$1.$ Convert the following function to a Heaviside equation.

$()=\{$

$0$; $<0$

$1$

$()$; $0<<$

$2$

$()$; $<<$

$3$

$()$; $>$

a$)$ f$($t$)=$ f$1($t$).$ua$($t$)+$ f$2($t$).$ua$-$b$($t$)+$ f$3($t$).$ub$($t$)$

b$)$ f$($t$)=$ f$1($t$).$u$($t$)+$ f$2($t$).$ua$($t$)+$ f$3($t$).$ub$($t$)$

c$)$ f$($t$)=$ f$1($t$).($u $($t$)-$ua$($t$))+$ f$2($t$).($ua$($t$)-$ub$($t$))+$ f$3($t$).$ub$($t$)$