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# Digital Logic PowerPoint PPT Presentation

Digital Logic. http://www.pds.ewi.tudelft.nl/~iosup/Courses/2011_ti1400_1.ppt. Outline. Basics of Boolean algebra and digital implementation Sum of products form and digital implementation Functional Units Repeated Operations Other Building Blocks. Unit of Information.

Digital Logic

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## Digital Logic

http://www.pds.ewi.tudelft.nl/~iosup/Courses/2011_ti1400_1.ppt

### Outline

Basics of Boolean algebra and digital implementation

Sum of products form and digital implementation

Functional Units

Repeated Operations

Other Building Blocks

### Unit of Information

• Computers consist of digital (binary) circuits

• Unit of information: bit(Binary digIT), e.g. 0and 1

• There are two interpretations of 0and 1:

• as data values

• as truth values (true andfalse)

### Bit Strings

• By grouping bits together we obtain bit strings

• e.g<10001>

which can be given a specific meaning

• For instance, we can represent non-negative numbers by bitstrings:

### Boolean Logic

• We want a computer that can calculate, i.e transform strings into other strings:

1 +2 = 3 <01>  <10> = <11>

• To calculate we need an algebrabeing able to use only two values

• George Boole (1854) showed that logic (or symbolic reasoning) can be reduced to a simple algebraic system

### Boolean algebra

• Rules are the same as school algebra:

• There is, however, one exception:

!

Commutative Law

Distributive Law

Associative Law

### Boolean algebra

• To see this we have to find out what the operations “+” and “.” mean in logic

• First the “.” operation: x.y(orx y)

• Suppose x means “black” and y means “cows”. Then, x.y means “black cows”

• Hence “.” implies the class of objects that has both properties. Also called ANDfunction.

### Boolean algebra

• The “+” operation merges independent objects: x + y (or x  y)

• Hence, ifx means “man” and y means “woman”

• Then x+y means “man or woman”

• Also called OR function

### Boolean algebra

• Now suppose both objects are identical, for example x means “cows”

• Then x.x comprises no additional information

• Hence

### Boolean algebra

• Next, we select “0” and “1” as the symbols in the algebra

• This choice is not arbitrary, since these are the only number symbols for which holds x2 = x

• What do these symbols mean in logic?

• “0” : Nothing

• “1”: Universe

• So 0.y = 0 and 1.y = y

### Boolean algebra

• Also, if x is a class of objects, then 1-x is the complement of that class

• It holds that x(1-x) = x -x2 = x-x =0

• Hence, a class and its complement have nothing in common

• We denote 1-x as x

### Boolean algebra

• A nice property of this system that we write any function f(x) as

• We can show this by observing that virtually every mathematical function can be written in polynomial form, i.e

### Boolean algebra

• Now

• Hence,

• Let b = a0 and a = a0 + a1

• Then we have

• From this it follows that

### Boolean algebra

• So

• More dimensional functions can be derived in an identical way:

• We apply this on the modulo-2 addition

### Binary multiplication

• Same for modulo-2 multiplication

### Functions

• Let X denote bitstring, e.g., <x4x3x2 x1>

• Any polynomial functionY=f(X)can be constructed using Boolean logic

• Also holds for functions with more arguments

• Functions can be put in table form or in formula form

### Gates

• We use basic components to represent primary logic operations (called gates)

• Components are made from transistors

x

x

x+y

x.y

y

y

OR

AND

x

x

INVERT

### Networks of gates

• We can make networks of gates

x

y

EXOR

### Outline

• Basics of Boolean algebra and digital implementation

• Sum of products form and digital implementation

• Functional Units

• Repeated Operations

• Other Building Blocks

simplify

f

x

y

### Minimization of expressions

• Logic expressions can often be minimized

• Saves components

• Example:

### Karnaugh maps (1)

x y

• Alternative geometrical method

v w

### Karnaugh maps (2)

Different drawing

y

w

v

x

### Don’t Cares

• Some outputs are indifferent

• Can be used for minimization

de Morgan’s Laws

### NAND and NOR gates

• NAND and NOR gates are universal

• They are easy to realize

### Outline

• Basics of Boolean algebra and digital implementation

• Sum of products form and digital implementation

• Functional Units

• Repeated Operations

• Other Building Blocks

### Delay

• Every network of gates has delays

transition time

1

input

0

1

propagation delay

output

0

time

Vcc

Gnd

nandgates

A

Y

B

Y

A,B

time

delay

### Functional Units

• It would be very uneconomical to construct separate combinatorial circuits for every function needed

• Hence, functional units are parameterized

• A specific function is activated by a special control stringF

A

Y

F

B

F

A

B

F

F

Y

### Outline

• Basics of Boolean algebra and digital implementation

• Sum of products form and digital implementation

• Functional Units

• Repeated Operations

• Other Building Blocks

### Repeated operations

• Y : = Y + Bi, i=1..n

• Cannot be done without intermediate storage of results

Y

F

B

F

### Registers

Y

F

B

F

= storage element

### SR flip flop

• Storage elements are not transient and are able to hold a logic value for a certain period of time

R

Qa

Qb

S

### Clocks

• In many circuits it is very convenient to have the state changed only at regular points in time

• This makes design of systems with memory elements easier

• Also, reasoning about the behavior of the system is easier

• This is done by a clock signal

clockperiod

Qn

C

Qn

D

### D flip flop

• D flip flop samples at clock is high and stores if clock is low

DQn+1

00

11

state change

### Edge triggered flip flops

• In reality most systems are built such that the state only changes at rising edge of the clock pulse

• We also need a control signal to enable a change

enables a state change

I

R/W

C

C

I

C

D

R/W

Q

O

O

time

### Outline

• Basics of Boolean algebra and digital implementation

• Sum of products form and digital implementation

• Functional Units

• Repeated Operations

• Other Building Blocks

### 4-bit register

I

R/W

I

I

I

C

C

D

C

D

C

D

C

D

Q

Q

Q

Q

O

O

O

O

A

B

Y = A if m=1

Y = B if m=0

m

MPLEX

Y

Y

Only output yA= 1, rest is 0

Decoder

A

a1

3

a1 a2#y

0 00

0 11

1 02

1 13

2

1

0

a2

### Decoder

Y

Only output yA= 1, rest is 0

Decoder

A

A

B

Y = A if m=1

Y = B if m=0

m

MPLEX

Y

b

y

a

m

• Questions?

Din

mplex

REG1

Dout

REG2

decoder

REG3

REG4

R/W

preset

MPLEX

R/W

REG

INC

0001

output

### Sequential circuits

• The counter example shows that systems have state

• The state of such systems depend on the current inputs and the sequence of previous inputs

• The state of a system is the union of the values of the memory elements of that system

code

S0

S1

S2

### State diagrams

• We call the change from one state to another a statetransition

• Can be represented as a state diagram

x=0

S0

S1

x=1

S2

y

Y

x

y

z

Z

x

z

x

z

Z

Q

D

Y

y

Q

D

### General scheme

Outputs

Inputs

Combinatorial Logic

Delay elements

### Procedure FST

• Make State Diagram

• Make State Table

• Give States binary code

• Put state update functions in Karnaugh Map

• Make combinatorial circuit to realize functions