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For a finite support signal x[n] = r[n] (u[n] – u[n – 11]) where r[n] is the discrete-time ramp function, the energy of x[n] is
- a)385-385
- b)3854-3851
- c)3857-3856
- d)3859-3852
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
For a finite support signal x[n] = r[n] (u[n] – u[n – 11])...
Given signal is, x[n] = r[n] (u[n] – u[n – 11])
Energy of x[n] is
Energy of x[n] is
Most Upvoted Answer
For a finite support signal x[n] = r[n] (u[n] – u[n – 11])...
Understanding the Signal x[n]
The signal x[n] is defined as x[n] = r[n] (u[n] - u[n - 11]), where:
- r[n]: Discrete-time ramp function, defined as r[n] = n for n >= 0.
- u[n]: Unit step function, where u[n] = 1 for n >= 0 and u[n] = 0 for n < />
This expression indicates that x[n] is a ramp function that is active for a limited duration (from n=0 to n=10).
Signal Duration
- The term (u[n] - u[n - 11]) restricts x[n] to the interval 0 ≤ n < 11.="" />
- Thus, x[n] = r[n] for n = 0, 1, 2, ..., 10 and x[n] = 0 otherwise.
Calculating x[n]
- In this interval, x[n] = 0, 1, 2, ..., 10.
Energy Calculation
The energy of a discrete signal x[n] is calculated using the formula:
- Energy, E = Σ |x[n]|²
For the defined range:
- E = |0|² + |1|² + |2|² + ... + |10|²
- E = 0 + 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100
Summing the squares
- The sum of squares can be calculated as:
- Sum = 0 + 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 = 385
Thus, the energy of the signal x[n] equals 385.
Conclusion
- The correct answer is option 'A', which states that the energy of x[n] is 385.
The signal x[n] is defined as x[n] = r[n] (u[n] - u[n - 11]), where:
- r[n]: Discrete-time ramp function, defined as r[n] = n for n >= 0.
- u[n]: Unit step function, where u[n] = 1 for n >= 0 and u[n] = 0 for n < />
This expression indicates that x[n] is a ramp function that is active for a limited duration (from n=0 to n=10).
Signal Duration
- The term (u[n] - u[n - 11]) restricts x[n] to the interval 0 ≤ n < 11.="" />
- Thus, x[n] = r[n] for n = 0, 1, 2, ..., 10 and x[n] = 0 otherwise.
Calculating x[n]
- In this interval, x[n] = 0, 1, 2, ..., 10.
Energy Calculation
The energy of a discrete signal x[n] is calculated using the formula:
- Energy, E = Σ |x[n]|²
For the defined range:
- E = |0|² + |1|² + |2|² + ... + |10|²
- E = 0 + 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100
Summing the squares
- The sum of squares can be calculated as:
- Sum = 0 + 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 = 385
Thus, the energy of the signal x[n] equals 385.
Conclusion
- The correct answer is option 'A', which states that the energy of x[n] is 385.
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For a finite support signal x[n] = r[n] (u[n] – u[n – 11]) where r[n] is the discrete-time ramp function, the energy of x[n] isa)385-385b)3854-3851c)3857-3856d)3859-3852Correct answer is option 'A'. Can you explain this answer?
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