What, mathematically, is the power spectrum of a signal? Given a signal $f(t)$ defined on $t\in(-\infty,\infty),$ what is the precise definition of the power spectrum of $f$, i.e., what is the mathematical operation that takes $f$ to the output of an ideal spectrometer applied to $f$?
Wikipedia defines the energy spectral density as
$$S(\omega)=|\hat{f}(\omega)|^2,$$
where $\hat{f}$ is the Fourier transform of $f$. But this cannot be what a spectrometer outputs, for at least two reasons:

*

*The Fourier transform (and hence $S$) are defined on negative frequencies, while spectrometers only output values at positive frequencies.

*The Fourier transform does not change over time, while the output of a spectrometer certainly changes over time.

I am guessing there must be characteristic timescales involved in the workings of the spectrometer, which must be taken into account.
Please note that I am looking for the mathematical description of how spectrometers generate an output from a given input, not the physical workings of the spectrometer.
 A: 

*

*The Fourier transform (and hence $S$) are defined on negative frequencies, while spectrometers only output values at positive frequencies.


For any real-valued signal, $f(\omega) = f(-\omega)^*$, which implies that the negative frequencies don't carry any information not already in the positive frequencies. Therefore, whether you say the domain is all frequencies or only positive frequencies is just a detail, with no real implications. (The power spectral density $S(\omega)$ typically only includes positive frequencies, but whether a factor of $2$ appears in the definition to compensate for the lack of negative frequencies is convention dependent. If you're unsure, consult the user manual for your spectrometer.)



*The Fourier transform does not change over time, while the output of a spectrometer certainly changes over time.


Roughly speaking, a spectrum analyzer looks at the output over a finite time window $\tau$ (which you can set by pushing appropriate buttons, see your user manual), and calculates the Fourier transform of that finite time series.
A "true" power spectral density (i.e. what the mathematicians use) is defined by taking an ensemble average, or equivalently by integrating over an infinite time. A spectrum analyzer produces an estimate of this power spectral density by integrating over only a finite time. This introduces statistical fluctuations into each bin, and also smears all features over a frequency width $\Delta \omega \sim 1 / \tau$ by the uncertainty principle.
Of course, in reality every specific instrument has other limitations, which can only be understood with physics. Frankly, it seems to me that your quest to understand physical phenomena while forbidding mention of the "physical workings" of any object involved is deeply misguided and will make everything unnecessarily difficult.
A: Power spectrum density is simply power of a signal per unit frequency. İt is often described by power/Hz. The process have power spectral density if:
Discrete time signal is a sequence of random variables with zero mean.
E{y(t)] = 0
y(t) is a second order stationary sequence. Second-order stationarity (also called weak stationarity) time series have a constant mean, variance and an autocovariance that doesn’t change with time. Other statistics in the system are free to change over time. This constrained version of strict stationarity is very common
r(k) is autocovariance sequence
[![enter image description here][1]][1]

*

*The signal must be Stationary process

*The spectrum is always real and non-negative.

*Spectrum of a real valued process is always even functions which is defined Lets build the power spectral density:
1-Autocovariance Sequence

