Wikipedia / Energy (signal processing)
Wikipedia / Energy (signal processing)
In signal processing, the energy of a continuous-time signal x(t) is defined as the area under the squared magnitude of the considered signal i.e., mathematically
.
And the energy of a discrete-time signal x(n) is defined mathematically as
Relationship to energy in physics
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:
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 would appear as volt2·seconds, which is not by Z, however, the dimensions of E would become volt2·seconds per ohm,
which is equivalent to joules, the SI unit for energy as defined in the physical sciences.
Spectral energy density
Similarly, the spectral energy density of signal x(t) is
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 represent the signal's spectral energy density (in volts2·second2 per meter2) as a function of frequency f (in hertz by Zo, 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.