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Tpde

The document discusses Z-transforms and difference equations. It provides the Z-transform formulas for some common functions like 1, an, n, cos(nθ), sin(nθ). It also describes the inverse Z-transform and methods for finding the inverse transform using partial fractions. Convolution of sequences is defined as the summation of the product of one sequence shifted over the other. The convolution theorem states that the Z-transform of the convolution of two sequences is equal to the product of the individual Z-transforms. Difference equations can be solved using Z-transforms by writing the difference equation in terms of the Z-transform and its time shifts.

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Abhishek
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93 views2 pages

Tpde

The document discusses Z-transforms and difference equations. It provides the Z-transform formulas for some common functions like 1, an, n, cos(nθ), sin(nθ). It also describes the inverse Z-transform and methods for finding the inverse transform using partial fractions. Convolution of sequences is defined as the summation of the product of one sequence shifted over the other. The convolution theorem states that the Z-transform of the convolution of two sequences is equal to the product of the individual Z-transforms. Difference equations can be solved using Z-transforms by writing the difference equation in terms of the Z-transform and its time shifts.

Uploaded by

Abhishek
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Z-TRANSFORMS AND DEFFERENCE EQUATION

Z-Transform of some basic functions:


1. z
Z 1 
z 1
2. z
Z  a n  
za
3. z
Z  n 
 z  1
2

4. 1  z 
Z    log  
n  z 1 
5.  1   z 
Z   z log  
 n  1  z 1 
6.  1  1  z 
Z   log  
 n  1 z  z 1 
7. 1 1
Z    ez
 n !
8. z ( z  cos  )
Z  cos n  
z  2 z cos   1
2

9. z sin 
Z sin n   2
z  2 z cos   1
Inverse Z-Transforms:
The inverse Z-transform of Z  f (n)   F ( z ) is defined as f ( n)  Z  F ( z )  .
1

The inverse Z-Transform of some basic functions:


1.  z  1  z 
Z 1   1 ; Z    (1)
n

 z  1  z  1
2.  z  1  z   1 
Z 1    a n
; Z    (a) n ; Z 1    a n 1
z a z a z a
3.  z 2

Z 1  2
 (n  1)a n
 ( z  a) 
For Eg.
1  z 
1) Z  2
 (n  1  1)a n 1  na n 1
 ( z  a) 
1  1 
2) Z  2
 (n  2  1)a n 2  (n  1)a n 2
 ( z  a) 
1  z2 
3) Z  2
 (n  1)1n  n  1
 ( z  1) 
1  z 
4) Z  2
 (n  1  1)1n  n
 ( z  1) 
1  1 
5) Z  2
 (n  2  1)1n  n  1
 ( z  1) 
4.  z2 
1 n
Z  2 2
 a n cos
z a  2
5.  z     n  n
Z 1  2 2
 a n cos(n  1)  a n cos     a sin
n

z a  2 2 2  2
Finding Inverse Z-transform by method of Partial Fractions:
Rules of Partial Fractions:
1. Denominator containing Linear factors:
f ( z) A B C
    ...
( z  a)( z  b)( z  c)... ( z  a) ( z  b) ( z  c)
2. Denominator containing factors ( z  a ) n :
f ( z) A B C D
    ... 
( z  a) n
( z  a) ( z  a) ( z  a)
2 3
( z  a) n
3. Denominator contains a quadratic factor of the form az 2  bz  c (where a,b,c are constants):
f ( z) A Bz
 2  2
az  bz  c az  bz  c az  bz  c
2

f ( z) Az  B
(Or)  2
az  bz  c az  bz  c
2

Residue Formulae
Case i: If z  a is pole of order 1 (or) simple pole then
 Re s F ( z ) z n 1   lim ( z  a) F ( z ) z n 1
z a z a

1 d m1
Case ii: If z  a is pole of order m then  Re s F ( z ) z n 1   lim ( z  a) m F ( z ) z n 1
z a m  1 z a dz m1
Convolution of two sequences:
If  f ( n) and  g (n) are any two sequences then its convolution is defined by
n
f ( n)  g ( n)   f ( k ) g ( n  k )
k 0
Convolution Theorem:
If Z  f (n)   F ( z ) and Z  g (n)   G ( z ) then Z  f (n)  g (n)   Z  f (n)   Z  g (n)   F ( z )  G ( z )
Note:
1) Z  f ( n)  g ( n)   F ( z )  G ( z )
f (n)  g (n)  Z 1  F ( z )  G ( z ) 
Z 1  F ( z )   Z 1 G ( z )   Z 1  F ( z )  G ( z )  Z 1  F ( z )   f (n) & Z 1 G ( z )   g (n)
Z 1  F ( z )  G ( z )   Z 1  F ( z )   Z 1 G ( z ) 
a n 1  1
2) 1  a  a 2  a 3  ...  a n 
a 1
Solutions of difference equation using Z-Transforms.
1. Z  yn   Z  y (n)   y ( z )
2. Z  yn 1   Z  y (n  1)   zy ( z )  zy (0)
3. Z  yn  2   Z  y (n  2)   z y ( z )  z y (0)  zy (1)
2 2

4. Z  yn 3   Z  y (n  3)   z y ( z )  z y (0)  z y (1)  zy (2)


3 3 2

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