The signal $(t)$ shown below, is fed into a PCM system to obtain a digital representation for this signal. The signal $(t)$ is nonzero only in the interval $0$ to $21$ milliseconds. The signal $(t)$ has a spectrum $W(f)$ which is also shown below. W$($f$)$ is non$-$zero in the frequency range $-83\mathrm{.}333$ to $83\mathrm{.}333$ Hz as shown on the graph $($Remember Hz corresponds to $\frac{1}{seconds}).$ The PCM system uses a Sampling Period $Ts$ which is the maximum Sampling Period allowed by the Sampling Theorem; and is wanted to result in a maximum PCM accuracy of $5\%.$

a$.$ Write the Sampling Time Instants for each sample $($Calculate Ts first$).$

b$.$ Write the Sampling Output $($Sample amplitudes at each Sampling time instant$)$ of the PCM system.

c$.$ Calculate the Number of Quantization Levels for the PCM system.

d$.$ Write the Quantizer output $($Quantization level for each sample$).$

e$.$ Draw the Encoding Table $($which includes codewords for each quantization level$)$ where the system uses GRAY CODING.

f$.$ Write the output of the PCM system $($bitstream$)$ for the given input signal.

g$.$ Calculate the Bitrate $($R$)$ for the bitstream.

Please write the answers regularly. Carefully indicate which value corresponds to which time instant.

$**$ Show all your calculations to get full points.