Complete Z-Transform Notes with Solved Problems & Theorems

Complete Z-Transform Notes with Solved Problems &
Theorems


Z-Transform

X ( Z )  Z [ x(t )]  Z  x(kT )    x  kT Z  k .....................(1)
k 0




Again, X ( Z )  Z [ x(k )]   x  k Z  k .............................(2)
k 0




The Z-transform defined by equation (1) and (2) is referred to as the one sided Z-transform.
In the one sided Z-transform, we assume x(t)=0 for t<0 or x(k)=0 for k<0. Note that here Z is
a complex variable.

The Z  transform of x(t ), where    t  , or of x  k  , where
k takes int eger values (k  0,  1,  2,......), is defined by


X ( Z )  Z [ x(t )]  Z  x(kT )    x  kT Z k
.....................(3)
k 




Again, X ( Z )  Z [ x(k )]   x  k Z
k 
k
.............................(4)




The Z-transform defined by equation (3) and (4) is referred to as the two sided Z-transform.
In the two sided Z-transform, the time function x(t) is assumed to be non-zero for t<0 and the
sequence x(k) is considered to have non-zero values for k<0. Both the one-sided and two-
sided Z-transforms are series in powers of Z-1. (The latter involves both positive and negative
powers of Z-1.)

Important Properties and Theorems of the Z Transform
We assume that the time function x(t ) is Z-transformable and that x(t ) is zero for t  0.

1. Multiplication by a Constant
2. Linearity of the Z Transform
3. Multiplication by a k

1

,4. Shifting Theorem
5. Complex Translation Theorem
6. Initial Value Theorem
7. Final Value Theorem



1. Multiplication by a Constant

If X  Z  is the Z-transform of x(t ), that is Z  x(t )  X  Z  then
Z  ax(t )  aZ  x(t )   aX  Z 
Where a is a constant.

To prove this, note that by definition
 
Z  ax(t )   ax(kT ) Z  k  a  x(kT ) Z  k  aX  Z 
k 0 k 0



2. Linearity of the Z Transform

The Z-transform possesses an important property: linearity.
If f (k ) and g (k ) are Z-transformable and  and  are scalars, then x(k ) formed by a
linear combination
x( k )   f ( k )   g ( k )

has the Z transform

X ( Z )   F ( Z )   G( Z )
Where F ( Z ) and G( Z ) are the Z transforms of f (k ) and g (k ) respectively.


Now, X ( Z )  Z  x  k    Z  f (k )   g (k ) 

   f (k )   g (k ) Z  k
k 0
 
   f (k )Z  k    g (k )Z  k
k 0 k 0

  Z  f (k )    Z  g (k ) 
  F  Z    G(Z )


3. Multiplication by a k

If X (Z ) is the Z transform of x(k ), then the Z transform of a k x(k ) can be given by
X (a 1Z ) .


2

, Z  a k x(k )   X (a 1Z )

  k


Now Now, Z  a k x(k )    a k x(k ) Z  k   x (k ) a 1Z   X (a 1Z )
k 0 k 0




4. Shifting Theorem

The shifting theorem presented here is also referred to as the real translation theorem.

If x(t )  0 for t  0 and x(t ) has the Z transform X (Z ), then
(a) Z  x(t  nT )   Z  n X ( Z )
 n 1

(b) Z  x(t  nT )   Z n  X ( Z )   x(kT ) Z  k 
 k 0 
Where n is zero or a positive int eger.

 
(a) Now, Z  x(t  nT )    x(kT  nT )Z  k  Z  n  x(kT  nT )Z  k n 
k 0 k 0

By defining m  k  n, the above equation can be written as follows:

Z  x(t  nT )  Z  n  x(mT )Z m

m  n

Since x(mT )  0 for m  0, we may change the lower limit of the summation from
m  n to m  0. Hence,

Z  x(t  nT )  Z  n  x(mT )Z  m  Z  n X ( Z )
m 0
n
Thus, multiplication of a Z-transform by Z has the effect of delaying the time function
x(t ) by time nT . (i.e., move the function to the right by time nT ).

(b) Z  x(t  nT )    x(kT  nT )Z  k
k 0

 Z n  x(kT  nT )Z  k  n 
k 0

 n 1 n 1

 Z n   x(kT  nT )Z  k  n    x(kT )Z  k   x (kT )Z  k 
 k 0 k 0 k 0 
  n 1

 Z n   x(kT )Z  k   x(kT )Z  k 
 k 0 k 0 
 n 1

 Z n  X ( Z )   x(kT ) Z  k 
 k 0 

For the number sequence x(k ), the above equation can be written as follows:


3