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.

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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


Sum of product form

simplify

f

x

y

Sum of product form


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


Arithmetic and logic unit

A

Y

F

B

F

A

B

F

F

Y

Arithmetic and Logic Unit


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

DQn+1

00

11


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


Basic storage element

enables a state change

I

R/W

C

C

I

C

D

R/W

Q

O

O

time

Basic storage element


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 00

0 11

1 02

1 13

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


Conditional change

x=0

S0

S1

x=1

S2

Conditional Change


Coding of state

Coding of State


Put in karnaugh map

Put in Karnaugh map

y

Y

x

y

z

Z

x

z


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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