dld 01-introduction

Overview
• Digital Systems and Computer Systems
• Information Representation
• Number Systems [binary, octal and
hexadecimal]
• Arithmetic Operations
• Base Conversion
• Decimal Codes [BCD (binary coded decimal),
parity]
• Gray Codes
3-May-16 CSE 225: Digital Logic Design 2

Digital System
• Takes a set of discrete information inputs and
discrete internal information (system state) and
generates a set of discrete information outputs.
System State
Discrete
Information
Processing
System
Discrete
Inputs Discrete
Outputs

Types of Digital Systems
• No state present
o Combinational Logic System
o Output = Function(Input)
• State present
o State updated at discrete times
=> Synchronous Sequential System
o State updated at any time
=>Asynchronous Sequential System
o State = Function (State, Input)
o Output = Function (State)
or Function (State, Input)

Digital System Example:
A Digital Counter (e. g., odometer, a meter that shows mileage traversed):
1 30 0 5 6 4
Count Up
Reset
Inputs: Count Up, Reset
Outputs: Visual Display
State: "Value" of stored digits
Synchronous or Asynchronous?

A Digital Computer Example
Synchronous or
Asynchronous?
Inputs:
Keyboard,
mouse,
modem,
microphone
Outputs: CRT,
LCD, modem,
speakers
Memory
Control
unit Datapath
Input/Output
CPU

Chapter 1 7
Signal
• An information variable represented by physical
quantity.
• For digital systems, the variable takes on discrete
values.
• Two level, or binary values are the most prevalent
values in digital systems.
• Binary values are represented abstractly by:
o digits 0 and 1
o words (symbols) False (F) and True (T)
o words (symbols) Low (L) and High (H)
o and words On and Off.
• Binary values are represented by values or ranges of
values of physical quantities

Chapter 1 8
Signal Examples Over Time
Analog
Asynchronous
Synchronous
Time
Continuous
in value &
time
Discrete in
value &
continuous
in time
Discrete in
value & time
Digital

5.0
4.0
3.0
2.0
1.0
0.0
Volts
HIGH
LOW
HIGH
LOW
OUTPUT INPUT
Signal Example – Physical Quantity: Voltage
Threshold Region

• What are other physical quantities represent 0 and 1?
o CPU Voltage
o Disk
o CD
o Dynamic RAM
Binary Values: Other Physical Quantities
Magnetic Field Direction
Surface Pits/Light
Electrical Charge

Binary Representation of Numbers
• 231.45 =
• Here, 10 is called the radix or base of the number
system
• In 10-base systems, there are 10 symbols (‘0’, ‘1’, … ,
‘9’)
• In n-base system, there are n symbols (‘0’, ‘1’, … , ‘n-1’)
• In binary (2-base) system, there are 2 symbols (‘0’, and
‘1’)
11

Class Room Practice
• Consider the binary number 101.011
• Calculate the decimal value of it.
• The reverse process—converting a decimal to
binary—is also straightforward.
• Divide a decimal number is exact sum of positive and
negative power of 2.
• Successively divide by 2 and ½.
12

Decimal to Binary
• Take decimal 14.875
• Hence, 14 = 1110
• For fractional part, we have to divide by ½ or
multiply by 2.
• 14.875 = 1110.111 (binary)
13

Conversion Between Bases
To convert from one base to another:
1) Convert the Integer Part
2) Convert the Fraction Part
3) Join the two results with a radix point

Conversion Details
• To Convert the Integral Part:
1. Repeatedly divide the number by the new radix and save
the remainders.
2. The digits for the new radix are the remainders in reverse
order of their computation.
3. If the new radix is > 10, then convert all remainders > 10 to
digits A, B, …
• To Convert the Fractional Part:
1. Repeatedly multiply the fraction by the new radix and save
the integer digits that result.
2. The digits for the new radix are the integer digits in order of
their computation.
3. If the new radix is > 10, then convert all integers > 10 to
digits A, B, …

Example: Convert 46.687510 To Base 2
• Convert 46 to Base 2
• Convert 0.6875 to Base 2:
• Join the results together with the radix point:
(46)10=(101110)2
(0.6875)10=(0.1011)2
(46.6875)10=(101110.1011)2

Checking the Conversion
• To convert back, sum the digits times their
respective powers of r.
• From the prior conversion of 46.687510
1011102 = 32 + 8 + 4 + 2
= 46
0.10112 = 1/2 + 1/8 + 1/16
= 0.5000 + 0.1250 + 0.0625
= 0.6875

Class Room Practice
• Convert decimal 17.375 to binary
• 17.375 = 10001.011 (binary)
CSE 257 Numerical Method 18

Additional Issue - Fractional Part
• Note that in this conversion, the fractional part
became 0 as a result of the repeated
multiplications.
• In general, it may take many bits to get this to
happen or it may never happen.
• Example: Convert 0.6510 to N2
o 0.65 = 0.1010011001001 …
o The fractional part begins repeating every 4 steps
yielding repeating 1001 forever!
• Solution: Specify number of bits to right of
radix point and round or truncate to this
number.

General Representation of Numbers
• Positive radix, positional number systems
• 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 =
 +
j = - m
j
j
i
i = 0
i rArA
(Integer Portion) + (Fraction Portion)
i = n - 1 j = - 1

General Representation of Numbers:
Example
• Number = 12345.6789
= (1 × 104
+ 2 × 103
+ 3 × 102
+ 4 × 101
+ 5 × 100
) +
(6 × 10−1 + 7 × 10−2 + 8 × 10−3 + 9 × 10−4)
Where,
A4=1 A3=2 A2=3 A1=4 A0=5 . A-1=6 A-2=7 A-3=8 A-9=9
and Radix, r =10
• Number = 10110.0110
= (1 × 24 + 0 × 23 + 1 × 22 + 1 × 21 + 0 × 20) + (0 ×
2−1 + 1 × 2−2 + 1 × 2−3 + 0 × 2−4)
Where,
A4=1 A3=0 A2=1 A1=1 A0=0 . A-1=0 A-2=1 A-3=1 A-9=0
and Radix, r =2

Number Systems – Examples
General Decimal Binary
Radix (Base) r 10 2
Digits 0 => r - 1 0 => 9 0 => 1
0
1
2
3
Powers of 4
Radix 5
-1
-2
-3
-4
-5
r0
r1
r2
r3
r4
r5
r -1
r -2
r -3
r -4
r -5
1
10
100
1000
10,000
100,000
0.1
0.01
0.001
0.0001
0.00001
1
2
4
8
16
32
0.5
0.25
0.125
0.0625
0.03125

Special Powers of 2
210 (1024) is Kilo, denoted "K"
220 (1,048,576) is Mega, denoted "M"
230 (1,073, 741,824)is Giga, denoted "G"

• Useful for Base Conversion
Exponent Value Exponent Value
0 1 11 2,048
1 2 12 4,096
2 4 13 8,192
3 8 14 16,384
4 16 15 32,768
5 32 16 65,536
6 64 17 131,072
7 128 18 262,144
19 524,288
20 1,048,576
21 2,097,152
8 256
9 512
10 1024
Positive Powers of 2

Commonly Occurring Bases
Name Radix Digits
Binary 2 0,1
Octal 8 0,1,2,3,4,5,6,7
Decimal 10 0,1,2,3,4,5,6,7,8,9
Hexadecimal 16 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
The six letters (in addition to the 10 integers) in
hexadecimal represent:

Decimal
(Base 10)
Binary
(Base 2)
Octal
(Base 8)
Hexadecimal
(Base 16)
00 00000 00 00
01 00001 01 01
02 00010 02 02
03 00011 03 03
04 00100 04 04
05 00101 05 05
06 00110 06 06
07 00111 07 07
08 01000 10 08
09 01001 11 09
10 01010 12 0A
11 01011 13 0B
12 01100 14 0C
13 01101 15 0D
14 01110 16 0E
15 01111 17 0F
16 10000 20 10
• Good idea to memorize!
Numbers in Different Bases

Octal Number System
• 8-base number system
• 8 symbols (0, 1, 2, 3, 4, 5, 6, 7)
• Base/radix is power of 2
• One-to-one relation between octal and binary
• A binary number could be converted easily to octal
• Group 3 bits and convert directly to octal

Octal (Hexadecimal) to Binary and Back
• Octal (Hexadecimal) to Binary:
o Restate the octal (hexadecimal) as three (four) binary digits
starting at the radix point and going both ways.
• Binary to Octal (Hexadecimal):
o Group the binary digits into three (four) bit groups starting at
the radix point and going both ways, padding with zeros as
needed in the fractional part.
o Convert each group of three bits to an octal (hexadecimal)
digit.

Binary to Octal to Decimal
• Instead of converting binary to decimal
• Faster process is binary to octal to decimal

Hexadecimal
• 16-base number system
• 16 symbols (0—9, A, B, C, D, E)
• Again radix is power of 2
• 4 bits to represent a hexadecimal number

Hexadecimal Conversion

Octal to Hexadecimal via Binary
• Convert octal to binary.
• Use groups of four bits and convert as above
to hexadecimal digits.
• Example: Octal to Binary to Hexadecimal
6 3 5 . 1 7 7 8
(110 011 101. 001 111 111)2
(1 9 D . 3 F 8)16

Complements
• Representation of negative number(e.g. -1 or 101) or
in Subtraction operation(e.g. 3-1 or 11-01).
• If base(radix) r, then there are 2types of
complements,
o r’s complement
o (r-1)’s complement
• If radix=10,
o 10’s complement
o 9’s complement
• If radix=2
o 2’s complement
o 1’s complement

The r’s complement
• Positive Number, N in base r with integer part of n
digits,
• Complement of N =
𝑟 𝑛
− 𝑁, 𝑖𝑓 𝑁 ≠ 0
0 , 𝑖𝑓 𝑁 = 0
• If radix=10, N=25.639
o 10’s complement of 25.639 10
= 102 − 25.639 = 74.361
• If radix=2, N=1011.01
= 24 − 1011.01 = 100.11

The (r-1)’s complement
• Positive Number, N in base r with integer part of n
digits and fraction part of m digits,
• Complement of N =
𝑟 𝑛
− 𝑟−𝑚
− 𝑁, 𝑖𝑓 𝑁 ≠ 0
0, 𝑖𝑓 𝑁 = 0
• If radix=10, N=25.639
= 102 − 10−3 − 25.639 =
74.360
• If radix=2, N=1011.01
= 24 − 2−2 − 1011.01 =
100.10

Comparison between 1’s
complement and 2’s complement
• Conversion to 1’s complement is easy, just flip each
bit(0→1, 1 →0)
• 1’s complement has positive zero(0000) and negative
zero(1111).
o As a result, n bit number can represent [−2 𝑛−1 − 1, 2 𝑛−1 − 1]
numbers.
o Example, for char(8-bit) n=8, number range[−27 − 1, 27 −

Binary Numbers and Binary Coding
• Flexibility of representation
o Within constraints below, can assign any binary combination
(called a code word) to any data as long as data is uniquely
encoded.
• Information Types
o Numeric
• Must represent range of data needed
• Very desirable to represent data such that simple,
straightforward computation for common arithmetic
operations permitted
• Tight relation to binary numbers
o Non-numeric
• Greater flexibility since arithmetic operations not applied.
• Not tied to binary numbers

• Given n binary digits (called bits), a binary code is a
mapping from a set of represented elements to a
subset of the 2n binary numbers.
• Example: A
binary code
for the seven
colors of the
rainbow
• Code 100 is
not used
Non-numeric Binary Codes
Binary Number
000
001
010
011
101
110
111
Color
Red
Orange
Yellow
Green
Blue
Indigo
Violet

• Given M elements to be represented by a binary code, the
minimum number of bits, n, needed, satisfies the following
relationships:
2n > M > 2(n – 1)
n = log2 M where x , called the ceiling
function, is the integer greater than or equal to
x.
• Example: How many bits are required to represent decimal
digits with a binary code?
Number of Bits Required

Number of Elements Represented
• Given n digits in radix r, there are rn distinct elements
that can be represented.
• But, you can represent m elements, m < rn
• Examples:
o You can represent 4 elements in radix r = 2 with n = 2 digits:
(00, 01, 10, 11).
o You can represent 4 elements in radix r = 2 with n = 4 digits:
(0001, 0010, 0100, 1000).

Binary Codes for Decimal Digits
Decimal 8,4,2,1 Excess3 8,4,-2,-1 Gray
0 0000 0011 0000 0000
1 0001 0100 0111 0100
2 0010 0101 0110 0101
3 0011 0110 0101 0111
4 0100 0111 0100 0110
5 0101 1000 1011 0010
6 0110 1001 1010 0011
7 0111 1010 1001 0001
8 1000 1011 1000 1001
9 1001 1100 1111 1000
There are over 8,000 ways that you can chose 10
elements from the 16 binary numbers of 4 bits. A few
are useful:

Binary Coded Decimal (BCD)
• The BCD code is the 8,4,2,1 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.
• Example: 1001 (9) = 1000 (8) + 0001 (1)
• How many “invalid” code words are there?
• What are the “invalid” code words?

Chapter 1 43
• What interesting property is common to
these two codes?
Excess 3 Code and 8, 4, –2, –1 Code
Decimal Excess 3 8, 4, –2, –1
0 0011 0000
1 0100 0111
2 0101 0110
3 0110 0101
4 0111 0100
5 1000 1011
6 1001 1010
7 1010 1001
8 1011 1000
9 1100 1111

Chapter 1 44
• What special property does the Gray code
have in relation to adjacent decimal digits?
Gray Code
Decimal 8,4,2,1 Gray
0 0000 0000
1 0001 0100
2 0010 0101
3 0011 0111
4 0100 0110
5 0101 0010
6 0110 0011
7 0111 0001
8 1000 1001
9 1001 1000

Chapter 1 45
B0
111
110
000
001
010
011100
101
B1
B2
(a) Binary Code for Positions 0 through 7
G0
G1
G2
111
101
100 000
001
011
010110
(b) Gray Code for Positions 0 through 7
Gray Code (Continued)
• Does this special Gray code property have any
value?
• An Example: Optical Shaft Encoder

Warning: Conversion or Coding?
• 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)

Binary Arithmetic
• Single Bit Addition with Carry
• Multiple Bit Addition
• Single Bit Subtraction with Borrow
• Multiple Bit Subtraction
• Multiplication
• BCD Addition

Single Bit Binary Addition with Carry
Given two binary digits (X,Y), a carry in (Z) we get the
following sum (S) and carry (C):
Carry in (Z) of 0:
Carry in (Z) of 1: Z 1 1 1 1
X 0 0 1 1
+ Y + 0 + 1 + 0 + 1
C S 0 1 1 0 1 0 1 1
Z 0 0 0 0
X 0 0 1 1
+ Y + 0 + 1 + 0 + 1
C S 0 0 0 1 0 1 1 0

• Extending this to two multiple bit examples:
Carries 0 0
Augend 01100 10110
Addend +10001 +10111
Sum
• Note: The 0 is the default Carry-In to the
least significant bit.
Multiple Bit Binary Addition

BCD Arithmetic
3-May-16
CSE 225: Digital Logic Design
50
Decimal digit Binary value BCD code BCD Code – Binary value
0 0000 0000 0
1 0001 0001 0
2 0010 0010 0
3 0011 0011 0
4 0100 0100 0
5 0101 0101 0
6 0110 0110 0
7 0111 0111 0
8 1000 1000 0
9 1001 1001 0
10 1010 0001 0000 6
11 1011 0001 0001 6
12 1100 0001 0010 6
13 1101 0001 0011 6
14 1110 0001 0100 6
15 1111 0001 0101 6
16 0001 0000 0001 0110 6
17 0001 0001 0001 0111 6
18 0001 0010 0001 1000 6
ForDecimaldigit>9

BCD Arithmetic
Given a BCD code,use binary arithmetic to add the digits:
8 1000 Eight
+5 +0101 Plus 5
13 1101 is 13 (> 9)
Note that the result is MORE THAN 9, so must be
represented by two digits!
To correct the digit, subtract 10 by adding 6 modulo 16.
8 1000 Eight
+5 +0101 Plus 5
13 1101 is 13 (> 9)
+0110 so add 6
carry = 1 0011 leaving 3 + cy
0001 | 0011 Final answer (two digits)
If the digit sum is > 9, add one to the next significant digit

BCD Addition Example

BCD Addition Example
• Add 2905BCD to 1897BCD showing carries
and digit corrections.
1897 0001 1000 1001 0111
2905 + 0010 1001 0000 0101

Chapter 1 54
Error-Detection Codes
• Redundancy (e.g. extra information), in the
form of extra bits, can be incorporated into
binary code words to detect and correct
errors.
• A simple form of redundancy is parity, an
extra bit appended onto the code word to
make the number of 1’s odd or even. Parity
can detect all single-bit errors and some
multiple-bit errors.
• 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.

Chapter 1 55
4-Bit Parity Code Example
• Fill in the even and odd parity bits:
• The codeword "1111" has even parity and the
codeword "1110" has odd parity. Both can be
used to represent 3-bit data.
Even Parity Odd Parity
Message - Parity Message - Parity
000 - 000 -
001 - 001 -
010 - 010 -
011 - 011 -
100 - 100 -
101 - 101 -
110 - 110 -
111 - 111 -

The End

dld 01-introduction

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