cell phones sir

7/31/2019 Cell Phones SIR

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Cell Phones:Signal-to-Interference

Ratio.

Objective Level Prerequisites Platform Instructor'sNotes Credits

Used with permission: Copyright 1998-2004 HowStuffWorks, Inc. All r

reserved.(See below for further restrictions.)

Objective: To develop a function that measures the power of the signal of a cell phone as a us

moves in a cellular network and then determine the position in the network when the signal is amaximum.

Level: This demo can be used at the precalculus or calculus level by using various componentsdeveloped in our presentation.

Prerequisites: For precalculus basic properties of graphs including increasing and decreasingaddition for calculus turning points and local extrema.

Platform:A graphing calculator or Excel to generate a graph that models the situation. Aninteractive Excel worksheet accompanies this demo together with an animation.

Instructor's Notes:The use of applications as part of mathematics course tries to point to therelevance of the topic to our lives. As instructors we want to incorporate applications, but oftenstudents may not be sufficiently acquainted with the topic to appreciate the impact of themathematics. Our students have cell phones, use them regularly, and probably experienced sodifficulties in using them. With a bit of general background information, cell phone communicatprovides a real application of mathematics to their lives.
http://www.mathdemos.org/mathdemos/cellsir/cellsir.html#Objectiveshttp://www.mathdemos.org/mathdemos/cellsir/cellsir.html#Level:http://www.mathdemos.org/mathdemos/cellsir/cellsir.html#Level:http://www.mathdemos.org/mathdemos/cellsir/cellsir.html#Prerequisiteshttp://www.mathdemos.org/mathdemos/cellsir/cellsir.html#Platform:http://www.mathdemos.org/mathdemos/cellsir/cellsir.html#Noteshttp://www.mathdemos.org/mathdemos/cellsir/cellsir.html#Noteshttp://www.mathdemos.org/mathdemos/cellsir/cellsir.html#Creditshttp://www.mathdemos.org/mathdemos/cellsir/cellsir.html#Level:http://www.mathdemos.org/mathdemos/cellsir/cellsir.html#Prerequisiteshttp://www.mathdemos.org/mathdemos/cellsir/cellsir.html#Platform:http://www.mathdemos.org/mathdemos/cellsir/cellsir.html#Noteshttp://www.mathdemos.org/mathdemos/cellsir/cellsir.html#Noteshttp://www.mathdemos.org/mathdemos/cellsir/cellsir.html#Creditshttp://www.mathdemos.org/mathdemos/cellsir/cellsir.html#Objectives


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A cellular network divides a region into cells (or zones) to process calls. Each cell has an antento receive and answer your call. The antenna is often on a pole with other communicationequipment called a base station. (For background information on cellular networks and cell phosignals click here.)

As you drive and use your cell phone (hopefully only as the passenger in the car) the networkdetermines which cell you are in, assigns you a communication channel, and monitors the strenof your signal using the cell base station. As you approach the boundary of your cell theneighboring cell's base station, which has also been monitoring your signal, readies itself to swyour channel to one in that cell. (This is sometimes called "handing off" the call.) Figure 1 illustrthis situation.

Figure 1.Used with permission: Copyright 1998-2004 HowStuffWorks, Inc.

All rights reserved.(See below for further restrictions.)

There are two base station antennas that are transmitting a signal of equal power to the phone;primary base station of the cell in which the car is moving and a secondary base station in theneighboring cell the car is approaching. The signal from the secondary station causes interferewith the signal from the primary station resulting a degradation of the cell phones capabilities. Tthe power of the signal received by the cell phone varies as the car moves along. We make thefollowing definitions:
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(This formula applies to both the primary base station as well as the secondary base station.)

Once we know the height of the antennas and the distance between base stations, both the powof the received signals and hence the signal-to-interference ratio can be computed. For purposof illustration we will consider the simplified model shown in Figure 2. The goal is to determinethe location x of the cell phone so that the signal-to-noise ratio is maximized.

Figure 2.

We won't specify the units on the distance and heights indicated in Figure 2 since this is a


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simplified model.

We first develop the function f(x) which measures the signal-to-interference ratio when the cellphone is located at a position x. Figure 3 shows coordinates assigned to positions in the simplif

network so we can compute the power of the received signals and f(x).

Note: In Figure 3 we drew the location of cell phone at the point (x,0) between the two basestations. In this case x is between 0 and 2. If the location of the cell phone is (x,0) for x less thazero, then it hasn't passed the primary base station. In this simplified model we must permit thephone to be either on either side of the primary base station.

Figure 3.


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Figure 4.

From Figure 3 we see that we have two right triangles as shown in Figure 4. The power of thesignal for each base station is the square of the length of the hypotenuse of the correspondingtriangle. We have

and

We develop two approaches to determine (or estimate) the position x of the cell phone so that tsignal-to-interference ratio is maximized.Approach 1 is graphical and suitable for a precalculus


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class whileApproach 2uses derivative properties from calculus.

Approach 1. Graph the function

over an interval that extends to the left of the primary base station and to the right of the secondbase station. A reasonable choice, based on Figure 4, is the interval [-1, 3]. Students can use tgraphing calculators or the Excel file which can be executed or down loaded by clicking here. Tform of the Excel file is shown in Figure 5. An animation generated from the Excel file can beviewed as a gif file by clicking here or as a Quicktime file with stop and start features byclicking here. (Requires the free Quicktime player.) The animation can be downloaded in gif anQuicktime formats in a zipped file by clicking here.

Using the graph, have students estimate the maximum height of the graph of f(x) and the value that determines this maximum value of the signal-to-interference ratio. On their calculators theycan use the MAXIMUM function (or the zoom feature) to get fairly accurate estimates. Using thExcel routine careful use of the slider can also produce accurate estimates. In either case studecan compare the measured estimate of the maximum height with the value of the function f(x)computed using their estimate of x.

The position of the cell phone that maximizes the signal-to-noise ratio may be a bit surprising.
http://www.mathdemos.org/mathdemos/cellsir/cell_func.xlshttp://www.mathdemos.org/mathdemos/cellsir/sirgraph_generation_80.gifhttp://www.mathdemos.org/mathdemos/cellsir/sirgraph_generation_80.movhttp://www.apple.com/quicktime/download/http://www.mathdemos.org/mathdemos/cellsir/sirgraph_animation.ZIPhttp://www.mathdemos.org/mathdemos/cellsir/cell_func.xlshttp://www.mathdemos.org/mathdemos/cellsir/sirgraph_generation_80.gifhttp://www.mathdemos.org/mathdemos/cellsir/sirgraph_generation_80.movhttp://www.apple.com/quicktime/download/http://www.mathdemos.org/mathdemos/cellsir/sirgraph_animation.ZIP


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Figure 5.

Approach 2. Use calculus optimization techniques to determine the value of x in [-1, 3] thatmaximizes

Procedural outline:

compute the derivative f '(x) set f '(x) = 0 and solve for x (not hard for this idealized problem) determine which of the solutions to f '(x) = 0 yields a maximum of f(x), thendetermine the maximum value

Following the outline we get


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Setting f '(x) = 0 is equivalent to setting the numerator equal to zero; that is,

We note that

The corresponding values of the signal-to-interference ratio are

It follows that the maximum of the signal-to-interference ratio is at

which is to the left of the primary base station. To see a sketch of f(x), click here.

A problem more realistic than that given in Figure 5 is to assume that the base towers are 500fand the two towers are a bout 10 miles apart. If we define a unit of distance to be 500 feet the 1miles is about 106 units (keeping to a whole number units). Algebraically this problem is a onlymoderately more intricate than the simplified problem.
http://www.mathdemos.org/mathdemos/cellsir/picture_f.gifhttp://www.mathdemos.org/mathdemos/cellsir/picture_f.gif


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