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Chapter 4 Combinational logic design principles
Switching algebra
Representations of the logic function How to make a best circuit in logic design

Principle of Switching algebra
Basic value of any variable or function: 1/0 basic operation in logic algebra:

AND (?) , OR(+), NOT( ’)

Axiom of switching algebra
if x≠1,then x =0; if x =1 ,then x≠0;

if x =0 ,then x’=1; if x =1 ,then x’=0; 0· 0=0 1+1=1 1· 1=1 0+0=0 0· 1=1· 0=0 1+0=0+1=1

Some important theorem

Some important theorem
Table 4-2

Important theorem
Shannon's expansion theorem:

F(x1,x2,...,xi,...xn) =xi'F(x1,x2,...,0,...xn)+xiF(x1,x2,...,1,...xn) =[xi+F(x1,x2,...,0,...xn)][xi'+F(x1,x2,...,1,...xn)
F(a,b)=a'b'F(0,0)+a'bF(0,1)+ab'F(1,0)+abF(1,1)

Important theorem
DeMorgan’s theorem: (x1+x2)’=x1’ ?x2’ (x1 ?x2)’=x1’+x2’

The perfect induction of the theorem
Use the truth table to prove the functions on both side are same !

?a ? b? ' ? a'?b'

Important theorem
Principle of Duality:

Any theorem or identity remains true if 0
and 1 are swapped , · and + are swapped !
F ?x) ? f ( x,0,1,?,?? ? F ( x) ? f ?x,1,0,?,??
D

All the axiom and theorem have their duality !

Be careful to keep the operation orders of the
functions.

Two kind of logic
Positive logic : 1 ( high level ) 0 (low level) Negative logic: 0 ( high level ) 1 (low level) If a logic relation exist in positive logic, it must be exist in negative logic. Relation between two logic:

FP ? FN '

XP ? XN '

Two kind of logic
Example: from positive to negative

positive logic form FP ? x p ? y p

FP ? x p ? y p ? FN ' ? xN '? yN '
negative logic form

FN ? ?xN '? yN '? ' ? xN ? yN

Some important theorem
Principle of Duality:
FN ?xN , yN ? ? F
D P

? xN , y N ?

Duality of the function is the negative logic form of the function .
FN ?xN , yN ? ? FPD ?x p ' , y p '?

Characteristics of XOR operation
A⊕B=A'B+AB' 1. Commultativity: A⊕B=B⊕A 2. Associativity: 3. Distributivity: A⊕(B⊕C)=(A⊕B)⊕C A(B⊕C)=(AB)⊕(AC)

4. Causality: A⊕B=C → A⊕C=B → B⊕C=A A⊕B⊕C⊕D=0 → 0⊕A⊕B⊕C=D

Characteristics of XOR operation
5. Variable and Constant A⊕A=0 A⊕A'=1 A⊕0=A A⊕1=A'
6. Multiple variable A0⊕A1⊕...⊕An=1 number of Ai with 1 is odd 0 number of Ai with 1 is even

Characteristics of XNOR operation
A☉B=A'B'+AB 1. Commultativity: A☉B=B☉A

2. Associativity:

A☉(B☉C)=(A☉B)☉C

3. No Distributivity: A(B☉C) ≠ (AB)☉(AC) 4. Causality: A☉B=C → A☉C=B → B☉C=A

Characteristics of XNOR operation
5. Variable and Constant A☉A=1 A☉A'=0 A☉1=A A☉0=A'
6. Multiple variable A0☉A1☉...☉An=1 number of Ai with 0 is even 0 number of Ai with 0 is odd

Relationship of XOR and XNOR
1. Even varibles for XOR and XNOR is complement: A⊕B=(A☉B)' A⊕B⊕C⊕D=(A☉B☉C☉D)'

2. Odd varibles for XOR and XNOR is equivalency: A⊕B⊕C=A☉B☉C

Relationship of XOR and XNOR
3. complement for one inverse variable A⊕B'=A☉B A⊕B=A☉B'

4. equivalency for two inverse variable A⊕B'=A'⊕B

A'☉B=A☉B'

Combinational logic

The output is determined only by its input.

Output can be changed when input changed.

Representations of logic functions
Example: NAND2

Truth table

Timing diagram

Logic equations

Logic circuits

Truth table input combination----output
(X,Y,Z)

Left: the input combinations in binary order Right: the output for the input

Logic design: Construct a Truth table
A device with majority judge function

output the majority input state .

Logic design: Construct a Truth table
Full adder add three input numbers to get their sum.

Construct a Truth table
4-bits prime-number detector when input is (1,2,3,5,7,11,13), the output is 1, otherwise the output is 0 .

Construct a Truth table
4-bit Binary to Gray code converter change binary input to Gray code output.


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