082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

712 Chapter 22 Difference equations and the z transform
F(z) = Z{f[k]} =
=
∞∑
f[k]z −k
k=0
∞∑
k=4
z −k
= 1 z 4 + 1 z 5 + 1 z 6 +···
= 1 z 4 (
1 + 1 z + 1 z 2 +···)
= 1 z 4 1
1−1/z
= 1 z 4 z
z −1
= 1 z 3 1
z −1
The same argument shows that
Z{f[k+1]}= 1 z 2 1
z −1
Itthen follows that
Z{f[k+1]}= 1 z 2 1
z −1 =zF(z)−zf[0]
since f[0] = 0.This illustratesthe first shift theorem.
22.10.3 Secondshifttheorem
Thefunction f (t)u(t)isdefined by
{ f(t) t0
f(t)u(t) =
0 t<0
whereu(t)istheunitstepfunction.Thefunction f (t−iT)u(t−iT ),whereiisapositive
integer,representsashifttotherightofisampleintervals.Supposethisshiftedfunction
issampled; thenweobtain
f[k−i]u[k−i]
k∈N
Thesecondshift theoremstates:
Z{f[k−i]u[k−i]} =z −i F(z) i ∈ N +
whereF(z)istheztransform of f[k].