Solved ELCT 222 Signals and Systems Computer Assignment 4

Fibonacci numbers 𝐹𝐹0, 𝐹𝐹1, 𝐹𝐹2, … satisfy the following recurrence relation: 𝐹𝐹𝑛𝑛+1 = 𝐹𝐹𝑛𝑛 + 𝐹𝐹𝑛𝑛−1 (𝑛𝑛 ≥ 1, 𝐹𝐹0 = 0, 𝐹𝐹1 = 1) The sequence begins with 0,1,1,2,3,5 …. Now, consider the polynomial defined by 𝑓𝑓(𝑡𝑡) ≔ 𝐹𝐹0 0! + 𝐹𝐹1 1! 𝑡𝑡 + 𝐹𝐹2 2! 𝑡𝑡2 + 𝐹𝐹3 3! 𝑡𝑡3 … = �𝐹𝐹𝑘𝑘 𝑘𝑘! ∞ 𝑘𝑘=0 𝑡𝑡𝑘𝑘 This is also known as exponential generating function (EGF) of Fibonacci sequence. 1. (10 pts) Show the 𝑛𝑛th Fibonacci number can be obtained as 𝐹𝐹𝑛𝑛 = 𝑑𝑑𝑛𝑛 𝑑𝑑𝑡𝑡𝑛𝑛 𝑓𝑓(𝑡𝑡)|𝑡𝑡=0 2. (10 pts) By using the recurrence relation, prove that the following relationships holds: 𝑑𝑑2𝑓𝑓(𝑡𝑡) 𝑑𝑑𝑡𝑡2 = 𝑑𝑑𝑑𝑑(𝑡𝑡) 𝑑𝑑𝑑𝑑 + 𝑓𝑓(𝑡𝑡) 3. (20 pts) Calculate the Laplace transform of 𝑓𝑓(𝑡𝑡) from the differential equation above. 4. (20 pts) Solve 𝑓𝑓(𝑡𝑡) by using partial fractions expansion of 𝐹𝐹(𝑠𝑠) and identify the zeros and poles of 𝐹𝐹(𝑠𝑠). 5. Plot 𝑓𝑓(𝑡𝑡) in MATLAB between for 𝑡𝑡 ∈ [−2,2]: a. (2 pts) By using EGF expression above by using terms 𝐹𝐹0, … , 𝐹𝐹5 b. (8 pts) By using the exact expression obtained by using the inverse Laplace transform. 6. (20 pts) Obtain a closed-form expression for 𝐹𝐹𝑛𝑛 by using the closed-form expression obtained from the inverse Laplace transform and 𝐹𝐹𝑛𝑛 = 𝑑𝑑𝑛𝑛 𝑑𝑑𝑡𝑡𝑛𝑛 𝑓𝑓(𝑡𝑡)|𝑡𝑡=0 (note that 𝑢𝑢(𝑡𝑡) = 1,𝑡𝑡 ≥ 0, 𝑢𝑢(𝑡𝑡) = 0,𝑡𝑡 < 0). 7. (10 pts) Express 𝐹𝐹19 by using the closed-form expression and calculate its value based on this expression in MATLAB.