07Decoders121

Review: Additional Boolean operations September 22, 2003 Decoders NAND (NOT-AND) NOR (NOT-OR) XOR (eXclusive OR) (xy)’ = x’ + y’ (x + y)’ = x’ y’ x  y = x’y + xy’ Operation: Expressions: Truth table: Logic gates:

XOR gates ,[object Object],[object Object],[object Object],September 22, 2003 Decoders x  y = x’ y + x y’

More XOR tidbits ,[object Object],[object Object],[object Object],September 22, 2003 Decoders x  (y  z) = x  (y’z + yz’) [ Definition of XOR ] = x’(y’z + yz’) + x(y’z + yz’)’ [ Definition of XOR ] = x’y’z + x’yz’ + x(y’z + yz’)’ [ Distributive ] = x’y’z + x’yz’ + x((y’z)’ (yz’)’) [ DeMorgan’s ] = x’y’z + x’yz’ + x((y + z’)(y’ + z)) [ DeMorgan’s ] = x’y’z + x’yz’ + x(yz + y’z’) [ Distributive ] = x’y’z + x’yz’ + xyz + xy’z’ [ Distributive ]

XNOR gates ,[object Object],[object Object],September 22, 2003 Decoders (x  y)’ = x’y’ + xy

Design considerations, and where they come from ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],September 22, 2003 Decoders

Decoders ,[object Object],[object Object],[object Object],[object Object],[object Object],September 22, 2003 Decoders

What is a decoder ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],September 22, 2003 Decoders

What a decoder does ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],September 22, 2003 Decoders

How can you build a 2-to-4 decoder? ,[object Object],[object Object],September 22, 2003 Decoders Q0 = S1’ S0’ Q1 = S1’ S0 Q2 = S1 S0’ Q3 = S1 S0

A picture of a 2-to-4 decoder September 22, 2003 Decoders

Enable inputs ,[object Object],[object Object],[object Object],[object Object],[object Object],September 22, 2003 Decoders

An aside: abbreviated truth tables ,[object Object],[object Object],September 22, 2003 Decoders

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Blocks and abstraction September 22, 2003 Decoders Q0 = S1’ S0’ Q1 = S1’ S0 Q2 = S1 S0’ Q3 = S1 S0

A 3-to-8 decoder ,[object Object],[object Object],[object Object],[object Object],[object Object],September 22, 2003 Decoders Q0 = S2’ S1’ S0’ Q1 = S2’ S1’ S0 Q2 = S2’ S1 S0’ Q3 = S2’ S1 S0 Q4 = S2 S1’ S0’ Q5 = S2 S1’ S0 Q6 = S2 S1 S0’ Q7 = S2 S1 S0

So what good is a decoder? ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],September 22, 2003 Decoders Q0 = S1’ S0’ Q1 = S1’ S0 Q2 = S1 S0’ Q3 = S1 S0

Design example: addition ,[object Object],[object Object],[object Object],September 22, 2003 Decoders 1 + 1 + 1 = 11 0 + 1 + 1 = 10 C(X,Y,Z) = m(3,5,6,7) S(X,Y,Z) = m(1,2,4,7)

[object Object],[object Object],Decoder-based adder September 22, 2003 Decoders C(X,Y,Z) = m(3,5,6,7) S(X,Y,Z) = m(1,2,4,7)

Using just one decoder ,[object Object],September 22, 2003 Decoders C(X,Y,Z) = m(3,5,6,7) S(X,Y,Z) = m(1,2,4,7)

Building a 3-to-8 decoder ,[object Object],[object Object],[object Object],[object Object],[object Object],September 22, 2003 Decoders Q0 = S2’ S1’ S0’ = m 0 Q1 = S2’ S1’ S0 = m 1 Q2 = S2’ S1 S0’ = m 2 Q3 = S2’ S1 S0 = m 3 Q4 = S2 S1’ S0’ = m 4 Q5 = S2 S1’ S0 = m 5 Q6 = S2 S1 S0’ = m 6 Q7 = S2 S1 S0 = m 7

Decoder expansion ,[object Object],September 22, 2003 Decoders

Modularity ,[object Object],[object Object],[object Object],September 22, 2003 Decoders

A variation of the standard decoder ,[object Object],[object Object],September 22, 2003 Decoders

Separated at birth? ,[object Object],[object Object],[object Object],September 22, 2003 Decoders Q3 = S1 S0 Q2 = S1 S0’ Q1 = S1’ S0 Q0 = S1’ S0’ Q3’ = (S1 S0)’ = S1’ + S0’ Q2’ = (S1 S0’)’ = S1’ + S0 Q1’ = (S1’ S0)’ = S1 + S0’ Q0’ = (S1’ S0’)’ = S1 + S0

Active-low decoder example ,[object Object],[object Object],[object Object],[object Object],September 22, 2003 Decoders

Summary ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],September 22, 2003 Decoders

07Decoders121

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