Prof. Rich Kozick
ELEC 320, Fall 1997

## Homework 9

Date Assigned: Wednesday, September 17, 1997
Date Due: Friday, September 19, 1997

2. Please solve parts (c) and (d) of Problems 2.8 in the text. The objective is to review the process by which differential equations are obtained to model RLC circuits. You do not need to solve the equations, but you should obtain the differential equation that relates the input signal to the output signal.

For more practice, try Problem 2.9, parts (c) and (d). (This is optional -- you do not have to hand in Problem 2.9.)

3. This is a fun problem that will give you some practice with difference equations and programming in MATLAB.

This classic mathematical problem 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, etc.
• How many rabbit pairs will there be after one year? Please write a MATLAB program that will compute the number of rabbit pairs after any number of months that is specified by the user. The MATLAB program fig1_29.m that you used for the loan system on Homework 7 is similar to the program you will need for the rabbit population system.

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

Thank you.