Problem $41$: $(18$ points$)$

A filter is a system that manipulates the frequency spectra of a signal in a desired fashion. For example, a lowpass filter will pass allow low frequency components to pass through and remove $($or filter out$)$ high frequency components. Filters play an important role in many areas of electrical engineering, including communication and control systems. Consider the frequency response functions for a set of four filters specified by their frequency response functions

$H_1()=$Arect$(\frac{}{2B})e^{-t_o}$

$H_2()=A[1-$rect$(\frac{}{2B})]e^{-t_o}$

$H_3()=A[rect(\frac{-{}_o}{2B})+$rect$(\frac{+{}_o}{2B})]e^{-t_o}$

$H_4()=\frac{A}{\frac{}{B}+1}$

where $A,B{}_o,$ and ${}_o$ are positive real constants.

$(8$ points$)$ Sketch the magnitude $(|H()|)$ and phase $(\frac{?}{{?}_H()})$ for each of the four filters.

$(2$ points$)$ Identify each of the filters as either a highpass filter, lowpass filter, or bandpass filter.

$(6$ points$)$ Find the impulse response functions for the frequency response functions $H_1()$ and $H_4()$ You may use the Fourier transform properties and elementary Fourier transform pairs derived in either lecture or in the problems sets.

$(2$ points$)$ If the impulse response function $h(t)$ of the filter is a causal signal, then the filter is said to be realizable. Which of the filters $H_1()$ and $H_4(),$ if either, are realizable?

$*******$PLEASE WRITE YOUR WORK FOR EVERY PART$******$HAND WRITTEN WOULD BE PREFERRED