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# 1 Frequency Domain Analysis 1. Given an input u(t) = cos(t) + 2 sin(5t) cos(5t) which is a sum of a lower frequency signal

## 1 Frequency Domain Analysis 1. Given an input u(t) = cos(t) + 2 sin(5t) cos(5t) which is a sum of a lower frequency signal and a higher-frequency noise. Determine the feasible range of time constant 7 for a first order low-pass filter 1 H(s) = TS+1 (1) such that the amplitude of high-frequency noise is reduced by 95%, while the lower- frequency signal amplitude is within 10% of the original amplitude. If you find this is not feasible, justify your answer. 2. For the same input as above, determine the feasible range of wn for the second-order Butterworth filter H(s) = s + 2(0.707)wns + w (2) 3. Consider the following three bode diagrams in Figure 1. Answer the questions for each one. Note that I have also uploaded "bodel.fig", "bode2.fig", and "bode3.fig" so that you can load the plots and get precise numbers. (a) What is the transfer function? (Your answers don't have to be exact, but they should be justified) (b) If this were the open-loop transfer function, what is the gain and phase margin? Is the system stable in closed-loop? (c) If the system is stable in closed-loop, is there a steady-state error in response to a step input? If there is a steady state error, what is it?

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