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.
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; 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
.gif)
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
_formula.gif)
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.
Copyright Information:
Use of Copyrighted Graphics. This demo includes
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Auxiliary Resources:
- A Morse code applet is available at
http://www.qsl.net/kf4kvg/cw.html
and a Java Morse Code Translator is available at
http://morsecode.scphillips.com/translator.html
Information about Samuel Morse is available at
http://www.usa-people-search.com/content-samuel-morse.aspx
- 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.
- Background information on cellular networks and
cellular phones was adapted from How Cell Phones Work at
http://www.brighthub.com/engineering/electrical/articles/3885.aspx, 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
