<< Chapter < Page Chapter >> Page >
(Blank Abstract)

Introduction

With the z-transform , the s-plane represents a set of signals ( complex exponentials ). For any given LTI system, some of these signals may cause the output of the system toconverge, while others cause the output to diverge ("blow up"). The set of signals that cause the system's output to convergelie in the region of convergence (ROC) . This module will discuss how to find this region of convergence forany discrete-time, LTI system.

The region of convergence

The region of convergence, known as the ROC , is important to understand because it defines the region wherethe z-transform exists. The z-transform of a sequence is defined as

X z n x n z n
The ROC for a given x n , is defined as the range of z for which the z-transform converges. Since the z-transform is a power series , it converges when x n z n is absolutely summable. Stated differently,
n x n z n
must be satisfied for convergence.

Properties of the region of convergencec

The Region of Convergence has a number of properties that are dependent on the characteristics of the signal, x n .

  • The ROC cannot contain any poles. By definition a pole is a where X z is infinite. Since X z must be finite for all z for convergence, there cannot be a pole in the ROC.
  • If x n is a finite-duration sequence, then the ROC is the entire z-plane, except possibly z 0 or z . A finite-duration sequence is a sequence that is nonzero in a finite interval n 1 n n 2 .As long as each value of x n is finite then the sequence will be absolutely summable. When n 2 0 there will be a z term and thus the ROC will not include z 0 . When n 1 0 then the sum will be infinite and thus the ROC will not include z . On the other hand, when n 2 0 then the ROC will include z 0 , and when n 1 0 the ROC will include z . With these constraints, the only signal, then, whose ROCis the entire z-plane is x n c δ n .

An example of a finite duration sequence.

The next properties apply to infinite duration sequences. As noted above, the z-transform converges when X z . So we can write

X z n x n z n n x n z n n x n z n
We can then split the infinite sum into positive-time and negative-time portions. So
X z N z P z
where
N z n -1 x n z n
and
P z n 0 x n z n
In order for X z to be finite, x n must be bounded. Let us then set
x n C 1 r 1 n
for n 0 and
x n C 2 r 2 n
for n 0 From this some further properties can be derived:

  • If x n is a right-sided sequence, then the ROC extends outward from the outermost pole in X z . A right-sided sequence is a sequence where x n 0 for n n 1 . Looking at the positive-time portion from the abovederivation, it follows that
    P z C 2 n 0 r 2 n z n C 2 n 0 r 2 z n
    Thus in order for this sum to converge, z r 2 , and therefore the ROC of a right-sided sequence is of theform z r 2 .

A right-sided sequence.
The ROC of a right-sided sequence.

  • If x n is a left-sided sequence, then the ROC extends inward from the innermost pole in X z . A left-sided sequence is a sequence where x n 0 for n n 2 . Looking at the negative-time portion from the abovederivation, it follows that
    N z C 1 n -1 r 1 n z n C 1 n -1 r 1 z n C 1 k 1 z r 1 k
    Thus in order for this sum to converge, z r 1 , and therefore the ROC of a left-sided sequence is of theform z r 1 .

A left-sided sequence.
The ROC of a left-sided sequence.

  • If x n is a two-sided sequence, the ROC will be a ring in the z-plane that is bounded on the interior and exterior bya pole. A two-sided sequence is an sequence with infinite duration in the positive and negativedirections. From the derivation of the above two properties, it follows that if -r 2 z r 2 converges, then both the positive-time and negative-time portions converge and thus X z converges as well. Therefore the ROC of a two-sided sequence is of the form -r 2 z r 2 .

A two-sided sequence.
The ROC of a two-sided sequence.

Examples

Lets take

x 1 n 1 2 n u n 1 4 n u n
The z-transform of 1 2 n u n is z z 1 2 with an ROC at z 1 2 .

The ROC of 1 2 n u n

The z-transform of -1 4 n u n is z z 1 4 with an ROC at z -1 4 .

The ROC of -1 4 n u n

Due to linearity,

X 1 z z z 1 2 z z 1 4 2 z z 1 8 z 1 2 z 1 4
By observation it is clear that there are two zeros, at 0 and 1 8 , and two poles, at 1 2 ,and -1 4 . Following the obove properties, the ROC is z 1 2 .

The ROC of x 1 n 1 2 n u n -1 4 n u n
Got questions? Get instant answers now!

Now take

x 2 n -1 4 n u n 1 2 n u n 1
The z-transform and ROC of -1 4 n u n was shown in the example above . The z-transorm of 1 2 n u n 1 is z z 1 2 with an ROC at z 1 2 .

The ROC of 1 2 n u n 1

Once again, by linearity,

X 2 z z z 1 4 z z 1 2 z 2 z 1 8 z 1 4 z 1 2
By observation it is again clear that there are two zeros, at 0 and 1 16 , and two poles, at 1 2 , and -1 4 . in ths case though, the ROC is z 1 2 .

The ROC of x 2 n -1 4 n u n 1 2 n u n 1 .
Got questions? Get instant answers now!

Graphical understanding of roc

Using the demonstration, learn about the region of convergence for the Laplace Transform.

Conclusion

Clearly, in order to craft a system that is actually useful by virtue of being causal and BIBO stable, we must ensure that it is within the Region of Convergence, which can be ascertained by looking at the pole zero plot. The Region of Convergence is the area in the pole/zero plot of the transfer function in which the function exists. For purposes of useful filter design, we prefer to work with rational functions, which can be described by two polynomials, one each for determining the poles and the zeros, respectively.

Questions & Answers

why should a firm close down when it's unable to pay it's variable cost?
ANDREW Reply
what is oligopolistic competitive market?
ngong
exchange of goods and services between countries is call
Hosea Reply
foreign trade
Peace
what is constant opportunity cost
Tiffany Reply
Constant opportunity cost means the value of sacrifice remains constant in every step.
Dipam
Gross Domestic Product GDP
Yusuf Reply
what is g d p
Jayapal Reply
gross daily performance
domingo
How best can a poor country respond to an economic crisis , what does it have to sacrifice.
Bah Reply
they should pay tax as progressive system and should make sacrifice for taxation of their income and land etc
Hamza
how have the nations tries to solve the problem of scarcity in their economies?
Amani Reply
total concentration on to reduce the per unit cost of commodity by technically or whatever
Hamza
explain what will happen to producer of green coconut now that we have to lockdown in the kingdom of tonga
Tuha Reply
the demand for coconut will decrease and supply increases which result in the decrease in the price of coconut and the coconut will be more elastic
Hamza
tonga is producing more long run economic good explain the meaning of the statement and its implication on the tonga economy
Tuha Reply
Demand is the various quantities of goods and services that consumer(s)are willing and able to purchase at a price within a time
Muhammad Reply
What is demand
Mc Reply
demand relates with the need of people for their satisfaction.
Mohd
demand is de amount of goods and services a consumer us willing to purchase at a given price over a given period of time
Osei
riDemand is the amount of good and services which consumers are willing and able to buy at a particular peroid of time and at a given price
Miguel
Yes that is the tire thing
samuel
Demand is the quantity of goods and services which consumers are willing and able to purchase at a given price over a period of time
Sundaybha
simply , Demand is the sum up of 1)- desire of the commodity 2)- purchasing power for that commodity
Hamza
demand is the amount of goods and service wen consumers are willing and able to buy them at a give time
Dora
Distinguish between cross elasticity and income elasticity of demand
Ruth Reply
Distinguish between cross elasticity and income elasticity of demand
Ruth
if change in the demand of the commodity with respect to change in demand of the substitute or other product called cross elasticity
Hamza
and. if change in the demand of the commodity due to change in the income . called income elasticity
Hamza
👍
Vipul
Cross elasticity of demand is the degree of responsiveness of quantity demanded of a commodity to a small change in price of another commodity whiles Income elasticity of demand is the degree of responsiveness of quantity demanded of a commodity to a small change in income of it's consumers
Afriyie
but these are book wordings
Hamza
income elasticity of demand shows how quantity demanded changes due to changes in income on the other hand cross elasticity refers to how the quantity demanded of a particular good alers given a change in the price of another good.
Keysie
what is the competitive demand
Adiza Reply
Income
Sanni
Competitive demand are those commodity dat are competitive in nature e.g the close up and my my toothpaste the increase in price of close up may bring abt decrease in demand of it and it will serve as increase in purchase of my my
Daniel
With regards to coal shortage and manicipal debts the what form of intervention do you think Eskom can put in place.
kedibone Reply
economic growth of Bhutan
Nima Reply
please, explain all the mathematics terms used in economics
nelson
The answer is: little more than high school algebra and graphs.
Tere
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Signals and systems' conversation and receive update notifications?

Ask