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What is the difference between the spectral density and the power spectral density, Or they are the same thing? In fact, I find this term in the book of GODA 2000.

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    $\begingroup$ Spectral Density , alone, means distribution of a quantity over frequency (a.k.a. the spectrum)... Power Spectral Density , therefore, refers to a distribution of power over frequency. $\endgroup$
    – Fat32
    Apr 9, 2022 at 16:26

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In signal processing applications, a Spectral Density can either be a Power Spectral Density or an Energy Spectral Density. When the total energy is finite or concentrated around a specific time interval, an Energy Spectral Density is used; but more typical is to show a Power Spectral Density which assumes the signal exists as given for all time as a stationary signal.

Note that the units need not be traditional "power" units such as Watts; given any complex waveform that exists for all time, squaring it would be referred to as a "power signal", and similarly squaring any complex waveform that converges when integrated over infinite time (has finite energy) would be referred to as an "energy signal".

Examples of this are $S_\phi(f)$ which is the power spectral density due to phase fluctuations vs time $x(t)=\phi(t)$, and $S_f(f)$ which is the power spectral density due to frequency fluctuations vs time $x(t)=\omega(t)$. The resulting power spectral densities $S_\phi(f)$ and $S_f(f)$ for the same waveform would appear very different.

Related to "Power" and "Energy" the following additional items may be helpful, interestingly showing at the end of this how a Power Spectral Density is the energy at each frequency.

First this table summarizes power and energy for both discrete and continuous time signals:

Power and Energy

Power is Energy delivered over a time interval, in other words energy normalized to time.

Consistent with this: Joules is a unit of energy, and Watts (a unit of power) is Joules/sec.

In signal processing we use the terms "Energy Signals" and "Power Signals" with the power term as the square of the complex waveform regardless of actual units of a time domain waveform. We can see the analogy to power from electrical circuits with a normalized resistance where power is defined as:

$$P(t)= \frac{V(t)^2}{R}$$

$$P(t) = I(t)^2 R$$

Here in units of Watts, given current in Amps, voltage in Volts, and resistance in Ohms.

Thus if the resistance is normalized to $R=1$, we get the power in either case. With the units of time included above, we also see intuitively how the general complex conjugate product is "instantaneous" power, given the actual power formulas above provide the power dissipated at any instant in time. This is consistent with "Instantaneous Power" as given in the table above.

Applicable to waveforms in signal processing we have the following related terms: "Energy Signals" and "Power Signals".

Energy Signal: A waveform that is non-zero for a finite duration. This is because such a waveform has finite total energy.

Power Signal: A waveform that extends with non-zero values to infinity, but converges when normalized to time. Such a waveform has infinite energy, but for the case that it's average power is bounded, it is referred to as a Power Signal. A sinewave extending for all time is an example of a power signal.

What gets really interesting is when we go to the frequency domain and specifically evaluate what the "Power Spectral Density" is in continuous time. Given a time domain power signal, such as white Gaussian noise with a total power dissipation (energy / time) of 1 Watt, given Parseval's theorem if we could integrate over the total white noise distribution in frequency, the result would be 1 Watt. This is an impossible condition and the power over any finite portion of that spectrum would be zero. Thankfully consistent with all physical processes, true white noise does not exist and there will always be an upper bandwidth limit. Thus when dealing with a "white-noise" process in the continuous time domain, the bandwidth of the measurement system must be specified. In that case, the 1W in the time domain dissipated as energy vs time, will be a measurable power spectral density in frequency with that 1W distributed evenly (as "white noise") out to the measurement bandwidth, resulting in a spectrum given in units of W/Hz. Watts/Hz is Power x Time which is an Energy quantity. Every value given in the frequency domain for the power spectral density is energy, and if we integrate the total energy in frequency, we get Power! So in the time domain we have Power as Energy over Time in seconds. In the frequency domain we have Energy as Power over bandwidth in Hz. The sum of the energies over frequency is total power, while the sum of the powers over time is total energy.

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  • $\begingroup$ For $N$ points, $P[n]$ is in Watts and is both physical and discrete instantaneous power, $E_\text{physical} \approx E_\text{discrete} \cdot T_\text{sampling}$ is in Joules, and $\bar{P}_\text{discrete} \approx \bar{P}_\text{physical}$ equals $E_\text{discrete} / N$ and is in Watts, agreed? The "approx" is due to stuff I discuss here (which I'm currently fixing) under "but it's inexact" $\endgroup$ Mar 5, 2023 at 14:04
  • $\begingroup$ $P[n]$ can only be in units of Watts if the sampling rate is 1 Hz and $x[n]$ is in units of amps or volts and $R=1$, or any other combination of units that truly leads to Joules/sec. Point is units need to be carefully specified, right? $\endgroup$ Mar 5, 2023 at 14:22
  • $\begingroup$ Ok so how to get $P_\text{physical}$ in Watts from $P_\text{discrete}[n]$ if $x[n]$ is volts, $R=1$, but $f_s$ is any? $\endgroup$ Mar 5, 2023 at 14:27
  • $\begingroup$ If $P(t)$ is instantaneous power, that holds pointwise, hence $P[n]$ is also instantaneous power, for all $f_s$? Doubling $f_s$ provides double the points, but each point's power is same $\endgroup$ Mar 5, 2023 at 14:34
  • $\begingroup$ @OverLordGoldDragon It depends on the units of the horizontal axis. When the sampling ratee is 1 Hz, then samples and seconds are equivalent, otherwise you need to scale by time to make them equivalent (just as in a discrete time integration or differentiation which is what you are doing). $\endgroup$ Mar 5, 2023 at 14:49

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