appendix c basics of digital logic part i. florida a & m university - department of computer and...

Appendix C

Basics of Digital LogicPart I

Florida A & M University - Department of Computer and Information Sciences

Modern Computer Digital electronics

operate with only two voltage levels – high and low other levels are temporary and occur while

transitioning between these values binary system matches the underlying abstraction

inherent in electronics

Use signals instead of voltage levels which are complements or inverses of one another

logically true : 1 (asserted) logically false : 0 (deasserted)

Logic Block Each input maybe 0 or 1 Each output maybe 0 or 1

Input(s)Logic block

Output(s)

Logic Blocks Combinational

without memory output depends on only on the current input completed in one machine cycle

Sequential Retains memory of previous state output depends on both the inputs and the logic block

Combinational Logic Block Defined by the values of the outputs for each

possible set of input values (truth table)

n inputs 2n combinations of input values

Truth Table - example Output y is 1 when inputs

w and x have the same value

Output z is 1 when both inputs are 0

Inputs Outputs

w x y z

1 1 1 0

1 0 0 0

0 1 0 0

0 0 1 1

Truth Table Completely describes combinational logic

function

Grow in size quickly

May not be easy to understand

Boolean Algebra Another approach to express logic function as

logic equations

All variables have the values 0 or 1

Three operators OR AND NOT (unary)

Boolean Algebra OR operator

result is 1 if either of the operands (variables) is 1 also known as logical sum notation: A + B

AND operator result is 1 only if both operands (variables) is 1 also known as logical product notation: A B

NOT operator Result is 1 only if operand is 0 Inversion or negation of the value Notation: A

Boolean Algebra Laws Identity : A + 0 = A A 1 = A

Zero and One : A 0 = 0 A + 1 = 1

Inverse:

Commutative: A B = B A A + B = B + A

Associative: A + (B + C) = (A + B) + C

A (B C) = (A B) C

Distributive: A (B + C) = (A B) + (A C)

A + (B C) = (A + B) (A + C)

1AA 0AA

Boolean Algebra Laws Idempotent : A + A = A A A = A

DeMorgan’s : BABA BABA

Boolean Algebra - example

y = wx +

Inputs Outputs

w x y z

1 1 1 0

1 0 0 0

0 1 0 0

0 0 1 1

Logic Gates

Used to build logic blocks by implementing basic logic functions

AND gate May have multiple inputs Output equal to the AND of all inputs

OR gate May have multiple inputs Output equal to the OR of all inputs

NOT gate inverter Single input

Logic Gates - example

y = wx +

Inputs Outputs

w x y z

1 1 1 0

1 0 0 0

0 1 0 0

0 0 1 1

What is logic gate implementation for y?

Logic Gate Implementation Algorithm

1. Determine the truth table for problem statement

2. Consider each output independently

a. Consider all nonzero entries for the output

b. Write the logic equation

c. Simplify the logic equation using the laws of

Boolean algebra (if possible)

3. Draw the digital logic gate implementation

of the simplified logic equation

Logical Function Find a logical expression for the logical function with

three inputs A, B, and C and four outputs W, X, Y and Z such that W is true if at least two of the inputs is true, X is true if exactly one of the inputs is true, Y is true if B is false, and Z is true if all three inputs are true.

Digital Logic Gate

Implementation

Logical Function – Truth Table

Inputs Outputs

A B C W X Y Z

1 1 1 1 0 0 1

1 1 0 1 0 0 0

1 0 1 1 0 1 0

1 0 0 0 1 1 0

0 1 1 1 0 0 0

0 1 0 0 1 0 0

0 0 1 0 1 1 0

0 0 0 0 0 1 0

Logic Function –Boolean Algebra

Find a logical expression for the logical function with three inputs A, B, and C and four outputs W, X, Y and Z such that W is true if at least two of the inputs is true, X is true if exactly one of the inputs is true, Y is true if B is false, and Z is true if all three inputs are true.

CBACBACBACBA Y

CBACBACBA X

CBACBACBA CBAW

Function – Simplified Boolean Algebra

Find a logical expression for the logical function with three inputs A, B, and C and four outputs W, X, Y and Z such that W is true if at least two of the inputs is true, X is true if exactly one of the inputs is true, Y is true if B is false, and Z is true if all three inputs are true.

CBACBACBA X

CBCABAW

Multiplexor / Selector

n inputs

1 output – one of the inputs

Control (selector) lines determine which input becomes the output

log2 n = # of required control

Multiplexor (2-1)

S0 A B Z

Multiplexor (4-1)

Don’t Cares Situations where one does not care what the value of

some output is Another output is true Subset of input combinations determines the values of the

outputs Make it easier to optimize implementation of a logic

function due to simpler expression Two types

Output : don’t care about its value for some input combination Input : output depends on only some of the inputs

Appear as x’s in truth table

Don’t Cares – example

Find a logical expression for the logical function with three inputs A, B, and C and three outputs W, Y and Z such that:

W is true if A or C are true, no matter the value of B,

Y is true if A or B is true, no matter the value of C, and

Z is true if exactly one of the inputs is true and don’t care about its value when both W and Y are true.

Inputs Outputs

A B C W Y Z

Inputs Outputs

A B C W Y Z

1 1 1 1

1 1 0 1

1 0 1 1

1 0 0 1

0 1 1 1

0 1 0 0

0 0 1 1

0 0 0 0

Inputs Outputs

A B C W Y Z

1 1 1 1 1

1 1 0 1 1

1 0 1 1 1

1 0 0 1 1

0 1 1 1 1

0 1 0 0 1

0 0 1 1 0

0 0 0 0 0

Inputs Outputs

A B C W Y Z

1 1 1 1 1 0

1 1 0 1 1 0

1 0 1 1 1 0

1 0 0 1 1 1

0 1 1 1 1 0

0 1 0 0 1 1

0 0 1 1 0 1

0 0 0 0 0 0

Inputs Outputs

A B C W Y Z

1 1 1 1 1 x

1 1 0 1 1 x

1 0 1 1 1 x

1 0 0 1 1 x

0 1 1 1 1 x

0 1 0 0 1 1

0 0 1 1 0 1

0 0 0 0 0 0

Inputs Outputs

A B C W Y Z

1 1 1 1 1 x

1 1 0 1 1 x

1 0 1 1 1 x

1 0 0 1 1 x

0 1 1 1 1 x

0 1 0 0 1 1

0 0 1 1 0 1

0 0 0 0 0 0

Inputs Outputs

A B C W Y Z

1 x x 1 1 x

0 1 1 1 1 x

0 1 0 0 1 1

0 0 1 1 0 1

0 0 0 0 0 0

Inputs Outputs

A B C W Y Z

1 x x 1 1 x

0 1 1 1 1 x

0 1 0 0 1 1

0 0 1 1 0 1

0 0 0 0 0 0

Array of Logic Elements Combinational operations must be performed on an

entire word (32 bits) Represent by showing that a given operation will happen

to entire collection of inputs Example:

A and B are both 32 bits Output, which is also 32 bits, gets A or B

What is the digital logic implementation?

•All multiplexers controlled by the same selector input

•0 selects A

•1 selects B

Arithmetic Logic Unit (ALU) Brawn of the computer Performs arithmetic operations

Addition Subtraction

Performs logical operations AND OR

Arithmetic Logic Unit (ALU) MIPS need 32-bit wide ALU

Connect 32 1-bit ALUs

ALU Logical Operations Operation value

selects logical operation

0 for a AND b 1 for a OR b

ALU Addition (Adder) Three inputs

two operands carryin : carryout of

neighbor adder Two outputs

sum carryout

Adder Implementation?Inputs Outputs

a b CarryIn Sum Carryout

0 0 0 0 0

0 0 1 1 0

0 1 0 1 0

0 1 1 0 1

1 0 0 1 0

1 0 1 0 1

1 1 0 0 1

1 1 1 1 1

Adder Implementation

baCarryInaCarryInb

CarryInbaCarryInbaCarryInbaCarryInbaCarryOut

CarryInbaCarryInbaCarryInbaCarryInbaSum

AND, OR, Adder implementation

32-bit ALU constructed from 32 1-bit ALUs

AND, OR, Adder implementation

32-bit ALU constructed from 32 1-bit ALUs

Adder Modification for Subtraction Add a 2:1 multiplexor that chooses between b and CarryIn of ALU0 set to 1 for subtraction Remember two’s complement

babababa )()1(1

NOR functionality Recall DeMorgan’s

slt Functionality RECALL: All bits except the least significant bit is 0, with

the least significant bit set according to comparison Expand multiplexor to 4:1 with new input Less

Less = 0 on ALU1 – ALU31 (a - b) < 0 a < b ALU31 (most significant bit ALU) has extra output line

Set adder output (also used for overflow detection) used only for slt input LESS on ALU0 (least significant bit ALU)

ALU Building Blocks

ALU0 – ALU30

Branch Instructions Branch if either two registers are equal or if they

are unequal Equality Test: subtract b from a and then test to

see if result is 0

REMEMBER: a – b = 0 a = b Implement:

Result0Result1Result32Result31Zero

MIPS ALU Combine CarryIn and Binvert into one single

control line Bnegate both set to 1 for subtraction both set to 0 for addition and logical operations

4 bit control lines Ainvert (1 bit) Bnegate (1 bit) Operation (selector) for multiplexor (2 bits)

MIPS ALU

Control Lines

FunctionAinvert Bnegate Operation

0 0 00 AND

0 0 01 OR

0 0 10 add

0 1 10 subtract

0 1 11 slt

1 1 00 NOR

appendix c basics of digital logic part i. florida a & m university - department of computer and...

Documents

basic electronics and digital logic computer science 240

gate+isro digital logic and computer organization...

digital logic circuits (part 1) computer architecture (fall...

cs 64: computer organization and design logic lecture...

logic design digital logic 1 - computer science at...

computer logic and digital design chapter 2 henry hexmoor...

ch01 an introduction to digital computer logic

logic design digital logic 1 - virginia...

engineering education: case of digital logic instruction a...

digital logic and computer design by m. morris mano

combinational logic design cs341 digital logic and computer...

advanced digital logic design – eecs 303 today’s … ·...

ece/engrd 2300 digital logic & computer organization ·...

ece 2300 digital logic & computer organization

comp541 digital logic and computer design

digital logic- computer architecture

computer logic and digital designrahimi/cs315/slides/chapter...

ece 2300 digital logic & computer organization · digital...

chapter # 1 digital logic circuits - psau · pdf filechapter...

chapter 1: introduction to boolean logic computing...