Signals and Systems
Z-Transform
(lecture 34, Date:-21/11/2023)
By: Dr. Sajai Vir Singh
Example 17: An LTI system is described by the following difference
equation 9 1
y ( n) y (n 1) y (n 2) x(n) 3x(n 1)
4 2
Determine
(i) The system functions and its poles and zeros.
(ii) Specify the ROC of H(z) and hence determine h(n) for the
following conditions.
(a) The system is stable.
(b) The system is causal.
(c) The system is anti-causal.
Sol: (i) 9 1
y (n) y (n 1) y (n 2) x(n) 3x(n 1)
4 2
Taking Z-transform on both side
9 1 1 2
Y ( z ) z Y ( z ) z Y ( z ) X ( z ) 3z 1 X ( z )
4 2
Y ( z) 1 3 z 1
H ( z)
X ( z ) 1 9 z 1 1 z 2
4 2
Poles-Zeroes calculation
Y ( z) z ( z 3) z ( z 3)
H ( z)
X ( z ) z 2 9 z 1 ( z 1 )( z 2)
4 2 4
Zeroes: Z=0, 3
Poles: Z=(1/4), 2
H ( z) ( z 3) A B
(ii)
z 1 1 ( z 2)
( z )( z 2) (z )
4 4
1
A( z 2) B( z ) ( z 3)
4
On solving above identity we get, A=(11/7) and B=(-4/7)
H ( z ) (11/ 7) (4 / 7)
So
z 1 ( z 2)
(z )
4
(11/ 7) (4 / 7)
H ( z)
1 1 (1 2 z 1 )
(1 z )
4
(a) The system is stable : ROC should include the unit circle
1
ROC ;, z 2
4
n
11 1 4 n
h(n) u (n) 2 u (n 1) System is non-causal
7 4 7
(b) The system is causal: ROC should be exterior of outermost circle.
ROC;, z 2
n
11 1 4 n System is non-stable as does not
h(n) u ( n) 2 u ( n) include the unit circle
7 4 7
(c) The system is anti-causal.
ROC should be interior of innermost circle.
1
ROC ;, z
4
n
11 1 4 n
h(n) u (n 1) 2 u (n 1)
7 4 7
System is non-stable as does not include the unit circle
One sided or unilateral Z-transform (right sided Z-
transform)
One sided or unilateral Z-Transform of a discrete time signal x(n) is
defined
Z [ x( n)] X ( z ) x[n]z n
n 0
Characteristic of one sided or unilateral Z-transform
X ( z ) is unique only for causal signal
ROC of X ( z ) is always the exterior of the circle so no need to define
the ROC of one sided Z-transform.
Example 18: Find the one sided Z-transform of following signal.
(i) x[n] {1, 2,5, 7, 0,1}
(ii) x[n] {1, 2,5, 7, 0,1}
Sol (i)
Z [ x( n)] X ( z ) x[n]z n 1 2 z 1 5 z 2 7 z 3 z 5
n 0
Sol (ii) Z [ x( n)] X ( z ) x[n]z n 5 7z 1 z 3
n 0
Properties of one sided Z-transform
All properties of one sided Z-transform are same as two sided Z-
transform with the exception of time shifting property.
Shifting properties:
Z
Case I: Time delay If x(n) X ( z )
then
k
x(n k ) z [ X ( z ) x( n) z n ], ,, , k 0
Z k
n 1
Proof:
Z [ x(n k )] x(n k ) z n Put nk p
n 0
p k
x( p) z p z k z k
p k
x( p) z p
X ( z)
1
z [ x( p) z
k p
x( p) z p ]
p k p 0
k k
z [ X ( z ) x ( p ) z ] z [ X ( z ) x ( n) z n ]
k p k
p 1 n 1
Special case : when k=1
Z [ x(n 1)] z 1 X ( z ) x(1)
Special case : when k=2
Z [ x(n 2)] z 2 X ( z ) x(1) z 1 x(2)
Case II: Time advance
Z
If x(n) X ( z )
then
k 1
z k [ X ( z ) x(n) z n ],,,, k 0
x(n k )
Z
n 0
Initial-Value and Final-Value Theorems (One-Sided ZT)
• Initial Value Theorem: x[0] lim X ( z )
z
Proof: lim X ( z ) lim x[n]z n lim x[0] x[1]z 1 ... x[0]
z z z
n 0
x[0]
• Final Value Theorem: lim x[n] x() lim( z 1) X ( z )
n z 1
Example 19: Find the initial value and final value (steady state) of
the signal x(n) whose z-transform is defined as
3z 2 2 z 4
X (z) 3
z 2 z 2 1.5 z 0.5
Sol: Initial Value x(0)
3 z 2
2z 4
x[0] lim X ( z ) lim( 3 )
z z z 2 z 1.5 z 0.5
2
3 z 1 2 z 2 4 z 3 0
lim(
) 0
z 1 2 z 1
1.5 z 0.5 z
2 3
1
Final Value of x(n)
3 z 2
2z 4
lim x[n] x() lim( z 1) X ( z ) lim( z 1)( 3 )
n z 1 z 1 z 2 z 1.5 z 0.5
2
(3z 2 2 z 4) (3z 2 2 z 4)
lim( z 1) lim 2
z 1 ( z 1)( z z 0.5) z 1 ( z z 0.5)
2
3z 2 2 z 4 5
2 10
z z 0.5 z 1 .5
Solution of difference equation for given excitation
(Response of the LTI system described by difference
equation)
Total response: y (n) yn (n) y fo (n)
Where yn (n) is called the natural or free response of the system. It
is due to only initial condition (y(-1), y(-2) etc.) of the system,
neglecting the applied input. While determining free response, put
input term to zero in difference equation and solve difference
equation for the given initial condition only.
Where y fo (n) is called the forced response of the system. It is due
to only the applied input to the system, neglecting the initial
condition. While determining force response, neglect initial
conditions of the system and solve difference equation for the given
input only.
Where y ( n) is called the total response. It is due to both input and
initial conditions considered, simultaneously.
