Map of content for digital logic — the mathematical and physical foundations of digital circuits, from individual transistors up to finite state machines. The path: switches → gates → combinational logic → memory → sequential logic → state machines.
Hardware foundations
The physical layer: switches, transistors, and how they implement Boolean functions.
- Analog vs digital signal — why digital matters.
- MOSFET — the universal switch.
- NMOS transistor and PMOS transistor — complementary switching elements.
- CMOS — the dominant logic family, built from complementary networks.
- Logic voltage levels and noise margin — what “0” and “1” actually mean electrically.
- Moore’s law — the doubling that drives the industry.
Number representation
How information is encoded in bits.
- Positional number representation — the general place-value system.
- Binary number system — base 2.
- Hexadecimal number system — the convenient compact form.
- Base Conversion — moving between bases.
- Signed binary number — representing negatives.
- Sign-and-magnitude representation — the simplest signed scheme.
- 1’s Complement — invert every bit.
- 2’s Complement Arithmetic — the standard.
- Unsigned Binary Subtraction — direct borrow vs subtraction-by-addition with complements.
- Radix Complement — the general theory.
- Overflow (binary arithmetic) — when results don’t fit.
Boolean algebra
The math of binary logic.
- Boolean Algebra — the laws of binary variables.
- Boolean Function — functions of binary inputs.
- Conjunction (AND), Disjunction (OR), Negation (NOT) — the basic operations.
- Exclusive Disjunction (XOR), Material Biconditional (XNOR) — the derived ones.
- De Morgan’s Laws — pushing negations through.
- Duality (Boolean algebra) — the two-form structure.
- Literal (Boolean) — counting variable occurrences.
- Venn diagram — visual reasoning about set/logic identities.
Canonical forms and minimization
Standardizing and reducing Boolean expressions.
- Minterms and Maxterms — atoms of canonical forms.
- Sum-of-products — disjunction of conjunctions.
- Product-of-sums — the dual.
- Karnaugh Map — visual minimization.
- Don’t Care Condition — flexibility in unused inputs.
- Multiple-output circuit — sharing terms across outputs.
- Multilevel synthesis — when two-level isn’t optimal.
- Gate Input Cost — measuring circuit complexity.
- Shannon’s expansion theorem — recursive decomposition.
Logic gates
Physical implementations of Boolean operations.
- Logic gate — overview and symbols.
- NAND gate and NOR gate — the cheap CMOS primitives.
- Universal Gate — why NAND or NOR alone is enough.
- Fan-in — practical limits on inputs per gate.
Combinational circuits
Logic without memory — output depends only on current inputs.
- Combinational circuit — the general concept.
- Iterative circuit — replicate one cell to scale a function across many bits.
- Adder — half, full, and ripple-carry; the arithmetic workhorse.
- Multiplexer — selecting one of many inputs.
- Demultiplexer — the reverse.
- Decoder — binary to one-hot.
- Encoder — one-hot to binary.
- One-hot encoding — the encoding itself.
- Arithmetic Logic Unit — combinational arithmetic and bitwise operations with status flags.
Programmable logic
Configurable hardware.
- Programmable logic device — PLAs and PALs.
- Field-programmable gate array — modern reconfigurable chips.
- Lookup table — the heart of FPGA logic blocks.
Sequential logic
Circuits with memory — output depends on history.
- Sequential circuit — combinational + storage with feedback.
- Latch (digital) — level-sensitive storage (SR, gated SR, gated D).
- Level and Edge-Triggering — when storage updates.
- D flip-flop — the standard edge-triggered element.
- T flip-flop — toggle behavior.
- JK flip-flop — the versatile one.
- Setup time, hold time, and clock frequency — timing constraints.
- Register (digital) — group of flip-flops.
- Counter (digital) — sync and async; up and down.
Finite state machines
Sequential circuits with structured behavior.
- Finite State Machine — the abstract model.
- Finite State Transducer — FSM that emits outputs, generalizing Mealy and Moore.
- Mealy machine — output depends on state and input.
- Moore machine — output depends only on state.
- State diagram — graphical FSM specification.
- State table — tabular form.
- State Equation — algebraic form.
- State Assignment — choosing binary codes for states.
- Sequence detector (FSM design example) — full design walkthrough.
Hardware description
Designing digital systems in code.
- VHDL — the language.
- VHDL entity — the interface.
- VHDL architecture — the implementation.
- VHDL signal assignment — the basic statement.
The natural continuation is Computer architecture — using digital logic to build a programmable processor. Many of the building blocks here (registers, ALU, decoders, FSMs) appear directly in CPU designs.