Problem $3(40$ points$)$:

Studying the impulse response of a system is critical in biomedical imaging applications, such as ultrasound or CT scanners, where it determines the system's ability to resolve fine details in an image.

For the system shown above, find:

$($a$)(20$ points$)$ the transfer function, which characterizes the relationship between the input and output signals in a medical imaging system.

$($b$)(20$ points$)$ the unit impulse response, which represents how the system reacts to a brief pulse of energy, simulating a single sound wave in ultrasound or an x$-$ray pulse in CT$.$ Plot the response using MATLAB for $\backslash (0<mtext></mtext><mi>\backslash </mi><mi>)</mi><mtext></mtext><mi>\backslash </mi><mi>(</mi>$ t