Question $3($ULO$4)$

Analyse a filter that is characterized by the following impulse response.

$\backslash [$

h$[$n$]=3($u$[$n$-1]-$u$[$n$-3])$

$\backslash ]$

It is supplied with an input $\backslash ($ x$[$n$]=\backslash \{1,0,1\backslash \}\backslash ).$

$($a$)$ Determine its output $\backslash (\backslash$mathrm$\{$y$\}[\backslash$mathrm$\{$n$\}]\backslash ).$

$($b$)$ Determine the output if both the filter and $\backslash (\backslash$mathrm$\{$x$\}[\backslash$mathrm$\{$n$\}]\backslash )$ are periodic.

$($c$)$ Use appropriate transformation method to obtain the frequency response equation of the filter. Draw or plot the magnitude response of the filter. Specify the filter type $($high pass, low pass, band pass or band stop$).$ List the pass and stop band frequencies.

Assume:

$-$ the passband is the region where the magnitude response is greater than $\backslash (70\backslash \%\backslash )$ of the maximum.

$-$ the stopband is the region where the magnitude response is less than $\backslash (25\backslash \%\backslash )$ of the maximum.

$($d$)$ A new filter is produced by cascading two of this filter as in Figure $2.$ Derive the equations of the resultant magnitude and phase responses.