Transcript Transistors and 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
• 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.
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Full Adder
Consider computing 7+6=13:
A combinational logic design
Now, consider one column of this addition:
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1-bit Full Adder
Truth table for a 1-bit adder:
A
0
0
0
0
1
1
1
1
Inputs | Outputs
B CarryIn | Sum CarryOut
0 0
|
0 1
|
1 0
|
1 1
|
0 0
|
0 1
|
1 0
|
1 1
|
Formulate a circuit for each output
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The Majority Circuit for CarryOut
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Odd-Parity Circuit for the Sum
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Putting It All Together: Full Adder
Add two bits and carry-in,
produce one-bit sum and carry-out.
A B Cin S Cout
0 0
0
0
0
0 0
1
1
0
0 1
0
1
0
0 1
1
0
1
1 0
0
1
0
1 0
1
0
1
1 1
0
0
1
1 1
1
1
1
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Four-bit Adder
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Multiplexer (MUX)
A device with multiple inputs and 1 output
Could 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
00
01
10
11
4-to-1 MUX
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4-to-1 MUX from Two 2-to-1 MUXs
S1
S0
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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).
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2-to4 Decoder
This circuit decodes a binary input into one of four possible
choices, or codes
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Decoder
n inputs, 2n outputs
• exactly one output is 1 for each possible input pattern
2-bit
decoder
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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
0
15
A = 0101001101010101
A[14:9] = 101001
A[2:0] = 101
May also see A<14:9>,
especially in hardware block diagrams.
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Multibit Values in Circuit Diagrams
A 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. 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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