Energy (signal processing)

In signal processing, the energy ${\displaystyle E_{s}}$ of a continuous-time signal x(t) is defined as the area under the squared magnitude of the considered signal i.e., mathematically

${\displaystyle E_{s}\ \ =\ \ \langle x(t),x(t)\rangle \ \ =\int _{-\infty }^{\infty }{|x(t)|^{2}}dt}$¹²

The units of ${\displaystyle E_{s}\,}$will be ${\displaystyle \left(\left[{\text{units of}}\ x(t)\right]^{2}\cdot {\text{s}}\right)}$.

And the energy ${\displaystyle E_{s}}$ of a discrete-time signal x(n) is defined mathematically as

${\displaystyle E_{s}\ \ =\ \ \langle x(n),x(n)\rangle \ \ =\sum _{n=-\infty }^{\infty }{|x(n)|^{2}}}$

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:

${\displaystyle E={E_{s} \over Z}={1 \over Z}\int _{-\infty }^{\infty }{|x(t)|^{2}}dt}$

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 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 ${\displaystyle E_{s}}$ would appear as volt²·seconds, which is not dimensionally correct for energy in the sense of the physical sciences. After dividing ${\displaystyle E_{s}}$ by Z, however, the dimensions of E would become volt²·seconds per ohm,

${\displaystyle {\frac {\rm {{V}^{2}}}{\rm {\Omega }}}{\rm {{s}={\rm {{W}{\rm {{s}={\rm {J}}}}}}}}}$

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

${\displaystyle \ E_{s}(f)=|X(f)|^{2}}$

where X(f) is the Fourier transform of x(t).

For example, if x(t) represents the magnitude of the electric field component (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 ${\displaystyle E_{s}(f)}$ 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 ${\displaystyle E_{s}(f)}$ 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 in SI units for spectral energy density.

As a consequence of Parseval's theorem³, one can prove that the signal energy is always equal to the summation across all frequency components of the signal's spectral energy density.⁴⁵