Q1. (a) A continuous time signal x(t) was sampled to obtain the discrete time signal x[n] = 4e for n 0, and x[n] = 0 otherwise. Write an expression for the continuous time signal x(t) and find the sampling frequency in Hz that was used to obtain the discrete sequence x[n]. [4 marks] (b) Suppose the signal x(t) from vibration measurement by a transducer device can be described by the equation: x(t) = 1 + cos(2000 t) + sin(6000mt) + cos(12000mt) V. Assume ADC device performs sampling of the signal at a rate of 0.2 ms and uses 12 bits for each sample representation. (i) With the support of a simple diagram, briefly explain how the sampling may be accomplished. [4 marks] (ii) Find the frequencies that are contained in the signal x(t). (iii) What is the Nyquist rate that must be used for sampling the [3 marks] signal x(t) for [3 marks] it to be adequately recovered from the samples. (iv) Find the number of samples that would be obtained after the ADC has sampled 60 seconds duration of the signal. Will you consider this number of samples the same if the Nyquist rate is used? Show this. [6 marks] (v) Find the discrete signal x[n] produced after the ADC sampling operation of x(t) (in simplest form or in terms of principal frequency). What is the discrete angular frequency of x[n]. [10 marks] (vi) Find the analog signal y(t) that is reconstructed from x[n] using an ideal interpolation, i.e., y(t) = y[nFs] =x[nFs]. [4 marks] (vi) Find the frequencies contained in the reconstructed signal y(t). [3 marks] (vii) Is the signal y(t) the same as the original signal x(t)? Give reasons why for your answer (as to why it is the same or not the same). [4 marks] (viii) How much memory space (MB) will be required to store the sampled data after the ADC sampling operation. [3 marks] (ix) Explain the difference between zero-order hold and first order hold method of signal reconstruction. Use diagram to support your answer. [6 marks]