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Logic Gates

Basic Logic Gate​

X AND y​

And operation takes two 0, 1 signals, only return 1 When both input is 1

X OR y​

or operation takes two 0, 1 signals, return 1 When either input is 1

NOTX​

unit operation, returns the opposite of the value.

NAND​

A ==NAND== gate is a fundamental digital logic gate that produces an output which is the inverse of the AND operation. NAND stands for "NOT AND" and it's considered a universal gate, meaning any other logic gate (AND, OR, NOT, XOR, etc.) can be constructed using only NAND gates.

==Universal Gate Property==

  • NAND gates can implement any Boolean function, making them extremely versatile. For example:

==Boolean Expression==

  • NOT gate: Connect both inputs of a NAND gate together
  • AND gate: Output of a NAND gate fed into a NOT gate (or another NAND)
  • OR gate: Apply NOT to each input, then feed into a NAND gate

(x NAND y) = NOT(x AND y)

XOR​

An XOR (Exclusive OR) gate is a digital logic gate that outputs true (1) only when the number of true inputs is odd. For a two-input XOR gate, this means the output is true if exactly one of the inputs is true, but not both.

  • XOR produces a true output when inputs differ
  • If inputs are the same (both 0 or both 1), the output is 0
  • output 1 if the two input are different.

==Boolean Expression==

  • XOR is typically written as A⊕B or A^B
  • It can also be expressed as (A AND NOT B) OR (NOT A AND B)

XOR gates have numerous practical applications:

  • Parity checking and generation in error detection systems
  • Binary addition (the sum bit in a half adder is an XOR of the inputs)
  • Comparators (to determine if two bits are different)
  • Encryption algorithms (XOR is used in many cryptographic protocols)
  • Random number generation

Boolean Identities​

Commutative Laws​

  • (x AND y) = (y AND x)
  • (x OR y) = (y OR x)

Associative Laws​

  • (x AND (y AND z)) = ((x AND y) AND z)
  • (x OR (y OR z)) = ((x OR y) OR z)

Distributive Laws​

  • (x AND (y OR z)) = ((x AND y) OR (x AND z)
  • (x OR (y OR z)) = ((x OR y) OR (y OR z))

De Morgan Laws​

  • NOT (x AND y) = NOT(x) OR NOT(y)
  • NOT(x OR y) = NOT(x) AND NOT(y)

Boolean Algebra​

NOT(NOT(x) AND NOT(x OR y)) = x OR y

Gate Logic​

  • a technique for implementing Boolean functions using logic gates
  • simple chip: well-defined funcitionality
  • Elementary(Nand, And, Or, Not)
  • Composite(Mux, Adder, ...)

Gate Interface/Gate Implementation​

  • one obstruction, maybe several different implementations

Circuit Implementations​

  • hardware specific architecture
  • light bulb will bright only if two switches are open(NAND)

HDL​

HDL (Hardware Description Language) is a specialized programming language used to describe the structure and behavior of electronic circuits and systems, particularly digital logic circuits. Unlike traditional programming languages that describe algorithms to be executed by a processor, HDLs describe the physical hardware structure and behavior.

  • interface and parts
  • textual description of chip
  • not unique, can be implemented in typically many ways
  • good documentation, good descriptive names, readable
  • functional declarative language, static description
  • procedural part is not part of the HDL code

test script​

  • script-based simulation
  • with/without output and compare files

simulation process​

  • load the hdl file into the hardware simulator
  • enter values into the chip's input pins
  • evaluate the chi's logic
  • inspect the resulting values