ENGR 254

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ENGR 254

• Lecture 6
• 2-2-09

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DeMorgan Symbol Equivalence
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Likewise for OR

• (be sure to check errata!) FIG 4-4

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DeMorgan Symbols
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Definitions (Sec. 4.1.6)

• Literal A variable or the complement of a
  variable. Example X, Y, Y, etc.
• Product term A single literal or product of
  literals. Example
• Sum-of-products expression A logical sum of
  product terms. Example
• Sum term A single literal of sum of literals.
  Example
• Product-of-sums expression A logical product of
  sum terms.
• Example

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Definitions

• Normal term A product or sum term in which no
  variable appears more than once.
• Example non-normal
• Example normal
• Minterm (n variables) A normal product term
  with n literals. Example
• Maxterm (n variables) A normal sum term with n
  literals. Example

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Truth table vs. minterms maxterms
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Combinational analysis

X Y Z F
0 0 0 0
0 0 1 1
0 1 0 1
0 1 1 0
1 0 0 0
1 0 1 1
1 1 0 0
1 1 1 1

0
1
2
3
4
5
6
7
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Signal expressions

• Multiply outF ((X Y) Z) (X Y Z)
  (X Z) (Y Z) (X Y Z)

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New circuit, same function

• F X Z Y Z X Y Z

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Add out logic function

• Circuit

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Shortcut Symbol substitution
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Different circuit, same function
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Another example
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Sum-of-products form
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Product-of-sums form
P-of-S preferred in CMOS, TTL (NAND-NAND)
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Brute-force design
row N3 N2 N1 N0 F 0 0 0 0 0 0
1 0 0 0 1 1 2 0 0 1 0
1 3 0 0 1 1 1 4 0 1 0
0 0 5 0 1 0 1 1 6 0 1
1 0 0 7 0 1 1 1 1 8 1
0 0 0 0 9 1 0 0 1 0 10
1 0 1 0 0 11 0 0 1 1 1 12
1 1 0 0 0 13 1 1 0 1
1 14 1 1 1 0 0 15 1 1 1 1
0

• Truth table --gt canonical sum (sum of minterms)
• Exampleprime-number detector
• 4-bit input, N3N2N1N0

F(N3, N2, N1, N0) SN3N2N1N0(1,2,3,5,7,11,13)
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Minterm list --gt canonical sum
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Algebraic simplification

• Theorem T8,
• Reduce number of gates and gate inputs

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Resulting circuit