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Mathematical Methods by Pearson
book

Mathematical Methods by Pearson

by E. Rukmangadachari
May 2024
Intermediate to advanced content levelIntermediate to advanced
437 pages
17h 41m
English
Pearson India
Content preview from Mathematical Methods by Pearson
Z-Transforms and Solution of Difference Equations 13-9
Proof By definition,
∞∞
−−
==
−−
−−
=
−−
⎛⎞⎛⎞
⋅=
⎜⎟⎜⎟
⎝⎠⎝⎠
=++ + +
+++++
++ ++ +
=++
++ ++
=〈 + + +
+
∑∑


00
1
00 01 10 02 11
2
20 0 1 1 2 2
0
01122
0
0
01122
() ()
()(
)(
)
(
)
(
nn
nn
nn
nn n
n
mnm n
nn n
n
n
mnm n
nn n
mn
uz vz uz vz
uv uv uv z uv uv
uv z uv uv uv
uv uv z
uv uv uv
uv uv z
Zuv uv uv
uv
by definition
=
=
++
⎛⎞
⎜⎟
⎝⎠
0
0
),
mn
m
mnm
m
uv
Zuv
Taking inverse Z-transforms of both sides
=
⋅=
1
0
(() ())
n
mnm
m
Zuzvz uv
(13.60)
Example 13.6
Evaluate
⎛⎞
⎜⎟
+
⎝⎠
1
(1)!
Z
n
. [JNTU 2001]
Solution By definition of Z-transforms
0
()
n
nn
n
Zu uz
=
〈〉=
Here
〈〉=
+
=
+

1
(1)!
11 1
,, ,
1! 2! ( 1)!
n
u
n
n
(1)
−−
=
−−
−−−
−+
⎛⎞
∴==+
⎜⎟
++
⎝⎠
+++ +
+
=+++
++
+

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Publisher Resources

ISBN: 9781299446557Publisher Website