**This course:** how do we *physically* implement computation? - Bridging the gap between computer programs → physical devices # Course roadmap *An introduction to the digital world!* 0. Devices, materials, atoms 1. **Digital design:** combinational and sequential circuits 2. **Computer architecture:** processors, caches, pipelining 3. **Computer systems:** operating systems, virtual memory, I/O - [I] How have we been able to design new generations of processors, chips, ... so quickly? - ==Abstractions!== There are well-defined abstractions between each of these layers that allow for parallel progress across all these fronts. ``` computer programs ---virtual machines--- computer sysems ---instruction set + memory--- computer architecture ---digital circuits--- digital design ---bits, logic gates--- devices, materials, atoms ``` (^ the horizontal separators are the abstractions.) ## Engineering abstractions > Good abstractions let us reason about behavior while shielding us from the details of the implementation. - Abstractions allow us to have the different layers of computer systems evolve independently! - [!] Anything *bad* about these abstractions? - These interfaces (e.g., instruction set) live very long, s.t. interfaces designed with a particular technology in mind may become suboptimal when the technology changes. - e.g., processors used in PCs are x86, which were designed in Intel in the 70s -- and we are still saddled with a lot of the design choices made back then! ## Course outline 1. **Module 1: Digital design** - Combinational and sequential circuits 2. **Module 2: Computer architecture** - Simple and pipelined processors - Caches and the memory hierarchy 3. **Module 3: Computer systems** - Operating system and virtual memory - Parallelism and synchronization ## CPUs! > *Course focus:* Systems with general-purpose processors (aka. CPUs - central processing unit). - [?] Why not GPUs, TPUs, network cards, etc.? 1. General-purpose processors (CPUs) are the basic building block of computer systems. 2. CPUs are the most sophisticated digital systems that exist today - understanding them will help you design all kinds of hardware (GPUs, TPUs, etc.)! - By the end of the term, you will have designed a processor from scratch! - Will be building toward a complete processor across each of the labs # The digital abstraction *Building digital systems in an analog world.* ## Analog vs. digital systems 2 ways of encoding information: - **Analog systems:** represent and process information using *continuous signals* - e.g., voltage, current, temperature, pressures, ... - Every possible value of the signal is valid! - **Digital systems:** represent and process information using *discrete symbols* - Typically binary symbols (bits) - Encoded using ranges of a physical quantity (e.g., voltage) - [I] ==Digital systems tolerate noise!== > We live in an analog world, but we want to encode information using digital systems. Need to figure out how to do this! ### Why don't we use analog systems? - Sensitive to noise! - The world is noisy! Also... - Manufacturing variations - Components degrade over time - So our goal is to take the analog, noisy world and extract from it clean, digital signal: ![[the digital abstraction.png|400]] ## Using voltages "digitally" > **Key idea:** Encode two symbols, "0" and "1" (1 bit) 1. Attempt 1: Threshold $V_{\text{TH}}$, where $V < V_{\text{TH}}$ interpreted as "0" and $V \geq V_{\text{TH}}$ interpreted as "1" - [!] *Issue:* it's really hard to distinguish between values that are very close to the threshold! 2. Attempt 2: *2 thresholds!* - $V \leq V_L$ interpreted as "0", $V_L < V< V_H$ "undefined," $V \geq V_H$ interpreted as "1"![[analog to digital - 2 thresholds.png|600]] - [I] We engineer things so that we avoid the undefined region. - What if we have noise that might push us into the undefined region? ==Noise margins!== ### Noise margins Imagine two logic gates connected by a wire: - **Output** = logic gate *sending* the signal - **Input** = logic gate down the line *receiving* the signal We implement *different specifications* (to add noise margins) for output/input because wires aren't perfect: as signal travels from the sender (output) to the receiver (input), it degrades. - Electrical interference, resistance, temperature changes... add **noise** to the voltage - [!] If the sender and receiver had the same standard for what constitutes a "1" or a "0", even tiny noise in voltage on the wire would cause the receiver to misread the signal. > **Solution:** noise margins! - Digital output: "0" if $V \leq V_{\text{OL}}$, "1" if $V \geq V_{\text{OH}}$ - Digital input: "0" if $V \leq V_{\text{IL}}$, "1" if $V \geq V_{\text{IH}}$ $$V_\text{OL} < V_\text{IL} < V_\text{IH} < V_\text{OH}$$ ![[noise margins.png|600]] > Digital systems *fight* noise! They actively remove noise. - Analog systems: noise accumulates - Digital systems: noise is canceled at each stage ![[digital vs. analog systems in fighting noise.png|500]] - [I] Intuitively, canceling noise requires *active components*, i.e., components that inject energy into the system. - *Intuition:* Imagine an output gate sends a perfect 5.0V signal (a logical "1") → as it travels down the wire, electrical resistance and noise degrade it → by the time it reaches the next gate, it has dropped to 4.2V. - If we want to "cancel" this noise and pass a perfect 5.0V to the next stage - where does that missing 0.8V come from? [Need to inject energy into the systems!] - [i] We need active components (transistors) that are connected to a power supply! - [?] How does the active gate know to rebuild the signal as a clean "0" or "1"? → decision-making rule = VTC! ### Interlude: What is a logic gate? > A logic gate is the fundamental building block of all digital computation: its sole purpose is to take one or more binary inputs (0/1) and apply a simple mathematical rule (AND, OR, ...) to produce a single binary output. - [I] A logic gate is essentially a tiny, automated decision-maker! - **AND gate:** Only outputs "1" if *both* Input A and Input B are "1" - **OR gate:** Outputs "1" if *either* Input A or Input B are "1" - **NOT gate:** Flips the input. - [*] By wiring *many* (billions!) of logic gates together in specific patterns, we build circuits that add numbers together, store memory, execute the instructions of a processor... :0 #### What is a logic gate physically composed of? A logic gate is a specific arrangement of transistors wired together: - **Materials:** semiconductor materials (usually silicon) → silicon is treated (doped) with other elements so its electrical properties can be manipulated. - **Transistors:** ^ using that silicon, we build transistors! - Transistor = a microscopic electrical switch (like a light switch that is flipped by *electricity itself*) → if you apply a small voltage to the transistor's "gate" terminal, it allows a larger current to flow through its other terminals. - [*] To build a logic gate, you wire a few transistors together. - e.g., building a physical AND gate: ~imagine two transistors wired in series along a single wire carrying voltage~ 1. If you only apply voltage to the first transistor, it acts as a *closed* switch, but the electricity stops at the second transistor: **output = 0** - "closed" - think of a transistor like a drawbridge (not like a door)! - Open switch: drawbridge is up → the physical wire is broken/disconnected. Electricity stops flowing! - Closed switch: drawbridge is down → physical wire is connected end-to-end. Electricity flows! 2. ^ same logic if you only apply voltage to the second transistor! **output = 0** 3. If you apply voltage to *both* transistors simultaneously, the circuit is completed, electricity flows all the way through, and the wire outputs a high voltage: **output = 1** ## Voltage transfer characteristic (VTC) > A **VTC** is a graph showing output voltage (y-axis) versus input voltage (x-axis) for a specific gate. - In an analog system, the VTC is a straight diagonal line: - If the input goes up by 10%, the output goes up by 10% - Noise accumulates! - In a digital system, the VTC is nonlinear: it is flat at the extremes and steep in the middle. - Example VTC: below is the VTC for a buffer logic gate! ![[buffer gate VTC.png|500]] - [!] VTC must avoid the shaded regions ("forbidden zones") which correspond to *valid* inputs but *invalid* outputs. - **More details about the VTC:** - ==Physical mechanism:== (physical breakdown of a logic gate) - *Power source* (energy in): every active logic gate is permanently connected to a main power supply (aka. $V_\text{DD}$, e.g. 5.0V) and ground (0.0V). $V_\text{DD}$ is like a massive, high-pressure water tower. - *Transistors* (valves): inside the logic gate are transistors that act as valves. The input wire ($V_\text{in}$) connects to the "handle" of these valves. - *Output* ($V_\text{out}$): this is the pipe coming out of the valves. - [I] Suppose the buffer receives a "1" input of 4.5V: 1. This voltage turns the physical transistor valve fully ON, directly connecting the output pipe to the 5.0V water tower. 2. The output floods with exactly 5.0V. - ^ this causes the largely flat VTC curve between $V < V_\text{IL}$ → output $V_\text{out}$ does not wiggle so long as the valve remains open ($V 1**! - $V_\text{OL} < V_\text{IL} < V_\text{IH} < V_\text{OH}$, so $V_\text{OH} - V_\text{OL} > V_\text{IH} - V_\text{IL}$ → so any curve that crosses this region must have climbed a tall vertical height over a relatively shorter horizontal distance → $$\Delta\text{output} / \Delta\text{input} > 1 \rightarrow \text{gain} > 1$$ - Passive components (e.g. resistors) can never have gain > 1 because they can only lose energy → **therefore, logic gates must be built using active components** (e.g. transistors connected to a power supply). - [b] **Gain** is a measure of amplification: it describes how much a system multiplies the signal you feed into it. In the context of the VTC curve: $$\text{voltage gain} = \frac{\Delta V_\text{out}}{\Delta V_\text{in}}$$ ### Buffer logic gate > A buffer outputs whatever you input: "0" → "0", "1" → "1" - Buffers are useless for logic, but essential for *physics*! - A piece of wire is a passive component: as signal travels down a long wire across a computer chip, resistance and interference cause the voltage to droop and pick up noise → if you string too many logical operations together, a "1" can eventually degrade into the "Undefined" zone and the computer crashes :"( - *Buffers fix this!* When you place a buffer in the middle of a long wire, it receives the degraded signal as its input and then triggers a fresh voltage directly from the power supply to then output a clean signal. - (In this lecture) - buffers are introduced because they make teaching the concept of a VTC easy (they have a simple VTC plot). ## Types of digital circuits 2 main classes: - **Combinational circuits:** - Do not have memory - Each output is a function of current input values - e.g., inverter (0→1, 1→0), AND - **Sequential circuits:** - Have memory (*state*) - Each output depends on current state + current inputs # Summary - Digital systems tolerate noise - Digital encoding - Valid voltage ranges for representing "0" and "1" - Undefined range avoids mistaking "0" for "1" and vice versa - Noise margins require tougher standards for outputs than for inputs ``` digital inputs → [circuit] → digital outputs ```

title	tags	date	read
Lecture 1 - Digital Abstraction		Jun 12, 2026	10 min
Lecture 2 - Combinatorial Devices and Boolean Algebra		Jun 12, 2026	1 min

6.191 MIT Course Notes