Textbook Chapter 3

CMPE12 – Summer 2008

Digital Logic: Boolean Algebra and Gates

CMPE12 – Summer 2008 – Slides by ADB 2

Basic Logic Gates

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Truth TableThe most basic representation of a logic functionLists the output for all possible input combinationsHow many rows of the truth table needed? 2#inputs

X Y …A B …

OutputsInputs

X Y …A B …

OutputsInputs

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Truth Table: InverterInverted signals are denoted with an overbarOr with a prime symbol

A’Y = A’A

OutputInput

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Truth Table: AND GateThe result of an AND operation is 1 if and only if all inputs are 1Depict AND by the multiplication symbol

A·BOr by lumping the signals together

ABWe don’t really build these gates…

Y = A · BA B

OutputInputs

CMPE12 – Summer 2008 – Slides by ADB 7

Truth Table: OR GateThe result of an OR operation is 1 if and only if any inputs are 1Depict OR by the addition symbol

A+B

Y = A + BA B

OutputInputs

CMPE12 – Summer 2008 – Slides by ADB 8

About the Little Circle…The little circle is what inverts

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Sum of ProductsHow do you get from a truth table to a logic expression?Sum of products is standard way of synthesizing simple circuitsProcedure:1. Find the rows with the ‘1’ output2. Write the product-form expression for the inputs

in that row (0=inverted, 1=normal)3. Combine the products in step 2 into a sum (OR

the results of step 2)

CMPE12 – Summer 2008 – Slides by ADB 10

Sum of Products1. Find the rows with the ‘1’

output2. Write the product-form

expression for the inputs in that row (0=inverted, 1=normal)

3. Combine the products in step 2 into a sum (OR the results of step 2)

101110000

011

YBA

CMPE12 – Summer 2008 – Slides by ADB 11

De Morgan’s Laws“Break the line, change the sign”Two laws:

A’ + B’ = (AB)’A’ B’ = (A+B)’

CMPE12 – Summer 2008 – Slides by ADB 12

De Morgan’s Laws

(A + B)’ = A’B’ conversely (AB)’ = A’ + B’

“Break the line, change the sign”

1 0

A

1 1

0 10 0

A·BA BA+BA+B A B

CMPE12 – Summer 2008 – Slides by ADB 13

De Morgan’s Laws

(A + B)’ = A’B’ conversely (AB)’ = A’ + B’

“Break the line, change the sign”

1 0

A

1 1

0 10 0

A+BA BABAB A B

CMPE12 – Summer 2008 – Slides by ADB 14

De Morgan’s LawsIn other words…Push the bubbles through!

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De Morgan’s Laws and SOPGenerate equivalent circuits

NAND/NANDNOR/NOR

We prefer NAND/NAND circuitsSame transistor count as NORNANDs are faster

CMPE12 – Summer 2008 – Slides by ADB 17

MaskingWant to look only at certain bits of a binary wordUse a mask to remove the uninteresting bitsExample:

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Axioms of Boolean Algebra0 · 0 = 1 + 1 = 1 · 1 = 0 + 0 =0 · 1 = 1 · 0 = 1 + 0 = 0 + 1 =1 + 0 = 0 + 1 = if x = 0 then x’ = if x = 1 then x’ =

CMPE12 – Summer 2008 – Slides by ADB 19

Single-Variable Theoremsx · 0 =x + 1 = x · 1 =x + 0 =x · x =x + x =x · x’ =x + x’ = (x’)’ =

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Properties of Boolean AlgebraCommutative

x · y = x + y =

Associativex · (y · z) = x + (y + z) =

Distributivex · (y + z ) =x + y · z =

CMPE12 – Summer 2008 – Slides by ADB 21

Properties of Boolean AlgebraAbsorption

x + x · y =x · (x + y) =

Combiningx · y + x · y’ = (x + y) · (x + y’) =

De Morgan’s Laws(x · y)’ = (x + y)’ =

Otherx + x’·y =x · (x’ + y) =

CMPE12 – Summer 2008 – Slides by ADB 22

Logic Minimization

01111011010110011110101001000000YCBA Example