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Consider the system with following input-output relation y [n] = (1 + (-1)n) x [n]
where, x[n] is the input and y[n] is the output. The system is
- a)invertible and time invariant
- b)invertible and time varying
- c)non-invertible and time invariant
- d)non-invertible and time varying
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
Consider the system with following input-output relation y [n] = (1 + ...
For two different inputs we have same output. Thus one to one mapping is not possible. Hence the systems is non invertible
Most Upvoted Answer
Consider the system with following input-output relation y [n] = (1 + ...
**Input-Output Relationship**
The given system has the following input-output relationship:
y [n] = (1 (-1)n) x [n]
where x[n] is the input and y[n] is the output.
**Invertibility**
A system is said to be invertible if different inputs produce different outputs. In other words, if two different inputs x1[n] and x2[n] result in the same output y[n], then the system is not invertible.
Let's consider two different inputs: x1[n] = 1 and x2[n] = (-1)n.
For x1[n] = 1,
y1[n] = (1 (-1)n) * 1
= 1
For x2[n] = (-1)n,
y2[n] = (1 (-1)n) * (-1)n
= -(-1)n
= (-1)n+1
For n even, y2[n] = (-1)n+1 = -1
For n odd, y2[n] = (-1)n+1 = 1
Therefore, for n even, y1[n] = y2[n], and for n odd, y1[n] = -y2[n]. This implies that the system is not invertible, as two different inputs x1[n] = 1 and x2[n] = (-1)n result in the same output for n even.
**Time Invariance**
A system is said to be time-invariant if a time shift in the input results in a corresponding time shift in the output. In other words, if x[n] produces y[n], then x[n - k] should produce y[n - k], where k is a constant.
Let's consider the input x1[n] = 1.
For x1[n] = 1,
y1[n] = (1 (-1)n) * 1
= 1
Now, let's consider the input x2[n] = 1, delayed by k samples.
For x2[n - k] = 1,
y2[n - k] = (1 (-1)n) * 1
= 1
Therefore, for both x1[n] = 1 and x2[n] = 1 delayed by k samples, the output y[n] remains the same. Thus, the system is time-invariant.
**Conclusion**
Based on the analysis, the given system is non-invertible and time-invariant. Therefore, the correct answer is option D: non-invertible and time varying.
The given system has the following input-output relationship:
y [n] = (1 (-1)n) x [n]
where x[n] is the input and y[n] is the output.
**Invertibility**
A system is said to be invertible if different inputs produce different outputs. In other words, if two different inputs x1[n] and x2[n] result in the same output y[n], then the system is not invertible.
Let's consider two different inputs: x1[n] = 1 and x2[n] = (-1)n.
For x1[n] = 1,
y1[n] = (1 (-1)n) * 1
= 1
For x2[n] = (-1)n,
y2[n] = (1 (-1)n) * (-1)n
= -(-1)n
= (-1)n+1
For n even, y2[n] = (-1)n+1 = -1
For n odd, y2[n] = (-1)n+1 = 1
Therefore, for n even, y1[n] = y2[n], and for n odd, y1[n] = -y2[n]. This implies that the system is not invertible, as two different inputs x1[n] = 1 and x2[n] = (-1)n result in the same output for n even.
**Time Invariance**
A system is said to be time-invariant if a time shift in the input results in a corresponding time shift in the output. In other words, if x[n] produces y[n], then x[n - k] should produce y[n - k], where k is a constant.
Let's consider the input x1[n] = 1.
For x1[n] = 1,
y1[n] = (1 (-1)n) * 1
= 1
Now, let's consider the input x2[n] = 1, delayed by k samples.
For x2[n - k] = 1,
y2[n - k] = (1 (-1)n) * 1
= 1
Therefore, for both x1[n] = 1 and x2[n] = 1 delayed by k samples, the output y[n] remains the same. Thus, the system is time-invariant.
**Conclusion**
Based on the analysis, the given system is non-invertible and time-invariant. Therefore, the correct answer is option D: non-invertible and time varying.
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Consider the system with following input-output relation y [n] = (1 + (-1)n) x [n]where, x[n] is the input and y[n] is the output. The system isa)invertible and time invariantb)invertible and time varyingc)non-invertible and time invariantd)non-invertible and time varyingCorrect answer is option 'D'. Can you explain this answer?
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Consider the system with following input-output relation y [n] = (1 + (-1)n) x [n]where, x[n] is the input and y[n] is the output. The system isa)invertible and time invariantb)invertible and time varyingc)non-invertible and time invariantd)non-invertible and time varyingCorrect answer is option 'D'. 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 the system with following input-output relation y [n] = (1 + (-1)n) x [n]where, x[n] is the input and y[n] is the output. The system isa)invertible and time invariantb)invertible and time varyingc)non-invertible and time invariantd)non-invertible and time varyingCorrect answer is option 'D'. 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 the system with following input-output relation y [n] = (1 + (-1)n) x [n]where, x[n] is the input and y[n] is the output. The system isa)invertible and time invariantb)invertible and time varyingc)non-invertible and time invariantd)non-invertible and time varyingCorrect answer is option 'D'. Can you explain this answer?.
Consider the system with following input-output relation y [n] = (1 + (-1)n) x [n]where, x[n] is the input and y[n] is the output. The system isa)invertible and time invariantb)invertible and time varyingc)non-invertible and time invariantd)non-invertible and time varyingCorrect answer is option 'D'. 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 the system with following input-output relation y [n] = (1 + (-1)n) x [n]where, x[n] is the input and y[n] is the output. The system isa)invertible and time invariantb)invertible and time varyingc)non-invertible and time invariantd)non-invertible and time varyingCorrect answer is option 'D'. 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 the system with following input-output relation y [n] = (1 + (-1)n) x [n]where, x[n] is the input and y[n] is the output. The system isa)invertible and time invariantb)invertible and time varyingc)non-invertible and time invariantd)non-invertible and time varyingCorrect answer is option 'D'. Can you explain this answer?.
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