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@@ -0,0 +1,128 @@ +# 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) |