MOBILE PHONES & MATHEMATICS

Mathematics has played an increasingly large role in the development of new technologies. That’s 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.

CHIP  graphs PIXEL  matrices, colours WAVES  functions CALCULATOR  algorithms WRITING  binary numeral system

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.

How do the voice calls travel from one device to another? And… How can this be described by mathematics?

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 it’s 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

f(x) = sin(2  f)t f(x) = sin(2  f)t + 1/3sin3(2  f)t f(x) = sin(2  f)t + 1/3sin3(2  f)t + 1/5sin5(2  f)t f(x) = sin(2  f)t + 1/3sin3(2  f)t + 1/5sin5(2  f)t + 1/7sin7(2  f)t + 1/9sin9(2  f)t + + 1/11sin11(2  f)t + 1/13sin13(2  f)t

> 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“

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:

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.

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 it’s still really famous and used in informatics. Text messages and ASCII

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)

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.

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.

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 Łόdź, 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.

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.

RADOM

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.

Mathematically speaking, this speech can be translated into a system of spheres’ equations:

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.

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.

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.

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