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# final - 2.171 Analysis and Design of Control Systems Prof...

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2.171 Analysis and Design of Control Systems Prof. David Trumper Final Exam Problem 1: This problem considers a system for which the impulse response h ( k ) is equal to 1 for N = 0 , 1 , ..., N 1, and equal to zero elsewhere. That is, N 1 h ( k ) = δ ( k i ) (1) i =0 The system has an input u and an output y . a) For this system, solve for the transfer function H ( z ). Clearly show your reasoning. b) Find a diﬀerence equation in terms of u and y which has this impulse response h ( k ). Explain why this type of system is often referred to as an N-point averager. (It may also be called a box-car ±lter, because of the shape of the impulse reponse.) c) Show that the frequency response of this system is given by H ( e j Ω ) = sin( N Ω / 2) e j Ω( N 1) / 2 (2) sin(Ω / 2) Sketch a Bode plot for this system for the value N = 4. Use a linear frequency and magnitude axis, and sketch from Ω = 0 to Ω = π . Be sure to include both magnitude and phase, and to label the axes

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final - 2.171 Analysis and Design of Control Systems Prof...

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