DIGITAL SYSTEMS

Real world systems and processes

Mostly continuous (at the macroscopic level): time, acceleration, chemical reactions

Sometimes discrete: quantum states, mass (# of atoms)

Mathematics to represent physical systems is continuous (calculus)

Mathematics for number theory, counting, approximating physical systems can be discrete

Representation of information

A. Continuous—represented analogously as a value of a continuously variable parameter1.position of a needle on a meter2.rotational angle of a gear3.amount of water in a vessel4.electric charge on a capacitor

B. Discrete—digitized as a set of discrete values corresponding to a finite number of states1. digital clock2. painted pickets3. on/off, as a switch

Representation of continuous processes

Analogous to the process itself

• Great Brass Brain—a geared machine to simulate the tides

• Slide rule—an instrument which does multiplication by adding lengths which correspond to the logarithms of numbers.

• Differential analyzer (Vannevar Bush)—variable-size friction wheels to simulate the behavior of differential equations

Vannevar Bush integratorTide calculator

Brass Brain was the equal of 100 mathematicians,weighted a mere 2500 lbs

Imagine the fearful gnashings of mathematicians in November, 1928 upon reading this account of the USGS's new "brass brain," which could "do the work of 100 trained mathematicians" in calculating tides:

The machine weighs 2,500 pounds. It is 11 feet long, 2 feet wide, and 6 feet high. Its whirring cogs are enclosed in a housing of mahogany and glass. Earthquakes, fresh-water floods, and strong winds that cannot be predicted affect the accuracy of the Brass Brain to a degree. Nevertheless 70% of the predicted tides agree within five minutes of the observed tide. The Coast and Geodetic Survey issues an annual bulletin in which it lists the forthcoming tides in 84 ports of the world. The report contains upwards of a million figures, all compiled by the Brass Brain. It has been estimated that the Brass Brain saves the government $125,000 each year in salaries of mathematicians who would be required to take its place.

From Instruments of Science: an historical encyclopediaGreat Brass Brain

“It remained in use until the late 1960s,when an IBM 7090 computer took over the job. Even when digital computers finally took over from analog instruments, the amount of arithmetic needed to properly evaluate the cosine series was so vast that the output had to be limited to simply times of high and low tide for any particular area. This was overcome only when, during the 1970s, digital computers became powerful enough. . .”

Discrete representations

What is it?? What is it??What is it??

Babbage difference engine to calculate polynomials

Electronic analog computers—circuitry connected to simulate differential equations

• Phonograph record—wiggles in grooves to represent sound oscillations

• Electric clocks• Mercury thermometers/barometers

Stereo phonograph record

Manipulation

Analog• adding the length-equivalents of logarithms to obtain a

multiply, e.g., a slide-rule• adjusting the volume on a stereo• sliding a weight on a balance-beam scale• adding charge to an electrical capacitor

Discrete• counting—push-button counter• digital operations—mechanical calculators• switching—open/closing relays• logic circuits—true/false determination

Marble binary counter

Marchant mechanical calculator

Analog vs. Discrete

Note: "Digital" is a form of representation for discrete

Analog• infinitely variable--information density high• limited resolution--to what resolution can you read a meter?• irrecoverable data degradation--sandpaper a vinyl record

Discrete/Digital• limited states--information density low, e.g., one decimal digit can represent only

one of ten values• arbitrary resolution--keep adding states (or digits)• mostly recoverable data degradation, e.g., if information is encoded as painted/not-

painted pickets, repainting can perfectly restore data

Digital systems• decimal

• not so good, because there are few 10-state devices that could be used to store information fingers. . .?

• binary• excellent for hardware; lots of 2-state devices: switches, lights,

magnetics• poor for communication: 2-state devices require many digits to

represent values with reasonable resolution• excellent for logic systems whose states are true and false. But

binary is king because components are so easy (and cheap) to fabricate.

• octal --base 8: • used to conveniently represent binary data; almost as efficient

as decimal

• hexadecimal--base 16: • more efficient than decimal; more practical than octal because

of binary digit groupings in computers

Decimal Binary21

Hexa-decimal24

0 0000 0

1 0001 1

2 0010 2

3 0011 3

4 0100 4

5 0101 5

6 0110 6

7 0111 7

8 1000 8

9 1001 9

10 1010 A

11 1011 B

12 1100 C

13 1101 D

14 1110 E

15 1111 F

Binary logic and arithmetic

Background• George Boole(1854) linked arithmetic, logic, and binary number systems by showing how

a binary system could be used to simplify complex logic problems

• Claude Shannon(1938) demonstrated that any logic problem could be represented by a system of series and parallel switches; and that binary addition could be done with electric switches

• Two branches of binary logic systems• Combinatorial—in which the output depends only on the present state of the

inputs• Sequential—in which the output may depend on a previous state of the inputs,

e.g., the “flip-flop” circuit

C

A

B

A B C

0 0 0

1 0 0

0 1 0

1 1 1

AND gate

C

A

B

A B C

0 0 0

1 0 0

0 1 0

1 1 1

AND gate

Simple “AND” Circuit

Battery

A B

C

OR gate

C

A

B

A B C

0 0 0

1 0 1

0 1 1

1 1 1

OR gate

C

A

B

A B C

0 0 0

1 0 1

0 1 1

1 1 1

A

Simple “OR” circuit

B

C

A B

NOT gate

A B

1 0

0 1

A B

NOT gate

A B

1 0

0 1

A

B

Simple “NOT” circuit

C

A

B

A B C

0 0 1

1 0 1

0 1 1

1 1 0

NAND gate

C

A

B

A B C

0 0 1

1 0 1

0 1 1

1 1 0

NAND gate

Simple “NAND” Circuit

Battery

A B

C

Control systems: e.g., car will start only if doors are locked, seat belts are on, key is turned

D S K I

0 0 0 00 0 1 00 1 0 00 1 1 01 0 0 01 0 1 01 1 0 01 1 1 1

Practice Problem

I = D AND S AND K

D

S

K

I

Control systems: e.g., car will start only if doors are locked, seat belts are on, key is turned

D S K I

0 0 0 00 0 1 00 1 0 00 1 1 01 0 0 01 0 1 01 1 0 01 1 1 1

Binary arithmetic: e.g., adding two binary digits

A B R C

0 0 0 0 0 1 1 0 1 0 1 01 1 0 1

A

BR

C

R = (A OR B) AND NOT (A AND B)

C = A AND B

Binary arithmetic: e.g., adding two binary digits

A B R C

0 0 0 0 0 1 1 0 1 0 1 01 1 0 1

AND rules OR rules

A*A = A A +A = AA*A' = 0 A +A' = 10*A = 0 0+A = A1*A = A 1 +A = 1A*B = B*A A + B = B+AA*(B*C) = (A*B)*C A+(B+C) = (A+B)+CA(B+C) = A*B+B*C A+B*C = (A+B)*(A+C)A'*B' = (A+B)' A'+B' = (A*B)‘ (DeMorgan’s theorem)

Notation: * = AND + = OR ‘ = NOT

Boolean algebra properties

Computing power has been growing at an exponential rate

Note: graph is a “semi-log” plot—the best way toindicate a function y(t)=aekt.