---
# System prepended metadata

title: Chapter 2-Boolean Algebra and Logic Gates
tags: [Digital IC Design]

---

---
tags: Introduction to Digital System
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# Chapter 2-Boolean Algebra and Logic Gates
## Boolean Algebra
1. Operations
    1. Two-Valued Boolean Algebra
        ![](https://i.imgur.com/9dglJo2.png)
    2. Properties:
        ![](https://i.imgur.com/lkQSpwO.png)
        * ※Complement: F'
2. Boolean Functions
3. Canonical and Standard Form
    1. Canonical Form
        1. Minterms and Maxterms:
            ![](https://i.imgur.com/3CnwNoX.png)
        2. Expressions
            ![](https://i.imgur.com/wuIqfNX.png)
            1. Sum of Minterms: The minterms whose sum defines the Boolean function are those that give the 1's of the function in a truth table.
                $f_1=x'y'z+xy'z'+xyz$
                $f_2=x'yz+xy'z+xyz'+xyz$
            2. Product of Maxterms:
                $f_1'=x'y'z'+x'yz'+x'yz+xy'z+xyz'\\\Rightarrow f_1''=f_1=(x+y+z)(x+y'+z)(x+y'+z')(x'+y+z')(x'+y'+z)$
        3. Boolean Algebraic Representation
            1. Sum of Minterms: $\sum$
            2. Product of Maxterms: $\prod$
        5. Disadvantage: It requires each term to contain all variables.
    2. Standard Forms
        
## Logic Gates
1. Gates
    ![](https://i.imgur.com/yknB84w.png)
    ![](https://i.imgur.com/QMGKYmg.png)
    * Extension to Multiple Inputs: A gate can be extended to multiple inputs if its binary operation is commutative and associative such as AND and OR.
2. Positive and Negative Logic
    ![](https://i.imgur.com/fn5ixzv.png)![](https://i.imgur.com/LV9cyrI.png)
3. Compositions of Logic Gates
    ![](https://i.imgur.com/yLb1dIM.png)
## Integrated Circuits:
1. Levels of Integration
    1. SSI: Small-Scale Integration Devices
    2. MSI: Medium-Scale Integration Devices
    3. LSI: Large-Scale Integration Devices
    4. VLSI: Very Large-Scale Integration Devices
2. Digital Logical Family:
    1. TTL(Transistor-Transistor Logic):Standard
    2. ECL(Emitter-Coupled Logic): High-spaced
    3. MOS(Metal-Oxide Semiconductor): High component density
    4. CMOS(Complementary Metal-Oxide Semiconductor: Low power consumption
3. Parameters for Digital Logic Families
    1. Fan-Out: The maximum number of ouputs driven by a single logic gate that don't impair the normal operation
    2. Fan-In: The number of inputs available in a gate
    3. Power Dissipation: Power consumption
    4. Propagation Delay: The length of time taken for a signal to reach its destination.
4. EDA
    1. Schematic Editor 
    2. HDL
    3. Logic Synthesis
5. Physical Realization of a Digital Circuit
    1. ASIC: Application-Specific Integrated Circuit
    2. FPGA: Field-Programmable Gate Array
    3. CPLD: Complex Programmable Logic Device 

**Reference**
NCKUCSIE 1101_Introduction to Digital System
Digital Design With an Introduction to The Verilog HDL, VHDL and System Verilog 6/e by M.Morris Mano Michael D. Ciletti