Learning notes of digital circuit (II)

Decimal system

• Abacus is a decimal counter , Any number can be used 0,1,2,3,4,5,6,7,8,9 this 10 One or more of the numbers , Arranged according to a certain rule to show , Its counting rule “ Dot into one ”.
• 10=1×10¹+0×10⁰, With 10 A counting system based on Radix .
• Such as 4587.29 It can be expressed as ：4587.29=4×10³+5×10²+8×10¹+7×10⁰+2×10^-1+9×10^-2, among 10³、10²、10¹、10⁰ Thousand bits respectively 、 Hundred bit 、 The right of ten digits and each digit , The weight of the number to the right of the decimal point is a negative power .
• Generally speaking , Any decimal number can be expressed as ：$$(N{)}_D=\sum _{i=-\infty }^{\infty }{K}_i\times 10^i$$
• D(Decimal) For decimal .
• Expand
  • hold 10 use R Instead of , You can get the expression of any decimal number ： $$(N{)}_D=\sum _{i=-\infty }^{\infty }{K}_i\times R^i$$
  • $K_i$ yes $i$ Coefficient of power , According to the cardinality $R$ Different , Its value is 0 To R-1 A different number . For example, for decimal numbers ,$R$ by 10, therefore $K_i$ The values for 0~9 common 10 Number .
• Digital circuits are not convenient for storing or processing decimal numbers . The basic idea of forming a digital circuit is to correspond the state of the circuit to the number . Binary only 0 And 1 Two numbers , It is represented by two states of digital circuit . Decimal 10 A digital number requires that the circuit have 10 A completely different state , This makes the circuit very complicated .

Binary system

• Binary , Only 0 And 1 Two numbers , Counting rule “ On the two into one ”, namely 1+1=10=1×2¹+0×2⁰.
• Binary is based on 2 A counting system based on Radix . Any binary number can be expressed as ：$$(N{)}_B=\sum _{i=-\infty }^{\infty }{K}_i\times 2^i$$
• $K_i$ Cardinal number “2” Of the $i$ Coefficient of power , It can be “0” perhaps “1”.
• Put the binary number (1010110)_B Convert to decimal ：
  • (1010110)_B=1×2⁶+0×2⁵+1×2⁴+0×2³+1×2²+1×2¹+0×2⁰=(86)_D
• advantage ：
  • The digital circuit is simple and reliable , Few components are used .
  • The basic operation rules are simple , Easy to operate .
• shortcoming ： There are too many digits , Not easy to write or read , Inconvenient to use .
• When entering the computer , The computer first converts decimal numbers into binary numbers that the digital system can accept ; At the end of the operation , Then convert binary numbers to decimal numbers , Output decimal results .

Four operations of binary system

• Addition operation
  • “ On the two into one ”, The operation rules are as follows ：0+0=0,0+1=1,1+0=1,1+1=10.
  • When you meet “1+1” And then move to the adjacent high position 1.
    
• Subtraction
  • “ Borrow one for two ”, The operation rules are as follows ：0-0=0,1-0=1,1-1=0,10-1=1.
  • When you meet “0-1” when , It is necessary to borrow from a high position 1 When 2 use .
    
• Multiplication
  • “ Multiply the numbers , Then add ”, The operation rules are as follows ：0×0=0,1×0=0,0×1=0,1×1=1.
    
• Division operations
  • “ Divide the numbers , And then subtract ”, The operation rules are as follows ：0÷1=0, 1÷1=1.
    

Binary waveform representation

• Binary data is usually represented by digital waveform , In binary 0~15
  
• The leftmost column indicates the bit weight of binary numbers （2⁰、2¹、2²）, Its lowest （Least Significant Bit, LSB）, highest （Most Significant Bit, MSB）, The last line marks the equivalent 0~15 The decimal number of .

Transmission of binary data

• Binary data is transferred from one place to another , It can be serial or parallel .
• Serial mode ：
  • A group of data is transmitted bit by bit under the control of a clock pulse .
  • The equipment required is simple , Only one conductor and a common grounding terminal are required .
  • Between two computers , Or the computer is connected to the network through the telephone line .
    
  • Under the control of clock pulse , The data is represented by the highest bit MSB To the lowest LSB In turn . One clock cycle is required for each bit transmitted , And it is completed at the falling edge of the clock pulse .
• Parallel transmission ：
  • Fast transmission speed , Transfer a set of binary data at the same time .
  • The equipment required is complex , Multiple transmission lines and other components are required .
  • Parallel transmission is a common technique in digital systems .
    
  • Pictured , need 8 Transmission line . transmission 8 The time required for bit data is the period of a clock pulse , Only serial transmission time 1/8.

Hexadecimal number

• Hexadecimal numbers have 16 Number ：0,1,2,3,4,5,6,7,8,9,A(10),B(11),C(12),D(13),E(14),F(15).
• Every 16 to one .
• (3A6.D)₁₆=3×16²+10×16¹+6×16⁰+13×16^-1.
• Hexadecimal common letters H Express , so (3A6.D)₁₆=(3A6.D)_H.

Number conversion

• Binary to decimal

  • Multiply the bits of binary number by bit weight and then sum .
  • (101.1)₂=1×2²+0×2¹+1×2⁰+1×2^-1=4+0+1+0.5=(5.5)₁₀

• Decimal to binary

  • Adopt Division 2 Surplus method , Divide decimal numbers by 2, And write down the remainder in turn , Divide until the quotient is 0, Finally, put all the remainder in reverse order , You can get binary numbers .
    
    

• Binary to hexadecimal

  • From the decimal point to the left 、 Right click 4 Bit grouping , Insufficient 4 Bit , The integer part can be added to the left of the highest bit 0 A filling , The decimal part is not enough 4 Bit , Add... To the right of the lowest position 0 A filling , Each group is replaced by its corresponding hexadecimal number , Write out the hexadecimal numbers in turn .
    
    

• Hexadecimal number to binary number

  • From left to right, use... For each number in the hexadecimal number to be converted 4 Bit binary number means .