# points) Consider the signal s(t) with Fourier Transform 10 1+?. S(a) figure below, we impulse sample

points) Consider the signal s(t) with Fourier Transform 10 1+ω. S(a) figure below, we impulse sample s) at a frequency o, rads/second, e signal sa(t). Can you find a finite sampling frequency o such that ly recover s(t) from so()? If so, find it. If not, explain why not. a) (5 pts) In ting in the can perfectly you s (t) sa(t) →| Impulse sample at- rate o b) (5 pts) Now suppose we filter the signal s() with an ideal low pass filter with frequency response H(o) and bandwidth B as shown in the figure below to produce the signal y(t), and then we impulse sample y() to produce ys(t). Find the filter bandwidth B so that the Fourier transform Y(o) of the filter output yt) is 0 for all frequencies ω where the value of S(o) is below S(0)/10, where S(0) denotes the DC value of S(0). H(o) (t) 」Impulse sample at yat) s(t) rate -B c) (5 pts) Using your value of B from part b, what is minimum value of the sampling rate co, that will allow the filter output y() to be perfectly recovered from its impulse sampled version ys()? d) (5 pts) What is the purpose of the filter H(o)? (One sentence answer please.) e) (10 pts) Suppose the sampling rate o, is double your answer in part (c). (This is called 2x over-sampling.) Carefully sketch the Fourier Transform Y&(0) of the impulse sampled output ys(0), showing all significant amplitudes and frequencies. 1) (5 pts) Now suppose that the sampled signal ystt) from part e) is the input to a recovery filter with frequency response H(0). Carefully sketch H(o) so that the output of the recovery filter is the signal y(), showing all significant amplitudes and frequencies. points) Consider the signal s(t) with Fourier Transform 10 1+ω. S(a) figure below, we impulse sample s) at a frequency o, rads/second, e signal sa(t). Can you find a finite sampling frequency o such that ly recover s(t) from so()? If so, find it. If not, explain why not. a) (5 pts) In ting in the can perfectly you s (t) sa(t) →| Impulse sample at- rate o b) (5 pts) Now suppose we filter the signal s() with an ideal low pass filter with frequency response H(o) and bandwidth B as shown in the figure below to produce the signal y(t), and then we impulse sample y() to produce ys(t). Find the filter bandwidth B so that the Fourier transform Y(o) of the filter output yt) is 0 for all frequencies ω where the value of S(o) is below S(0)/10, where S(0) denotes the DC value of S(0). H(o) (t) 」Impulse sample at yat) s(t) rate -B c) (5 pts) Using your value of B from part b, what is minimum value of the sampling rate co, that will allow the filter output y() to be perfectly recovered from its impulse sampled version ys()? d) (5 pts) What is the purpose of the filter H(o)? (One sentence answer please.) e) (10 pts) Suppose the sampling rate o, is double your answer in part (c). (This is called 2x over-sampling.) Carefully sketch the Fourier Transform Y&(0) of the impulse sampled output ys(0), showing all significant amplitudes and frequencies. 1) (5 pts) Now suppose that the sampled signal ystt) from part e) is the input to a recovery filter with frequency response H(0). Carefully sketch H(o) so that the output of the recovery filter is the signal y(), showing all significant amplitudes and frequencies.