Digital System Design Digital Systems and Binary Number

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Digital System Design Digital Systems and Binary Number

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Title: Digital System Design Digital Systems and Binary Number

1
Digital System DesignDigital Systems and Binary
Number

Assoc. Prof. Pradondet Nilagupta
pom_at_ku.ac.th

2
Acknowledgement

This lecture note is modified from Engin112
Digital Design by Prof. Maciej Ciesielski, Prof.
Tilman Wolf, University of Massachusetts Amherst
and original slide from publisher

3
Digital Hardware Systems
Digital Systems
Digital vs. Analog Waveforms
Analog values vary over a broad range
continuously
Digital only assumes discrete values
4
Digital Circuits

Digital circuit can be represented by a black-box
with inputs on one side, and outputs on the other.

The input/output signals are discrete/digital in
nature, typically with two distinct voltages (a
high voltage and a low voltage).
In contrast, analog circuits use continuous
signals.
5
Digital Hardware Systems

Digital Binary System
Two discrete values
yes, on, 5 volts, current flowing, "1"
no, off, 0 volts, no current flowing, "0
Advantage of binary systems
rigorous mathematical foundation based on logic
its easy to implement

both the door must be open and the car running
before I can back out
IF the garage door is open AND the car is
running THEN the car can be backed out of the
garage
the preconditions must be true to imply the
conclusion
6
Binary Bit and Group Definitions

Bit - a single binary digit
Nibble - a group of four bits
Byte - a group of eight bits
Word - depends on processor 8, 16, 32, or 64
bits
LSB - Least Significant Bit (on the right)
MSB - Most Significant Bit (on the left)

7
Binary Representation of Information

Information divided into groups of symbols
26 English letters
10 decimal digits
50 states in USA
Digital systems manipulate information as 1s
0s
The mapping of symbols to binary value is known
as a code
The mapping must be unique

8
Digital Systems

Digital systems operate on discrete elements of
information
Numbers (e.g., pocket calculator) -gt digits -gt
digital
Letters (e.g., word processor)
Pictures (e.g., digital cameras)
For a digital systems to operate on a continuous
data, it needs to quantize (digitize) that data
first
Covert data into digital representation
Topics
How are numbers represented in digital systems
How computer performs basic arithmetic operations

9
Numbers

Common numbering system is base10
Why?
Numbers in base 10
Ten different digits 0, 1, 2, 3, 4, 5, 6, 7, 8,
9
Number is represented by a sequence of digits an
an-1 a1 a0
Value of number is an10nan-110n-1a1101a0
100
Positional notation
General equation
May contain a decimal point
Negative index for digits after decimal point
Examples
1234.56 if ambiguous, write (1234.56)10
Leading zeros cause no problems 00001234.56

10
Number Systems

General form, with base r
Base r is also called radix
In decimal system r 10 in binary r 2
Coefficients in positional notation are 0,1,,
r-1.
What is the range of values of an n-bit number in
radix r ?
Minimum value 0
Maximum value rn-1
Number of different values rn

11
Positional Number Systems

Numeric value is represented by a series of
digits
Number of digits used is fixed by radix
Digits multiplied by a power of the radix
Digit order determines radix powers
Very large numbers can be represented
Can also represent fractional values.

12
Positional Integer Number Values
Given a digit series of The full expression
for the represented value is
13
Positional Fractional Number Values
Given a digit series of The full expression
for the represented value is
14
Binary Numbers

Base 2 number use only two digits 0, 1
Why?
Digits need to be represented in a system
Electronic systems typically use voltage levels
Representing 10 different voltages reliably is
difficult
Binary decision is much easier (On, Off)
Binary representation is ideal
Minimal number of digits
Easily represented in voltages

15
Examples for Binary Numbers

What value is represented by (01001)2?
Leading zero makes no difference
(1001)2 translates into 123022021120800
1(9)10
Same process for numbers with decimal point
What is the value of (1001.1001)2?
(1001.1001)2 12302202112012-102-20
2-312-480011/2001/16(9.5625)10
Important its NOT (9.9)10!
Can you count binary?
How far can you count with 10 fingers?

16
Binary Number Terminology

Base is also called radix
Binary numbers are made of binary digits (bits)
Groups of four bits are called nibbles
E.g., (1101)2
Groups of eight bits are called bytes
E.g., (01001101)2
What is the range of values of an n-bit binary
number?
Minimum value 0
Maximum value 2n-1
Number of different values 2n
Powers of 2 are important in ECE

17
Powers of 2

You must memorize all powers of 2 up to 216!
Other important powers of 2
Trick to simplify estimation
21010241000103
Example 232423041094 billion
Prefixes
kilo (103 210), Mega (106 220), Giga (109
230), Tera (1012 240),
In computer systems typically based on powers of 2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1 2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384 32768 65536
8 16 24 32 64
256 65536 16777216 4294967296 18446744073709551616
18
Memory Size

Definition of Mega and Giga can make a
difference!
Disk drive manufacturers define it conveniently
How many more MP3 songs could you store on this
80GB disk if GB was based on the real computer
systems definition?
Difference is 8023080109 5,899,345,920 bytes
128kbit MP3 coding 16kB per second
5,899,345,920/16,384 360000 s. 100 hours of
music

19
Other Number Systems

Octal number system
Digits 0, 1, 7
Aggregates 3 digits of binary system (i.e., 3-bit
number)
E.g., (57)8 (47)10
Hexadecimal number system
16 digits require 6 new digit symbols 0, 9, A,
B, C, D, E, F
Aggregates 4 digits of binary system (1 nibble or
half byte)
E.g., (1F)16 (31)10
Conversion between system

20
Number Base Conversions

Conversion from base r to decimal
Expansion to power series and
addition of terms
Conversion from decimal to base r
Divide number and successive quotients by r
Sequence of remainders is base r number
Example convert (41)10 to binary (r2)
41 20 ? 2 1
20 10 ? 2 0
10 5 ? 2 0
5 2 ? 2 1
2 1 ? 2 0
1 0 ? 2 1

a0 1 a1 0 a2 0 a3 1 a4 0 a5 1
(101001)2
21
Number Base Conversions

Conversion to/from octal and hexadecimal
Easier if done via binary
3 or 4 bit sequences correspond to digit
Example

(2414)10(100101101110)2

Direct conversion possible, too
2414 301 ? 8 6
301 37 ? 8 5
37 4 ? 8 5
4 0 ? 8 4 gt (4 5 5 6)8

22
Number Systems - Summary

Base r numbers
Binary
Octal
Hexadecimal
Conversion between number systems
Summation of power series
Division with remainders
Powers of 2
Computer systems based on powers of 2
kilo 1024
Mega 1024 1024
Giga 1024 1024 1024
Next
Computer arithmetic
Signs, complements

23
Computer Arithmetic

Computer arithmetic
Addition, subtraction
Multiplication
Signed numbers
Complements

24
Binary Addition

Binary addition works like normal addition
Stay within 0,1
Carries as usual
Example

Carry 111111
111101 10111
1010100

Addition of multiple numbers possible
Carry gets a bit more difficult

25
Binary Subtraction

Subtraction same as normal subtraction
Borrows as usual
Example

10 0
0 10
111101 - 10111
0
1
1
0
0
1
26
Binary Multiplication

Binary multiplication
Same as normal multiplication
Multiplication a lot easier in binary domain
Example

111101 1010
000000
111101
000000
111101
1001100010
27
Signed Numbers

How are signed numbers handled in base 10?
Plus or minus sign placed in front of number
Can we do that for binary numbers?
Sign needs to be represented in digital system
Only choice are 0 and 1
0 indicates
1 indicates
Examples on five-bit numbers
01101 11101 00000 10000
13 13 0 0
Signed Magnitude representation

28
Arithmetic with Signed Magnitude

Addition example 1
Plus signs are leading zeros -gt no problem
01001 (9)10
00010 (2)10
01011 (11)10
Addition example 2
What happens with negative numbers?
01001 (9)10
10010 (2)10
11011 ??? (7)10
Problem negative numbers
Sign can turn addition into subtraction

29
Signed Magnitude - Problems

Need to consider many cases to implement addition
correctly
Decision process for (A) (B)
If A ? B then
If A lt B then

magnitude calculation
A
-
B (AB), (A-B), -
- (A-B), (AB), -
sign of result
A
-
B (AB), (B-A),
- (B-A), - (AB), -
30
Signed Magnitude Representation

Arithmetic with signed magnitude is difficult
Two representations of zero
Different cases for addition and subtraction

1000
31
Complements

Complements allow easier arithmetic
Representation of negative numbers a bit more
involved
Two types of complements
Radix complement r s complement
Decimal 10s complement
Binary 2s complement
Diminished radix complement (r1)s complement
Decimal 9s complement
Binary 1s complement

32
Diminished Radix Complement

Given a number N in base r having n digits, the
(r1)s complement of N is defined as
(r n 1) N
Example for 6-digit decimal numbers
9s complement is (r n1)N (1061)N
999999N
9s complement of 546700 is 999999546700
453299
Example for 7-digit binary numbers
1s complement is (r n 1) N (271)N
1111111N
1s complement of 1011000 is 11111111011000
0100111
Observation
Subtraction from (rn1) will never require a
borrow
Diminished radix complement can be computed
digit-by-digit
For binary 1 0 1 and 1 1 0
Flips 0s to 1s and 1s to 0 (bit
complementation)

33
Ones Complement Representation

1s complement is simple to compute
Bit complementation
Still, two representations of zero

1000
34
Radix Complement

The rs complement of an n-digit number N in base
r is defined as
r n N for N ? 0 and 0 for N 0
Radix complement is diminished radix complement
1 (r n 1) N 1 r n N
Example for 6-digit decimal numbers
10s complement is r n N 106N 1000000N
10s complement of 546700 is 1000000546700
453300
Rule Leave least significant 0s unchanged,
subtract first nonzero least significant digit
from 10, subtract all higher significant digits
from 9.
Example for 7-digit binary numbers
2s complement is r nN 27N 10000000N
2s complement of 1011000 is 100000001011000
0101000
Rule Leave least significant 0s and first 1
unchanged, replace 1s with 0s and 0s with 1s
in all higher significant digits.

35
Twos Complement Representation

Arithmetic with 2s complement is most efficient
Bit complementation 1
Single representation of zero!

0
1000
36
Complements - Summary

Complement of complement is original number
Diminished radix complement
(r n1)((r n1)N) r n1r n1 N N
Radix complement
r n(r nN) r nr n N N
Representation of zero
Radix complement 0
Diminished radix complement r n 1 and its
complement 0
Radix points
Remove radix point
Compute complement
Put radix point back at same relative position

37
Comparison ( n4 )
Decimal
Signed Magnitude
Ones Complement
Twos Complement
-8 -
-
1000 -7
1111
1000 1001 -6
1110
1001
1010 -5 1101
1010
1011 -4
1100 1011
1100 -3
1011
1100
1101 -2 1010
1101
1110 -1
1001 1110
1111 0
0000 or 1000 0000
or 1111 0000
1 0001
0001
0001 2 0010
0010
0010 3
0011
0011 0011 4
0100
0100
0100 5 0101
0101
0101 6
0110 0110
0110 7
0111
0111 0111
38
Subtraction with Complements

Subtraction M N
Add minuend M to rs complement of subtrahend N
M N M (rn N) M N r n
What about the additional r n ?
If M ? N
Then M N r n ? r n
With n bits only numbers lt r n can be expressed
n1st digit is ignored (drop the carry out bit)
If M lt N
Then M N r n r n (N M) r s
complement of (N-M)
r s complement signifies negative number
So, (N M) M N

39
Arithmetic with Radix Complement

Subtraction works as addition of complement
Addition/subtraction of signed numbers
Negative numbers are expressed as r s complement
Simple addition yields correct result
No need to distinguish different cases
Addition/subtraction of unsigned numbers
Can be performed on same hardware as signed
numbers
Most modern digital systems use 2s complement

40
Twos Complement Overflow
Consider two 8-bit 2s complement numbers. I can
represent the signed integers -128 to 127 using
this representation.What if I do (1) (127)
128. The number 128 is OUT of the RANGE
that I can represent with 8 bits. What happens
when I do the binary addition?
127 7F 1 01------------------
- 128 / 80 (this is actually -128 as a
twos complement
number!!! - the wrong answer!!!)How do I know
if overflowed occurred? Added two POSITIVE
numbers, and got a NEGATIVE result.
41
Detecting Twos Complement Overflow
Twos complement overflow occurs is Add two
POSITIVE numbers and get a NEGATIVE result
Add two NEGATIVE numbers and get a POSITIVE
resultI CANNOT get twos complement overflow if
I add a NEGATIVE and a POSITIVE number
together. The Carry out of the Most Significant
Bit means nothing if the numbers are twos
complement numbers.
42
Overflow Conditions
Add two positive numbers to get a negative
number or two negative numbers to get a positive
number
43
Example Overflow
0 1 1 1 0 1 0 1 0 0 1 1 1 0 0 0
1 0 0 0 1 0 0 1 1 1 0 0 1 0 1 1 1
5 3 -8
-7 -2 7
Overflow
Overflow
0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 1
1 1 1 1 1 1 0 1 1 0 1 1 1 1 0 0 0
5 2 7
-3 -5 -8
No overflow
No overflow
Overflow when carry in to sign does not equal
carry out
44
Floating Point Numbers (I)

Fixed point numbers have limited range.
To represent very large or very small numbers, we
use floating point numbers (cf. scientific
numbers).
Examples
0.23 x 1023 (very large positive number)
-0.1239 x 10-10 (very small negative number)

45
Floating Point Numbers (II)

Floating point numbers have three parts
mantissa, base, and exponent
Base is usually fixed for each number system.
Therefore, needs only mantissa and exponent.

46
Floating Point Numbers (III)

Mantissa is usually in normalised form
(base 10) 23 x 1021 normalised to 0.23 x
1023
(base 10) -0.0017 x 1021 normalised to
-0.17 x 1019
(base 2) 0.01101 x 23 normalised to
0.1101 x 22
A 16-bit floating point number may have 10-bit
mantissa and 6-bit exponent.
More bits in exponent gives larger range.
More bits for mantissa gives better precision.

47
Arithmetic with Floating Point Numbers (I)

Arithmetic is more difficult for floating point
numbers.
MULTIPLICATION
Steps (i) multiply the mantissa
(ii) add-up the exponents
(iii) normalise
Example
(0.12 x 102)10 x (0.2 x 1030)10
(0.12 x 0.2)10 x 10230
(0.024)10 x 1032 (normalise)
(0.24 x 1031)10

48
Arithmetic with Floating Point Numbers (II)

ADDITION
Steps (i) equalise the exponents
(ii) add-up the mantissa
(iii) normalise

Example
(0.12 x 103)10 (0.2 x 102)10
(0.12 x 103)10 (0.02 x 103)10 (equalise
exponents)
(0.12 0.02)10 x 103 (add
mantissa)
(0.14 x 103)10
Can you figure out how to do perform SUBTRACTION
and DIVISION for (binary/decimal) floating-point
numbers? (Hint remember sometimes, the mantissa
might need normalizing)

49
Weighted and Unweighted Codes

Most numeric number representations are in a
class known as Weighted Codes where
Binary integers and fractions are special case
where weights are powers of 2
Unweighted codes are codes that cannot be
assigned a weight value for each bit

50
Binary Coded Decimal

Four bits are used to represent each decimal
digit
In each 4-bit group, 6 values are not used
Many possible codes, natural BCD (equivalent
binary digits) most common
BCD is not as efficient as binary
BCD is easy to convert to/from decimal (it really
is decimal with different symbols)
BCD add/subtract circuits are complex

51
BCD Code Examples
Weighted codes
Unweighted code
Digit 0 1 2 3 4 5 6 7 8 9
8421 0000 0001 0010 0011 0100 0101 0110 0111 1000
1001
84-2-1 0000 0111 0110 0101 0100 1011 1010 1001 10
00 1111
XS3 0011 0100 0101 0110 0111 1000 1001 1010 1011
1100
The 8421 or natural BCD code is the most common
BCD code in use
52
Other Decimal Codes
Weighted codes
Unweighted code
Decimal digit BCD 8421 2421 Excess-3 8 4 -2 -1
0 0000 0000 0011 0000
1 0001 0001 0100 0111
2 0010 0010 0101 0110
3 0011 0011 0110 0101
4 0100 0100 0111 0100
5 0101 1011 1000 1011
6 0110 1100 1001 1010
7 0111 1101 1010 1001
8 1000 1110 1011 1000
9 1001 1111 1100 1111
The 8421 or natural BCD code is the most common
BCD code in use
53
BCD Addition
Case 1
Case 2
0001 1 0101 5 (0) 0110 (0) 6
0110 6 0101 5 (0) 1011 (1) 1
WRONG!
Case 3
1000 8 1001 9 (1) 0001 (1) 7
Note that for cases 2 and 3, adding a factor of 6
(0110) gives us the correct result.
54
BCD Addition (cont.)

BCD addition is therefore performed as follows
1) Add the two BCD digits together using normal
binary addition
2) Check if correction is needed
a) 4-bit sum is in range of 1010 to 1111
b) carry out of MSB 1
3) If correction is required, add 0110 to 4-bit
sum to get the correct result BCD carry out 1

55
BCD Negative Number Representation

Similar to binary negative number representation
except r 10.
BCD sign-magnitude
MSD (sign digit options)
MSD 0 (positive) not equal to 0 negative
MSD range of 0-4 positive 5-9 negative
BCD 10s complement
-N ? 10r - N 9s complement 1
BCD 9s complement
invert each BCD digit (0?9, 1 ? 8, 2 ? 7,3 ? 6,
7 ? 2, 8 ? 1, 9 ? 0)

56
Negative BCD Numbers

84-2-1 and XS3 codes allow for easy digit
inversion.
XS3 code is also easy to implement
Addition is like binary
Correction factor is -3 or 3

57
Decimal Arithmetic

Everything needs to be 4-bit aligned
represented by 0 (0000)
represented by 9 (1001)
Signed magnitude representation or complements
Signed magnitude hardly used
10s complement most common
Example 375 (240)
Negative numbers represented by 10s complement
10s complement of 240 is 104 240 9760
Addition of all digits and discard of end carry
Sign of result automatically correct

0 375
9 760
0 135
58
Mechanical encoder with sequential sector
numbering
At boundaries can get a code different than
either adjacent sector
59
Gray Codes

Grey codes are minimum change codes
From one numeric representation to the next, only
one bit changes
Primary use is in numeric input encoding apps.
where we expect non-random input values changes
(I.e. value n to either n-1 or n1)
Milling machine table position
Rotary shaft position

60
Rotary Encoder

an electro-mechanical device used to convert the
angular position of a shaft or axle to a digital
code, making it a sort of transducer. These
devices are used in robotics, in top-of-the-line
photographic lenses, in computer input devices
(such as optomechanical mice and trackballs), and
in rotating radar platforms.

61
Standard binary encoding
Sector Contact 1 Contact 2 Contact 3 Angle
1 off off off 0 to 45
2 off off on 45 to 90
3 off on off 90 to 135
4 off on on 135 to 180
5 on off off 180 to 225
6 on off on 225 to 270
7 on on off 270 to 315
8 on on on 315 to 360
the drawback that if the disc stops between two
adjacent sectors, or the contacts are not
perfectly aligned, it can be impossible to
determine the angle of the shaft.
62
Example problem

off- on- on (starting position)
on- on- on (first, contact 1 switches on)
on- on- off (next, contact 3 switches off)
on- off- off (finally, contact 2 switches off)
The sequence are 4,8,7,and then 5.
The shaft appears to have jumped from sector 4 to
sector 8, then gone backwards to sector 7, then
backwards again to sector 5, which is where we
expected to find it.
In many situations, this behavior is undesirable
and could cause the system to fail.
For example, if the encoder were used in a robot
arm, the controller would think that the arm was
in the wrong position, and try to correct the
error by turning it through 180, perhaps causing
damage to the arm.

63
Gray encoding
Sector Contact 1 Contact 2 Contact 3 Angle
1 off off off 0 to 45
2 off off on 45 to 90
3 off on on 90 to 135
4 off on off 135 to 180
5 on on off 180 to 225
6 on on on 225 to 270
7 on off on 270 to 315
8 on off off 315 to 360
In the previous example, the transition from
sector 4 to sector 5, like all other transitions,
involves only one of the contacts changing its
state from on to off or vice versa. This means
that the sequence of incorrect codes shown in the
previous illustration cannot happen here
64
Gray Code for decimal Digits
Gray Code0 00001 00012
00113 00104 01105
11106 10107 10118
10019 1000
A Gray code changes by only 1 bit for adjacent
values. This is also called a thumbwheel code
because a thumbwheel for choosing a decimal digit
can only change to an adjacent value (4 to 5 to
6, etc) with each click of the thumbwheel. This
allows the binary output of the thumbwheel to
only change one bit at a time this can help
reduce circuit complexity and also reduce signal
noise.
65
Gray Codes (cont.)
66
Binary Code to Gray Code Conversion

Convert n bit Binary code into n bit Gray Code
Bit i of the Gray Code is obtained by comparing
bits i and i1 of the Binay Code - Add extra 0
to the left of the Binary code- If bits i and
i1 in the Binary code are the same then bit i of
the Gray code is 0 , else bit i is 1.
Example Binary 10011
010011Gray Code 11010
Exercise - Convert the Binary Code 11101 into
Gray code - Convert the Gray
Code 1101 !!
Answer - The gray Code is 10011
- The Binary Code is 1001

67
Alphanumeric Representation

Binary codes used to represent alphabetic and
numeric characters
Two most common are
ASCII, 7 bit code, 128 symbols
EBCDIC, 8 bit code, 256 symbols
Problems can arise when comparing symbol values
(collation)
Comparing A to a in ASCII system yields
different results in an EBCDIC system.

68
ASCII CODE

The ASCII code (American Standard Code for
Information Interchange) is a 7-bit code for
Character data. Typically 8 bits are actually
used with the 8th bit being zero or used for
error detection (parity checking).8 bits 1
Byte. A 01000001 41
00100110 267 bits can only represent 27
different values (128). This enough to represent
the Latin alphabet (A-Z, a-z, 0-9, punctuation
marks, some symbols like ), but what about other
symbols or other languages?

69
ASCII CODE
b6b5b4
b3b2b1b0 000 001 010 011 100 101 110 111
0000 NUL DLE SP 0 _at_ P p
0001 SOH DC1 ! 1 A Q a q
0010 STX DC2 2 B R b r
0011 ETX DC3 3 C S c s
0100 EQT DC4 4 D T d t
0101 ENQ NAK 5 E U e u
0110 ACK SYN 6 F V f v
0111 BEL ETB 7 G W g w
1000 BS CAN ( 8 H X h x
1001 HT EM ) 9 I Y I y
1010 LF SUB J Z j z
1011 VT ESC K k
1100 FF FS , lt L \ l
1101 CR GS - M m
1110 S0 RS . gt N n
1111 S1 US / ? O _ o DEL
70
UNICODE

UNICODE is a 16-bit code for representing
alphanumeric data. With 16 bits, can represent
216 or 65536 different symbols.16 bits 2 Bytes
per character.
0041-005A A-Z0061-4007A a-z
Some other alphabet/symbol ranges
3400-3d2d Korean Hangul Symbols3040-318F
Hiranga, Katakana, Bopomofo,
Hangul4E00-9FFF Han (Chinese, Japenese,
Korean)UNICODE used by Web browsers, Java, most
software these days.

71
Error Detection and Error Correction Code

the transmission of data over a channel may
result in error due to interference, noise, etc.
hence, data will be corrupted
some schemes have been devised to detect and
correct the error
it is usually accomplished by inserting extra
bits in the data being transmitted

72
Single error detection with parity

simplest method for error detection
insert an additional parity bit for transmission
even parity may be generated by
for a 4-bit data
e.g. for a 4-bit data 0110
odd parity may be obtained by
e.g.

73
Parity Bit

ASCII code may have an extra bit appended to
detect data transmission errors
P 0 if the number of 1s in the character is
even, else P 1 (even parity)
P 0 if the number of 1s in the character is
odd, else P 1 (odd parity)
If any single bit changes, parity will be wrong
at receive end

Even parity Odd parity ASCII A
1000001 01000001 11000001 ASCII T
1010100 11010100 01010100
74
Parity Code Example

Concatenate a parity bit to the ASCII code for
the characters 0, X, and to produce both
odd-parity and even-parity codes.

Character ASCII Odd-Parity ASCII Even-Parity ASCII
0 0110000 10110000 00110000
X 1011000 01011000 11011000
0111100 10111100 00111100
75
Binary Storage and Registers