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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
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
Full Adder

Consider computing 7+6=13:

A combinational logic design

Now, consider one column of this addition:

1 bit full adder
1-bit Full Adder

Truth table for a 1-bit adder:

Formulate a circuit for each output

putting it all together full adder
Putting It All Together: Full Adder
  • Add two bits and carry-in, produce one-bit sum and carry-out.
multiplexer mux
Multiplexer (MUX)
  • A device with multiple inputs and 1 output
  • Could be used to allocate a resource to one of multiple clients:
slide9
MUX
  • 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
multiplexer mux1
Multiplexer (MUX)
  • n-bit selector and 2n inputs, one output
    • output equals one of the inputs, depending on selector

00

01

10

11

4-to-1 MUX

decoder
Decoder
  • 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
2-to4 Decoder

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

decoder1
Decoder
  • n inputs, 2n outputs
    • exactly one output is 1 for each possible input pattern

2-bit

decoder

representing multi bit values
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.

0

15

A = 0101001101010101

A[2:0] = 101

A[14:9] = 101001

multibit values in circuit diagrams
Multibit Values in Circuit Diagrams
  • A 4-to-1 mux, selecting one byte out of a 32-bit value...
combinational vs sequential
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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