In signal processing, the energy
Es
Es = \langlex(t),x(t)\rangle =
infty | |
\int | |
-infty |
{|x(t)|2}dt
Unit of
Es
And the energy
Es
Es = \langlex(n),x(n)\rangle =
infty | |
\sum | |
n=-infty |
{|x(n)|2}
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:
E={Es\overZ}={1\overZ}
infty | |
\int | |
-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 volts) of an electrical signal propagating across a transmission line, then Z would represent the characteristic impedance (in ohms) of the transmission line. The units of measure for the signal energy
Es
Es
\rm{V | |
2}{\rm{\Omega}} |
\rm{s}=\rm{W}\rm{s}=\rm{J}
which is equivalent to joules, the SI unit for energy as defined in the physical sciences.
Similarly, the spectral energy density of signal x(t) is
Es(f)=|X(f)|2
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
Es(f)
Es(f)
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.