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The energy of the signal ,x(t) =e power -100 u(t) contained in the band mode f > then equal to 50 pi hz is?
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The energy of the signal ,x(t) =e power -100 u(t) contained in the ban...
Calculation of Energy of the Signal x(t)


  • The given signal is x(t) = e^(-100u(t))

  • To calculate the energy of this signal, we need to find its Fourier Transform.

  • The Fourier Transform of x(t) is X(f) = 1/(j2πf - 100), where j is the imaginary unit.

  • The energy of the signal can be calculated using Parseval's Theorem:



E = ∫(|X(f)|^2) df


  • Since the signal x(t) is causal, its Fourier Transform X(f) is zero for f < />

  • Therefore, we need to integrate the square of the magnitude of X(f) over the range f > 50π Hz.

  • Substituting X(f) in the above equation, we get:



E = ∫(1/(4π^2f^2 + 10^4)) df from 50π to ∞


  • Integrating this expression gives the energy of the signal in the frequency range f > 50π Hz.

  • After performing the integration, we get:



E = 0.00044 Joules

Explanation


  • The given signal x(t) is a decaying exponential function with a time constant of 1/100 seconds.

  • The Fourier Transform of the signal is a Lorentzian function centered at 100 Hz.

  • The energy of the signal in the frequency range f > 50π Hz can be calculated by integrating the square of the magnitude of its Fourier Transform over this frequency range.

  • Parseval's Theorem relates the energy of a signal to the square of its Fourier Transform.

  • By substituting the Fourier Transform of the signal in the Parseval's Theorem, we can calculate the energy of the signal in the frequency range f > 50π Hz.

  • The energy of the signal is a measure of the total power contained in the signal over this frequency range.
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The energy of the signal ,x(t) =e power -100 u(t) contained in the band mode f > then equal to 50 pi hz is?
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