ft[s_,t_,v_]:=FourierTransform[s,t,v,FourierParameters->{0,-2Pi}];

s=Exp[-Pi t^2];
Plot[s,{t,-4,4},PlotRange->All]
ft[s,t,v]

s=Exp[-a Abs[t]];
Plot[s/.a->2,{t,-4,4},PlotRange->All]
ft[s,t,v]

$Assumptions=T>0;
s=UnitBox[t/(2T)];
Plot[s/.T->3,{t,-5,5}]
ft[s,t,v]

s=(1-Abs[t]/T)UnitBox[t/(2T)];
Plot[s/.T->3,{t,-5,5}]
ft[s,t,v]//FullSimplify

s=Exp[-a Abs[t]]* Sign[t];
Plot[s/.a->2,{t,-4,4},PlotRange->All]
ft[s,t,v]

s=Sign[t];
Plot[s,{t,-3,3}]
ft[s,t,v]

s=UnitStep[t];
Plot[s,{t,-3,3}]
ft[s,t,v]

s=1/t;
ft[s,t,v]