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Consider a two-sided discrete-time signal (neither left sided, nor right sided). The region of convergence (ROC) of the z-transform of the sequence is: 1) All region of z-plane outside a unit circle (in z-plane) 2) All region of z-plane inside a unit circle (in z-plane) 3) Ring in z-plane Which of the above is/are correct?

Consider a two-sided discrete-time signal (neither left sided, nor right sided). The region of convergence (ROC) of the z-transform of the sequence is: 1) All region of z-plane outside a unit circle (in z-plane) 2) All region of z-plane inside a unit circle (in z-plane) 3) Ring in z-plane Which of the above is/are correct? Correct Answer 3 only

Concept:

Properties of ROC of Z-Transform:

i) The ROC of Z-transform X(z) is in the form of a circle centered about the origin.

ii) The ROC of z-transform X(z) never contains any poles.

iii) X(n) is a right-sided signal and if the ROC contains r = r0 , then all the circles for which r0 < r < ∞ will also be present in the ROC

iv) If X(n) is a left-sided signal & if the circle (r = r0) is in the ROC then all values of r for which 0 < r < r0 will also be present in ROC.

v) For a right-sided signal, the ROC is always outer to the outermost pole excluding infinity

vi) For a left-sided signal, the ROC is always left to the innermost non-zero pole

vii) For a two-sided signal, the ROC is between the two circles (poles) but does not contain any pole

viii) For finite duration absolutely summable signal, the ROC is entire Z-plane except possibly for Z = 0 and/or Z = ∞

Application:

From the above exclamation of point (vi), we can conclude that for a two-sided signal, the ROC of Z-transform service is a ring in the Z-plane.

Related Questions

The transfer function of a discrete time LTI system is given by
$$H\left( z \right) = {{2 - {3 \over 4}{z^{ - 1}}} \over {1 - {3 \over 4}{z^{ - 1}} + {1 \over 8}{z^{ - 2}}}}$$
Consider the following statements:
S1 : The system is stable and causal for
$$ROC:\left| z \right| > {1 \over 2}$$
S2 : The system is stable but not causal for
$$ROC:\left| z \right| S3 : The system is neither stable nor causal for
$$ROC:{1 \over 4} Which one of the following statements is valid?
If a + b + c + d = 4, then find the value of $$\frac{1}{{\left( {1 - a} \right)\left( {1 - b} \right)\left( {1 - c} \right)}}$$     + $$\frac{1}{{\left( {1 - b} \right)\left( {1 - c} \right)\left( {1 - d} \right)}}$$     + $$\frac{1}{{\left( {1 - c} \right)\left( {1 - d} \right)\left( {1 - a} \right)}}$$     + $$\frac{1}{{\left( {1 - d} \right)\left( {1 - a} \right)\left( {1 - b} \right)}}$$     is?
If a + b + c + d = 4, then the value of $$\frac{1}{{\left( {1 - a} \right)\left( {1 - b} \right)\left( {1 - c} \right)}}$$     + $$\frac{1}{{\left( {1 - b} \right)\left( {1 - c} \right)\left( {1 - d} \right)}}$$     + $$\frac{1}{{\left( {1 - c} \right)\left( {1 - d} \right)\left( {1 - a} \right)}}$$     + $$\frac{1}{{\left( {1 - d} \right)\left( {1 - a} \right)\left( {1 - b} \right)}}$$     is?
The value of the expression $$\frac{{{{\left( {a - b} \right)}^2}}}{{\left( {b - c} \right)\left( {c - a} \right)}} + $$   $$\frac{{{{\left( {b - c} \right)}^2}}}{{\left( {a - b} \right)\left( {c - a} \right)}} + $$    $$\frac{{{{\left( {c - a} \right)}^2}}}{{\left( {a - b} \right)\left( {b - c} \right)}}$$   = ?
The Hamiltonian of a particle is given by $$H = \frac{{{p^2}}}{{2m}} + V\left( {\left| {\overrightarrow {\bf{r}} } \right|} \right) + \phi \left( { + \left| {\overrightarrow {\bf{r}} } \right|} \right)\overrightarrow {\bf{L}} .\overrightarrow {\bf{S}} ,$$        where $$\overrightarrow {\bf{S}} $$ is the spin, $$V\left( {\left| {\overrightarrow {\bf{r}} } \right|} \right)$$  and $$\phi \left( {\left| {\overrightarrow {\bf{r}} } \right|} \right)$$  are potential functions and $$\overrightarrow {\bf{L}} \left( { = \overrightarrow {\bf{r}} \times \overrightarrow {\bf{p}} } \right)$$   is the angular momentum. The Hamiltonian does not commute with
$$\frac{{{{\left( {4.53 - 3.07} \right)}^2}}}{{\left( {3.07 - 2.15} \right)\left( {2.15 - 4.53} \right)}} + \, $$     $$\frac{{{{\left( {3.07 - 2.15} \right)}^2}}}{{\left( {2.15 - 4.53} \right)\left( {4.53 - 3.07} \right)}} + \,\, $$     $$\frac{{{{\left( {2.15 - 4.53} \right)}^2}}}{{\left( {4.53 - 3.07} \right)\left( {3.07 - 2.15} \right)}}$$     is simplified to :
Each item consists of six sentences of a passage. The first and sixth sentences are given in the beginning as S1 and S6. The middle four sentences in each have been jumbled up and labelled as P, Q, R and S. You are required to find the proper sequence of the four sentences and mark your response accordingly on the Answer Sheet. S1 : He no longer dreamed of storms, nor of women, nor of great occurrences, nor of great fish, nor fights, nor contests of strength, nor of his wife.

S6 : He urinated outside the shack and then went up the road to wake the boy.

P : He never dreamed about the boy.

Q : He only dreamed of places and of the lions on the beach now.

R : He simply woke, looked out through the open door at the moon and unrolled his trousers and put them on.

S : They played like young cats in the dusk and he loved them as he loved the boy.

The correct sequence should be
Consider the differential equation $$\frac{{{{\text{d}}^2}{\text{y}}\left( {\text{t}} \right)}}{{{\text{d}}{{\text{t}}^2}}} + 2\frac{{{\text{dy}}\left( {\text{t}} \right)}}{{{\text{dt}}}} + {\text{y}}\left( {\text{t}} \right) = \delta \left( {\text{t}} \right)$$      with $${\left. {{\text{y}}\left( {\text{t}} \right)} \right|_{{\text{t}} = 0}} = - 2$$   and $${\left. {\frac{{{\text{dy}}}}{{{\text{dt}}}}} \right|_{{\text{t}} = 0}} = 0.$$
The numerical value of $${\left. {\frac{{{\text{dy}}}}{{{\text{dt}}}}} \right|_{{\text{t}} = 0}}$$   is
The quark content of $$\sum {^ + } ,\,{K^ - },\,{\pi ^ - }$$   and p is indicated: $$\left| {\sum {^ + } } \right\rangle = \left| {uus} \right\rangle ;\,\left| {{K^ + }} \right\rangle = \left| {s\overline u } \right\rangle ;\,\left| \pi \right\rangle = \left| d \right\rangle ;\,\left| p \right\rangle = \left| {uud} \right\rangle $$
In the process, $${\pi ^ - } + p \to {K^ - } + \sum {^ + } ,$$     considering strong interactions only, which of the following statements is true?
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