order digit of its (R-1)'s complement
Example
- 10's complement of 83510 is 16410 + 1 = 16510
- 2's complement of 10102 is 01012 + 1 = 01102
SIGNED NUMBERS Representation
Need to be able to represent both positive and negative numbers
- Following 3 representations
Signed magnitude representation
Signed 1's complement representation
Signed 2's complement representation
Example: Represent +9 and -9 in 7 bit-binary number
Only one way to represent +9 ==> 0 001001
Three different ways to represent -9:
In signed-magnitude:
1 001001
In signed-1's complement: 1 110110
In signed-2's complement: 1 110111
SIGNED NUMBERS Representation
-Range of Signed 2’s
complement : -2n-1 to 2n-1-1
-Range of Signed 1’s
complement: -2n-1 +1 to 2n-1-1
-Range of Signed Magnitude:
-2n-1 +1 to 2n-1-1
Decimal
Signed-2’s
complement
Signed-1’s
complement
Signed
Magnitude
+7
+6
+5
+4
+3
+2
+1
+0
-0
-1
-2
-3
-4
-5
-6
-7
-8
0111
0110
0101
0100
0011
0010
0001
0000
1111
1110
1101
1100
1011
1010
1001
1000
0111
0110
0101
0100
0011
0010
0001
0000
1111
1110
1101
1100
1011
1010
1001
1000
-
0111
0110
0101
0100
0011
0010
0001
0000
1000
1001
1010
1011
1100
1101
1110
1111
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ARITHMETIC ADDITION:
SIGNED MAGNITUDE
[1] Compare their signs
[2] If two signs are the same, ADD the two magnitudes - Look out for an overflow
[3] If not the same , compare the relative magnitudes of the numbers and
then SUBTRACT the smaller from the larger --> need a subtractor to add
[4] Determine the sign of the result
6+9
6 0110
+) 9 1001
15 1111 -> 01111
6 + (- 9)
9 1001
-) 6 0110
- 3 0011 -> 10011
Overflow
9 + 9 or (-9) + (-9)
9
1001
+) 9
1001
overflow (1)0010
-6 + 9
9 1001
- ) 6 0110
3 0011 -> 00011
-6 + (-9)
6 0110
+) 9 1001
-15 1111 -> 11111
ARITHMETIC ADDITION:
SIGNED 2’s COMPLEMENT
Add the two numbers, including their sign bit, and discard any carry out of
leftmost (sign) bit - Look out for an overflow
Example
6 0 0110
+) 9 0 1001
15 0 1111
-6
+) 9
3
6 0 0110
+) -9 1 0111
-3 1 1101
-9
1 0111
1 0111
+) -9
-18 (1)0 1110
1 1010
0 1001
0 0011
(cn-1  cn)
9
+) 9
18
0 1001
0 1001
1 0010
overflow
(cn-1  cn)
Fixed Point Representations
ARITHMETIC ADDITION:
SIGNED 1’s COMPLEMENT
Add the two numbers, including their sign bits.
- If there is a carry out of the most significant (sign) bit, the result is
incremented by 1 and the carry is discarded.
Example
+)
+)
+)
6
-9
-3
-9
-9
0 0110
1 0110
1 1100
1 0110
1 0110
(1)0 1100
1
0 1101
end-around carry
-6
1
1001
0
1001
+) 9
(1) 0(1)0010
1
+)
3
0
0011
not overflow (cn-1
+)
9 0
1001
9 0
1001
1 (1)0010
overflow
(cn-1  cn)
 cn) = 0
COMPARISON OF REPRESENTATIONS
* Easiness of negative conversion
S + M > 1’s Complement > 2’s Complement
* Hardware
- S+M: Needs an adder and a subtractor for Addition
- 1’s and 2’s Complement: Need only an adder
* Speed of Arithmetic
2’s Complement > 1’s Complement(end-around C)
Fixed Point Representations
ARITHMETIC SUBTRACTION
Arithmetic Subtraction in 2’s complement
Take the complement of the subtrahend (including the sign bit)
and add it to the minuend including the sign bits.
(A)-(-B) =(A)+ B
(A)- B=(A)+( -B)
Example : (-6) – (-13) = +7 in 8-bit representation
11111010 – 11110011 = 11111010 + 00001101 = 00000111 (+7)
Binary codes
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Binary codes are used to represent
discrete elements of information
The codes must be in binary that the
computer can only hold 0’s or 1’s.
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Cont. Binary Codes
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Given n binary digits (called bits), there
are 2n distinct elements that can be
represented.
But, you can represent m elements, m <
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Examples:
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2n
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You can represent 4 elements with n = 2
bits: (00, 01, 10, 11).
You can represent 4 elements with n = 4
digits: (0001, 0010, 0100, 1000).
This second code is called a "one hot" code.
Non-numeric Binary Codes
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Example: A
binary code
for the seven
colors of the
rainbow
Code 100 is
not used
Color
Red
Orange
Yellow
Green
Blue
Indigo
Violet
Binary Code
000
001
010
011
101
110
111
External Representations
DECIMAL CODES - Binary
Codes for Decimal Digits
Decimal BCD(8421)
0
1
2
3
4
5
6
7
8
9
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
2421
84-2-1
Excess-3
0000
0001
0010
0011
0100
1011
1100
1101
1110
1111
0000
0111
0110
0101
0100
1011
1010
1001
1000
1111
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
Note: 8,4,2,-2,1,-1 in this table is the weight
associated with each bit position.
d3 d2 d1 d0: symbol in the codes
BCD: d3 x 8 + d2 x 4 + d1 x 2 + d0 x 1
 8421 code.
2421: d3 x 2 + d2 x 4 + d1 x 2 + d0 x 1
84-2-1: d3 x 8 + d2 x 4 + d1 x (-2) + d0 x (-1)
Excess-3: BCD + 3
Binary Coded Decimal (BCD)
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The BCD code is the 8,4,2,1 code.
8, 4, 2, and 1 are weights
BCD is a weighted code
This code is the simplest, most
intuitive binary code for decimal digits
and uses the same powers of 2 as a
binary number, but only encodes the
first ten values from 0 to 9.
Chapter 1
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Warning: Conversion or Coding?
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Do NOT mix up conversion of a
decimal number to a binary
number with coding a decimal
number with a BINARY CODE.
1310 = 11012 (This is
conversion)
 13  0001|0011 (This is
coding)
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BCD Addition
BCD addition of two numbers involve following rules:1) Maximum value of the sum for two digits = 9 (max digit 1)
+ 9 (max digit 2) + 1 (previous addition carry) = 19
2) If sum of two BCD digits is less than or equal to 9 (1001)
without carry then the result is a correct BCD number.
3) If sum of two BCD digits is greater than or equal to 10
(1010) the result is in-correct BCD number. Perform steps
4 for correct BCD sum.
4) Add 6 (0110) to the result.
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Example -BCD addition
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Add 599 and 984 using BCD numbers
1
2
3
599 0101 1001 1001
+984 1001 1000 0100
1110 10001 1101
Binary Sum 1, 2 and 3 are greater than 1010.
So, from Step 3 and 4 add ‘6’ to the sum.
carry
Result
+6
1
1
1110 10001 1101
0110
0110 0110
1 0101 1000 0011
End Carry
Answer. Result of BCD addition is 1583
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Exercise: add 599 + 484 using BCD addition
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GRAY CODE
– Decimal
ALPHANUMERIC CODES - ASCII Character
Codes
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ASCII means “American Standard Code for Information
Interchange”
The ASCII code assigns binary patterns for
- Numbers 0 to 9
- All the letters of English alphabet, uppercase and lowercase
- Many control functions (E.g. BS = Backspace, CR = Carriage
Return) and punctuation marks.
The ASCII system uses 7 bits to represent each code
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ASCII Code
PARITY BIT Error-Detection Codes
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parity, an extra bit appended onto the
code word to make the number of 1’s
odd or even.
A code word has even parity if the
number of 1’s in the code word is even.
A code word has odd parity if the
number of 1’s in the code word is odd.
4-Bit Parity Code Example
Even Parity
Message
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- Parity
000 001 -
0
010 011 -
1
100 101 -
1
110 111 -
0
1
1
0
0
Odd Parity
Message
- Parity
000 001 010 011 100 101 110 111 -
1
0
0
1
0
1
1
0
The codeword "1111" has even parity and the codeword
"1110" has odd parity. Both can be used to represent 3-bit
data.
PARITY BIT GENERATION
Parity Bit Generation
For b6b5... b0(7-bit information); even parity bit beven
beven = b6  b5  ...  b0
For odd parity bit
bodd = beven  1 = beven
Error Detecting codes
PARITY GENERATOR AND PARITY CHECKER
Parity Generator Circuit (even parity)
b6
b5
b4
b3
b2
b1
beven
b0
Parity Checker
beven
b6
b5
b4
b3
b2
b1
b0
Even Parity
error indicator