Building Functions from Logic Gates

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Building Functions from Logic GatesWe'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

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Consider computing 7+6=13:

Now, consider one column of this addition:

A combinational logic design


1-bit Full Adder

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Inputs | Outputs A B CarryIn | Sum CarryOut 0 0 0 | 0 0 1 | 0 1 0 | 0 1 1 | 1 0 0 | 1 0 1 | 1 1 0 | 1 1 1 |

Truth table for a 1-bit adder:

Formulate a circuit for each output


The Majority Circuit for CarryOut

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Odd-Parity Circuit for the Sum

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Putting It All Together: Full AdderAdd two bits and carry-in, produce one-bit sum and carry-out. A B Cin S Cout

0 0 0 0 00 0 1 1 00 1 0 1 00 1 1 0 11 0 0 1 01 0 1 0 11 1 0 0 11 1 1 1 1


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Four-bit Adder


Multiplexer (MUX)

A device with multiple inputs and 1 outputCould be used to allocate a resource to one of multiple clients:

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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

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2-to-1 MUX

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Multiplexer (MUX)n-bit selector and 2n inputs, one output

• output equals one of the inputs, depending on selector

4-to-1 MUX

00 01 10 11


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

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S0S1


DecoderGeneral 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).

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2-to4 Decoder

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


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Decodern inputs, 2n outputs

• exactly one output is 1 for each possible input pattern

2-bitdecoder


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Representing Multi-bit ValuesNumber bits from right (0) to left (n-1)

• just a convention -- could be left to right, but must be consistentUse 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] = 101A[14:9] = 101001

015


Multibit Values in Circuit DiagramsA 4-to-1 mux, selecting one byte out of a 32-bit value...

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MUX example -- what does this circuit do?

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Combinational vs. SequentialCombinational Circuit

• always gives the same output for a given set of inputsex: adder always generates sum and carry,

regardless of previous inputsSequential 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 counteradvances when you push the buttonoutput depends on previous state

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

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Building Functions from Logic Gates