Consider the Fibonacci sequence  generated by the difference equation f[n] = f[n − 1] + f[n − 2], n ≥ 0 with initial conditions f[ − 1] = 1, f[ − 2] = −1.

(a) Find the Z-transform of f[n], or F(z). Find the poles φ₁, and φ₂ and the zeros of F(z). How are the poles connected? How are they  related to the ”golden-ratio”?

(b) The Fibonacci difference equation has zero input, but its response  is a sequence of ever-increasing integers. Obtain a partial fraction expansion of F(z) and find f[n] in terms of the poles φ₁ and φ₂,  and show that the result is always integer. Use MATLAB to implement the inverse in term of the poles.