GPS

Global Positioning System (GPS)

Introduction

We chose the practical assignment: Global Positioning System, because we thought it was an interesting topic. Many people we know have a TomTom nowadays, and we didn’t really know how they worked, so therefore we wanted to find this out.
The investigation of the Global Positioning System is very important, because it influences a lot of things in daily life. The system is used on a very broad scale. Most people do not even notice how the system works, but it dóes have to be made. This means that there must have been someone who invented the system. We are going back to the beginning, and re-invent the GPS. We want to do this, so we can give a clear view on how the system works, etc.
Also, an experiment is done, to see whether we are able to find out our position, by using our own system.
There are so many things in life for which the GPS is used to find out the position, or even the shortest routes. Route describers, maritime transport, business people, air traffic, logistics, scouts, military purposes, etc., have so many advantages from this system, that an investigation can lead to a lot of knowledge.
With the outcome of this investigation, only a few can be done. It is a simple version, and a test, to see whether it is really as logical as said.
However, the outcome of the investigation may be used for different school years. Pupils of secondary schools can get a bit of knowledge about the GPS system, and they can get familiar with the system.

Theory
What is GPS?

The Global Positioning System (GPS) is a satellite-based navigation system made up of a network of 24 satellites placed into orbit around the earth by the U.S. Department of Defence. GPS was originally intended for military applications, but in the 1980s, the government made the system available for civilian use. The system is now used by business men, in air traffic, logistics, and many more. GPS works in any weather condition, any place in the world, 24 hours a day.

History of GPS

Before the invention of the GPS, other techniques of navigation were used, like the compass and the sextant. The needle of the compass always points towards the north. So even if people didn't know where they were, at least they knew in what direction they were travelling. The sextant measures the exact angles of stars, the moon and the sun above the horizon by the use of adjustable mirrors. Early sextants could only measure the latitude, and sailors were still not able to work out their longitude.
In Great Britain, a group of well-known scientists was formed in the seventeenth century, called the Board of Longitude. They offered a substantial cash reward to any person who could find a way of working out the longitude of a ship within thirty miles. In 1761, a man named John Harrison developed the chronometer. This invention lost ,or gained, only about one second a day. Sextants and chronometers were used together to provide travellers with the latitude and longitude.
Radio-based navigation systems were developed in the early twentieth century, and were used in World War II. As this technology advanced, both ships and airplanes used ground-based radio-navigation systems. The disadvantage of using a system that uses ground-generated radio waves, is that a choice has to made between a high-frequency system that is accurate, but does not cover a wide area, and a low-frequency system that covers a wide area, but is not very accurate.
When Sputnik (satellite) was launched into space by Russians on October 4th, 1957, it became clear that "artificial stars" could be used for navigation. The evening after the launch, researchers of the Massachusetts Institute of Technology determined the orbit of the Russian satellite, by nothing that the Sputnik's radio signal increased as it approached and decreased as it left. So the fact that a satellite's position could be tracked from the ground was the first step in recognizing that a subject's whereabouts on the ground could be determined using radio signals from the satellite.
The U.S. Navy experimented with satellite navigation. In the mid-sixties the Transit System was developed for submarines carrying Polaris nuclear missiles. This system has six satellites that circled the earth in polar orbits. In measuring the Doppler shift of the radio signals the submarines could locate its position within fifteen minutes.
The Global Positioning System, GPS, was designed, built, operated and maintained by the U.S. Department of Defence. It used to be known as the Navistar Global Positioning System and was first brainstormed at the Pentagon in 1973 as they were looking for a satellite system that was error-proof. In 1978 the first operational GPS satellite was launched. By the mid-1990s the system was fully operational with 24 satellites.

Who use GPS?

GPS is usable everywhere, except at places where it is impossible to receive the signals, for example inside most buildings, caves, other subterranean locations, and underwater. The most common airborne applications are for navigation by general aviation and commercial aircrafts. At sea, GPS is also used for navigation by recreational boaters, commercial fishermen, and professional mariners. Land-based applications are more diverse. The scientific community uses GPS for its precise timing capability and position information.
Recreational uses of GPS are almost as varied as the number of recreational sports available. GPS is popular among hikers, hunters, snowmobilers, mountain bikers, and cross-country skiers, just to name a few. It is used by anyone who needs to keep track of where he or she is, needs to find his or her way to a specified location, or needs to know what direction and how fast he or she is travelling.
GPS is now common in cars as well. Some basic systems provide emergency roadside assistance, with the push of a button (by transmitting your current position to a dispatch center). More different systems that can show your position on a street map are also available, for example: Tomtom. Nowadays, these systems allow a driver to keep track of where he or she is, and suggest the best route to reach a location.

The GPS satellite system

The 24 satellites that make up the GPS space segment, are orbiting around the earth at about 12,000 miles (19,300km) above us. They are constantly moving, making two complete orbits in less than 24 hours. These satellites are travelling at speeds of about 7,000 miles (11,300km) an hour.
GPS satellites are powered by solar energy. They have backup batteries on board to keep them running during a solar eclipse, when there is no solar power. Small rocket boosters on the satellites keep them flying in the correct path.
Here are some other interesting facts about the GPS satellites:
• The first GPS satellite was launched in 1978.
• The first time all 24 satellites were operating, was in 1994
• Each satellite is built to last about 10 years. Replacements are constantly being built and launched into orbit.
• A GPS satellite weighs approximately 2,000 pounds (907 kilograms) and is about 17 feet (5.81 meters) wide with the solar panels extended.
• The transmitter power is only 50 watts or less

How do these satellites work?

GPS satellite system works with the concept of trilateration.

How trilateration works:

This is the example of 2D Operating Mode (latitude and longitude); Imagine you are somewhere in the United States and you are totally lost and you don't have a clue where you are. You find a friendly-looking person and ask, "Where am I?" and the person says to you, "You are 625 miles from Boise, Idaho." This is a piece of information, but it is not really that useful by itself. You could be anywhere on a circle around Boise that has a radius of 625 miles.

So you ask another person, and he says, "You are 690 miles away from Minneapolis, Minnesota." This is helpful -- if you combine this information with the Boise information, you have two circles that intersect. You now know that you are at one of the two points, but you don't know which one.

If a third person tells you that you are 615 miles from Tucson, Arizona, you can figure out which of the two points you are at:

With three known points, you can see that you are near Denver, Colorado!
Trilateration is a basic geometric principle that allows you to find one location if you know the distance from other, already known, locations. The geometry behind this is very easy to understand in two dimensional spaces.
This same concept works in three dimensional (3D) (latitude, longitude and altitude) spaces as well, but you're dealing with spheres instead of circles. You also need four spheres instead of three circles to find your exact location. The heart of a GPS receiver is the ability to find the receiver's distance from four (or more) GPS satellites. The receiver can calculate its exact location and altitude on Earth. If the receiver can only find three satellites, then it can use an imaginary sphere to represent the Earth and it can give you location information, but no altitude information.

For a GPS receiver to find your location, it has to determine two things:
• The location of at least three satellites above you
• The distance between you and each of those satellites

Sources of GPS signal errors

• Ionosphere and troposphere delays — The satellite signal slows as it passes through the atmosphere. The GPS system uses a built-in model that calculates an average amount of delay to partially correct this type of error.
• Signal multipath — This occurs when the GPS signal is reflected off objects such as tall buildings or large rock surfaces before it reaches the receiver. This increases the travel time of the signal, therefore causing errors.
• Receiver clock errors — A receiver's built-in clock is not as accurate as the atomic clocks onboard the GPS satellites. Therefore, it may have slight timing errors.
• Orbital errors — Also known as ephemeris errors, these are inaccuracies of the satellite's reported location.
• Number of satellites visible — The more satellites a GPS receiver can "see," the more accurate it is. Buildings, terrains, electronic interference, or sometimes even dense foliage can block signal reception, causing position errors or possibly no position definition at all. GPS units will not work indoors, underwater or underground.
• Satellite geometry/shading — This refers to the relative position of the satellites at any given time. Ideal satellite geometry exists when the satellites are located at wide angles relative to each other. When the satellites are located in a line or in a tight grouping, it may result in bad calculations.
• Intentional degradation of the satellite signal — Selective Availability (SA) is an intentional degradation of the signal once imposed by the U.S. Department of Defence. SA was intended to prevent military adversaries from using the highly accurate GPS signals. The government turned off SA in May 2000, which significantly improved the accuracy of civilian GPS receivers.

Here you can see how much these errors can differ the position.

Source Uncorrected Error Level
Ionosphere 0-30 meters
Troposphere 0-30 meters
Measurement Noise 0-10 meters
Ephemeris Data 1-5 meters
Clock Drift 0-1.5 meters
Multipart 0-1 meter
Selective Availability 0-70 meters

Introduction to Latitude and Longitude

Lines of latitude circle the earth in an east/west direction while lines of longitude circle in a north/south direction from pole to pole. Degrees of latitude start at zero at the equator and increase to either ninety degrees north or ninety degrees south at the poles. There is an equal distance between each line of latitude (69 miles/111 km). Also, there is an equal degree of rotation the between lines of longitude. The distance between lines of longitude varies from 69 miles at the equator, to zero degrees at the poles.
This has to do with GPS because the system (located in for example your car), gets the information on your “exact” location in coordinates. Inside the device there is something that converts these coordinates into a place on the map.

Theoretical introduction

The GPS system is to be copied in these experiments. In these tests and calculations, satellites are excluded because the school does not have the rights to get this kind of data from a satellite, logical, of course.

In the experiments, a sound wave will be used instead of the usual radio waves. To use it in a setup which is used in real life, the speed of sound needs to be known first. This can be achieved with the use of two microphones; make a sound near the one on which the trigger norm is installed and measure the difference in time and distance from microphone 1 to microphone 2. With this, the speed of sound can be calculated. If the experiment is done more than one or two times, most of all three times; the average of all these different speeds of sound gives a more accurate number of the real speed of sound. A deviation on this speed can also be calculated by repeating this experiment more than once.

The second and third experiment is the closest to the real GPS as we know it. With the “real GPS”, the 2 dimensional part is meant, so only two coordinates are given; used in cars and by walkers, for example in a big forest to find out your exact location. There is also a 3 dimensional system, but this is not used in everyday life that often; only in planes and devices in which the height is also important. That is the third experiment
To get back on the 2 dimensional GPS; the experimental setup isn’t really the same as the 2 D GPS. In this system, they make use of intersecting spheres while in this experiment it is only about the intersection circles.
Three microphones are placed (it is fixed and their distance to one another is measured) in a room; another microphone, in this experiment called Microphone 4, is placed somewhere in the room as well (this is the one on which the trigger norm is installed). Very close to this so-called M4, a sound is produced, so the graph’s zero is confirmed. Then, in different graphs, the results of the other microphones are put. By measuring the time it takes for the sound to come from M4 to the other microphones (when the pressure is above the x-axis) the distance can be calculated with the speed of sound found in experiment 1. We will do de same in the third experiment, but than there is a z-axis too.
Then, the mathematical part takes place; the intersection of the circles. It is done with the use of the formula of Pythagoras (explained in the theory). The outcome will most certainly be a little area of which the average coordinates are calculated, and so the positioning of the source of the sound.

Experiment 1

The speed of sound
Introduction
Sound is a wave which is created by vibrating objects and propagated through a medium which transports energy through the medium without permanently transporting matter from one location to another. In a wave, particles of the medium are temporally displaced and then return to their original position.
So, to pass on a sound wave, you first need the medium. It is simply a series of interconnected and interacting particles. Secondly, there is an original source of the wave, some vibrating object capable of disturbing the first particle of the medium. The vibrating object which creates the disturbance could be anything, like you voice or two sticks that are being smacked together. Thirdly, the sound wave is transported from one location to another by means of the particle interaction. If the sound wave is moving through the air, one air particle is displaced from its equilibrium position, and exerts a push or pull on its nearest neighbours, causing them to be displaced from their equilibrium position. This particle interaction continues throughout the entire medium, with each particle interacting and causing a disturbance of its nearest neighbours. Since a sound wave is a disturbance which is transported through a medium via the mechanism of particle interaction, a sound wave is characterized as a mechanical wave.

Aim
What is the speed of sound?

Hypothesis
The speed of sound is supposed to be around 333 m/s; this is assumed because you can calculate the distance in kilometres of a thunderstorm by dividing the seconds between the lightning and the thunder by 3. The speed of light is negligible, and you take this as a start for counting.
Take into account that this is outside with a temperature of about 10 or 15 ºC and indoors it is 20 to 25 ºC. Therefore, the speed of sound indoors may be a bit faster because the molecules (to the molecule model) have a bigger distance between each other and bump into each other more often when it is warmer. This makes it easier to quickly pass on the sound waves from molecule to molecule so the sound travels a bit faster.

The Binas also tells you that at T = 293K (25°C) the speed of sound is 0.343 * 103 m s-1 which is the same as 343 m/s.

Experimental section
Materials
- two microphones
- a computer with the software Coach5
- spring rule
- wires
- two metal sticks

Method
1. Connect the microphones to the computer with the program Coach5 on it
2. Install a pressure (in Pa) / time (in ms) graph and the measuring speed of 5000 times per second
3. install the trigger function for one of the two microphones.
4. Place the two microphones as far as possible from each other
5. Measure distances between the microphones with a spring rule
6. Smack one of the sticks on the other one close to one microphone
7. Let Coach5 put the results of the two microphones in a pressure(in Pa)/time(in ms) graph
8. Repeat step 6 and 7 at least three times in the same arrangement (NB. Still smack the sticks close to one microphone)
9. Calculate the speed with the formula V = S / T for each hit
10. Calculate the average speed of sound

Scheme of setup:
Speed of time
Time analogue 1 (sec) Time analogue 2
0.542 0.539
0.8412 0.8378
0.732 0.7288
0.430 0.4266

Processing of the results
For a calculation of velocity V = S / T
(Actually, the formula is V = ∆S / ∆T, but because S0 and T0 are always 0, you can take S1 and T1 as ∆S and ∆T because ∆S = S1 – 0 = S1 and ∆T = T1 – 0 = T1)

1. V1 ?
S = 1 m
T = 0.003s
V1 = s/t = 333 m/s

2. V2 ?
S = 1 m
T = 0.0034s
V2 = s/t =294 m/s

3. V3 ?
S = 1 m
T = 0.0032 s
V3 = s/t = 313 m/s

4. V4 ?
S = 2.77 m
T = 0,003 s
V4 = s/t = 333 m/s

V average : (V1 + V2 + V3 + V4) / 4
V average = (333 + 294 + 313 + 333) / 4 = 320 m/s

Diversion of V average = V average - Vexp if Vexp < V average
Or diversion of V average = Vexp - V average if Vexp > V average

Exp. ∆S (in m) ∆T (in s) V(∆S / ∆T) (in m/s) Diversion of the Vaverage (in m/s)
1 1.00 0.003 333 13
2 1.00 0.0034 294 26
3 1.00 0.0032 313 7
4 1.00 0.003 333 13
average 1.00 0.00315 320 14.75

Conclusion
The speed of sound (V average) is 320 ± 15 m/s.
The rest of the calculations have been made with this speed (320)

Discussion
The hypothesis is quite correct. The difference between 343 m/s and 320 ± 15 m/s is not very much.
This difference may be because the air had a different composition which affected the sound wave.
But mostly, the diversion of the answer compared to the hypothesis is to blame on the short distance between the two microphones, 1.00m. Because you work with such small numbers, the accuracy is not very high.
The magnitude of the errors that affected the results is 15 m/s (the average of the average diversion from the average speed of sound). This is in percentages (15 / 320) *100% = 5,3 %. That’s quite good.

Improvements for additional research:
The experiment could better be done in a bigger room to higher the accuracy. But in this case it was impossible for us to do that.
The cooperation was clear, there was one person who smacked the sticks, one who pushed the button on the computer to wait until the trigger norm was reached and the other one took the measurements.

Experiment 2

2D GPS
Introduction
To get to know the place of the source of a noise, you need 4 microphones. At first you need to measure the time it takes for the sound to travel from the source (microphone 4) to the other microphones. With the use of this time and the speed of sound you can calculate the distance from the microphone to the source of the sound. If you want to know the point where the sound is made, you should intersect the circles around each microphone. The radius of each circle is the distance from the microphone to the source of the sound.

Firstly, the r (the radius) of the measured circle around the microphone was calculated:
V * t1 = r1
Because 4 microphones were used, of which the 4th microphone functioned as trigger,
the r was calculated 3 times per group of measurements. So R1, R2 and R3 are the radius around the microphones 1, 2 and 3. For the processing of the results, the next formulas were made:
R2 = (x–a)2 + (y-b)2 Of which (a,b) are the (x,y) coordinates of the centre of the circle with radius r.
Example: R2 = 4,2 m coordinates mic. 2 (3.1;3.3)
4.22 = (x – 3.1)2 + (y-3.3)2 
0 = x2 + y2 - 6.2x - 6.6y + 2.86

Becaues one microphone always had coordinates around (0,0), the next thing was done (in abstract form an example, see appendix for calculations with filled in numbers)

R1 = x2 + y2 + ax + by + c = 0
R2 = x2 + y2 – q = 0
R3 = x2 + y2 +dx + ey +f = 0

From R2 follows: x2 + y2 = q  fill q in in R1 and R3
R1q = ax + by + c + q = 0
R2q = dx + ey + f + q = 0

Next, the terms of R1q were multiplied with d and all terms of R3q were multiplied with a, so the x could be removed.

R1q = dax + dby + dc + dq = 0
R3q = adx + aey + af + aq = 0

Then R1q – R3q = (db – ae)y + (dc – af) + (dq – aq) because all letters were known, except for the y, the y could be calculated.
The y was filled in in R1q or R3q and then the x was calculated. 
The x and y were filled in in R1 & R2 & R3, getting the same result, which is the margin (in meters) for each axis of the answer, and that is how much the answer can deviate from reality. Now the answer can be given as (x,y) with a margin of m.
See appendix for calculated measurements.

Aim
Where is the source of the noise?

Hypothesis
The noise is on the place where the three circles of distance around the microphones meet. This will almost certainly be an area and not a single point because the speed of sound, used to calculate the r, has a diversion of 15 m/s and the accuracy of these measurements of this experiment will not be very high either.

Experimental section
Materials
□ four microphones
□ a computer with the software Coach5
□ spring rule
□ wires
□ two metal sticks

Method
1. Connect the microphones to the computer with the program Coach5 on it
2. Install a pressure (in Pa) / time (in ms) graph and the measuring speed of 5000 times per second
3. install the trigger norm on microphone 4
4. Place the two microphones as far as possible from each other
5. Measure distances between the microphones with a spring rule
6. Smack one of the sticks on the other close to the microphone which you named 4 (the microphone which measures the trigger norm)
7. Let Coach5 put the results of the two microphones in a pressure(in Pa)/time(in ms) graph
8. Calculate the r (as S) with the formula S = T * V sound
9. Calculate the coordinates of the places where the circles intersect.
10. Calculate the average x and y coordinates.

Results
Definition of place 2d
Measurement no. Time A1 sec A2 sec A3 sec A4 sec
1 0.0526 0.525 0.536 0
2 0.003 0.003 0.007 0
3 0.001 0.0025 0.002 0
4 0.0035 0.004 0.0145 0
5 0.003 0.0015 0.013 0
6 0.002 0.003 0.008 0
7 0.005 0.002 0.0092 0
8 0.0051 0.002 0.00655 0
9 0.005 0.002 0.0055 0
You can find the graphs in the attachment.

Processing of the results
2d Measurements 1,2,3:
Microphone 1 (0,0)
Microphone 2 (3.1;3.3)
Microphone 3 (1.6;-1.2)
Source: (0.6;-0.2)

R1 = 0.64 m.
R2 = 4.2 m.
R3 = 1.44 m.

According to the calculations (witch you can find in the attachment), the source is situated on (0.65;-0.4), with a margin of 0.02 m for each axis. However, the real source is situated on (0.6;-0.2) 
Distance between real source and measured source: √((Δx)2 + (Δy)2) = √(0.052 + 0.092)= 0.1m
The margin was already 0.02m, so therefore the calculation error is 0.1 - 0.02 = 0.08m
The calculation error is 0.08m

2d Measurements 3,4,5:
Microphone 1 (0.94;1.26)
Microphone 2 (0;0)
Microphone 3 (5.4;-1.3)
Source: (0;2.2)
R1 = 1 m.
R2 = 1.4 m.
R3 = 4.9 m.

According to the calculations (witch you can find in the attachment), the source was situated on (0.3;1.2) with a margin of 0.5 for each axis.
However, the real source was situated on (0;2.2) 
Distance between real source and measured source: √((Δx)2 + (Δy)2) = √(0.32 + 12)= 1 m.
The margin was already 0.5m, so therefore the calculation error is 1- 0.5 = 0.5 m.
The calculation error is 0.5m

2d Measurements 7,8,9:
Microphone 1 (0;0)
Microphone 2 (0.85;1.6)
Microphone 3 (0.2;3.9)
Source: (0.2;1.6)

R1 = 1.6 m.
R2 = 0.65 m.
R3 = 2.3 m.

According to the calculations (witch you can find in the attachment), the source is situated on (0.2;1.6)), with a margin of 0.02 m for each axis. And indeed, the source was situated on that place. Therefore the calculation error is 0.

Total calculation error on average: (0.08 + 0.5 + 0)/3 = 0.2 m
The average calculation error of the measurements is 20cm.
Conclusion
The aim was: What is the source of the sound? (2d)?
For measurement 1,2,3 that is: (0.65;-0.4) with a margin of 0.02m and a calculation error of 0.08m
For measurement 4,5,6 that is: (0.3;1.2) with a margin of 0.5m and a calculation error of 0.5 m
For measurement 7.8.9 that is: (0.2;1.6) with a margin of 0.02 m and a calculation error of 0
The average calculation error of the measurements is 20 cm
Discussion
The hypothesis is right. There is quite a difference between the real numbers which were actually measured and the theoretical numbers. There also was a little area where the noise could have been produced.
This might have been the case because the more you calculate, the more you round of numbers, the bigger the final error becomes.
And this wouldn’t be the only reason; because of the small distances used, the outcomes were very small as well and they were hard to be measured correctly which caused a lot of significance inaccuracies. This also gives quite an error.
The final reason could be that the source of the noise wasn’t found on one single spot, but also a bit around it. It is impossible to create a noise on a single place; it always is a small area in which it is produced.

Improvements for additional research:
• The place were we hit (made the sound) was not very stable, it always moved a little. We tried to make this as stable as possible, but there is always a little bit of movement.
• Different microphones were used. This could also cause a little mistake, because every microphone has it own specifications.
• We did not use state of the art microphones, software and cables.
• Also, different cables were used to connect the microphones to the computer, but this will only give a tiny difference.
• Also, we measured on different days, with different weather conditions. We tried to do as much as possible on one day so we would have the same conditions. We measured with hygrometers and thermometers, but there is always a small difference. (pressure difference)
• More measurements taken leads to better results and smaller calculation errors.

Experiment 3

3D GPS

Introduction
To get to know the place of the source of a noise, you need 4 microphones. At first you need to measure the time it takes for the sound to travel from the source (microphone 4) to the other microphones. With the use of this time and the speed of sound you can calculate the distance from the microphone to the source of the sound. If you want to know the point where the sound is made, you should intersect the circles around each microphone. The radius of each circle is the distance from the microphone to the source of the sound.

Firstly, the r (the radius) of the measured circle around the microphone was calculated:
V * t1 = r1
Because 4 microphones were used, of which the 4th microphone functioned as trigger,
the r was calculated 3 times per group of measurements. So R1, R2 and R3 are the radius around the microphones 1, 2 and 3. For the processing of the results, the next formulas were made:
R2 = (x–a)2 + (y-b)2 + (z-c)2 Of which (a,b,c) are the (x,y,z) coordinates of the centre of the circle with radius r.
Example: R2 = 2.3 m coordinates mic. 2 (0;0;0)
2,32 = (x – 0)2 + (y-0)2 + (z-0)2 
0 = x2 + y2 +z2 - 5,29

Becaues one microphone always had coordinates around (0;0;0) this time it was microphone 2, the next thing was done (in abstract form an example, see appendix for calculations with filled in numbers)

R1 = x2 + y2 + z2 + ax + by + cz + w= 0
R2 = x2 + y2 + z2 – q = 0
R3 = x2 + y2 + z2 + dx + ey + v = 0

From R2 follows: x2 + y2 + z2 = q  fill q in in R1 and R3
R1q = ax + by + cz + w + q = 0
R2q = dx + ey + v + q = 0

Next, the terms of R1q were multiplied with d and all terms of R3q were multiplied with a, so the x could be removed.

R1q = dax + dby + dcz + dq + dw= 0
R3q = adx + aey + aq + av = 0

Then R1q – R3q = (db – ae)y + (dcz – afz) + (dq – aq) + (dw-av) because all letters were known (because z was really 0 because only R1 had a z coordinate and therefore z can only be 0) , except for the y, the y could be calculated.
The y was filled in in R1q or R3q and then the x was calculated. 
The x and y and z were filled in in R1 & R2 & R3, getting the same result, which is the margin (in meters) for each axis of the answer, and that is how much the answer can deviate from reality. Now the answer can be given as (x,y,z) with a margin of m.
See appendix for calculated measurements.

Aim
Where is the source of the noise? (3d)

Hypothesis
The noise is on the place where the three bolls of distance around the microphones meet. This will almost certainly be an area and not a single point because the speed of sound, used to calculate the r, has a diversion of 15 m/s and the accuracy of these measurements of this experiment will not be very high either.

Experimental section
Materials
□ four microphones
□ a computer with the software Coach5
□ spring rule
□ wires
□ two metal sticks

Method
1. Connect the microphones to the computer with the program Coach5 on it
2. Install a pressure (in Pa) / time (in ms) graph and the measuring speed of 5000 times per second
3. install the trigger norm on microphone 4
4. Place the two microphones as far as possible from each other
5. Measure distances between the microphones with a spring rule
6. Smack one of the sticks on the other close to the microphone which you named 4 (the microphone which measures the trigger norm)
7. Let Coach5 put the results of the two microphones in a pressure(in Pa)/time(in ms) graph
8. Calculate the r (as S) with the formula S = T * V sound
9. Calculate the coordinates of the places where the circles intersect.
10. calculate the source of the noise

Results
Defintion of place 3d
Measurement no. Tijd A1 sec A2 sec A3 sec A4 sec
1 0.0051 0.0049 0.0032 0
2 0.013 0.0085 0.0045 0
3 0.007 0.0071 0.0049 0
4 0.006 0.006 0.007 0

Processing of the results
3d Definition:
Microphone 1 (1.6;0.6;0.7)
Microphone 2 (0;0;0)
Microphone 3 (0;4.6;0)
Source: (0.2;2.3;0)

R1 = 2.25 m.
R2 = 2.3 m.
R3 = 2.2 m

According to the calculations, the source is situated on (0.88;2.35;0) with a margin of 1 m.
Distance between real source and measured source: √((Δx)2 + (Δy)2 + (Δz)2) = √(0.682 + 0.052 + 0 2) = 0.68m
However, the 0.68m is already in the margin of 1m, so therefore the calculation error is also 0 in this calculation.

Conclusion
The aim was: What is the source of the sound? (3d)
That is (0.88;2.35;0) with a margin of 1 meter and a calculation error of 0.

Discussion
The hypothesis is right. There is quite a difference between the real numbers which were actually measured and the theoretical numbers. There also was a little area where the noise could have been produced.

This might have been the case because the more you calculate, the more you round of numbers, the bigger the final error becomes.
And this wouldn’t be the only reason; because of the small distances used, the outcomes were very small as well and they were hard to be measured correctly which caused a lot of significance inaccuracies. This also gives quite an error.
The final reason could be that the source of the noise wasn’t found on one single spot, but also a bit around it. It is impossible to create a noise on a single place; it always is a small area in which it is produced.

Improvements for additional research:
• The place were we hit (made the sound) was not very stable, it always moved a little. We tried to make this as stable as possible, but there is always a little bit of movement.
• Different microphones were used. This could also cause a little mistake, because every microphone has it own specifications.
• We did not use state of the art microphones, software and cables.
• Also, different cables were used to connect the microphones to the computer, but this will only give a tiny difference.
• Also, we measured on different days, with different weather conditions. We tried to do as much as possible on one day so we would have the same conditions. We measured with hygrometers and thermometers, but there is always a small difference. (pressure difference)
• More measurements taken leads to better results and smaller calculation errors.

Bibliography

Some sites were only used to inform ourselves on the matter of GPS and others were used in this paper.
http://www.tycoelectronics.com/gps/basics.asp
http://www.cmtinc.com/gpsbook/
http://www.mindcontrolforums.com/gps1.htm (trilateration information)
http://aboriginal-center.uwaterloo.ca/~aklassen/Some%20Errors.htm
http://www.garmin.com/aboutGPS/glossary.html
http://www.garmin.com/manuals/GPSGuideforBeginners_Manual.pdf
http://www.scholieren.com/werkstukken/12350
http://www.gpsinformation.org/dale/theory.htm
http://www.physicsclassroom.com/Class/sound/U11L1a.html