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Consider a discrete time signal given by
x[n] = (-0.25)n u[n]+ (0.5)n u[-n -1]
The region of convergence of its Z-transform would be
x[n] = (-0.25)n u[n]+ (0.5)n u[-n -1]
The region of convergence of its Z-transform would be
- a)the region inside the circle of radius 0.5 and centered at origin.
- b)the region outside the circle of radius 0.25 and centered at origin.
- c)the annular region between the two circles, both centered at origin and having radii 0.25 and 0.5.
- d)the entire Z plane.
Correct answer is option 'C'. Can you explain this answer?
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Consider a discrete time signal given byx[n] = (-0.25)n u[n]+(0.5)n u[...
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Consider a discrete time signal given byx[n] = (-0.25)n u[n]+(0.5)n u[...
To determine the region of convergence (ROC) of the given discrete-time signal's Z-transform, let's analyze the signal and its Z-transform.
Given signal: x[n] = (-0.25)^n u[n] (0.5)^n u[-n -1]
1. Start by expressing the signal in terms of the unit step function u[n]:
x[n] = (-0.25)^n (0.5)^n u[n] u[-n -1]
2. The Z-transform of a discrete-time signal x[n] is defined as:
X(z) = Σ x[n] z^(-n)
3. Substitute the expression of x[n] into the Z-transform equation:
X(z) = Σ (-0.25)^n (0.5)^n u[n] u[-n -1] z^(-n)
4. Simplify the expression by considering the properties of the unit step function:
X(z) = Σ (-0.25)^n (0.5)^n z^(-n) for n ≥ 0
= Σ (-0.25)^n (0.5)^n z^(-n) for n < />
Now, let's analyze the ROC of the Z-transform.
- ROC for n ≥ 0:
For the first term of the sum, (-0.25)^n (0.5)^n z^(-n), the condition for convergence is |(-0.25)(0.5)z^(-1)| < 1,="" which="" simplifies="" to="" |z|="" /> 0.5. Therefore, the ROC for n ≥ 0 is the region outside the circle of radius 0.5 and centered at the origin.
- ROC for n < />
For the second term of the sum, (-0.25)^n (0.5)^n z^(-n), the condition for convergence is |(-0.25)(0.5)z^(-1)| < 1,="" which="" simplifies="" to="" |z|="" />< 4.="" therefore,="" the="" roc="" for="" n="" />< -1="" is="" the="" region="" inside="" the="" circle="" of="" radius="" 0.25="" and="" centered="" at="" the="" />
Since the Z-transform converges for both n ≥ 0 and n < -1,="" the="" overall="" roc="" is="" the="" intersection="" of="" the="" rocs="" for="" these="" two="" />
- Intersection of ROCs:
The intersection of the region outside the circle of radius 0.5 and centered at the origin (ROC for n ≥ 0) and the region inside the circle of radius 0.25 and centered at the origin (ROC for n < -1)="" forms="" an="" annular="" region="" between="" the="" two="" circles,="" both="" centered="" at="" the="" origin="" and="" having="" radii="" 0.25="" and="" />
Therefore, the correct option is (c) the annular region between the two circles, both centered at the origin and having radii 0.25 and 0.5.
Given signal: x[n] = (-0.25)^n u[n] (0.5)^n u[-n -1]
1. Start by expressing the signal in terms of the unit step function u[n]:
x[n] = (-0.25)^n (0.5)^n u[n] u[-n -1]
2. The Z-transform of a discrete-time signal x[n] is defined as:
X(z) = Σ x[n] z^(-n)
3. Substitute the expression of x[n] into the Z-transform equation:
X(z) = Σ (-0.25)^n (0.5)^n u[n] u[-n -1] z^(-n)
4. Simplify the expression by considering the properties of the unit step function:
X(z) = Σ (-0.25)^n (0.5)^n z^(-n) for n ≥ 0
= Σ (-0.25)^n (0.5)^n z^(-n) for n < />
Now, let's analyze the ROC of the Z-transform.
- ROC for n ≥ 0:
For the first term of the sum, (-0.25)^n (0.5)^n z^(-n), the condition for convergence is |(-0.25)(0.5)z^(-1)| < 1,="" which="" simplifies="" to="" |z|="" /> 0.5. Therefore, the ROC for n ≥ 0 is the region outside the circle of radius 0.5 and centered at the origin.
- ROC for n < />
For the second term of the sum, (-0.25)^n (0.5)^n z^(-n), the condition for convergence is |(-0.25)(0.5)z^(-1)| < 1,="" which="" simplifies="" to="" |z|="" />< 4.="" therefore,="" the="" roc="" for="" n="" />< -1="" is="" the="" region="" inside="" the="" circle="" of="" radius="" 0.25="" and="" centered="" at="" the="" />
Since the Z-transform converges for both n ≥ 0 and n < -1,="" the="" overall="" roc="" is="" the="" intersection="" of="" the="" rocs="" for="" these="" two="" />
- Intersection of ROCs:
The intersection of the region outside the circle of radius 0.5 and centered at the origin (ROC for n ≥ 0) and the region inside the circle of radius 0.25 and centered at the origin (ROC for n < -1)="" forms="" an="" annular="" region="" between="" the="" two="" circles,="" both="" centered="" at="" the="" origin="" and="" having="" radii="" 0.25="" and="" />
Therefore, the correct option is (c) the annular region between the two circles, both centered at the origin and having radii 0.25 and 0.5.
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Consider a discrete time signal given byx[n] = (-0.25)n u[n]+(0.5)n u[-n -1]The region of convergence of its Z-transform would bea)the region inside the circle of radius 0.5 and centered at origin.b)the region outside the circle of radius 0.25 and centered at origin.c)the annular region between the two circles, both centered at origin and having radii 0.25 and 0.5.d)the entire Z plane.Correct answer is option 'C'. Can you explain this answer?
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Consider a discrete time signal given byx[n] = (-0.25)n u[n]+(0.5)n u[-n -1]The region of convergence of its Z-transform would bea)the region inside the circle of radius 0.5 and centered at origin.b)the region outside the circle of radius 0.25 and centered at origin.c)the annular region between the two circles, both centered at origin and having radii 0.25 and 0.5.d)the entire Z plane.Correct answer is option 'C'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about Consider a discrete time signal given byx[n] = (-0.25)n u[n]+(0.5)n u[-n -1]The region of convergence of its Z-transform would bea)the region inside the circle of radius 0.5 and centered at origin.b)the region outside the circle of radius 0.25 and centered at origin.c)the annular region between the two circles, both centered at origin and having radii 0.25 and 0.5.d)the entire Z plane.Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider a discrete time signal given byx[n] = (-0.25)n u[n]+(0.5)n u[-n -1]The region of convergence of its Z-transform would bea)the region inside the circle of radius 0.5 and centered at origin.b)the region outside the circle of radius 0.25 and centered at origin.c)the annular region between the two circles, both centered at origin and having radii 0.25 and 0.5.d)the entire Z plane.Correct answer is option 'C'. Can you explain this answer?.
Consider a discrete time signal given byx[n] = (-0.25)n u[n]+(0.5)n u[-n -1]The region of convergence of its Z-transform would bea)the region inside the circle of radius 0.5 and centered at origin.b)the region outside the circle of radius 0.25 and centered at origin.c)the annular region between the two circles, both centered at origin and having radii 0.25 and 0.5.d)the entire Z plane.Correct answer is option 'C'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about Consider a discrete time signal given byx[n] = (-0.25)n u[n]+(0.5)n u[-n -1]The region of convergence of its Z-transform would bea)the region inside the circle of radius 0.5 and centered at origin.b)the region outside the circle of radius 0.25 and centered at origin.c)the annular region between the two circles, both centered at origin and having radii 0.25 and 0.5.d)the entire Z plane.Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider a discrete time signal given byx[n] = (-0.25)n u[n]+(0.5)n u[-n -1]The region of convergence of its Z-transform would bea)the region inside the circle of radius 0.5 and centered at origin.b)the region outside the circle of radius 0.25 and centered at origin.c)the annular region between the two circles, both centered at origin and having radii 0.25 and 0.5.d)the entire Z plane.Correct answer is option 'C'. Can you explain this answer?.
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Consider a discrete time signal given byx[n] = (-0.25)n u[n]+(0.5)n u[-n -1]The region of convergence of its Z-transform would bea)the region inside the circle of radius 0.5 and centered at origin.b)the region outside the circle of radius 0.25 and centered at origin.c)the annular region between the two circles, both centered at origin and having radii 0.25 and 0.5.d)the entire Z plane.Correct answer is option 'C'. Can you explain this answer?, a detailed solution for Consider a discrete time signal given byx[n] = (-0.25)n u[n]+(0.5)n u[-n -1]The region of convergence of its Z-transform would bea)the region inside the circle of radius 0.5 and centered at origin.b)the region outside the circle of radius 0.25 and centered at origin.c)the annular region between the two circles, both centered at origin and having radii 0.25 and 0.5.d)the entire Z plane.Correct answer is option 'C'. Can you explain this answer? has been provided alongside types of Consider a discrete time signal given byx[n] = (-0.25)n u[n]+(0.5)n u[-n -1]The region of convergence of its Z-transform would bea)the region inside the circle of radius 0.5 and centered at origin.b)the region outside the circle of radius 0.25 and centered at origin.c)the annular region between the two circles, both centered at origin and having radii 0.25 and 0.5.d)the entire Z plane.Correct answer is option 'C'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice Consider a discrete time signal given byx[n] = (-0.25)n u[n]+(0.5)n u[-n -1]The region of convergence of its Z-transform would bea)the region inside the circle of radius 0.5 and centered at origin.b)the region outside the circle of radius 0.25 and centered at origin.c)the annular region between the two circles, both centered at origin and having radii 0.25 and 0.5.d)the entire Z plane.Correct answer is option 'C'. Can you explain this answer? tests, examples and also practice GATE tests.
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