Digital’Logic’Review - California State University...

Digital Logic ReviewLecture 2

ECE 3220 Digital Design with VHDL

California State University

Lecture Topics

n Basic Logic ReviewBasic GatesBoolean AlgebraDe Morgan’s LawStandard Forms of Boolean Expressions

§ Sum of Products§ Product of Sums

Synthesis using NAND / NOR Gates


Textbook References

n Stephen Brown and Zvonko Vranesic, Fundamentals of Digital Logic with VHDL Design, 2nd or 3rd Edition

Chapter 2 Introduction to Logic Circuits


x 1 = x 0 =

(a) Two states of a switch

S

x

(b) Symbol for a switch

Switch or Relay Logic

n A Binary Switch

(c) Simple connection to a battery

S

Battery Light x


(a) The logical AND function (series connection)

S Power supply

S

S

Power supply S

(b) The logical OR function (parallel connection)

Light

Light x1 x2

x1

x2

Let y represent the state of the light (y=0 means off, y=1 means on).

y = x1. x2

y = x1+x2

Basic Logic Gates


S

Power supply S Light

S X1

X2

X3

L(x1, x2, x3) = (x1 + x2) x3

Basic Logic Gates

n A serial to parallel connection


S Light Power supply

R

x

Basic Logic Gates

n An inverting circuit

L(x) = x’

This is a perfectly valid way to implement a Boolean function, and is effectively how NMOS circuits work (with N-channel transistors acting as the switch).

The switch when closed short circuits the light. The resistor plays two roles here, 1) it provides a path for the current to flow to the light when the switch is open, and 2) limits the current that flows through the switch when it is closed. The resistor is called a pullup.


Basic Concepts

n Simple logic gates§ AND à 0 if one or more inputs is 0§ OR à 1 if one or more inputs is 1§ NOT§ NAND = AND + NOT 1 if one or more inputs is 0§ NOR = OR + NOT 0 if one or more input is 1§ XOR implements exclusive-OR function

n NAND and NOR gates require fewer transistors than AND and OR in standard CMOS (Inverters = 2 transistors, 2-input NAND/NOR = 4 transistors, 2-input AND/OR = 6 transistors)

n Functionality can be expressed by a truth tableA truth table lists output for each possible input combination


Basic Logic Gates


Complete Set of Gates

n Complete setsA set of gates is complete

if we can implement any logical function using only the type of gates in the set

Some example complete sets{AND, OR, NOT} Not a minimal complete set{AND, NOT}{OR, NOT}{NAND}{NOR}

Minimal complete setA complete set with no redundant elements.


NAND as a Complete Set

n Proving NAND gate is universal


Logic Functions

n Logical functions can be expressed in several ways:Truth tableLogical expressionsGraphical formHDL code

n Example:Majority function

Output is one whenever majority of inputs is 1We use 3-input majority function


Logic Functions (cont’d)

Truth table

A B C F0 0 0 00 0 1 00 1 0 00 1 1 11 0 0 01 0 1 11 1 0 11 1 1 1

Logical expression formF = A B + B C + A C

Graphical schematic form


Boolean Algebra

Name AND version OR versionIdentity x.1 = x x + 0 = xComplement x. x’ = 0 x + x’ = 1Commutative x.y = y.x x + y = y + xDistribution x. (y+z) = xy+xz x + (y. z) = (x+y) (x+z)Idempotent x.x = x x + x = xNull x.0 = 0 x + 1 = 1

n Boolean identities


Boolean Algebra (cont’d)

n Boolean identities (cont’d)

Name AND version OR versionInvolution x = (x’)’ ---Absorption x. (x+y) = x x + (x.y) = xAssociative x.(y. z) = (x. y).z x + (y + z) = (x + y) + z


De Morgan’s Theorem

x 1 x 2

x 1

x 2

x 1 x 2

x 1 x 2

x 1

x 2

x 1 x 2

(a)

(b)

(x. y)’ = x’ + y’

(x + y)’ = x’ . y’


There are two standard forms in which all Boolean expressions can be written:

1. Sum of Products (SOP)2. Product of Sums (POS)

Standard Forms of Boolean Expressions


Sum of Products (SOP)

n Product Termn Logical product = AND operationn A product term is the ANDing of literalsn Examples: A.B, A'.B.C, A.C', B.C'.D', A.B.C.D

n “Sum of”n Logical sum = OR operationn The sum of products is the ORing of product terms.



n The distributive laws are used to multiply out a general Boolean expression to obtain the sum of products (SOP) form.

n The distributive laws are also used to convert a Boolean expression in POS form to one in SOP form.

n A SOP expression is realized using a set of AND gates(one for each product term) driving a single OR gate (for the sum).


Example of SOP

Designing a SOP form from the following truth table


Example of SOP

f = (x1’.x2’. x3) + (x1.x2’.x3’) + (x1.x2’.x3) + (x1.x2.x3’)

u There are 4 rows in the truth table where the function takes on the value 1.

u Thus, the SOP representation will contain 4 minterms.



n SOP form expressions are often written using the following shorthand notation:

n Direct implementation of the SOP form expressions result in AND-OR circuits:

f (x1, x2, x3) = Σ m(1, 4, 5, 6)

A B

F(A,B)=A’B + AB’


Product of Sums (POS)

n SumTermn Logical sum = OR operationn A sum term is the ORing of literalsn Examples: A+B, A'+B+C, A+C', B+C'+D'

n “Product of”n Logical product = AND operationn The product of sums is the ANDing of sum terms.



n The distributive laws are used to factor a general Boolean expression to obtain the product of sums (POS) form.

n The distributive laws are also used to convert a Boolean expression in SOP form to one in POS form.

n A POS expression is realized using a set of OR gates (one for each sum term) driving a single AND gate (for the product).


Example of POS

Designing a POS form from the following truth table


Example of POS

f = (x1+ x2+ x3) (x1+ x2’ + x3) (x1 + x2’ + x3’) (x1’ + x2’ + x3’)

u There are 4 rows in the truth table where the function takes on the value 0.

u Thus, the POS representation will contain 4 maxterms.



n POS form expressions are often written using the following shorthand notation:

f (x1, x2, x3) = Π M(0, 2, 3, 7)

n Direct implementation of the POS form expressions result in OR-AND circuits:

A B

F(A,B)=(A’+B) (A+B’)



n NAND / NOR gates are simpler to realize in CMOS technology than AND/OR gates

A B Z 0 0 1 0 1 0 1 0 0 1 1 0

A B Z 0 0 1 0 1 1 1 0 1 1 1 0

NOR NAND

NAND / NOR gates are simpler to realize in CMOS technology than AND/OR gates



n Using NAND gates to implement a SOP:

Using NAND gates to implement a sum-of-products.

x 1x 2x 3x 4x 5

x 1x 2x 3x 4x 5

x 1x 2x 3x 4x 5

NAND-NAND network



f

f

(a) SOP implementation

(b) NAND implementation

x1

x3

x2

x3

x2

x1



n Using NOR gates to implement a POS:

Using NOR gates to implement a product-of sums.

x 1x 2

x 3x 4x 5

x 1x 2

x 3x 4x 5

x 1x 2

x 3x 4x 5

NOR-NOR network



x1

f

(a) POS implementation

(b) NOR implementation

NOR gate logic was used in building the Apollo spacecraft guidance system in the 1960’s. (see http://klabs.org/history/ech/agc_schematics/index.htm)

f

x3

x2

x1

x3

x2

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