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Building Functions from Logic Gates

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- 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.

Consider computing 7+6=13:

A combinational logic design

Now, consider one column of this addition:

Truth table for a 1-bit adder:

Formulate a circuit for each output

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

- 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

- A MUX has two sets of inputs:
- Simplest example is a 2-to-1 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

S0

S1

- 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).

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

2-bit

decoder

- 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

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

- Combinational Circuit
- always gives the same output for a given set of inputs
- ex: adder always generates sum and carry,regardless of previous inputs

- always gives the same output for a given set of 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”