Digital circuit and system design (1)

Chapter 1 Fundamentals of Digital Circuits

1.1 Overview of digital circuits

1.1.1 Overview of digital circuits

Digital circuit : A circuit that uses digital signals to perform arithmetic and logical operations, called digital circuits or digital systems. Digital circuits have logic operations and logic processing functions, also known as digital logic circuits.
Digital signal : The change in time and amplitude is a discrete signal. That is, discrete in time, and integer changes in amplitude (low level represents logic 0, high level represents logic 1).

1.1.2 Development and Classification of Digital Circuits

Electron tube-semiconductor discrete component-small scale integrated circuit (SSI)-medium scale integrated circuit (MSI)-large scale integrated circuit (SSI)-very large scale integrated circuit (VLSI)

1.1.3 Materials and processes of integrated circuits

Materials : mainly silicon materials, but also compound semiconductor materials such as gallium arsenide.
Process :

1. TTL: Transistor transistor logic
2. CMOS: Complementary Metal Oxide Semiconductor

Note : TTL digital logic devices came out relatively early. With the development of high-speed, low-power CMOS technology, the dominant position of TTL has a tendency to be replaced by CMOS devices .

1.1.4 Analysis methods and testing techniques of digital circuits

1. Analysis methods of digital circuits.
   Basic analysis methods:
   function tables, truth tables, logic expressions, waveform diagrams, etc.
   Simulation software: EWB, Quartus II

2. The testing method of digital circuit
Digital multimeter, oscilloscope, logic analyzer

Questions:
1. Why is digital circuit also called switch circuit?
2. Which instruments are commonly used for testing digital circuits?

1.2 Introduction to the digital system

Base conversion:

A three-digit binary number represents an octal number; a
four-digit binary number represents a hexadecimal number

Convert decimal to binary: integer part-divide by two and round ; decimal part-multiply by two and round off
Decimal to hexadecimal (octal):
Method 1:
Integer part-divide by 16 (8) and take the remainder of the
decimal part- -Multiply by 16 (8) to round off
Method 2:
Convert to binary first, with the help of the relationship between binary and hexadecimal (octal).

BCD encoding

The process of arranging a number of binary digits according to a certain rule to represent given information is called encoding.
Binary coded Decimal (BCD-Binary coded Decimal)
BCD code: A four-digit binary represents a decimal number.

• 8421 BCD code
  8 a3+4 a2+2 a1+1 a0
• 5421 BCD code
  5 a3+4 a2+2 a1+1 a0
• Remaining 3 codes (8421BCD+0011)

Cycle code (Gray code)

Cyclic code is a kind of reliability code.
Composition feature: The highest position on both sides of the symmetry axis is symmetrical reversed, and the other low positions are equal .
The characteristic of cyclic code: only one bit is different between two adjacent code groups . Two adjacent codes are also called unit distance codes.

Odd/Even coding (*)

Odd/even code structure:
information bit | test bit (or test bit | information bit)
Odd code: the sum of the number of 1 in the information bit and the test bit is an odd number.
Even code: The sum of the number of 1 in the information bit and the test bit is an even number.

Reliability of odd/even coding:
Before adding odd/even test bit, the probability of a bit error in the code group is 1/2.
For example: 8421BCD code 1001 becomes 1000 or 0001, and no error will be found.
After adding the odd/even test bit, when an error occurs in a bit in the code group, the odd/even will change, so that the error can be found.
For example, if the odd 8421BCD code 10011 becomes 10001 or 00011, the error can be found.
Only when the two bits are wrong at the same time, the original odd/evenness remains unchanged, so that the error cannot be found.
For example: Odd 8421 code 01011 becomes 10011 or 01000, and no error can be found.

Signed number encoding

How does the number system represent positive and negative numbers?
Usually the highest digit is used as the sign bit, 0 means a positive number, 1 means a negative number, and the remaining bits are numerical bits.

Original code

The original code representation method of the binary number is: sign bit + value bit
eg: the true value is +36 and -36 respectively, if it is represented by the original code of 8-bit word length, it can be written as:
[36]Original=00100100 [ -36] Original=10100100

One's complement

The inverse code of a positive number: the same as the original code, sign bit + value bit
. The inverse code of a negative number: sign bit is "1" + the value of the original code is inverted by bit.
eg: The true value is +36 and -36 respectively, if used It can be expressed by the inverse code of 8-bit word length:
[36]Original=00100100 [36]Reverse=00100100
[-36]Original=10100100 [36]Reverse=11011011

Complement

The complement of a positive number and 0: the same as the original code, the sign bit + the value bit
The inverse of a negative number: the inverse of a negative number +1
eg: The true value is +36 and -36 respectively, if you use an 8-bit complement To express it, it can be written as:
[36]Original=00100100 [36]Supplement=00100100
[-36]Original=10100100 [36]Reverse=11011011 [-78]Supplement=11011100

Note:
(1) The decimal value range represented by the original binary code, one's complement code and the complement code of n-bit word length is:
original code: -(2 ^n-1 -1) — +(2 ^n-1 -1)
reverse Code: -(2 ^n-1 -1) — +(2 ^n-1 -1)
Complement code: -2 ^n-1 — +(2 ^n-1 -1) (excluding -0)

Chapter 2 Fundamentals of Logical Algebra

2.1 Three basic operations of logical algebra

2.2 Formulas and rules of logical algebra

2.2.1 Logic Algebraic Formula

证明加对乘的分配律：A+BC=(A+B)(A+C)
证明： 
	右边![在这里插入图片描述](https://img-blog.csdnimg.cn/20200924085359414.png?x-oss-process=image/watermark,type_ZmFuZ3poZW5naGVpdGk,shadow_10,text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzM3OTU1NzA0,size_16,color_FFFFFF,t_70#pic_center)
	= AA+AC+BA+BC
		= A*1+AB+AC+BC
		= A(1+B+C)+BC
		= A+BC
		= 左边
		得证！

2.2.2 Chemical formula

证：AB+A‘C+BC=AB+A‘C
	证明：左边 = AB+A'C+(A+A')BC
			=AB+A'C+ABC+A'BC
			=AB(1+C)+A'C(1+B)
			=AB+A'C
			=右边
		得证！

证：AB+A'C+BCD=AB+A'C
	证明：左边 = AB+A'C+(A+A')BCD
			= AB+ A'C+ABCD+A'BCD
			= AB(1+CD)+A'C(1+BD)
			= AB+A'C
			= 右边
		得证！

证：(A+B)(A'+C)(B+C) = (A+B)(A'+C)
	证明：等式两边同时取非，可得
		[(A+B)(A'+C)(B+C)]' = [(A+B)(A'+C)]'
		(A+B)'+(A'+C)'+(B+C)' = (A+B)'+(A'+C)'
		A'B'+AC'+B'C' = A'B'+AC'
	得证：AB+A'C+BC = AB+A'C    （1）
	又因为 (1)左边 = AB+A'C+(A+A')BC
				= AB+A'C+ABC+A'BC
				= AB(1+C)+A'C(1+B)
				= AB+A'C
	得证！

2.2.3 Three important rules

若F=(A+B')[C+(D'+E)']',则试着写出其反函数？
解：F'=(A'B)+[C'(DE')']'

2.3 Compound logic operations and compound gates

2.3.1 Commonly used coincidence logic operations and compound gates

2.3.2 The equivalent symbols of logic gates

2.3.3 Open collector gate and tri-state logic gate

2.4 Common forms of logical function expressions

2.4.1 Common forms

2.4.2 Two standard forms of logic functions

2.5 Simplification method of logic function

2.5.1 Algebraic Simplification

2.5.2 Karnaugh map simplification

1. The composition of the Karnaugh map

2. Karnaugh map representation of logical functions

3. The merging law of Karnaugh map

4. Karnaugh map method to simplify the logic function

(1) Simplify the function to the simplest AND OR formula

(2) Reduce the function to the simplest OR and formula

2.5.3 Irrelevant logistic function and its simplification