Building functions from logic gates
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Building Functions from Logic Gates. We've already seen how to implement truth tables using AND, OR, and NOT -- an example of combinational logic . Combinational Logic Circuit output depends only on the current inputs stateless Sequential Logic Circuit

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Building Functions from Logic Gates

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Building Functions from Logic Gates

  • We've already seen how to implement truth tablesusing AND, OR, and NOT -- an example of combinational logic.

  • Combinational Logic Circuit

    • output depends only on the current inputs

    • stateless

  • Sequential Logic Circuit

    • output depends on the sequence of inputs (past and present)

    • stores information (state) from past inputs

  • We'll first look at some useful combinational circuits,then show how to use sequential circuits to store information.

Full Adder

Consider computing 7+6=13:

A combinational logic design

Now, consider one column of this addition:

1-bit Full Adder

Truth table for a 1-bit adder:

Formulate a circuit for each output

The Majority Circuit for CarryOut

Odd-Parity Circuit for the Sum

Putting It All Together: Full Adder

  • Add two bits and carry-in, produce one-bit sum and carry-out.

Four-bit Adder

Multiplexer (MUX)

  • A device with multiple inputs and 1 output

  • Could be used to allocate a resource to one of multiple clients:


  • A 2n-to-1 multiplexer (MUX) sends one of 2n input lines to a single output line

    • A MUX has two sets of inputs:

      • 2n data input lines

      • n select lines used to pick one of the 2n data inputs

  • Simplest example is a 2-to-1 MUX

2-to-1 MUX

Multiplexer (MUX)

  • n-bit selector and 2n inputs, one output

    • output equals one of the inputs, depending on selector





4-to-1 MUX

4-to-1 MUX from Two 2-to-1 MUXs




  • General example:

  • Assume that some information is encoded in n bits

  • For each encoding, we want to activate the (one) correct output line

  • The general idea: given an n-bit input

  • Detect which of the 2n combinations is represented

  • Produce 2n output, only one of which is “1”

  • A n-to-2n decoder takes an n-bit input and produces 2n outputs. The n inputs represent a binary number that determines which one of the 2n outputs is “true” (i.e., 1).

2-to4 Decoder

This circuit decodes a binary input into one of four possible choices, or codes


  • n inputs, 2n outputs

    • exactly one output is 1 for each possible input pattern



Representing Multi-bit Values

  • Number bits from right (0) to left (n-1)

    • just a convention -- could be left to right, but must be consistent

  • Use brackets to denote range:D[l:r] denotes bit l to bit r, from left to right

  • May also see A<14:9>, especially in hardware block diagrams.



A = 0101001101010101

A[2:0] = 101

A[14:9] = 101001

Multibit Values in Circuit Diagrams

  • A 4-to-1 mux, selecting one byte out of a 32-bit value...

MUX example -- what does this circuit do?

Combinational vs. Sequential

  • Combinational Circuit

    • always gives the same output for a given set of inputs

      • ex: adder always generates sum and carry,regardless of previous inputs

  • Sequential Circuit

    • stores information

    • output depends on stored information (state) plus input

      • so a given input might produce different outputs,depending on the stored information

    • example: ticket counter

      • advances when you push the button

      • output depends on previous state

    • useful for building “memory” elements and “state machines”

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