digital and analog computers presentation

Analog/Digital 500.101
I 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

II Representation of information
A. Continuous—represented analogously as a value of a continuously variable parameter
1.position of a needle on a meter
2.rotational angle of a gear
3.amount of water in a vessel
4.electric charge on a capacitor
B. Discrete—digitized as a set of discrete values corresponding to a finite number
of states
1. digital clock
2. painted pickets
3. on/off, as a switch

III Representation of continuous processes
Analogous to the process itself
1.Great Brass Brain—a geared machine to simulate the tides
2.Slide rule—an instrument which does multiplication by adding lengths
which correspond to the logarithms of numbers.
3.Differential analyzer (Vannevar Bush)—variable-size friction wheels to
simulate the behavior of differential equations
Vannevar Bush integrator
Tide 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 encyclopedia
Great 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 calculuate polynomials

Electronic analog computers—circuitry connected to simulate differential equations
1.Phonograph record—wiggles in grooves to represent sound oscillations
2.Electric clocks
3.Mercury thermometers/barometers
Stereo phonograph record

IV Manipulation
A. Analog
1. adding the length-equivalents of logarithms to
obtain a multiply, e.g., a slide-rule
2. adjusting the volume on a stereo
3. sliding a weight on a balance-beam scale
4. adding charge to an electrical capacitor
B. Discrete
1. counting—push-button counters
2. digital operations—mechanical calculators
3. switching—open/closing relays
4. logic circuits—true/false determination
Marble binary counter
Marchant mechanical calculator

V Analog vs. Discrete
Note: "Digital" is a form of representation for discrete
A. Analog
1. infinitely variable--information density high
2. limited resolution--to what resolution can you read a meter?
3. irrecoverable data degradation--sandpaper a vinyl record
B. Discrete/Digital
1. limited states--information density low, e.g., one decimal digit
can represent only one of ten values
2. arbitrary resolution--keep adding states (or digits)
3. mostly recoverable data degradation, e.g., if information is
encoded as painted/not-painted pickets, repainting can
perfectly restore data

VI Digital systems
A. decimal--not so good, because there are few 10-state devices
that could be used to store information fingers. . .?
B. 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.
C. octal --base 8: used to conveniently represent binary data;
almost as efficient as decimal
D. hexadecimal--base 16: more efficient than decimal;
more practical than octal because of binary digit groupings
in computers
Decimal
Binary
21
Hexa-
decimal
24
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

VII Binary logic and arithmetic
A. Background
1. George Boole(1854) linked arithmetic, logic, and binary
number systems by showing how a binary system
could be used to simplify complex logic problems
2. 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
3. Two branches of binary logic systems
a) Combinatorial—in which the output depends
only on the present state of the inputs
b) 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

3. 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 0
0 0 1 0
0 1 0 0
0 1 1 0
1 0 0 0
1 0 1 0
1 1 0 0
1 1 1 1

3. 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 0
0 0 1 0
0 1 0 0
0 1 1 0
1 0 0 0
1 0 1 0
1 1 0 0
1 1 1 1
I = D AND S AND K
D
S
K
I

Binary arithmetic: e.g., adding two binary digits
A B R C
0 0 0 0
0 1 1 0
1 0 1 0
1 1 0 1

Binary arithmetic: e.g., adding two binary digits
A B R C
0 0 0 0
0 1 1 0
1 0 1 0
1 1 0 1
A
B
R
C
R = (A OR B) AND NOT (A AND B)
C = A AND B

AND rules OR rules
A*A = A A +A = A
A*A' = 0 A +A' = 1
0*A = 0 0+A = A
1*A = A 1 +A = 1
A*B = B*A A + B = B+A
A*(B*C) = (A*B)*C A+(B+C) = (A+B)+C
A(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 to
indicate a function y(t)=aekt.

digital and analog computers presentation

Recommended

Recommended

More Related Content

Similar to digital and analog computers presentation

Similar to digital and analog computers presentation (20)

Recently uploaded

Recently uploaded (10)

digital and analog computers presentation