Question C$1[28$ marks$]$

A full rectified sinusoidal signal given by $x(t)=|sin(5t)|$ is filtered by a low$-$pass filter with the frequency response given in Fig C$1$ below to produce an output signal $y(t).$

Figure C$1$: Low pass filter response

$($i$)$ Obtain the exponential Fourier series coefficients of $x(t).$

$[7$ marks$]$

$($ii$)$ Plot the power spectral density of $x(t)$ in the frequency range $-20Hzf20Hz.$

$[7$ marks$]$

$($iii$)$ Plot the power spectral density of the output signal $y(t).$

$[7$ marks$]$

$($iv$)$ Determine the time domain expression for $y(t).$

$[7$ marks$]$