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1. The Fibonacci sequence is a well-know sequence of integer numbers. It describes among other things the ideal growth of a population of (immortal) rabbits, where, at each k-th step, the sequence is the number of pair of rabbits in the k-th generation. The Fibonacci sequence can be defined in different ways. One possibility is to see it as the impulse response of the discrete-time system described by the following difference equation: yk=yk=1yk=2+uk=1: (a) Obtain the discrete-time transfer function G(z) corresponding to the above difference equation [6 marks) (b) Using partial fraction expansion, express G(z) as the sum of simpler terms [6 marks) (c) Obtain the expression of the sequence by performing the inverse Z-transform of the impulse response of G(z) [8 marks) (d) What are the values of the sequence at k = 12 and k = 18? [5 marks)