(See below for further restrictions.)

## Objectives

To develop a function that measures the power of the signal of a cell phone as a user moves in a cellular network and then determine the position in the network when the signal is a maximum.

## Level

This demo can be used at the precalculus or calculus level by using various components developed in our presentation.

## Prerequisites

For precalculus basic properties of graphs including increasing and decreasing; in addition for calculus turning points and local extrema.

## Platform

A graphing calculator or Excel to generate a graph that models the situation. An interactive 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 the relevance of the topic to our lives. As instructors we want to incorporate applications, but often students may not be sufficiently acquainted with the topic to appreciate the impact of the mathematics. Our students have cell phones, use them regularly, and probably experienced some difficulties in using them. With a bit of general background information, cell phone communication provides a real application of mathematics to their lives.

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

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

There are two base station antennas that are transmitting a signal of equal power to the phone; the primary base station of the cell in which the car is moving and a secondary base station in the neighboring cell the car is approaching. The signal from the secondary station causes interference with the signal from the primary station resulting a degradation of the cell phones capabilities. Thus the power of the signal received by the cell phone varies as the car moves along. We make the following definitions:

(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 power of the received signals and hence the signal-to-interference ratio can be computed. For purposes of illustration we will consider the simplified model shown in Figure 2. The goal is to determine the 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 simplified model.

We first develop the function f(x) which measures the signal-to-interference ratio when the cell phone is located at a position x. Figure 3 shows coordinates assigned to positions in the simplified 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 base stations. In this case x is between 0 and 2. If the location of the cell phone is (x,0) for x less than zero, then it hasn't passed the primary base station. In this simplified model we must permit the cell phone to be either on either side of the primary base station.

 Figure 3.

 Figure 4.

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

and

We develop two approaches to determine (or estimate) the position x of the cell phone so that the signal-to-interference ratio is maximized. Approach 1 is graphical and suitable for a precalculus class while Approach 2 uses 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 secondary base station. A reasonable choice, based on Figure 4, is the interval [-1, 3]. Students can use their graphing calculators or the Excel file which can be executed or down loaded by clicking here. The form of the Excel file is shown in Figure 5. An animation generated from the Excel file can be viewed as a gif file by clicking here or as a Quicktime file with stop and start features by clicking here. (Requires the free Quicktime player.) The animation can be downloaded in gif and Quicktime 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 of x that determines this maximum value of the signal-to-interference ratio. On their calculators they can use the MAXIMUM function (or the zoom feature) to get fairly accurate estimates. Using the Excel routine careful use of the slider can also produce accurate estimates. In either case students can 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.

 Figure 5.

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

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), then determine the maximum value

Following the outline we get

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 500ft tall and the two towers are a bout 10 miles apart. If we define a unit of distance to be 500 feet the 10 miles is about 106 units (keeping to a whole number units). Algebraically this problem is a only moderately more intricate than the simplified problem.

Use of Copyrighted Graphics. This demo includes two graphics displays that are the copyrighted by HowStuffWorks, Inc. We have permission to include these within the demo. We do not have the right to redistribute, license, or authorize use of this copyrighted material to any other party. Violation of this policy or authorization is in violation of US and International copyright laws. Please contact HowStuffWorks, Inc. at www.HowStuffWorks.com if you want to use these graphics in your work or include them on your web site.

Auxiliary Resources:

1. A Morse code applet is available at http://www.babbage.demon.co.uk/morse.html and a Java Morse Code Translator is available at http://www.scphillips.com/morse/trans.html

2. The simplified problem used in this demo appeared in An Introduction to Optimization, Second Edition, by Edwin K.P. Chong and Stanislaw H. Zak, John Wiley and Sons, 2001.

3. Background information on cellular networks and cellular phones was adapted from How Cell Phones Work by Marshall Brian and Jeff Tyson which is available at http://www.verizon.com/learningcenter, the document Understanding Cell Phone Coverage Areas available from the Federal Communication Commission, Consumer & Government Affairs Bureau at http://www.fcc.gov/cgb/consumerfacts/cellcoverage.html and How Mobile Phone Networks Work, available at http://www.sitefinder.radio.gov.uk/mobilework.htm