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What is the energy density spectrum of the signal x(n)=anu(n), |a|<1?
- a)1/(1+2acosω+a2 )
- b)1/(1-2acosω+a2 )
- c)1/(1-2acosω-a2 )
- d)1/(1+2acosω-a2 )
Correct answer is option 'B'. Can you explain this answer?
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What is the energy density spectrum of the signal x(n)=anu(n), |a|<...
Given x(n)= anu(n), |a|<1 The auto correlation of the above signal is rxx(l)=1/(1-a2 ) a|l|, -8< l <8 According to Wiener-Khintchine Theorem, Sxx(?)=F{ rxx(l)}= [1/(1-a2)].F{a|l|} = 1/(1-2acos?+a2 )
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Most Upvoted Answer
What is the energy density spectrum of the signal x(n)=anu(n), |a|<...
The energy density spectrum of the signal x(n) = anu(n) is:
E(f) = |a|^2/2πf, 0 < f="" />< />
where E(f) is the energy density at frequency f and u(n) is the unit step function.
To derive this, we first find the Fourier transform of x(n) as:
X(f) = ∑n=0^∞ anu(n) e^(-j2πfn)
= ∑n=0^∞ an e^(-j2πfn)
= a/(1 - e^(-j2πf))
Using Parseval's theorem, the energy in the signal can be calculated as:
E = (1/2π) ∫_-∞^∞ |X(f)|^2 df
= (1/2π) ∫_0^∞ |a|^2/(1 + 4π^2f^2) df
= |a|^2/2π ∫_0^∞ 1/(1 + (2πf)^2) df
= |a|^2/2
Therefore, the energy density at frequency f is given by:
E(f) = |X(f)|^2/2π
= |a|^2/(2π(1 + 4π^2f^2))
= |a|^2/2πf, 0 < f="" />< />
This result shows that the energy density spectrum of x(n) is a constant multiple of 1/f, which is a characteristic of a power-law spectrum.
E(f) = |a|^2/2πf, 0 < f="" />< />
where E(f) is the energy density at frequency f and u(n) is the unit step function.
To derive this, we first find the Fourier transform of x(n) as:
X(f) = ∑n=0^∞ anu(n) e^(-j2πfn)
= ∑n=0^∞ an e^(-j2πfn)
= a/(1 - e^(-j2πf))
Using Parseval's theorem, the energy in the signal can be calculated as:
E = (1/2π) ∫_-∞^∞ |X(f)|^2 df
= (1/2π) ∫_0^∞ |a|^2/(1 + 4π^2f^2) df
= |a|^2/2π ∫_0^∞ 1/(1 + (2πf)^2) df
= |a|^2/2
Therefore, the energy density at frequency f is given by:
E(f) = |X(f)|^2/2π
= |a|^2/(2π(1 + 4π^2f^2))
= |a|^2/2πf, 0 < f="" />< />
This result shows that the energy density spectrum of x(n) is a constant multiple of 1/f, which is a characteristic of a power-law spectrum.
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What is the energy density spectrum of the signal x(n)=anu(n), |a|<1?a)1/(1+2acosω+a2)b)1/(1-2acosω+a2)c)1/(1-2acosω-a2)d)1/(1+2acosω-a2)Correct answer is option 'B'. Can you explain this answer?
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