MODD / Mr. Hansen 1/10/2006

Name: _________________________

Final Exam

A calculator is permitted. Please raise your hand if anything is unclear.

All questions are “essay style” or fill-in-the-blank questions requiring thought and clear expression.
Complete sentences are not necessarily required, but clarity and legibility are.
Illegible or ill-thought-out responses may not earn full credit.

Point values are in parentheses following each question number.

1. (10) We discussed several important mathematicians during the semester. Which one is your personal favorite? Describe briefly what he or she accomplished and how those accomplishments are relevant to the mathematics of digital data.

2. (10) Give an example of a 4-byte hexadecimal constant as it might be stored on a Macintosh:

__________________________

Now rewrite that value as it might be stored on a Windows/Intel PC: __________________________

The terms that describe the byte ordering of the two types of systems are _________ - __________ for Macs and most Linux or UNIX systems, _________ - __________ for Windows/Intel PCs. Can byte ordering create compatibility problems (yes or no)? ________

3. (10) A left shift of the binary number 00010110 produces the binary value 00101100, which is _______ times larger than the original when interpreted as an unsigned integer. In the same manner, a left shift by one position in a radix-n number system that has a leading 0 will result in a multiplication by a factor of ________ . In the space below, demonstrate doing the left-shift process in base ten, starting with some base-ten number that has a leading 0.

4. (20) Shannon’s Theorem, which we unfortunately did not have time to discuss, gives a maximum data rate (in bps) as a function of the available bandwidth of a channel and the S/N ratio. As the channel bandwidth increases, the maximum data rate increases, and as the channel bandwidth decreases, the maximum data rate decreases. As the S/N ratio increases, the maximum data rate with error correction increases, and as the S/N ratio decreases, the maximum data rate with error correction decreases. Use these facts to explain

(a) why your cell phone garbles fewer packets of transmitted data when you have a fresh battery (as opposed to an almost dead battery), and















(b) why you may experience more “dead zones” during times of the day when cell phone usage is at its peak. Try to use the word “multiplexing” in your answer to part (b).

5. (25) I have designed a compression scheme for monochrome bitmaps. I will not say whether it is a good scheme or a bad scheme; that is for you to decide later. What name (beginning with the letter “P”) is used to mean “scheme” for compression, encryption, or error correction/detection? ___________________

At any rate, here is the scheme. The bitmap dimensions are fixed at 12 ´ 12. Each hex digit (nybble) of data represents a pair of bits followed by a true (binary excess 1) repeat count. Here is exactly what this means. For the first two bits, 11 means two black pixels, 10 means a black followed by a white, 01 means a white followed by a black, and 00 means two white pixels. The next two bits indicate the number of true repeats: 00 denotes 0 repeats (i.e., 1 total occurrence), 01 denotes 2 occurrences, 10 denotes 3 occurrences, and 11 denotes 4 occurrences.

For example, the nybble 0xE equals binary 1110, which breaks into a pixel code (11) followed by a repeat count (10). This means that a pair of black pixels are to be recorded 3 times.

What does the nybble 0x6 represent? Write your answer here and raise your hand to make sure I know that you understand what you are doing: _________________________________

Now, on the grid provided, decode the 12 ´ 12 monochrome bitmap represented by these hex digits:

4D 48 40 C8 58 74 84 8C 58 4D 48 4F FF 1C 1C 8C 48 C4 8C 48 C4 8C 48 C4 1C 1C

How many bytes are in my representation above? _____________________

Spare copies of grid in case you mess up the first one:

Now encode the bitmap above as a straight uncompressed bitmap using the conventional method of 1=black, 0=white. Give your answer in hex:

_______________________________________________________________________

How many bytes are in your representation? __________________

What can you conclude about Mr. Hansen’s scheme? _____________________________

________________________________________________________________________

6. (25) Imagine that the year is 2025. Mr. Hansen is nearing retirement, and you are in the middle of your career as the manager of a division of a government intelligence agency or a contractor to a government intelligence agency. Your staff is trying to design a communications system that can be used by human agents who have infiltrated Al Qaeda. The requirements of the system are as follows:

• The system must operate with extremely low power. Therefore, the S/N ratio of the transmitted signal from the agents to their U.S. receivers will be low. This may make it difficult for the messages to get through accurately.
• The system must be extremely accurate. The maximum acceptable error rate at the receiving end, following any decoding that may be necessary, is only one data error in every 10¹⁰ bytes.
• The system must be extremely efficient so that even relatively long messages can be sent in only a brief burst of data. This is so our agents run less risk of being detected by Al Qaeda and shot.
• The system must be secure, in the sense that the agents who have penetrated Al Qaeda must not be able to have their transmissions be readable even if Al Qaeda were to intercept a key update.

Write several paragraphs (using reverse side also) in which you discuss some of the engineering issues that your division must grapple with as you design this communications system. I am not looking for a technical discussion with formulas and numbers. I am looking for more of a qualitative discussion that indicates that you obtained some value from that MODD class that you took 2 decades earlier. You may need to be creative and speculate on some technologies that did not exist in the year 2005 or were only partially formed then.