### Unit 3: Fundamentals of Digital Logic Design

We will begin this unit with an overview of digital components, identifying the building blocks of digital logic. We will build on that foundation by writing truth tables and learning about more complicated sequential digital systems with memory. This unit serves as background information for the processor design techniques we learn in later units.

**Completing this unit should take you approximately 9 hours.**

Upon successful completion of this unit, you will be able to:

- describe the evolution of physical components used to implement Boolean logic in the design of digital processors and computers;
- design a simple digital circuit from a truth table or a K map;
- design a 4 bit adder; and
- design a simple sequential circuit using state diagrams and state transition tables.

### 3.1: Beginning Design: Logic Gates, Truth Table, and Logic Equations

Study these important design principles, which are applicable to any type of design, particularly computer system design, software, and hardware. Consider these principles alongside the other design considerations we use as a guide to computer system design.

Read the introduction, look at the different logic families under "Electronic Gates" and study carefully the sections "Symbols" and "Truth Tables." Scan the rest of the article. Truth tables list the total possible input combinations of 1's and 0's and the corresponding outputs for each gate. Logic devices are physical implementations of Boolean logic and are built from components, which have gotten larger and more complex over time, for example: relays and transistors, gates, registers, multiplexors, adders, multipliers, ALUs (arithmetic logic units), data buses, memories, interfaces, and processors. These devices respond to control and data signals specified in machine instructions to perform the functions for which they were designed.

This section on ladder logic is optional. If you would like to know more, read these sections.

Read this chapter on Karnaugh mapping, which is a tabular way for simplifying Boolean logic. There are several ways for representing Boolean logic: algebraic expressions, which use symbols and Boolean operations; Venn diagrams, which use distinct and overlapping circles; and tables relating inputs to outputs (for combinational logic) or tables relating inputs and current state to outputs and next state (for sequential logic). When designing sequential logic, some of the components are memory devices. Cost and processing time are considerations in using memory devices, which can be expensive. To reduce the cost or processing time the logic can be simplified. This simplification can be done using algebraic rules to manipulate the symbols and operations, analysis of the areas inside the circles for Venn diagrams, or Karnaugh maps for input/output tables.

### 3.2: Combinational Logic

Read this chapter, which describes the design of several components using logic gates, including adders, encoders and decoders, multiplexers, and demultiplexers. This chapter also mentions ladder logic. If you are not familiar with ladder logic, you may also optionally read Chapter 6.1 and 6.2 as a reference. Note that ladder logic is not generally used in computer design and may be omitted. For the one input decoder, note that for a 0 input, the D0 output is a 1 and when the input is a 1, the D1 output is a 1. All the other decoders work the same way. One output line is a 1and the rest are 0's, indicating which binary number has been placed on the input lines. So an input in binary of the number 6 would cause D6 to be a 1.

### 3.3: Flip-Flops, Latches, and Registers

Read this chapter, which discusses how logic gates are connected to store bits (0s and 1s). Combinational circuits, described in the previous section, do not have memory. Using logic gates, latches and flip flops are designed for storing bits. Groups of flip flops are used to build registers which hold strings of bits. For each storage device in Chapter 10, focus on the overview at the beginning of the section and the review of the device's characteristics at the end of its section. While you do not absolutely need to know the details of how latches and flip flops work, you might find the material of interest. We strongly recommend that you read the details of the design of each storage device, because it will give you a stronger background.

### 3.4: Sequential Logic Design

Read this chapter on sequential circuits. Combinatorial circuits have outputs that depend on the inputs. Sequential circuits and finite state machines have outputs that depend on the inputs AND the current state, values stored in memory.

### 3.5: Case Study: Design of a Finite State Machine (FSM) to Control a Vending Machine

Read this article for an example of a finite state machine design of a simple vending machine.A sequential circuit is also called a sequential machine or a finite state machine (FSM) or a finite state automaton. This case study gives an example of the design of a sequential circuit. Each state is assigned a distinct set of 1's and 0's, not using more variables than necessary. Thus 4 states require 2 variables, 00, 01, 10 and 11. A binary table represents the input/output behavior of the circuit. We use a sequential circuit, because the output also depends on the state. Recall that state requires memory (that is, flip flops). Thus, a binary table with entries that give the output and next state for given inputs and current state represents the design of the machine. A finite state machine diagram can also represent the design: circles represent states; arrows represent transitions next to states; and inputs and outputs label the arrows (sometimes written as input/ output). Finally, Boolean equations can also represent the design. Lastly, Karnaugh maps or Boolean logic rules can be used to simplify (or minimize) the equations, and thus the design.

### Unit 3 Assessment

- Receive a grade
Take this assessment to see how well you understood this unit.

- This assessment
**does not count towards your grade**. It is just for practice! - You will see the correct answers when you submit your answers. Use this to help you study for the final exam!
- You can take this assessment as many times as you want, whenever you want.

- This assessment