Task A $($Handwritten$/$MATLAB$)$: Provide solutions steps and MATLAB codes.

Consider that you have a continuous$-$time audio signal with amplitude and frequency pairs $(A,f)$ as follows:

$(A,f(Hz))(1,500)(1,600)(1\mathrm{.}5,750)(1\mathrm{.}5,850)$

This audio signal is the input to a microphone. The microphone scales the magnitude of the input by a factor of two and adds noise to the audio signal. The components of noise have amplitude$-$frequency pairs as follows:

$(A,f(Hz))(1,200)(1,300)(1,350)$

A$.1$

Consider the microphone as a system with input and output waveforms:

$(0\mathrm{.}5$ pt$)$ What is the fundamental period of the input audio signal?

$(0\mathrm{.}5$ pt$)$ Plot the original input and noisy output waveforms on the same window using the subplot command.

$(0\mathrm{.}5$ pt$)$ If you were to sample the noisy output of the microphone, what is the sampling rate that you recommend? Justify your recommendation.

$(0\mathrm{.}5$ pt$)$ Test for different sampling rates, and provide a graphical comparison between the sampled signals.

A$.2(1$ pt$)$

Is the microphone considered to be an LTI system? Justify your answer.

A$.3(1$ pt$)$

Design a filter circuit to reject the added noise.

What is the transfer function corresponding to this filter? What are its zeros and poles, if any?

A coil consists of $200$ turns of wire wrapped around a square frame of sides $0\mathrm{.}25$ m $.$ The coil is centered at the origin with each of its sides parallel to the x or y axis. Find the induced emf across the open$-$circuited ends of the coil if the magnetic field is given by

$(10$ marks$)$

a$)$ marks$)$

b$)$ marks$)$