memo

para = U ->  UnitStep

example 5.11    p.74

q[t_] := t U[t] /. para

qt = q[t] - 2 q[t - a] + q[t - 2 a]

Simplify[LaplaceTransform[qt, t, s]]

Fig.5.16

f[t_] := 4  Sin[1/2 Pi t]

zua = Plot[f[t], {t, 0, 6}]

fb = f[t - 1] U[t - 1] /. para
zub = Plot[fb, {t, 0, 6}]

fc = f[t] U[t - 1] /. para
zuc = Plot[fc, {t, 0, 6}]

fd = 
      f[t] (U[t] - U[t - 2] + U[t - 4] - U[t - 6] + U[t - 8] - 
              U[t - 10]) /. para
zud = Plot[fd, {t, 0, 10}]

Show[GraphicsArray[{{zua, zub, zuc, zud}}]]