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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.

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digital logic

Digital Logic

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

outline
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
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
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
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
Boolean algebra
  • Rules are the same as school algebra:
  • There is, however, one exception:

!

Commutative Law

Distributive Law

Associative Law

boolean algebra1
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 algebra2
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 algebra3
Boolean algebra
  • Now suppose both objects are identical, for example x means “cows”
  • Then x.x comprises no additional information
  • Hence
boolean algebra4
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 algebra5
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 algebra6
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 algebra7
Boolean algebra
  • Now
  • Hence,
  • Let b = a0 and a = a0 + a1
  • Then we have
  • From this it follows that
boolean algebra8
Boolean algebra
  • So
  • More dimensional functions can be derived in an identical way:
binary addition
Binary addition
  • We apply this on the modulo-2 addition
binary multiplication
Binary multiplication
  • Same for modulo-2 multiplication
functions
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
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
Networks of gates
  • We can make networks of gates

x

y

EXOR

outline1
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
Minimization of expressions
  • Logic expressions can often be minimized
  • Saves components
  • Example:
karnaugh maps 1
Karnaugh maps (1)

x y

  • Alternative geometrical method

v w

karnaugh maps 2
Karnaugh maps (2)

Different drawing

y

w

v

x

don t cares
Don’t Cares
  • Some outputs are indifferent
  • Can be used for minimization
nand and nor gates

de Morgan’s Laws

NAND and NOR gates
  • NAND and NOR gates are universal
  • They are easy to realize
outline2
Outline
  • Basics of Boolean algebra and digital implementation
  • Sum of products form and digital implementation
  • Functional Units
  • Repeated Operations
  • Other Building Blocks
delay
Delay
  • Every network of gates has delays

transition time

1

input

0

1

propagation delay

output

0

time

packaging
Packaging

Vcc

Gnd

making functions

nandgates

A

Y

ADD

B

Y

A,B

time

delay

Making functions
functional units
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
outline3
Outline
  • Basics of Boolean algebra and digital implementation
  • Sum of products form and digital implementation
  • Functional Units
  • Repeated Operations
  • Other Building Blocks
repeated operations
Repeated operations
  • Y : = Y + Bi, i=1..n
  • Repeated addition requires feedback
  • Cannot be done without intermediate storage of results

Y

F

B

F

registers
Registers

Y

F

B

F

= storage element

sr flip flop
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
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

d flip flop

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

edge triggered flip flops

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
outline4
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
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

some basic circuits

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
decoder

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

multiplexer

A

B

Y = A if m=1

Y = B if m=0

m

MPLEX

Y

b

y

a

m

Multiplexer
end of lecture
End of Lecture
  • Comments?
  • Questions?
memory
Memory

Din

mplex

REG1

Dout

REG2

decoder

Address

REG3

REG4

R/W

counter
Counter

preset

MPLEX

R/W

REG

INC

0001

output

sequential circuits
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
state diagrams

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
Scheme

x

z

Z

Q

D

Y

y

Q

D

general scheme
General scheme

Outputs

Inputs

Combinatorial Logic

Delay elements

procedure fst
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
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