Rich Kozick
Spring, 1997

## EE 329: Fun Assignment

Date Assigned: Tuesday, February 18, 1997
Date Due: Friday, February 21, 1997

Item 2 is an interesting problem that you should work on for Friday. It will give you some practice with difference equations and block diagrams for discrete-time systems. Nothing needs to be handed in on Friday.

1. Continue reading Chapter 3 in the textbook. Also continue to review other books and notes on discrete-time systems and Z transforms that you may have from previous courses. In particular, please review partial fraction expansions.

2. Here's a classic mathematical problem that goes back to Leonardo Fibonacci (? - ca 1250). To get a neat formulation, we're going to make the extreme assumptions that every pair of rabbits matures in one month, and produces a pair of baby rabbits the month after reaching maturity and every month thereafter. Start with one pair of baby rabbits at the beginning of Month 0. At the beginning of Month 1 this pair matures, but there will still be only one pair of rabbits. By the beginning of Month 2, however, there will be two pairs: the original pair, plus one new baby pair born to that original pair. By the beginning of Month 3, there will be only one more pair, for a total of three pairs, because the baby pair is not yet able to reproduce. By the beginning of Month 4, however, there will be a total of five pairs, three from the preceding month, plus two more born to the pairs that were mature that preceding month.
• Derive a difference equation that specifies r(n), the number of rabbit pairs at month n, in terms of r(n-1) and r(n-2). Note that r(0) = 1, r(1) = 1, r(2) = 2, r(3) = 3, r(4) = 5, ...
• Draw a block diagram of the system.
• How many rabbit pairs will there be after one year? You can do this manually, or you might write a simple Matlab program.
• Do you think this system is stable?
• Can you derive a closed-form expression for r(n) that is a function of n only? (We will reconsider this question after we discuss the Z transform.)

(This problem is from K. Steiglitz, A Digital Signal Processing Primer, Addison-Wesley, 1996, page 195.)

3. You might look at some of the Z transform notes and demos located at http://image-1.rose-hulman.edu/~yoder/bookcd/visible/chapters/8ztrans/overview.htm
Thank you.