Please answer all of these in detail, Thank you.

QUESTION 3 (30 MARKS] An analog vibration sensor installed on a water pump generates a continuous signal Xe(t) = 0.5 + 0.5 cos(500n t) This signal is input to a microcontroller and sampled every 2.5 ms, generating a sampled and digitized discrete-time sequence x[n]. This signal is further processed to obtain some insights into the condition of the water pump. (a) Evaluate the frequency transforms of x () and x. Determine their frequency domain equations and sketch both spectrums. Finally, determine the equation for the discrete-time signal, x[n]. [16 marks] (b) Is the sampled signal, x[n], aliased? Explain your answer and show proof. [6 marks] (c) What needs to be done to ensure that a signal is not aliased due to sampling? [2 marks] (d) The signal x[n] is passed through an ideal digital low-pass filter, of frequency response H(e ), with a cut-off frequency of @ = 0.25x rad., to obtain an output signal y[n]. Sketch the spectrum of the output, Y(e"). [6 marks]