MOBILE PHONES & MATHEMATICS. Mathematics has played an increasingly large role in the development of...
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MOBILE PHONES & MATHEMATICS
MOBILE PHONES & MATHEMATICS. Mathematics has played an increasingly large role in the development of new technologies. That’s because scientists exploit
Mathematics has played an increasingly large role in the
development of new technologies. Thats because scientists exploit
physics phenomena to construct devices and to make them work
properly, and physics phenomena ARE ALL DESCRIBED MATHEMATICALLY.
Cell phones in particular operate by principles of
ELECTROMAGNETICS.
In the past telephones were used instead of mobile phones, and
calls were carried through wires and cables connecting your home
phone to a huge communications web. Today physical wires no longer
connect the offices together for each phone call. That system was
incredibly expensive. Instead, a fiber-optic line carries a
digitized version of your voice. Your voice (along with thousands
of others) becomes a stream of bytes flowing on a fiber-optic line
between offices. The difference in cost between "a pair of copper
wires carrying a single conversation" and "a single fiber carrying
thousands and thousands of conversations" is phenomenal.
FIBER-OPTIC LINES = strands of optically pure glass as thin as a
human hair that carry digital information over long distances. BYTE
= (1) group of adjacent binary digits, treated as a unit from the
computer. The most common size of byte consists of 8 binary digits.
(2) A group of binary digits used to encode a character.
Slide 5
How do the voice calls travel from one device to another? And
How can this be described by mathematics?
Slide 6
A voice call is a digitized signal carried by WAVES which, in
order to be analyzed, must be decomposed in sub waves. To do so its
used the Fourier Tansform, a powerful mathematical tool for the
analysis of NON PERIODIC functions that decompose a signal process
into its SINE and COSINE COMPONENTS. Fourier Transform
> A voice call is a non-periodic and non-sinusoidal wave
which cannot be defined by a function with parameters. Through the
FT: Is possible to obtain the sine and cosine components, described
by the Fourier series, an infinite series in which the terms are
constants multiplied by sine and cosine functions of integer
multiples of the variable. It is an analog wave representing the
vibrations created by your voice. For example, here is a graph
showing the analog wave created by saying the word "hello
Slide 11
The Anti Transform Fourier is instead usefull to turn again the
sub waves into the signal, that is the replying voice coming from
the first cell phone:
Slide 12
Binary numeral system Every internal operation in every
computer, as well as every other digital device such as cell
phones, DVRs and DVD players, and mobile data devices, is a
binary-number operation. The binary numeral system is an easy way
(the only one that technological devices can interpret) to
represent informations. So, for instance, every time we push a key
on the keyboard, we send an input to the mobile phone which is
converted in an information through the binary code.
Slide 13
Slide 14
The word ASCII is an acronym for "American Standard Code for
Informations Interchange", and it is a character-encoding scheme
originally based on the English alphabet and proposed by the IBM
engineer Bob Bemer. ASCII code represents text in computers,
communications equipment and other devices that use texts. Today
ASCII has been overtaken by other character-encoding like UTF-8,
but its still really famous and used in informatics. Text messages
and ASCII
Slide 15
In the ASCII character set, each binary value between 0 and 127
is given a specific character. ASCII includes definitions for: 128
characters 33 non printing control characters; 95 printable
characters. In other words in cell phones memory every single text
character (visible or not) is stored as a numeral code and it
occupies 7bit (7 binary numbers). When we write a message where we
ask some friend of ours: > we are actually writing 14 numeral
codes that correspond to the image of the letters or to operations
like space. > (60) (60) (72) (111) (119) (31) (97) (114) (101)
(31) (121) (111) (117) (63) (62) (62)
Slide 16
Slide 17
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Slide 19
G lobal P ositioning S ystem When people talk about "a GPS,"
they usually mean a GPS receiver. The Global Positioning System
(GPS) is actually a constellation of 27 Earth-orbiting satellites
(24 in operation and three extras in case one fails). The U.S.
military developed and implemented this satellite network as a
military navigation system, but soon opened it up to everybody
else. Each of these make two complete rotations every day. The
orbits are arranged so that at any time, anywhere on Earth, there
are at least four satellites "visible" in the sky.
Slide 20
A GPS receiver's job is to locate four or more of these
satellites, figure out the distance to each, and use this
information to deduce its own location. This operation is based on
a simple mathematical principle called trilateration. Trilateration
in three- dimensional space can be a little tricky, so we'll start
with an explanation of simple two- dimensional trilateration.
Slide 21
Imagine to be somewhere in Poland, in an unknown city. You ask
a friendly local where you are, and he/she replies that you are 100
km far from Warszawa. You could be anywhere on a circle around
Warszawa that has a radius of 100 km. If you ask a second person,
and if he/she answer that you are distant 130 km from d, you can
combine these two informations and obtain two circles that
intersect. Now you know you must be at one of these two
intersection points.
Slide 22
Slide 23
If a third person tells you that you are 110 km from Lublin,
you can eliminate one of the two possibilities, because the third
circle will only intersect with one of these points. You now know
exactly where you are - Radom. This same concept works in three-
dimensional space, as well, but you're dealing with spheres instead
of circles.
Slide 24
RADOM
Slide 25
Slide 26
If you know you are 10 miles from satellite A in the sky, you
could be anywhere on the surface of a huge, imaginary sphere with a
10-mile radius. If you also know you are 15 miles from satellite B,
you can overlap the first sphere with another, larger sphere. The
spheres intersect in a perfect circle. If you know the distance to
a third satellite, you get a third sphere, which intersects with
this circle at two points. The Earth itself can act as a fourth
sphere -- only one of the two possible points will actually be on
the surface of the planet, so you can eliminate the one in space.
Receivers generally look to four or more satellites, however, to
improve accuracy and provide precise altitude information.
Slide 27
Mathematically speaking, this speech can be translated into a
system of spheres equations:
Slide 28
In order to make this simple calculation, then, the GPS
receiver has to know two things: 1) The distance between you and
each of those satellites. 2) The location of at least three
satellites above you; The GPS receiver figures both of these things
out by analyzing high-frequency, low-power radio signals from the
GPS satellites. The receiver can figure out how far the signal has
traveled by timing how long it took the signal to arrive. In the
next section, we'll see how the receiver and satellite work
together to make this measurement.
Slide 29
1) We saw that a GPS receiver calculates the distance to GPS
satellites by timing a signal's journey from satellite to receiver.
At a particular time (let's say midnight), the satellite begins
transmitting a long, digital pattern called a random code. The
receiver begins running the same digital pattern also exactly at
midnight. When the satellite's signal reaches the receiver, its
transmission of the pattern will lag a bit behind the receiver's
playing of the pattern. The length of the delay is equal to the
signal's travel time. The receiver multiplies this time by the
speed of light to determine how far the signal traveled. Assuming
the signal traveled in a straight line, this is the distance from
receiver to satellite.
Slide 30
Slide 31
2) In order for the distance information to be of any use, the
receiver also has to know where the satellites actually are. This
isn't particularly difficult because the satellites travel in very
high and predictable orbits. The GPS receiver simply stores an
almanac that tells it where every satellite should be at any given
time. Things like the pull of the moon and the sun do change the
satellites' orbits very slightly, but the Department of Defense
constantly monitors their exact positions and transmits any
adjustments to all GPS receivers as part of the satellites'
signals.