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

710 Chapter 22 Difference equations and the z transform
22.10.1 Linearity
If f[k] andg[k] are two sequences then
Z{f[k] +g[k]} = Z{f[k]} + Z{g[k]}
This statement simply says that to find theztransform of the sum of two sequences we
canaddtheztransformsofthetwosequences.Ifcisaconstant,whichmaybenegative,
and f[k]isasequence, then
Z{cf[k]} =cZ{f[k]}
Together these two properties mean thattheztransform isalinear operator.
Example22.17 Find theztransform of e −k +k.
Solution From Table 22.2 wehave
Z{e −k z
} =
z−e −1
and
z
Z{k} =
(z −1) 2
Therefore,
Z{e −k +k} =
z
z−e −1 +
z
(z −1) 2
Example22.18 Findtheztransform of3k.
Solution FromTable 22.2 wehave
z
Z{k} =
(z −1) 2
Therefore,
Z{3k} =3Z{k} =3×
z
(z −1) = 3z
2 (z −1) 2
Example22.19 Findtheztransform ofthe function f (t) = 2t 2 sampledatt =kT,k ∈ N.
Solution Thesequenceofsampledvalues is
f[k] = 2(kT) 2 = 2T 2 k 2
TheztransformofthissequencecanbereaddirectlyfromTable22.2usingthelinearity
properties. We have
Z{2T 2 k 2 } =2T 2 Z{k 2 } = 2T2 z(z +1)
(z −1) 3