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

Individual Project

Dr. Claude Shannon (1916 – 2001), “the father of information theory,” observed that the maximum error-free capacity in bits per second (bps) obtainable in a communication channel can be found by the Shannon-Hartley equation:
C(SNR)= B log_2⁡〖(1+SNR)〗
Above, B, is the bandwidth of the channel in Hertz and SNR is the signal-to-noise ratio of the channel. Since SNR ≤  0 does not make sense in this situation, assume that the formula below is correct:
SNR=  (Signal Strength in watts)/(Noise Strength in watts )>0
This is a “pure” number with no unit labels on it. The value of C(SNR) is called Shannon’s Capacity Limit or channel capacity. This is the theoretical upper limit for the bits per second through the channel with a specific SNR value and a specific given channel frequency, B.
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.
1.0    In the table below, based on the first letter of your last name, choose a bandwidth for your communication channel. Write your maximum error-free channel capacity function.
(Note: The actual value of B will be your chosen value times 1,000,000 since the table values are in MegaHertz (MHz).)
First letter of your last name    Possible values for B in MHz
A–F    100-199
G–L    200-299
M–R    300-399
S–Z    400–499

2.)    Calculate C\'(SNR), the derivative of your channel capacity function with respect to SNR. Interpret the meaning of it in terms of channel capacity.

3.)    Generate a graph of this function C(SNR)using Excel or another graphing utility. (There are free downloadable programs like Graph 4.4.2 or Mathematics 4.0; or, there are also online utilities such as 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 in maximum error-free channel capacity with respect to SNR, for SNR=30?

5.)    What is the equation of the tangent line to the graph of C(SNR), when SNR=30?

6.)    Research the Internet or Library to find a reasonable SNR (signal and noise values should both be in watts) and bandwidth in Hertz for a CAT6 Ethernet 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 value that you found?  NOTE: The conversion formula for converting S/N in decibels to SNR, a pure number as is required in the Shannon-Hartley equation is:  SNR = 10^(dB/10).  For example, if the cable has an S/N of 20 dB, then SNR = 10^(20/10) = 100 as a pure number SNR value.

7.)    At what value of SNR will C(SNR)=C^\' (SNR)? 
(Note: You cannot solve this equation algebraically using ordinary techniques. You will need to use an equation solver like Mathematics 4.0 and the fact that log_2⁡w=log⁡w/log⁡2  or log_2⁡w=ln⁡w/ln⁡2  by the Change-of-Base formula for logarithms. Or, you may solve this equation by graphing both C(SNR) and C\'(SNR) 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 algebraic methods.)


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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Shannon-Hartley Equation

C= B log2 (1+S/N)

Where:

C= Maximum capacity of channel in bits per second

B= Bandwidth of channel in hertz,

S= Signal power in watts

N= Noise power in watts

S/N, otherwise known as SNR = Signal to Noise Ratio

 

Writing the error-free channel capacity function

 

Using Letter H, the chosen Bandwidth is 250 MHz

using B = 250 * 1,000,000 =250,000,000

C= 250,000,000 log2 (1+SNR)

 

Calculating the derivative of the channel capacity function

First, to avoid calculating logs to the base 2

C= 1/log10 2 B log10 (1 + SNR) = 3.32 B log 10 (1+ SNR)

= 3.32*250,000,000 ln (SNR + 1)

Applying constant multiple rule (C.f(x))' = C. (f(x))' with C= 830,000,000 and f(x) = ln (SNR + 1)

830,000,000 ln ( SNR + 1) = 830,000,000(ln(SNR +1))'

Function ln (SNR +1) is a composition f(g(x)) of two functions

f(u) = ln (u) and g(x) = SNR + 1

Applying the chain rule: (f(g(x)))' = d /du(f(u) . (g(x))'

 

 

The derivative of natural logarithm is

 

 

 

Returning to old variable

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