# MATH233 Unit 4 Individual Project

Question

MATH233 Unit 4 Individual Project

Dr. Claude Shannon (1916 – 2001), “the father of information theory,” observed that the maximum error-freecapacity in bits per second (bps) obtainable in a communication channel can be found by theShannon-Hartley equation:

.png”>

Above, .png”>, is the bandwidth of the channel in Hertz and.png”> is the signal-to-noiseratio of the channel. Since.png”> 0 does not make sense in this situation, assume that the formulabelow is correct:

.png”>

This is a “pure” number with no unit labels on it.The value of.png”> is called Shannon’s Capacity Limitor channel capacity. This is the theoretical upper limit for the bits per second through the channel with a specific.png”>value and a specific given channel frequency,.png”>.

Be sure to show your work details for all calculations and explain in detail how the answers were determined for critical thinking questions. Round all value answers to three decimals.

1. In the table below, based on the first letter of your lastname, choose a bandwidth for your communication channel.Write your maximum error-free channel capacity function.

(Note: The actual value of.png”> will be your chosen value times 1,000,000 since the table values are inMegaHertz (MHz).)

First letter of your lastname

Possible values for.png”> in MHz

A–F

100.png”>199

G–L

200.png”>299

M–R

300.png”>399

S–Z

400–499

2. Calculate.png”> the derivative of your channel capacity function with respect to.png”> Interpret the meaning of it in terms of channel capacity.

3. Generate a graph of this function.png”>using Excel or another graphing utility. (There are free downloadable programs like.padowan.dk/”>Graph 4.4.2 or.en.uptodown.com/”>Mathematics 4.0; or, there are also online utilities such as.desmos.com/”>this site and many others.) Insert the graph into the Word document containing your answers and work details. Be sure to label and number the axes appropriately.

4. For your function what is the instantaneous rate of change inmaximum error-free channel capacity with respect to SNR, for.png”>?

5. What is the equation of the tangent line to the graph of.png”>, when.png”>?

6. Research the Internet or Library to find a reasonable.png”>(signal and noise values should both be in watts) and bandwidth in Hertz for a CAT6 coaxial cable. Be sure to list creditable sources for your research. Based on your research, what would be the theoretical channel capacity for the CAT6 cable’s valuethat you found?

7. At what value of.png”> will.png”>

(Note:You cannot solve this equation algebraically using ordinary techniques. Youwill need to use an equation solver like Mathematics 4.0 and the fact that.png”> or .png”> by the Change-of-Base formula for logarithms. Or, you may solve this equation by graphing both.png”> and.png”> on the same graph to see where these graphs intersect. Alternatively, you may investigate the Lambert W-Function. These are some examples of how you can approximately solve equations when the solution cannot be found easily with usual algebraicmethods.)

References

Desmos. (n.d.). Retrieved from https://www.desmos.com/

Graph 4.4.2. (n.d.). Retrieved from the Graph Web site: http://www.padowan.dk/

Mathematics 4.0. (n.d.). Retrieved from the Microsoft Web site: http://microsoft-mathematics.en.uptodown.com/

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