[final review notes] digital logic

Chapter one: Fundamentals of digital logic

• Binary system 、 octal 、 Conversion between decimal and hexadecimal
• Original code 、 Inverse code 、 Conversion between complements

8421BCD code 、 Gray code 、 The conversion between the remainder three code base

Define concepts ：

8421 code ：8421 Code is the most commonly used BCD code
, Is one of the most commonly used decimal codes . In this coding method , Of each binary code “1” All represent a fixed value . Will each “1” The binary numbers represented can be added up to get the decimal number it represents . Because every bit in the code looks from left to right “1” They represent numbers “8”“4”“2”“1”, Therefore, the name 8421 code . Each of them “1” The decimal number represented is called the weight of this digit . Because everyone's power is fixed , therefore 8421 Code is constant weight code .

more than 3 code ： more than 3 The code is in 8421 On the basis of , Add... To every code 0011（3） Code , His main advantage is that when adding decimal numbers , It can generate carry signal correctly , It also brings convenience to subtraction

Gray code ： Make any two adjacent codes have only one binary bit in different states , rest 3 Binary bits must have the same state , The benefits of this coding are , Change from one code to the next adjacent code , Only one person's state changes , It is beneficial to get better decoding waveform . Gray code is a kind of cyclic code

• frequently-used BCD code

Decimal number	8421 code	more than 3 code	Gray code
0	0000	0011	0000
1	0001	0100	0001
2	0010	0101	0011
3	0011	0110	0010
4	0100	0111	0110
5	0101	1000	1110
6	0110	1001	1010
7	0111	1010	1000
8	1000	1011	1100
9	1001	1100	1101

8421 Code to decimal conversion

example 1：
(3)₁₀ =(0011)₈₄₂₁

because ：3 Can be 8、4、2、1 in 4 Of the numbers 2 and 1 Add to get , So you just need to be in the corresponding 2 and 1 Write below 1 that will do , rest 4 Place complement 0

In turn to see , use 8421 Code marks binary numbers every four digits , contain 1 Add the corresponding numbers above the position of to get 10 Hexadecimal number

example 2：
(712)₁₀ =(0111 0001 0010)₈₄₂₁
because ： Let's look at the individual figures ,7 Can be 8、4、2、1 in 4 Of the numbers 4+2+1 get 、1 Can be 1 get 、2 Can be 2 get , So just write... Under the corresponding number 1, The rest are the same 0 that will do

7：

1：

2：

therefore (712)₁₀ =(0111 0001 0010)₈₄₂₁

In turn to see , use 8421 Code marks binary numbers every four digits , contain 1 Add the corresponding numbers above the position of to get 10 Hexadecimal number

8421 Code and remainder code conversion

It was said that , more than 3 The code is in 8421 On the basis of , Add... To every code 0011（3） Code , So we are 8421 Code based on the operation can

example 1：
(1001)₈₄₂₁ =(1100) _{The remaining three yards}

example 2：
(0111 0001 0010)₈₄₂₁ =(1010 0100 0101) _{The remaining three yards}

Gray code conversion

First write the binary number , Like binary 0110, The corresponding four digit gray code is ： First in the right 0 And the second from the right 1 Do XOR （ Same as 0, Different for 1）, In this way, I get 1 As the first place of gray code , By analogy , The highest word remains the same as binary （ Here is 0）, In this way, the gray code is 0101.

(0110)₈₄₂₁ =(0101) _{Gray code}