Nonimaging optics

2014-01-29T09:11:56Z

131.155.69.232: /* Theory */

{{Unreferenced|date=December 2006}}

In [[signal processing]], the '''energy''' <math>E_s</math> of a continuous-time signal ''x''(''t'') is defined as

:<math>E_{s} \ \ = \ \ \langle x(t), x(t)\rangle \ \ = \int_{-\infty}^{\infty}{|x(t)|^2}dt</math>

Energy in this context is not, strictly speaking, the same as the conventional notion of [[energy]] in [[physics]] and the other sciences. The two concepts are, however, closely related, and it is possible to convert from one to the other:
:<math>E = {E_s \over Z} = { 1 \over Z } \int_{-\infty}^{\infty}{|x(t)|^2}dt </math>
:where ''Z'' represents the magnitude, in appropriate units of measure, of the load driven by the signal.

For example, if ''x''(''t'') represents the [[electric potential|potential]] (in [[volt]]s) of an electrical signal propagating across a transmission line, then ''Z'' would represent the [[characteristic impedance]] (in [[ohm]]s) of the transmission line. The units of measure for the signal energy <math>E_s</math> would appear as volt2·seconds, which is ''not'' dimensionally correct for energy in the sense of the physical sciences. After dividing <math>E_s</math> by ''Z'', however, the dimensions of ''E'' would become volt2·seconds per ohm, which is equivalent to [[joule]]s, the [[SI]] unit for energy as defined in the physical sciences.

==Spectral energy density==

Similarly, the '''[[Spectral density|spectral energy density]]''' of signal x(t) is

:<math>\ E_s(f) = |X(f)|^2 </math>
where ''X''(''f'') is the [[Fourier transform]] of ''x''(''t'').

For example, if ''x''(''t'') represents the magnitude of the [[electric field]] component (in [[volts]] per meter) of an optical signal propagating through [[free space]], then the dimensions of ''X''(''f'') would become volt·seconds per meter and <math>E_s(f)</math> would represent the signal's spectral energy density (in volts2·second2 per meter2) as a function of frequency ''f'' (in [[hertz]]). Again, these units of measure are not dimensionally correct in the true sense of energy density as defined in physics. Dividing <math>E_s(f)</math> 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 in [[SI]] 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.

==See also==
* [[Signal processing]]
* [[Parseval's theorem]]
* [[Inner product]]

[[Category:Signal processing]]

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Nonimaging optics