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

22.11 Inversion of z transforms 715
EXERCISES22.10
1 UseTable 22.2 to find theztransformsof
(a)3(4) k +7k 2 ,k 0
(b)3e −k sin4k−k,k 0
2 Find theztransformsofthe following continuous
functionssampledatt =kT,k ∈ N:
(a) t 2 (b) 4t (c) sin2t
(d) u(t −4T )
(e) e 3t
3 Find theztransformof (k −3)u[k −3]bydirectuse
ofthe definition oftheztransform.Hence verifythe
resultofExample 22.24.
4 Provethat theztransformofe −at f (t)isF(e aT z).
5 Prove the firstandsecond shifttheorems.
6 Use the complex translation theoremto findthez
transformsof
(a) ke −bk
(b) e −k sink
7 If f[k] = 4(3) k find Z{f[k]}.Usethefirstshift
theorem to deduce Z{f[k +1]}.Show that
Z{f[k+1]}−3Z{f[k]}=0.
8 Write down the firstfive terms ofthe sequence
definedby f[k] = 4(2) k−1 u[k −1],k 0.Finditsz
transformdirectly, andalso byusingthe second shift
theorem.
Solutions
1 (a)
(b)
2 (a)
(c)
(e)
3z 7z(z +1)
1
+
z −4 (z −1) 3
3
z 2 (z −1) 2
3ze −1 sin4
z 2 −2ze −1 cos4 +e −2 − z
ze b
(z −1) 2 6 (a)
(ze b −1) 2
T 2 z(z +1) 4Tz
ezsin1
(z −1) 3 (b)
(z −1) 2 (b)
e 2 z 2 −2ezcos1+1
zsin2T 1
z 2 (d)
4z
−2zcos2T +1 z 3 (z−1) 7
z −3 , 12z
z −3
z
z−e 3T 8 0, 4,8,16,32.
4
z −2
22.11 INVERSIONOFzTRANSFORMS
Just as it is necessary to invert Laplace transforms we need to be able to invertztransforms
and as before we can make use of tables of transforms, partial fractions and the
shift theorems.Invery complicatedcasesmore advanced techniquesarerequired.
Example22.26 IfF(z) = z +3 ,find f[k].
z −2
Solution Note that wecan writeF(z)as
z +3
z −2 = z
z −2 + 3
z −2
Thereasonforthischoiceisthatquantitieslikethefirstonther.h.s.appearinTable22.2.
Fromthis table we find
Z{2 k } = z
z −2