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

22.7 Definition of the z transform 699
Example22.8 Find theztransformof the sequence defined by
{
1 k=0
f[k]=
0 k≠0
Solution Z{f[k]} =
This sequence is sometimes called the Kronecker delta sequence, often denoted by
δ[k].
∞∑
f[k]z −k
k=0
=f[0]+ f[1]
z
=1 + 0 z + 0 z 2 + 0 z 3 +···
=1
HenceF(z) = 1.
+ f[2]
z 2 + f[3]
z 3 +···
Example22.9 Findtheztransformofthe sequencedefined by
f[k]=1
k∈N
Solution Z{f[k]} =
This istheunit stepsequence, oftendenotedbyu[k].
∞∑
f[k]z −k
k=0
=1 + 1 z + 1 z 2 + 1 z 3 +···
This is a geometric progression with first term 1 and common ratio 1 . The progression
z
converges if |z| > 1 inwhich case the sum toinfinity is
1
1−1/z = z
z −1
We see thatF(z)has the convenient closed form
F(z) =
for |z| > 1.
z
z −1
Note that the process of taking theztransform converts the sequence f[k] into the continuousfunctionF(z).
Example22.10 Findtheztransformofthesequencedefinedby f[k] =k,k ∈ N.Thissequenceiscalled
theunit ramp sequence.