Seminar Report On

Discrete time system and Z transform

PRESENTED BY:-VIKAS KUMAR MANJHIElectrical EnggReg No. 1201214203

GUIDED BY:-Er. SMRUTI PRAGYAN DALAI

CONTENTS System Type of system Z transform Region of convergence properties Some common Z transform pair Inverse Z transform Advantage and disadvantage Conclusion

SYSTEM

A system responds to applied input signals and its response is described in terms of one or more output signals

A DT system transforms DT inputs into DT output

Types of Systems• Causal & Noncausal

• Linear & Non Linear

• Time Variant &Time-invariant

• Static & Dynamic

Causal & Noncausal Systems

• Causal system : A system is said to be causal if the present value of the output signal depends only on the present and/or past values of the input signal.

• Example : y[n]= ][nXnm

Causal & Noncausal Systems Contd.

Noncausal system : A system is said to be noncausal if the present value of the output signal depends only on the future values of the input signal.• Example: y[n]=x[n^2]

Linear & Non Linear Systems• A system is said to be linear if it satisfies the

principle of superposition• For checking the linearity of the given system,

firstly we check the response due to linear combination of inputs

• Then we combine the two outputs linearly in the same manner as the inputs are combined and again total response is checked

• If response in step 2 and 3 are the same,the system is linear otherwise it is non linear.

8

Time-Invariant and Time variant Systems

• Time-Invariant (shift-invariant) Systems– A time shift at the input causes

corresponding time-shift at output• Example– Square

]nn[xT]nn[y]}n[x{T]n[y oo

2]n[x]n[y

2

o

21

][n-ny givesoutput the

][y isoutput input the Delay the

o

o

nnxDelay

nnxn

Time variant systems A system is said to be time variant if time is not independent example y[n]=nx[n]

][)(n-ny givesoutput the

][y isoutput input the Delay the

o

1

oo

o

nnxnnDelaynnxn

Static & Dynamic Systems

• A static system is memory less system• It has no storage devices• its output signal depends on present values of the

input signal• For example y[n]=nx[n]

Static & Dynamic Systems Contd.• A dynamic system possesses memory• It has the storage devices• A system is said to possess memory if its output

signal depends on past values and future values of the input signal

• Example : y[n]=x[n]+x[n-1]

Z-transform

• Z transform covert a discrete time signal into a complex frequency domain representation

where n is integer time index

n

nznxX(z) ][

Region of Convergence (ROC)

• ROC: The set of values of z for which the z-transform converges

13

Re

Im • Example: z-transform converges for values of 0.5<r<2

Properties of The ROC of Z-Transform

• The signal is right sided ,ROC is outside the circle whose radius is largest pole in magnitude

• The signal is left sided signal , ROC is inside the circle whose radius is smallest pole in magnitude

• The signal is two sided ,ROC is bounded between largest and smallest pole radius

14

Right-Sided Exponential Sequence

• For Convergence we require

• Hence the ROC is defined as

• Inside the ROC series converges to

15

0n

n1

n

nnn azznuazX nuanx

0n

n1az

az1az n1

azz

az11azzX

0n1

n1

Re

Im

a 1o x

• Region outside the circle of radius a is the ROC

• Right-sided sequence ROCs extend outside a circle

Z-transform of left-sided sequence

Z-Transform Properties: Linearity

• Notation

• Linearity

ROC is common of both the sequence

– Example:

•Both sequences are right-sided

•Both sequences have a pole z=a

•Both have a ROC defined as |z|>|a|

17

xZ RROC zXnx

21 xx21

Z21 RRROC zbXzaXnbxnax

N-nua-nuanx nn

Z-Transform Properties: Time Shifting

• Here no is an integer– If positive the sequence is shifted right– If negative the sequence is shifted left

– Example

18

xRROCzXonzZonnx

1z 1112 ]2[

zznX

Some common Z-transform pairs

SEQUENCE TRANSFORM ROC

1z

0m ifor

0m if 0except z All

1z111

z

111

z

mz

mn

nu

nu

nua n

1

)(11

1 aZ

a|z |

The Inverse Z-Transform

• It is used to covert Z domain in time domain• The inverse Z transform is determine by following method

– Inspection method– Long division method– Partial fraction expansion

• Inspection MethodMake use of known z-transform pairs such as

Example: The inverse z-transform of

20

aaz

nua Zn

z 1

11

nu21nx 2

1z z2

111zX

n

1

Partial fraction method

21

21z :ROC

z211z4

111zX

11

1

2

1

1

z211A

z411AzX

1

41

211

1zXz411A 1

41z

11

2

21

411

1zXz211A 1

21z

12

• ROC extends to infinity – Indicates right sided sequence

22

21z

z2112

z4111zX

11

nu41-nu2

12nxnn

Long division method

• Long division to obtain Bo

23

1z z1z2

11z1

z21z2

31zz21zX

11

21

21

21

1z5 2z3z

21z2z1z2

3z21

1

12

1212

11

1

z1z211

z512zX

12

11

z1A

z211A2zX

9zXz211A

21z

11

8zXz1A1z

12

• ROC extends to infinity– Indicates right-sided sequence

24

1z z18

z21192zX 1

1

n8u-nu219n2nx

n

Advantage and Disadvantage

• Advantage

Z transform is used to analysis of discrete systems

Z transform is used for the digital signals

• Disadvantage Z transform can not apply in continuous signal

Conclusion

• We have concluded that Z transform is useful for the manipulation of discrete data sequence and has acquired a new significance in the formation and analysis of discrete time system

• It is used extensively today in the areas of applied mathematics and digital signal

processing

THANK YOU