Ss Ztransform

Z-Transform (Part-I)

Nishant ParikhElectrical Engineering Department

Shankersinh Vaghela Bapu Institute of Technology,Gandhinagar-382650

January 20, 2015

Fourth Semster Bachelor of Engineering () Singals and Systems January 20, 2015 1 / 24

Z-Transform

Z -transform provides a valuable technique for analysis and design ofdiscrete time signals and discrete time LTI systems.Z -Plane: The Z -transform has real and imaginary parts like fouriertransform. A plot of imaginary part versus real part is called asZ -plane or complex Z -plane.Advantages of Z-transform:

1 Discrete time signals and LTI systems can be completely characterizedusing Z -transform.

2 The stability of LTI system can be determined using Z -transform.3 Mathematical calculations are reduced using Z -transform. ForExample: convolution operation is transformed into simplemultiplication operation.

4 By calculating Z -transform of given signal, DFT and FT can bedetermined.

5 Entire family of digital �lters can be obtained from one proto-typedesign using Z -transform.

6 The solution of di¤erential equation can be simpli�ed usingZ -transform.


Basics of Z-transform

Two Types: single sided Z -transform and Double sided Z -transform

De�nition: Z-transform: A Z-transform of discrete time signal x(n)is de�ned as,

X(Z) =∞

∑n=0x(n)Z�n (1)

Here, "Z" is a complex variable. If the limit of summation are from 0to ∞, it is known as single sided Z -transform and if the limit is from�∞ to ∞, it is called as double sided Z -transform.

x(n)Z$ X (Z )

Here, the arrow is bidirectional. This is because we can obtain x(n)from X (Z ) using inverse Z -transform.The Z -transform is also denoted as,

X (Z ) = Z fx(n)g

Here, x(n) and X (Z ) are called Z -transform pairs.Fourth Semster Bachelor of Engineering () Singals and Systems January 20, 2015 3 / 24

Region of Convergence

Region of Convergence: The region of convergence (ROC) of X(Z)is set for all the values of Z for which X(Z) attains a �nite value.

Every time when we �nd the Z -transform, we must indicate its ROC.

Signi�cance of ROC:

1 ROC will decide whether a system (�lter) is stable or unstable.2 ROC also determines the type of sequence such as Causal ornon-causal, Finite or in�nite.


Examples

Obtain the Z-transform of following �nite duration sequences,1 x(n) = f1, 2, 4, 5, 0, 7g2 x(n) =

�1, 2, 4, 5, 0, 7

"

�3 x(n) =

�1, 2, 4

", 5, 0, 7

�The �rst sequence is causal, since x(n) is present only for positivevalues of n. Thus, for a causal �nite sequence ROC is entire Z -planeexcept jZ j = 0.

The second sequence is anticausal. This is because it is present onlyfor negative values of �n�. Thus, for anticausal �nite durationsequence, ROC is entire Z -plane except jZ j = ∞.Third sequence is bothsided sequence. This is because the signal ispresent for both positive and negative values of �n�. Thus, forbothsided sequence the ROC is entire Z -plane except jZ j = 0 andjZ j = ∞.


Examples


�1, 2, 4, 5, 0, 7

"

�3 x(n) =

�1, 2, 4

", 5, 0, 7

�The �rst sequence is causal, since x(n) is present only for positivevalues of n. Thus, for a causal �nite sequence ROC is entire Z -planeexcept jZ j = 0.The second sequence is anticausal. This is because it is present onlyfor negative values of �n�. Thus, for anticausal �nite durationsequence, ROC is entire Z -plane except jZ j = ∞.

Third sequence is bothsided sequence. This is because the signal ispresent for both positive and negative values of �n�. Thus, forbothsided sequence the ROC is entire Z -plane except jZ j = 0 andjZ j = ∞.


Examples


�1, 2, 4, 5, 0, 7

"

�3 x(n) =

�1, 2, 4

", 5, 0, 7

�The �rst sequence is causal, since x(n) is present only for positivevalues of n. Thus, for a causal �nite sequence ROC is entire Z -planeexcept jZ j = 0.The second sequence is anticausal. This is because it is present onlyfor negative values of �n�. Thus, for anticausal �nite durationsequence, ROC is entire Z -plane except jZ j = ∞.Third sequence is bothsided sequence. This is because the signal ispresent for both positive and negative values of �n�. Thus, forbothsided sequence the ROC is entire Z -plane except jZ j = 0 andjZ j = ∞.


Answers

1 X (Z ) = 1+ 2Z +

4Z 2 +

5Z 3 +

7Z 5 , ROC: Entire Z-plane except

jZ j = 0 (The given sequence is a causal �nite duration sequence.)

2 X (Z ) = Z 5 + 2Z 4 + 4Z 3 + 5Z 2 + 7, ROC: Entire Z-plane exceptjZ j = ∞ (The given sequence is anticausal �nal duration sequence.)

3 X (Z ) = Z 2 + Z + 4+ 5Z +

2Z 2 +


jZ j = 0 and jZ j = ∞ (The given sequence is bothsided sequencebecause x(n) is present for both positive and negative values of �n�.


Answers

1 X (Z ) = 1+ 2Z +

4Z 2 +

5Z 3 +


jZ j = 0 (The given sequence is a causal �nite duration sequence.)2 X (Z ) = Z 5 + 2Z 4 + 4Z 3 + 5Z 2 + 7, ROC: Entire Z-plane exceptjZ j = ∞ (The given sequence is anticausal �nal duration sequence.)

3 X (Z ) = Z 2 + Z + 4+ 5Z +

2Z 2 +




Answers

1 X (Z ) = 1+ 2Z +

4Z 2 +

5Z 3 +


jZ j = 0 (The given sequence is a causal �nite duration sequence.)2 X (Z ) = Z 5 + 2Z 4 + 4Z 3 + 5Z 2 + 7, ROC: Entire Z-plane exceptjZ j = ∞ (The given sequence is anticausal �nal duration sequence.)

3 X (Z ) = Z 2 + Z + 4+ 5Z +

2Z 2 +




Summary of Z-transform Pairs

Sr. No. Discrete Time Signal Z-Transform1 δ(n) 12 δ(n� k) Z�k

3 δ(n+ k) Z k

4 u(n) ZZ�1

5 nu(n) Z(Z�1)2

6 αnu(n) ZZ�α

7 �αnu(�n� 1) ZZ�α

8 αnu(n) + βnu(�n� 1) ZZ�α �

ZZ�β

9 cosω�nu(n) Z 2�Z cosω�Z 2�2Z cosω�+1

10 sinω�nu(n) Z sinω�Z 2�2Z cosω�+1

11 an cosω�n Z 2�aZ cosω�Z 2�2aZ cosω�+a2

12 an sinω�nu(n) aZ sinω�Z 2�2aZ cosω�+a2

13 u(�n) 11�Z

14 nαnu(n) aZ(Z�a)2Fourth Semster Bachelor of Engineering () Singals and Systems January 20, 2015 7 / 24

Z-Transform of Unit Impulse

Unit Impulse δ(n)δ(n) = 1 (only at n = 0)δ(n) = 0 (otherwise)

X (Z ) =∞∑

n=�∞x(n)Z�n

Here, x(n) = δ(n)

) X (Z ) =∞∑

n=�∞δ(n)Z�n

) X (Z ) = δ(0)Z�0 = 1

ROC is entire Z-plane.


Z-Transform of delayed Unit Impulse

Delayed Unit Impulse δ(n� k)δ(n� k) = 1 (only at n = k and k > 0)δ(n� k) = 0 (otherwise)

) X (Z ) =∞∑

n=�∞δ(n� k)Z�n

) X (Z ) = 1 � Z�k = Z�k

ROC is entire Z -plane except Z = 0 because at Z = 0, X (Z ) = ∞.


Z-Transform of Advanced Unit Impulse

Advanced Unit Impulse δ(n+ k)δ(n+ k) = 1 (only at n = �k and k > 0)δ(n+ k) = 0 (otherwise)

) X (Z ) =∞∑

n=�∞δ(n+ k)Z�n

) X (Z ) = 1 � Z k = Z k

ROC is entire Z -plane except Z = ∞ because at Z = ∞, X (Z ) = ∞.


Z-Transform of Unit Step

Unit Step u(n)u(n) = 1 (n � 0)u(n) = 0 (otherwise)

) X (Z ) =∞∑n=0u(n)Z�n

) X (Z ) =∞∑n=01 � Z�n =

∞∑n=0

�Z�1

�n) X (Z ) = 1

1�Z�1 if��Z�1�� < 1 �

*∞∑n=0An = 1

1�A , jAj < 1�

) X (Z ) = ZZ�1 , ROC:

��Z�1�� < 1 i.e. jZ j > 1


Z-Transform of Unit Ramp

Unit Ramp r(n)r(n) = n (n � 0)r(n) = 0 (otherwise)

It can also be described as r(n) = nu(n)

X (Z ) =∞∑n=0nu(n)Z�n

) X (Z ) =∞∑n=0nZ�n =

∞∑n=0n�Z�1

�n) X (Z ) = Z�1

(1�Z�1)2 if��Z�1�� < 1 �

*∞∑n=0nAn = A

(1�A)2 , jAj < 1�

) X (Z ) = Z(Z�1)2 , ROC:

��Z�1�� < 1 i.e. jZ j > 1


Z-Transform of Right Handed Exponential Sequence(Causal Exponential Sequence)

x(n) = αnu(n) = αn, (n � 0)x(n) = 0, (n < 0)

X (Z ) =∞∑n=0

αnZ�n

) X (Z ) =∞∑n=0

�αZ�1

�n) X (Z ) = 1

1�αZ�1 , if��αZ�1�� < 1 �

*∞∑n=0An = 1

1�A , jAj < 1�

) X (Z ) = ZZ�α ,ROC:

��αZ�1�� < 1) jαj < jZ j ) jZ j > jαj


Properties of Z-TransformLinearity

If x(n) = a1x1(n) + a2x2(n) then X (Z ) = a1X1(Z ) + a2X2(Z )

Proof: X (Z ) =∞∑

n=�∞[a1x1(n) + a2x2(n)]Z�n

=∞∑

n=�∞a1x1(n)Z�n +

∞∑

n=�∞a2x2(n)Z�n

= a1∞∑

n=�∞x1(n)Z�n + a2

∞∑

n=�∞x2(n)Z�n

= a1X1(Z ) + a2X2(Z )

ROC: The combined ROC is the overlap or intersection of theindividual ROC�s of X1(Z ) and X2(Z ).


Continue.....Time Shifting

If x(n) Z$ X (Z ) then x(n� k) Z$ Z�kX (Z ).

Z fx(n)g = X (Z ) =∞∑

n=�∞x(n)Z�n

) Z fx(n� k)g =∞∑

n=�∞x(n� k)Z�n

) Z fx(n� k)g =∞∑

n=�∞x(n� k)Z�(n�k ) � Z�k

) Z fx(n� k)g = Z�k∞∑

n=�∞x(n� k)Z�(n�k ) (*Def.ofZ-Transform)

) Z fx(n� k)g = Z�kX (Z )


Continue....Scaling

If x(n) Z$ X (Z ) , ROC: r1 < jZ j < r2then anx(n) Z$ X (Za ),ROC: jaj r1 < jZ j < jaj r2

Z fanx(n)g =∞∑

n=�∞anx(n)Z�n

=∞∑

n=�∞x(n)

�a�1Z

��n=

∞∑

n=�∞x(n)

�Za

��n) Z fanx(n)g = X

�Za

�ROC: The ROC of X(Z) is r1 < jZ j < r2To obtain ROC of X

�Za

�, replace �Z�by Z

a ,

)ROC of X�Za

�: jaj r1 < jZ j < jaj r2


Continue....Time Reversal

If x(n) Z$ X (Z ) , ROC: r1 < jZ j < r2then x(�n) Z$ X

�Z�1

�, ROC: 1r1 < jZ j <

1r2

Z fx(�n)g =∞∑

n=�∞x(�n)Z�n

Put l = �n, ) n! �∞) l ! ∞ and n! ∞) l ! �∞

Z fx(�n)g =�∞∑n=∞

x(l)Z l =∞∑

n=�∞x(l)

�Z�1

��lComparing this equation with the de�nition of Z -Transform we get,

Z fx(�n)g = X�Z�1

�ROC: ROC of x(n) is the inverse of that of x(�n). This means that,if Z0 belongs to the ROC of x(n) then 1

Z0is in the ROC for x(�n).


Continue....Di¤erentiation

If x(n) Z$ X (Z ) then nx(n) Z$ �Z dX (Z )dZ

X (Z ) =∞∑

n=�∞x(n)Z�n

) dX (Z )dZ = d

dZ

�∞∑

n=�∞x(n)Z�n

�) dX (Z )

dZ =∞∑

n=�∞

ddZ [x(n)Z

�n ] =∞∑

n=�∞(�n) � x(n)Z�n�1

) dX (Z )dZ =

∞∑

n=�∞(�n)x(n)Z�n � Z�1 = �Z�1

∞∑

n=�∞[nx(n)]Z�n

) �Z dX (Z )dZ =∞∑

n=�∞[nx(n)]Z�n

) �Z dX (Z )dZ = Z fnx(n)g

) nx(n) Z ! �Z dX (Z )dZ (ROC remain same for both the transform)


Continue....Convolution

If x1(n)Z$ X1(Z ) and x2(n)

Z$ X2(Z )

then x1(n) � x2(n) Z$ X1(Z ) � X2(Z )According to de�nition of convolution, x(n) =

x1(n) � x2(n) =∞∑

k=�∞x1(k)x2(n� k)

) X (Z ) = Z�

∞∑

k=�∞x1(k)x2(n� k)

�=

∞∑

n=�∞

�∞∑

k=�∞x1(k)x2(n� k)

�� Z�n

) X (Z ) =∞∑

k=�∞x1(k)

�∞∑

n=�∞x2(n� k)Z�n

�=

∞∑

k=�∞x1(k)

�∞∑

n=�∞x2(n� k)Z�(n�k ) � Z�k

�) X (Z ) =

�∞∑

k=�∞x1(k) � Z�k

� �∞∑

n=�∞x2(n� k)Z�(n�k )

�=

X1(Z ) � X2(Z )ROC is atleast the intersection of X1(Z ) and X2(Z ).


Continue....Initial Value Theorem

If x(n) is a causal sequence then its initial value is given by,x(0) = lim

Z!∞X (Z ).

Proof: X (Z ) =∞∑

n=�∞x(n)Z�n

) X (Z ) =∞∑n=0x(n)Z�n (*causal sequence)

Expanding the summation,

) X (Z ) = x(0)Z 0 + x(1)Z�1 + x(2)Z�2 + x(3)Z�3 + ....limZ!∞

X (Z ) = limZ!∞

x(0) + limZ!∞

x(1)Z�1 + limZ!∞

x(2)Z�2 + ....

) limZ!∞

X (Z ) = x(0)

) Initial Value = x(0) = limZ!∞

X (Z )


Continue....Final Value Theorem

If x(n) Z$ X (Z ) then x (∞) = limZ!1

[(Z � 1)X (Z )] . (causal sequence)

Z fx(n)� x(n� 1)g =∞∑

n=�∞[x(n)� x(n� 1)]Z�n

) Z fx(n)� x(n� 1)g =∞∑n=0

[x(n)� x(n� 1)]Z�n (* causal)

) Z fx(n)g � Z fx(n� 1)g =∞∑n=0x(n)Z�n �

∞∑n=0x(n� 1)Z�n

) X (Z )� Z�1X (Z ) =∞∑n=0x(n)Z�n �

∞∑n=0x(n� 1)Z�n

(* Shifting Property)


Continue....Final Value Theorem

From Previous Slide,

X (Z )� Z�1X (Z ) =∞∑n=0x(n)Z�n �

∞∑n=0x(n� 1)Z�n

(* Shifting Property)

) X (Z )�1� Z�1

�=

∞∑n=0x(n)Z�n �

∞∑n=0x(n� 1)Z�(n�1) � Z�1

) X (Z )�1� Z�1

�=

∞∑n=0x(n)Z�n � Z�1 �

∞∑n=0x(n� 1)Z�(n�1)

) limZ!1

X (Z )�1� Z�1

�= lim

Z!1∞∑n=0x(n)Z�n � lim

Z!1Z�1 �

∞∑n=0x(n� 1)Z�(n�1)

) limZ!1

X (Z )�1� Z�1

�= X (∞)


Examples

(1) Find Z -Transform of x(n) = anu(n) + δ (n� 5) (LinearityProperty)

(2) It is given that: x1(n) =�1", 2, 3, 4, 0, 1

�.

Using time shifting property �nd Z -Transform of

x2(n) =�1, 2, 3

", 4, 0, 1

�.(Time Shifting Property)

(3) Express the Z -Transform of y(n) =n∑

k=�∞x(k) in terms of

X (Z ).(Time Shifting Property)

(4) Obtain the Z -Transform of x(n) = an cosω0nu (n) .(ScalingProperty)

(5) Determine Z -Transform and draw ROC of the followingsignal x(n) = (2)n+2 u(n� 1). Is the signal causal? (ScalingProperty)


Examples (Continue....)

(6) Determine Z -Transform and ROC of x(n) =� 12

�nu (�n)

(Time Reversal Property)

(7) Obtain the Z -Transform of x(n) = nanu(n). (Di¤erentiationProperty)

(8) Find the linear convolution of x1(n) and x2(n) usingZ�Transform.

x1(n) =�1, 2, 3

", 4�and x2(n) =

�1, 2, 0

", 2, 1

�. (Convolution Property)

(9) Determine the value of signal at n = 0 and n = ∞ if X (Z )= 2Z 2+0.25

(Z+0.25)(Z�1) .


Ss Ztransform

Documents

values of z

z fxnghere

inverse z

basics of z

advantages of z

characterizedusing z

entire z plane

complex z plane