_Signal_FAQ06_Energy_&_Power_Spectrum

_Signal_FAQ06_Energy_&_Power_Spectrum - , where ( 29 (...

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What is the relationship between energy and power spectra? Let ( 29 t x 1 be a signal with a spectrum of ( 29 ( 29 [ ] t x f X 1 1 F = . The power (or, more strictly, average power) of the signal is the mean square value: ( 29 ( 29 = = - 2 2 2 1 2 1 1 1 Lim T T T dt t x T t x p . Since power is the rate of energy change or = T e p T 1 1 Lim , the energy of the signal is ( 29 - = 2 2 2 1 1 Lim T T T dt t x e . Since ( 29 ( 29 - - = df f X dt t x 2 1 2 1 , the energy of the signal can also be found in the frequency domain using ( 29 - = df f X e 2 1 1 . We can therefore regard ( 29 ( 29 2 1 1 f X f E = as the energy density spectrum or energy spectrum, as ( 29 2 1 1 f f df f E = Energy of ( 29 t x 1 from 1 f to 2 f Hz, and integrating the energy spectrum over all frequencies gives the total energy.
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Similarly, the power of the signal can be found using = T e p T 1 1 Lim ( 29 ( 29 ( 29 - - - = = = df f P df T f E df f E T T T 1 1 1 Lim 1 Lim
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Unformatted text preview: , where ( 29 ( 29 T f E f P T 1 1 Lim ∞ → = can be regarded as the power spectral density or power spectrum. Again, this is in the sense that ( 29 ∫ 2 1 1 f f df f P = Power of ( 29 t x 1 from 1 f to 2 f Hz, and integrating the power spectrum over all frequencies gives the total power. Obviously, the power spectrum ( 29 ( 29 T f E f P T 1 1 Lim ∞ → = is proportional to the energy spectrum ( 29 f E 1 and both are proportional to the square of the magnitude spectrum ( 29 2 1 f X . In most applications, it is the shape of the spectrum that is important, and from this point of view, there is really very little difference between the energy and power spectra. Both will have the same shape except for a scaling factor....
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This note was uploaded on 08/02/2009 for the course ECE EE2009 taught by Professor Prof.c.c.ko during the Fall '07 term at National University of Singapore.

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_Signal_FAQ06_Energy_&_Power_Spectrum - , where ( 29 (...

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