circuit simplification truth table minimized logic gates
TRANSCRIPT
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Circuit Simplification
Truth Table Minimized Logic Gates
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Where We Are…
• Can represent a circuit's output with Boolean expression:
AB + AC
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Where We Are…
• We can simplify boolean expressions
AB + ACA(B + C)
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Where We Are…
• Thus we can simplify circuits
AB + AC
A(B + C)
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Truth Table
• Truth table defines Boolean function– When in doubt, check the truth table…
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Proof By Exhaustion
• Proof by exhaustion : prove equivalence by comparing truth tables
• Ex: =
A B
0 0 1
0 1 1
1 0 1
1 1 0
A B
0 0 1
0 1 1
1 0 1
1 1 0
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DeMorgan's Theorems
+ If you don't have both of A and B…you do not have A or do not have B
If you don't have either of A or B…you do not have A and you do not have B
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DeMorgan's
• Break up solid bar by switching operation:
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Samples
B
= B DeMorgan's
= B Commutative
= (1 + B) Distributive
= (1) 1+ anything = 1
= 1 · anything = self
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Samples
·A
= ()·A DeMorgan's
= (A)·A Double negative cancel
= A·A Associative
= A·A Commutative
= A Anything self = self
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Sum Of Products
• Can convert any truth table to sum of products expression– Each 1 in output is a product of inputs (AND)– Any of them work (OR them together)
X = B + A + AB
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Multiple Inputs
• Each product shows EVERY input– Either raw or inverted
Ex:Each product hasA or B or C or
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SOP Circuit
• Can build circuit from SOP expression:– Inputs feed layer of AND gates• Some inputs inverted first
– OR gate takes ANDs
X = B + A + AB
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Implication 1
Can build circuit from SOP expression:• Any circuit can be build with NOT, AND & OR
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And…
• Can make NOT, AND & OR with only NAND
NOT AND OR
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Implication 2
• Any circuit can be build with NAND gates only…
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Implication 2
• Replace NOT with NAND
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Implication 2
• Put inversion bubbles out of AND & into OR– 2 negations cancel out
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Implication 2
• Convert + …. to – De Morgan's Theorem
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SOP Circuit
• Can build circuit from SOP expression…
… but can often do better…
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Simplification
• Can simplify things…
X = B + A + AB = B + A( + B) = B + A() = B + A = B + A = A + B
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Multiple Outputs
• Multiple outputs– Each output separate function/circuit
IN OUT
A B X Y
0 0 0 1
0 1 0 0
1 0 0 0
1 1 1 1
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Multiple Outputs
• Multiple outputs– Each output separate function/circuit
X = ABIN OUT
A B X Y
0 0 0 1
0 1 0 0
1 0 0 0
1 1 1 1
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Multiple Outputs
• Multiple outputs– Each output separate function/circuit
X = AB
Y = AB +
IN OUT
A B X Y
0 0 0 1
0 1 0 0
1 0 0 0
1 1 1 1
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Logisim
• Build simplified circuit from truth table…– Window Combinational Analysis
• Build truth table for circuit…– Project Analyze Circuit