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Chapter: Mathematics (maths) - Z-Transforms and Difference Equations

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Z - Transforms and its Properties

Let {fn} be a sequence defined for n = 0,1,2,…….,then its Z-transform F(z) is defined as F(z)

Z - Transforms and its Properties

 

Definition

 


Let {fn} be a sequence defined for n = 0,1,2,…….,then its Z-transform F(z) is defined as  


whenever the series converges and it depends on the sequence {fn}.

 

The inverse Z-transform of F(z) is given by Z-1{F(z)} = {fn}.

 

Note: If {fn}   is   defined   for   n   =   0,   ±   1,   ±   2,   …….,



Properties of Z-Transforms

 

1.     The Z-transform is linear.

 

 

i.e,  if F(z) = Z{fn} and G(z) = Z{gn}, then

 

Z{afn + bgn} = aF(z) + bG(z).

 


 


4. If Z{fn} = F(z), then

 

Z{fn+k}= zk{ F(z) –f0 –(f 1 / z ) - … - ( fk-1 / zk-1) } (k > 0)

 

 

Proof


In Particular,

 

(i) Z{f n+1} = z {F(z) - f0

 

(ii) Z{f n+2}= z2 { F(z) –f0 –(f1/z) } 

 

Corollary

 

If  Z{fn} = F(z), then Z{fn–k} = z-k F(z).

 

(5) Initial value Theorem

 


Proof

 

We know that F (z) = f0 + f1 z-1 + f2z-2 + . . .

 

Taking limits as z ® ¥on both sides, we get



(6) Final value Theorem





SOME STANDARD RESULTS

 

1.                 Z{an} = z / (z-a), for |z| > |a|. 

 

Proof

By definition, we have


In particular, we have

 

Z{1} = z / (z-1), (taking a = 1).

 

and      Z{(-1)n} = z / (z +1), (taking a = -1).

 

 

2.  Z{nan} =  az /(z-a)2

 

Proof: By property, we have

 


Similarly, we can prove

 

Z{n2an} = {az(z+a)}/ (z-a)3





Equating the Real and Imaginary parts, we get



Equating the Real and Imaginary parts, we get




Table of Z –Transforms







Example 2

Find the Z–transform of

1.     n(n-1)

2.     n2 + 7n + 4

3.     (1/2)( n+1)(n+2)






 

Example 3                                                        

Find the Z- transforms of 1/n   and  1/n(n+1)


= - log (1 –1/z ) if  |1/z| < 1 

 

= - log (z-1 / z) 

 

=   log (z/z-1), if | z | >1. 






Example 4

 

Find the Z- transforms of

 

(i) cos np/2 

(ii)   sin np/2






Example 5

 

Show that Z{1/ n!} = e1/z and hence find Z{1/ (n+1)!} and  Z{1/ (n+2)!}


We know that Z{fn+1} = z { F(z) –f0}

 

Therefore,



Example 6

 

Find the Z- transforms of the following





 

 

 

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