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

state


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 +

z =

Inputs Outputs

w x y z

1 1 1 0

1 0 0 0

0 1 0 0

0 0 1 1

xw

xw


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 +

z =

Inputs Outputs

w x y z

1 1 1 0

1 0 0 0

0 1 0 0

0 0 1 1

xw

xw

z

x

w

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.

A

B

C

W

X

Y

Z

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.

CBAZ

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.

CBAZ

B Y

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

0

1 1 1

1 0 1

0 1 0

0 0 0

1

1 1 1

1 0 0

0 1 1

0 0 0


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

1 1 1

1 1 0

1 0 1

1 0 0

0 1 1

0 1 0

0 0 1

0 0 0







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

1 x x 1 1 x

1 x x 1 1 x

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?


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

b

babababa )()1(1


NOR functionality Recall DeMorgan’s

BABA


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

ALU31


ALU


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


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

Date post:	19-Jan-2016
Category:	Documents
Upload:	georgia-wood
View:	212 times
Download:	0 times

Appendix C Basics of Digital Logic Part I. Florida A & M University - Department of Computer and...

Documents