Derive the z-transforms for the time functions listed below. Do not use any z-transform tables. Use the plan f (t) → f *(t) → F*(s) → F(z), followed by converting F(z) into closed form making use of the fact that 1/(1 - z^-1) = 1 + z^-1 + z^-2 + z^-3 + ∙ ∙ ∙ . Assume ideal sampling.

a. e^-atu(t)

b. u(t)

c. t²e^-atu(t)

d. cos ωt u(t)