1 Digital Logic iosup/Courses/2011_ti1400_1.ppt.

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Digital Logic

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

Outline

1. Basics of Boolean algebra and digital implementation

2. Sum of products form and digital implementation3. Functional Units4. Repeated Operations5. Other Building Blocks

TU-DelftTI1400/11-PDS

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Unit of Information

• Computers consist of digital (binary) circuits• Unit of information: bit (Binary digIT), e.g. 0 and 1• There are two interpretations of 0 and 1:

• as data values• as truth values (true and false)


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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:

0123

<00><01><10><11>

<10><00><01><11>


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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 algebra being able to use

only two values• George Boole (1854) showed that logic (or symbolic

reasoning) can be reduced to a simple algebraic system


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Boolean algebra

• Rules are the same as school algebra:

• There is, however, one exception: !

x y y x

x y y x

x(y z) x y x z

(x y) z x (y z)

Commutative Law

Distributive Law

Associative Law

x x x


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Boolean algebra

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

• First the “.” operation: x.y (or x 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 AND function.


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Boolean algebra

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

• Hence, if x means “man” and y means “woman”• Then x+y means “man or woman”• Also called OR function


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Boolean algebra

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

• Then x.x comprises no additional information• Hence

x x x 2 x

x x x


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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


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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 xx


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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

f (x) ax b(1 x)

f (x) a0x a1x a2x 2 ...


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Boolean algebra

• Now• Hence,

• Let b = a0 and a = a0 + a1

• Then we have• From this it follows that

f (x) a0 a1x

f (x) ax b(1 x)

f (1) a

f (0) b


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Boolean algebra

• So

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

f (x) f (1)x f (0)x

f (x, y) f (1,1)x y f (1,0)x y f (0,1)x y f (0,0)x y


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Binary addition

• We apply this on the modulo-2 addition

x y

0 0 01 0 10 1 11 1 0

x y x y y x


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Binary multiplication

• Same for modulo-2 multiplication

x y

0 0 01 0 00 1 01 1 1

x y x y


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Functions

• Let X denote bitstring, e.g., <x4 x3 x2 x1 >

• Any polynomial function Y=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


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Gates

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

• Components are made from transistors

xy

OR

x+y xy

x.y

x x

AND

INVERT


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Networks of gates

• We can make networks of gates

x

y

EXOR

x y

x y x y


Outline

1. Basics of Boolean algebra and digital implementation2. Sum of products form and digital implementation3. Functional Units4. Repeated Operations5. Other Building Blocks


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Sum of product form

x y f

0 0 01 0 10 1 11 1 1

xy

f

simplify

f x y x y x y


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Minimization of expressions

• Logic expressions can often be minimized• Saves components• Example:

f x y z x y z x y z x y z

f x y (z z) y z(x x)

f x y 1 y z1f x y y z


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Karnaugh maps (1)

• Alternative geometrical method

00 01 11 10

00 1 1 0 1

01 1 1 0 1

11 0 0 0 0

10 1 1 1 0

x yv w

f v x v w x v w y v x y


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Karnaugh maps (2)

w

v

y

x

Different drawing


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Don’t Cares

• Some outputs are indifferent• Can be used for minimization

x y f

0 0 01 0 10 1 d1 1 1


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NAND and NOR gates

• NAND and NOR gates are universal• They are easy to realize

de Morgan’s Laws

x y x y

x y x y


Outline



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Delay

• Every network of gates has delays

input

output

transition time

propagation delay

1

0

1

0time


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Packaging

Vcc

Gnd


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Making functions

time

A,B Y

delay

A

BYADD

nand gates


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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

string F


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Arithmetic and Logic Unit

F add subtract compare or

f1 f0 0 0 0 1 1 0 1 1

A

BYF

F

F

A B

Y

F


Outline



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Repeated operations

• Y : = Y + Bi, i=1..n

• Repeated addition requires feedback• Cannot be done without intermediate storage of

results

BYF

F


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Registers

BYF

F = storage element


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SR flip flop

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

S R Qa Qb

0 0 0/1 1/0

0 1 0 1

1 0 1 0

1 1 0 0

R

S

Qa

Qb


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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

clock period


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D flip flop

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

Qn

Qn

C

D

D Qn+1

0 0

1 1


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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

state change


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Basic storage element

O

C

I

R/W

time

C D

Q

C

I

O

R/W

enables a state change


Outline



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4-bit register

C

R/W

C D

Q

I

O

C D

Q

I

O

C D

Q

I

O

C D

Q

I

O


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Some basic circuits

Y

MPLEXm

A B

Y = A if m=1Y = B if m=0

Y

Decoder

A

Only output yA= 1, rest is 0


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Decoder

Y

Decoder

A

Only output yA= 1, rest is 0

a2

a1

0

1

2

3

a1 a2 #y0 0 00 1 11 0 21 1 3


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Multiplexer

Y

MPLEXm

A B

Y = A if m=1Y = B if m=0

y

m

a

b


End of Lecture

• Comments?• Questions?


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Memory

REG1

REG2

REG3

REG4

mplex

decoder

Address

Din

Dout

R/W


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Counter

MPLEX

INC

0001

R/W

preset

output

REG


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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


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State diagrams

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

• Can be represented as a state diagram

S0 S1

S2

state

0 0 S00 1 S11 0 S20 0 S0

code


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Conditional Change

S0 S1

S2

x=0

x=1

Presentstate

Nextx=0

Statex=1

S0 S1 S2

S1 S2 S2

S2 S0 S0


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Coding of State

Presentstate

yz

Nextx=0YZ

Statex=1YZ

00 01 10

01 10 10

10 00 00


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Put in Karnaugh map

0 0 d 1

1 0 d 1x

y

z

1 0 d 0

0 0 d 0x

y

z

Y

Z

Y x y z

Z x y z (x z)y


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Scheme

DQ

DQ

x

Y

Z

z

y

x y

x y z

x z

(x z)y


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General scheme

Combinatorial Logic

Delay elements

Inputs Outputs


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Procedure FST

1. Make State Diagram2. Make State Table3. Give States binary code4. Put state update functions in Karnaugh Map5. Make combinatorial circuit to realize functions

1 Digital Logic iosup/Courses/2011_ti1400_1.ppt.

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