Digital Logic

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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.

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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
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

Packaging

Vcc

Gnd

nandgates

A

Y

B

Y

A,B

time

delay

Making functions
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
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

D Qn+1

0 0

1 1

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
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

Some basic circuits

a1

3

a1 a2 #y

0 0 0

0 1 1

1 0 2

1 1 3

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

Multiplexer
End of Lecture
• Questions?
Memory

Din

mplex

REG1

Dout

REG2

decoder

REG3

REG4

R/W

Counter

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
Scheme

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