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# Digital Hardware and Logic Gates
## Digital Hardware Concepts
### Digital Hardware
- Hardware (HW) means electronic circuits
- Digital means every wire (or signal) can be in one of two (binary) states
- Electrically, the states are two different voltage levels (e.g. 5V and 0V)
- We abstractly define two logic values (1 and 0) to be associated with these two voltage levels
- We also use the terms "true" and "false"
- Abstraction is key to much of this course!
### Representing Information
- Every signal means something (abstractly)
- A single signal can represent some state, or be used as a means of control
- Multiple signals can be grouped together to create a binary value
- Each signal is a bit (binary digit)
- The same multi-bit value can mean different things in different contexts...
### Putting values in context
- The 6-bit value 101000 might mean
- 40, as an unsigned number, or address
- -24, if viewed as a 6-bit signed number
- "store byte", as an opcode (instruction)
- Sometimes, leading 0's are implied
- 101000 as an 8-bit unsigned number is still 40
- But it's not still -24 as an 8-bit signed number (it's also 40)
- With no context, the default is to interpret as an unsigned (i.e. no negatives) number
---
## Counting in Binary
- Insight into unsigned numbers
- When we count, we start from 0, add 1 at each step:
- 0+1 = 1
- 1+1 = 2, but we can't express 2
- In decimal, 9+1 = 10
- Carry into next position and go back to 0
- Same with binary, but that happens at 1+1
- 1+1 = 10, 10+1 = 11, 11+1 = 100, etc.
- What is 111?
### Counting from 0 to 15
- You will need to know this range of binary numbers
- Leading 0's: 3 as a 4-bit number is 0011
|Decimal|Binary|
|--|----|
|0 |0 |
|1 |1 |
|2 |10 |
|3 |11 |
|4 |100 |
|5 |101 |
|6 |110 |
|7 |111 |
|8 |1000|
|9 |1001|
|9 |1001|
|10|1010|
|11|1011|
|12|1100|
|13|1101|
|14|1110|
|15|1111|
---
## ASCII codes
- Another context for representing letters
- A table can be found [here](https://www.asciitable.com/)
- ASCII codes are 7-bit values (between 0 and 127)
---
## Logic building blocks
### Logic Circuit Building Blocks
- Logic circuits are functions that generate an output based on a set of inputs
- Inputs and output are binary/logical values
- Three fundamental operations
- AND (output is true only if all inputs are true)
- OR (output is true if any of the inputs are true)
- NOT (output is the inverse of the input)
- Also called INV (for inverter)
- **All** logic functions are made by a combination of these three operations
### Logic in algebraic form
- Note: variable names are generic; they have no inherent meaning
- AND
- Expressed as a product (multiplication)
- `Y = X1 * X2` (or just `X1X2`)
- OR
- Expressed as a sum (addition)
- `Y = X1 + X2`
- NOT
- `Y = !X1`
### Logic Gates
- Because these are HW circuits, it's useful to have a schematic abstraction
- We call these logic "gates"
- The symbols for AND, OR, and INV are below
- AND and OR can have more than 2 inputs
- Same symbol
- Note the overbar used at the output of the inverter
- Just another way of expressing `!` (or `~`)
### Logic Circuits
- If we want a function that is different than just AND or OR, or we have more than two signals to work with, we can combine gates into a larger circuit
- Much of what we'll be covering is how to define the circuit needed to implement a given logic function
---
[Truth tables, minterms, and simple synthesis ->](2.md)
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