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What is the value of discrete time signal x(n) at n=0 whose Fourier transform is represented as below?
- a)ωc.π
- b)-ωc/π
- c)ωc/π
- d)None of the mentioned
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
What is the value of discrete time signal x(n) at n=0 whose Fourier tr...
Explanation: We know that,
At n = 0
Therefore, the value of the signal x(n) at n=0 is ω_c/π.
Most Upvoted Answer
What is the value of discrete time signal x(n) at n=0 whose Fourier tr...
Given Information:
The Fourier transform of a discrete-time signal x(n) is represented as follows: X(e^(jω)) = c/ω, where ω is the frequency variable.
To find the value of x(n) at n=0:
To find the value of x(n) at n=0, we need to compute the inverse Fourier transform of X(e^(jω)).
Inverse Fourier Transform:
The inverse Fourier transform of X(e^(jω)) is given by: x(n) = (1/2π) ∫[X(e^(jω)) * e^(jωn) dω], where the integration is over the range of ω from -π to π.
Substitute the given Fourier transform:
Substitute X(e^(jω)) = c/ω into the inverse Fourier transform equation.
x(n) = (1/2π) ∫[(c/ω) * e^(jωn) dω]
Calculate x(0):
To find the value of x(0), substitute n=0 into the equation.
x(0) = (1/2π) ∫[(c/ω) dω]
x(0) = c/2π ∫[1/ω dω]
x(0) = c/2π * ln|ω| evaluated from -π to π
x(0) = c/2π * (ln|π| - ln|-π|)
x(0) = c/2π * (ln(π) - ln(-π))
x(0) = c/2π * (ln(π) - ln(π))
x(0) = c/2π * 0
Final Result:
x(0) = 0
Therefore, the value of discrete-time signal x(n) at n=0 is 0.
The Fourier transform of a discrete-time signal x(n) is represented as follows: X(e^(jω)) = c/ω, where ω is the frequency variable.
To find the value of x(n) at n=0:
To find the value of x(n) at n=0, we need to compute the inverse Fourier transform of X(e^(jω)).
Inverse Fourier Transform:
The inverse Fourier transform of X(e^(jω)) is given by: x(n) = (1/2π) ∫[X(e^(jω)) * e^(jωn) dω], where the integration is over the range of ω from -π to π.
Substitute the given Fourier transform:
Substitute X(e^(jω)) = c/ω into the inverse Fourier transform equation.
x(n) = (1/2π) ∫[(c/ω) * e^(jωn) dω]
Calculate x(0):
To find the value of x(0), substitute n=0 into the equation.
x(0) = (1/2π) ∫[(c/ω) dω]
x(0) = c/2π ∫[1/ω dω]
x(0) = c/2π * ln|ω| evaluated from -π to π
x(0) = c/2π * (ln|π| - ln|-π|)
x(0) = c/2π * (ln(π) - ln(-π))
x(0) = c/2π * (ln(π) - ln(π))
x(0) = c/2π * 0
Final Result:
x(0) = 0
Therefore, the value of discrete-time signal x(n) at n=0 is 0.
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What is the value of discrete time signal x(n) at n=0 whose Fourier transform is represented as below?a)ωc.πb)-ωc/πc)ωc/πd)None of the mentionedCorrect answer is option 'C'. Can you explain this answer?
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