- Energy (signal processing)
In
signal processing , the energy E_s of a continuous-time signal "x"("t") is defined as:E_{s} = langle x(t), x(t) angle = int_{-infty}^{infty}{|x(t)|^2}dt
energy in this context is not, strictly speaking, the same as the conventional notion of
energy inphysics and the other sciences. The two concepts are, however, closely related, and it is possible to convert from one to the other::E = {E_s over Z} = { 1 over Z } int_{-infty}^{infty}{|x(t)|^2}dt :where "Z" represents the magnitude, in appropriate units of measure, of the load driven by the signal.For example, if "x"("t") represents the potential (in
volt s) of an electrical signal propagating across a transmission line, then "Z" would represent the characteristic impedance (inohm s) of the transmission line. The units of measure for the signal energy E_s would appear as volt2-seconds, which is "not" dimensionally correct for energy in the sense of the physical sciences. After dividing E_s by "Z", however, the dimensions of "E" would become volt2-seconds per ohm, which is equivalent tojoule s, theSI unit for energy as defined in the physical sciences.pectral Energy Density
Similarly, the spectral energy density of signal x(t) is
:E_s(f) = |X(f)|^2 where "X"("f") is the
Fourier transform of "x"("t").For example, if "x"("t") represents the magnitude of the
electric field component (involts permeter ) of an optical signal propagating throughfree space , then the dimensions of "X"("f") would become volt-seconds per meter and E_s(f) would represent the signal's spectral energy density (in volts2-second2 per meter2) as a function of frequency "f" (inhertz ). Again, these units of measure are not dimensionally correct in the true sense of energy density as defined in physics. Dividing E_s(f) by "Z"o, the characteristic impedance of free space (in ohms), the dimensions become joule-seconds per meter2 or, equivalently, joules per meter2 per hertz, which is dimensionally correct inSI units for spectral energy density.Parseval's Theorem
As a consequence of
Parseval's theorem , one can prove that the signal energy is always equal to the summation across all frequency components of the signal's spectral energy density.ee also
*
Signal processing
*Parseval's theorem
*Inner product
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