Binary Encoding

Binary encoding is a numerical encoding method consisting of two digits, 0 and 1. It is widely used in computer science and electronic engineering because the electronic components in a computer processor can only understand 0 and 1.

In binary encoding, each digit represents a weight, which is a power of 2. For example, a binary number 1101 represents:

1 × 2³ + 1 × 2² + 0 × 2¹ + 1 × 2⁰ = 8 + 4 + 0 + 1 = 13

Binary encoding can be used to represent various data types, such as numbers, characters, images, audio, video, etc. Inside a computer, all data is stored and processed in the form of binary encoding. For example, the binary encoding of a letter or symbol in a computer can be represented using standards such as ASCII code or Unicode code.

1. The Definition of Encoding

Encoding is defined as using binary numbers with a certain number of bits to represent decimal digits, letters, symbols and other information.

Through encoding, we can convert raw data into a format that can be processed and stored by computers. Common encoding methods include ASCII code and Unicode. In encoding, each binary digit represents a certain weight, usually a power of 2. Encoding can be used to represent various data types such as numbers, characters, images, audio, and video. In computers, all data is stored and processed in the form of binary encoding.

2. Source Code

Source code is an encoding formed by one or more 0s and 1s arranged in a specific order, such as 10, 01, 00, 11, etc.

When given specific rules, the source code can be converted into a well-defined encoding. For example, if the weight between the binary source code words is set to 2ⁿ, an unsigned binary number is formed. If positive and negative sign rules are added to the unsigned number, a signed number is formed. If a decimal point rule is added, a special binary fraction encoding is formed.

3.One’s Complement

The concept of Two’s Complement in Chinese is very simple. It refers to a binary encoding method for integers in which each bit of the original code is flipped to obtain one’s Complement. Flipping a bit means changing 1 to 0 and 0 to 1.

For example, one’s complement code of “4’b1010” is “4’b0101”, and the one’s complement code of “8’b1000_1111” is “8’b0111_0000”.

4.Bias Code and Two’s Complement

The encoding of a binary number using the usual sequence of bits is called the natural code. For example, 0000 represents 0 and 1111 represents 15. Sometimes, a bias is added to the natural code to serve some specific purpose, which results in a biased code.

In some cases, a bias may also form due to direct current bias effect in circuit processing, which requires the study of how to eliminate the bias. The most commonly used biased code is the two’s complement, which is formed by subtracting 2^N-1 (half of the whole number system) from the natural code. Here, N represents the number of bits in the binary system. For example, in a 4-bit binary system, the natural code is 0000-1111, and the bias is -2^N-1=-23=-8, resulting in the two’s complement of a signed number. Please check following table for natural code and bias code of commonly used Integers .

table 1. natural code and bias code of commonly used Integers 

5. BCD code

BCD code, also known as binary coded decimal, uses a 4-bit binary number (b₃b₂b₁b₀) to represent the decimal digits 0 to 9. It is commonly referred to as the 8421 BCD code, where the weights of each digit position are 8, 4, 2, and 1, respectively.

The main reason for using BCD code is that binary or hexadecimal numbers are not intuitive when inputting, outputting, or displaying, such as (C8)₁₆ or (11001000)₂. We cannot immediately determine the size of the number. However, if the decimal number 200 is given, we can quickly respond to the size of the number.

It is important to note that not every 4-bit binary number corresponds to a BCD digit. There are six 4-bit binary numbers (1010), (1011), (1100), (1101), (1110), and (1111) that have no corresponding BCD code.

Table2.  BCD Code Table

 

6.What is the Difference of BCD Code and Hexadecimal ?

Common points:

• Both BCD code and hexadecimal number use 4-bit binary digits as their basic unit (bit).
• They can represent the same numerical values from 0 to 9.

Differences:

• BCD code has a radix of 10, while hexadecimal has a radix of 16.
• BCD code only has the basic digits 0-9, while hexadecimal has 0-9 and the additional letters A-F.
• BCD code has a weight of 10, while hexadecimal has a weight of 16.

7. ASCII encoding

To facilitate the use of computers, 7-bit binary numbers are encoded. The ASCII (American Standard Code for Information Interchange) code can easily handle numbers, letters, words, and articles. Later, it was extended to 256 Unicode and other codes. However, ASCII encoding is indeed the foundation of computer encoding.

Standard or lower ASCII characters and codes

Char	Dec	Binary	Char	Dec	Binary	Char	Dec	Binary
!	033	00100001	A	065	01000001	a	097	01100001
“	034	00100010	B	066	01000010	b	098	01100010
#	035	00100011	C	067	01000011	c	099	01100011
$	036	00100100	D	068	01000100	d	100	01100100
%	037	00100101	E	069	01000101	e	101	01100101
&	038	00100110	F	070	01000110	f	102	01100110
‘	039	00100111	G	071	01000111	g	103	01100111
(	040	00101000	H	072	01001000	h	104	01101000
)	041	00101001	I	073	01001001	i	105	01101001
*	042	00101010	J	074	01001010	j	106	01101010
+	043	00101011	K	075	01001011	k	107	01101011
,	044	00101100	L	076	01001100	l	108	01101100
–	045	00101101	M	077	01001101	m	109	01101101
.	046	00101110	N	078	01001110	n	110	01101110
/	047	00101111	O	079	01001111	o	111	01101111
0	048	00110000	P	080	01010000	p	112	01110000
1	049	00110001	Q	081	01010001	q	113	01110001
2	050	00110010	R	082	01010010	r	114	01110010
3	051	00110011	S	083	01010011	s	115	01110011
4	052	00110100	T	084	01010100	t	116	01110100
5	053	00110101	U	085	01010101	u	117	01110101
6	054	00110110	V	086	01010110	v	118	01110110
7	055	00110111	W	087	01010111	w	119	01110111
8	056	00111000	X	088	01011000	x	120	01111000
9	057	00111001	Y	089	01011001	y	121	01111001
:	058	00111010	Z	090	01011010	z	122	01111010
;	059	00111011	[	091	01011011	{	123	01111011
<	060	00111100	\	092	01011100	|	124	01111100
=	061	00111101	]	093	01011101	}	125	01111101
>	062	00111110	^	094	01011110	~	126	01111110
?	063	00111111	_	095	01011111		127	01111111
@	064	01000000	`	096	01100000

Extended ASCII characters and codes

Extended ASCII uses eight instead of seven bits, which adds 128 additional characters.

This gives extended ASCII the ability for extra characters, such as special symbols, foreign language letters, and drawing characters as shown below.

Extended or Higher ASCII characters and codes