EC3354-SIGNALS AND SYSTEMS PPGIT, Coimbatore

UNIT-IV
PART-A
1. Define unilateral z transform?
The unilateral z transform of the signal x(n) is given as

The unilateral and bilateral z transform are same for the causal system.

2. What is region of convergence?

The region of convergence is specified for the z transform, where it
converges.

3. State the convolution property of the z transform?

The convolution property of the z transform states that
z z

If x1(n)x1(z) and
x2(n)x2(z) Then
x1(n)*x2(n) 
x1(z)*x2(z)

4. What is the z transform of A(n-m).

By time shifting property,
-m
Z{ A(n-m).}=AZ since z{(n)}=1.

-1.
5. What is the inverse z transform of 1/1-aZ ?
From the standard z transform
-1 n
pairs, IZT{1/1-aZ }.= a
u(n), ROC |z|>|a|.

n
6. What is the z transform of (1/2) u(-n).
u(-n)=1 for n0

X(z)=  (1/2)n z-n.
n=-
EC3354-SIGNALS AND SYSTEMS PPGIT, Coimbatore

-

=  (1/2)n zn.
n=0

=  (1/2)-n zn.
n=0

=  [(1/2)-1 z].n
n=0

-1
=1/1-(1/2) z for |z|<1/2
7. Determine the Z transform of x(n)=(n)-0.95(n-6)

-k
Z{(n)}=1 and z{x(n-k)} =z x(z).
Then the z transform will be
-6
X(z)=1-0.95 z .

8. If X(w) is DTFT of x(n) find the DTFT of x*(-n)?

By symmetry properties of DTFT,
DTFT
x*(-n)  x*(w)

9. State the methods to find the inverse z transform.

1. Partial fraction expansion.
2. Contour integration.
3. Power series expansion and convolution.

10. Obtain the inverse z transform of

x(z)=1/z-a, |z|>a

Given function is ,x(z)=1/z-a

-1 -1
= z /1-az .
By time shifting property,
n
X(n)=a u(n-1)
EC3354-SIGNALS AND SYSTEMS PPGIT, Coimbatore

11. How unit sample response of discrete time system is defined?

The unit sample response of the discrete time system is output of the system to
unit sample sequence i.e.
T[(n)]=h(n)
-
Also h(n)=z 1{H (z)}.
12. Write any 4 properties of ZT
a) Linearity
b) Time shifting
c) Convolution
d) Multiplication
13. Write the time reversal property of ZT.
Time reversal;
If x (z) = Z{x(n)} then
-1
Z{x (-n)} = X (Z )

Proof:
∞
Z { x (-n)} = ∑ x (-n)
-n
z n=-
∞
Let l= -n, then
-1
Z{x (-n)} = X (Z )

PART-B
1. Explain-Properties of ZT.
Properties of Z transform:

1. Linearity

2. Time shifting

3. Multiplication by exponential sequence

4. Time reversal
EC3354-SIGNALS AND SYSTEMS PPGIT, Coimbatore

5. Multiplication by n

6. Convolution

7. Time expansion

8. Conjugation

9. Complex Convolution theorem

10. Parseval’s relation

11. Correlation

12. Initial value theorem

13. Final value theorem

1. Linearity:
The linearity property of Z Transform states that , weighted sum of two
signals is equal to the weighted sum of individual Z transform
If
X1(z) = Z{X1(n)} and X2(z) = Z{X2(n)}

Then
Z {a X1(n)+bX2(n)} = a X1(z)+b X2(z)

Proof:
∞
Z {a X1(n) +b X2(n)} = ∑ {a X1(n) +b
-n
X2(n)} z n=-∞

∞ ∞
-n -n
=a ∑ X1(n) z + b ∑ X2(n) z
n=-∞ n=-∞

Z {a X1(n) +b X2(n)} = a X1(z)+ b X2(z)

2. Time shifting:

If X (z) = Z{x(n)} and the initial conditions for x(n) are zero, then

-m
Z{x (n-m)} = Z X (z)

Where m is the positive or negative integer

Proof:
∞
-n
Z{x (n-m)} = ∑ x (n-m) z
EC3354-SIGNALS AND SYSTEMS PPGIT, Coimbatore

n=-∞

Let n-m= p then n=p+m, now

∞
-
Z{x (n-m)} = ∑ x (p) z
(m+p)
n=-
∞
∞
-m -p
Z{x (n-m)} = Z ∑ x (p) z
n=-∞

-m
Z{x (n-m)} = Z X (z)

3. Multiplication by an exponential sequence:

If X (z) = Z{x (n)} then

n -1
Z {a x (n)} = X (a z)
Proof: ∞
n n -n
Z {a x (n)} = ∑ a x (n) z
n=-∞

∞
-1 -n
= ∑ x (n) (a z)
n=-∞

n -1
Z {a x (n) = X (a z)

4. Time reversal;
If x (z) = Z{x(n)} then
-1
Z{x (-n)} = X (Z )

Proof:
∞
-n
Z { x (-n)} = ∑ x (-n) z
n=-∞
Let l= -n, then
-1
Z{x (-n)} = X (Z )
EC3354-SIGNALS AND SYSTEMS PPGIT, Coimbatore

5. Multiplication by n:

If z{x (n)} = x (z) then

Z {n x(n)} = -z d/dt{X(z)]

∞
-n
X (z) = ∑ x (n) z n=-∞
∞
-n
Z {n x (n)} = ∑ n x (n) z n=-∞

∞
-n
= Z ∑ n x (n) z n=-∞

∞
-(n+1)
= Z ∑ x (n) [n z ] n=-∞

∞
-(n)
= Z ∑ x (n) [d/dz [ z ]] n=-∞

∞
-(n)
= - Z d/dz∑ x (n) z n=-∞

Z {n x(n)} = -z d/dt{X(z)]

6. Convolution:

One of the important properties of ZT used in analysis of discrete time system is

convolution property. According to this property, the Z-transform of convolution of two
signals is equal to the multiplication of their Z transform

Z{ x(n) *h(n)}= X(Z)H(Z)

Where x (n) *h (n) denotes the linear convolution of the sequences

Proof:
Let y (n)= x (n) *h (n) , then we have
∞
y(n)= ∑ x(k) h (n-
k) k=-∞
EC3354-SIGNALS AND SYSTEMS PPGIT, Coimbatore

Taking Z transform on both sides

∞ ∞
y(z)= ∑ [ ∑ x(k) h (n-k)]
-n
z n=-∞ k=-∞

On interchanging the order of summation and we have

∞ ∞
-n-k -
y(z)= ∑ h (n-k)] z ∑ x(k) z
k
n=-∞ k=-∞

On replacing (n-k) = l , we have

∞ ∞
-l -k
y(z)= ∑ h (l)] z ∑ x(k) z
l=-∞ k=-∞

y(z)= X(z) H(z)

7. Time expansion:
Time expansion property of the ZT states that

If z{x (n)} = x (z) then

k
Z { xk(n)} = X(z )
8. Conjugation:

If z{x (n)} = x (z) then

*
Z { x* (n)} = X*(z )

9. Complex Convolution theorem:

The ZT of the product of two sequences can be related to the ZT of the individual
sequences through complex convolution theorem. This theorem states that if
X3(n) = X1(n) X2(n)

Then
-1
X3(z) = 1/2пj ∫ X1(v) X2(z/v) v dv
C

10. Parseval’s relation:

Let usn consider two complex valued sequences X1(n) and X2(n) . Parseval’s
relation states that
EC3354-SIGNALS AND SYSTEMS PPGIT, Coimbatore

∞
∑ X1(n) X2*(n) = 1/2пj ∫ X1(v) X2*(1/v*) dv
k=-∞ C

11. Correlation:

If X1(z) = Z{X1(n)} and X2(z) = Z {X2(n)]

Then Rx1x2(Z) = X1(z) X2(z-1)

12. Initial value theorem:

If X+ (Z) = Z[x(n)] then x(0)= Lt X+(z)
Z ∞

13. Final value theorem

If X+ (Z) = Z[x(n)] then x (∞)= Lt X+(z) (Z-1)

Z 1

2. Explain-Properties of DTFT.
Properties of DTFT:

1. Periodicity
2. Time Shifting
3. Frequency Shifting
4. Differentiation in frequency domain
5. Time reversal
6. Differentiation in frequency domain
7. Convolution in time domain
8. Convolution in frequency domain
9. Correlation theorem
10. Modulation theorem
11. Parseval’s theorem

1. Periodicity:

The Discrete time Fourier transform X (ω) is periodic in ω with the period of

2п Satisfying the following condition

X (ω) = X (ω + 2пk)

2. Time Shifting:

If F{x(n)} = X(ω)
EC3354-SIGNALS AND SYSTEMS PPGIT, Coimbatore

-jωk
Then F [X(n-k)}= X(k) e
EC3354-SIGNALS AND SYSTEMS PPGIT,
Coimbatore

3. Frequency Shifting:

If F{x(n)} = X(ω)

-jωn j(ω- ω0)

Then F {x(n) e }= X[ e ]

4. Time reversal

If F{x (n)} = X (ω)

F{x (-n)} = X (-ω)

5. Differentiation in frequency domain:

If F{x (n)} = X (ω)

F {n x (n)} = j d/dω [X (ω)]

6. Convolution in time domain:

If F{X1(n)} = X1(ω)

F{X2(n)} = X2(ω)

F{X1(n) * X2(n)} = X1(ω ) X2(ω)

7. Convolution in frequency domain:

If F{X1(n)} = X1(ω)

F{X2(n)} = X2(ω)

jθ j(ω-θ)
F{X1(n) X2(n)} = X1(ω ) * X2(ω) = 1/2п ∫ X1(e ) X2 (e ) dθ

2п
EC3354-SIGNALS AND SYSTEMS PPGIT, Coimbatore

8. Correlation theorem:

If F{X1(n)} = X1(ω)

F{X2(n)} = X2(ω)

Then,

F {γ x1x2 (l) } = Гx1x2(ω) = X1(ω) X2(ω)

9. Modulation Theorem

F{x(n)}= X(ω)

Then
j(ω +ω0 ) j(ω -ω0 )
F(x(n) cos ωo n ) = ½{ X{e + X{e )

10. Parseval’s theorem:

If F{x (n)}= X (ω)

∞
2 2
E = ∑ |x(n)| = 1/2п |x(ω)| dω n=-
∞

3.Find the ZT of the signal x (n) = [sin ωo n] u(n).

∞
-n
X (z) = ∑ x (n) z n=-
∞

∞
-n
= ∑ [sin ωo n] u (n)z n=-
∞

∞
-n
= ∑ [sin ωo n] z n=0
jωo n -jωo n -n
= ∑ {[e +e ]/2j } z
n=0

-1 -1 -2
X (z) = (Sin ωo) Z / {1-2(Cos ωo Z +Z }}
EC3354-SIGNALS AND SYSTEMS PPGIT, Coimbatore

4.Find the ZT of the signal x (n) = [Cos ωo n] u(n).

∞
-n
X (z) = ∑ x (n) z
n=-∞

∞
-n
= ∑ [Cos ωo n] u (n)z
n=-∞

∞
-n
= ∑ [Cos ωo n] z
n=0

∞
jωo n -jωo n -n
= ∑ {[e -e ]/2 } z
n=0

-1 -1 -2
X (z) = 1- (Cos ωo) Z / {1-2(Cos ωo Z +Z }}

5. Determine the ZT of the following.

n
X (n) = r sin [ωo n] u(n)
-1 -1 -2
Z {sin ωo n} = (Sin ωo) Z / {1-2(Cos ωo Z +Z }}

By using the multiplication by an exponential sequence property of ZT we can write Z

n -1
{a u (n)} = X (a Z)
n -1 -1 2 -2
Z {r sin ωo n} = r (Sin ωo) Z / {1-2r(Cos ωo Z +r Z }}

6.Determine the ZT of the following.

n
X (n )= r cos [ωo n] u(n)
-1 -1 -2
Z {Cos ωo n} = 1- (Cos ωo) Z / {1-2(Cos ωo Z +Z }}

By using the multiplication by an exponential sequence property of ZT we can write

n -1
Z {a u (n)} = X (a Z)
n -1 -1 2 -2
Z {r cos ωo n} = 1- (Cos ωo) r Z / {1-2r (Cos ωo Z +r Z }}