# Energy (signal processing)

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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

- Unit of will be (unit of signal)
^{2}. second.

And the **energy** of a discrete-time signal *x*(*n*) is defined mathematically as

## Contents

## Relationship to energy in physics[edit]

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) of the transmission line; the units of measure for the signal energy would appear as volt^{2}·seconds, which is *not* dimensionally correct for energy in the sense of the physical sciences. After dividing by *Z*, however, the dimensions of *E* would become volt^{2}·seconds per ohm, which is equivalent to joules, the SI unit for energy as defined in the physical sciences.

## Spectral energy density[edit]

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 become volt·seconds per meter and would represent the signal's spectral energy density (in volts^{2}·second^{2} per meter^{2}) 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 by *Z*_{o}, the characteristic impedance of free space (in ohms), the dimensions become joule-seconds per meter^{2} or, equivalently, joules per meter^{2} per hertz, which is dimensionally correct in SI units for spectral energy density.

## Parseval's theorem[edit]

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.