QB3 a) List and explain two methods that can be used to implement Infinite Impulse Response (IIR) digital filters based upon analogue prototypes. Discuss the advantages and problems for each of the methods. (6 marks) b) An analogue filter is given by the following transfer function: 30s H() $2 + 30s + 200 i) Design a digital version of the analogue filter by employing the impulse invariance method assuming the sampling interval T, and obtain its z-transfer function. (6 marks) ii) Is this filter low-pass, high-pass, or band-pass? Explain your answer. (3 marks) c) A controller is an algorithm that permits to modify the behaviour of a system in a desired way. One such algorithm is given by y(t) = ax(t) + B placed where x(t) is the input signal and y(t) is the output signal. i) Calculate the transfer function of the controller. Determine the impulse response of the controller. (3 marks) ii) Give a digital version of the controller using the impulse invariance method in terms of a z-transfer function H(2). (3 marks) iii) Given a = B = 2, for the digital version of the controller, determine the input-output relationship in terms of a difference equation. (4 marks)