EEM16 Lecture 2 UCLA

8/14/2019 EEM16 Lecture 2 UCLA

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EEM16/CSM51A:Logic Design of Digital Systems

Lecture #2Boolean Algebra and Combinational

System Design

Prof. Danijela Cabric

Fall 2013


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(c) 2005-2012 W. J. Dally

Agenda

ReviewThe world is digital

Analog at the edges, hardware for demanding problems, 100x per decadeDigital signals

Encode discrete states in a continuous signal Reject noise

Representations Binary, set, continuous, compound

Boolean Algebra (0,1, , ) discussion section Axioms, properties, duality

Logic equations express binary functionsCombinational logic

Output is a function only of current input


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Agenda

How to implement combinational logic by hand

Given a description of a logic function

Generate a gate-level circuit that realizes that function

Everyone needs to do this once

To understand how its doneDemystifies what the synthesis tools do

Better understanding of what synthesis tools can and cant do

In practice you will rarely have to do this by hand

General practice is:

Design using VerilogSimulate with test cases

Generate gates with synthesis program


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What is involved in creating a digital system?

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Implementation

Specification

Synthesis(Design)

Analysis


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High-level and Binary-level Specifications

High-level: system is described by a function on finite sets, z = F(x)Binary-level: all variables are binary, i.e. {0, 1} zb = F b(x b )

where zb and x b are vectors of binary variables and, F b( ) is a binary function

Mapping between the two via coding and decoding functionsCoding: maps each distinct value of x to a different combination of bits of x bDecoding: maps each distinct value of z to a different combination of bits of z b

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Example: A mod3 system

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Example (contd.)

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High Level Specification

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Specification at Binary Level

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Switching Functions

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Several ways to describe a combinational logic function

Example Majority Circuit1. English Language Description

Outputs 1 if more inputs are 1 than are 0

2. Logic equation q = (a b) (a c) (b c)

3. Truth table

abc q000 0001 0010 0011 1100 0101 1110 1111 1


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Boolean Algebra

Boolean Algebra: algebra over 2 elements: {0,1} 3 operators: AND ( ), OR ( ), NOT ( )

AND Operation: a b = 1 iff a=1 and b=1

OR Operation: a b = 1 if a=1 or b=1

NOT Operation: a = 1 iff a=0

a b a&b a|b0 0 0 00 1 0 11 0 0 11 1 1 1

a ~a0 1


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Note on Notation

For AND logic function:

use or & or .

For OR logic function: use or | or +


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Axioms of Boolean Algebra

Axioms: a set of mathematical statements that we assert to be true.

Identity:

0 x = 0

1 x = 1Idempotence:

1 x = x

0 x = x

Negation: 0 = 1 1 = 0


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Expression Evaluation

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Useful Boolean Properties(all can be derived from the axioms)

Communica t ive x y = y x x y = y x

Associa t ive x (y z) = (x y) z x (y z) = (x y) z

Distr ibu t ivex (y z) = (x y) (x z) x (y z) = (x y) (x z)

Idempotence x x = x x x = x

A b s o r p t io n x (x y) = x x (x y) = x

Co m b in in g (x y) (x y) = x (x y) (x y) = x

DeMorgan

s (x y) = x y (x y) = x y


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DeMorgan

s Law

(x y) = x y (x y) = x y

Proof by perfect induction

x y (x y) x y

0 0 1 1

0 1 1 1

1 0 1 1

1 1 0 0


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Graphical Representations as Gate Symbols

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DeMorgan Graphically


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Graphical Representations as Gate Symbols(contd.)

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Applying Boolean Properties

f(a,b,c) = (a c) (a b c) ( a b c) (a b c)

f(a,b,c) = (a c) (a b c) ( a b c) (a b c)

(a b c) (a b c)

f(a,b,c) = (a c) ( a b c) (a b c) (a b c) (a b c)

f(a,b,c) = (a c) (b c) (a b)

Apply Absorption Property

Apply Combining Property


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Tabular Representation of Switching Functions:Truth Tables

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Incomplete Switching Functions

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Minterm

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For a Boolean function of n variables x1, , xn,a product term in which each of the nvariables appears once (either complemented,or uncomplemented) is called a minterm

There are 2n minterms of n variables

Example:


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Indexing Minterms

Minterm m j indexed by integer j= i=0,,n-1 x i 2 i

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Minterm Functions

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Example: E(x2, x1, x0) = m1+m2+m6 = m(1,2,6)


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Truth Table

F(d,c,b,a) is true if input d,c,b,a is prime

No dcba q0 0000 01 0001 12 0010 13 0011 14 0100 0

5 0101 16 0110 07 0111 18 1000 09 1001 0

10 1010 0

11 1011 112 1100 013 1101 114 1110 015 1111 0


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F(d,c,b,a) is true if input d,c,b,a is prime

No dcba q

0 0000 01 0001 12 0010 13 0011 14 0100 05 0101 1

6 0110 07 0111 18 1000 09 1001 0

10 1010 011 1011 1

12 1100 013 1101 114 1110 015 1111 0

dcba

mf )13,11,7,5,3,2,1(

Equation

Example: Table Sum of Products


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Example: Table Sum of Products(Minterms)

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EEM16 Lecture 2 UCLA

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