# Do energy of signal and power spectrum (in db/hz) indicate towards same signal property?

If I calculate energy of signal by squaring the magnitude of signal sample and power spectrum using spectrogram (db/hz) what kind of property of signal do they indicate and are they point towards same characterstic. That is higher energy signal will always have higher peak spectral power and vice versa. Also what is there practical significance.

## 2 Answers

On a straight forward approach, energy of a signal given by $$E = \int \limits_{-\infty}^{+\infty}|x^2(t)|dt$$ is a constant number whereas PSD can be obtained from FFT which is a function of frequency. PSD is the square of the absolute value of FFT.

In either case the value of energy or PSD gives an insight of the amplitude/strength of a signal.

if you undo the conversion to dB, so that the power spectrum is in terms of power/Hz rather than dB/Hz (which is not a unit we use for anything), then when you integrate the power spectrum (in energy/Hz) over all frequency (or up to Nyquist for sampled signals), then you have the power of the signal, which, if you get your units lined up, will be equal to the mean-square of the samples of the signal.

• Power spectrum should be in Watts/Hz, so if you integrate it you get power (not energy). What you said is true for the energy spectrum (Joules/Hz), which, if you integrate it, gives you the energy. – Matt L. Sep 7 '18 at 8:11
• watts/Hz only if you know the "gain" coefficients from the numerical sample value to the A/D or D/A converters and know what resistance load it's applied to. and that is if it's a so-called "power signal" rather than an "energy signal". if it's the latter, how do we consider the magnitude-square of the spectrum? – robert bristow-johnson Sep 7 '18 at 8:52
• Matt, power, energy, whatever. i'm just trying to frame it with Parseval's theorem (without mentioning the term). – robert bristow-johnson Sep 7 '18 at 8:54
• Sure, units are always an issue in DSP. But we should be a bit careful when it comes to statements like "integrating the power spectrum gives you the energy", because such statements might get students into trouble when their prof is a bit picky (as s/he should be ...). – Matt L. Sep 7 '18 at 9:00
• It's your answer after all, just trying to be critical in a constructive way. What I would still change is "... power spectrum (in *energy*/Hz)". I might just silently leave out the actual unit of the power spectrum to avoid confusion. – Matt L. Sep 7 '18 at 9:10