Talk:Spectral density

message: What is the definition of the capital F function used in the definition of the cross spectral density? I don't think it's the F defined just above it which is the integrated spectrum. 208.69.128.41 (talk) 22:48, 26 January 2015 (UTC)
thread_title:  Cross spectral density definition 

message: Physics is not the only field in which this concept appears, although I am not ready to write an article on its use in mathematics or statistics.  Is any other Wikipedian? Michael Hardy 20:11 Mar 28, 2003 (UTC)
:Yes, I am.  All the fields in which it appears are using the methods of statistical analysis.  So the concept is fundamentally statistical.  The different applications could be preceded by some fundamental sections in which the statistical concepts are explained, using only the signal processing case as an example, or using only a statistical time series example as an example.66.167.204.242 (talk) 07:13, 18 April 2014 (UTC)
thread_title: Other fields

message: I removed the line defining the SPD as the FT of the autocorrelation. The relationship to the autocorrelation is in the "properties" section. As it stands now, the SPD is defined as the absolute value of the FT of the signal, and the relationship to the autocorrelation follows. We could use either as a definition, and the other as a derived property,
but I think the present set-up is best, because it goes directly to the fact that the SPD is a measure of the distribution among frequencies. If we start by defining it as the FT of the autocorrelation, we have to introduce the autocorrelation which may be confusing, then define the SPD, then show that its the square of the FT of the signal, in order to show that its a measure of distribution among frequencies. This way is better. PAR 19:44, 21 July 2005 (UTC)
replies: text: I disagree. The definition you removed was the correct and only sensible definition of PSD. What you put instead I have relabeled as an energy spectral density, and I've put back a definition of PSD in terms of autocorrelation function of a stationary random process. I think I need to go further, making that definition primary, since that's what the article is supposed to be about. Comments? Dicklyon 19:47, 13 July 2006 (UTC)
replies: text: It is common and mathematically well-founded to DEFINE the Power Specral Density (PSD, and where on Earth did "SPD" come from?) as the Fourier Transform of the autocorrelation function. All of the others, such as the ones given in the article now, are simply mathematical Hand Waving with no foundation in the fact. I despise Hand Waving because it is more confusing that it is anything else.98.81.0.222 (talk) 22:18, 8 July 2012 (UTC)

text: The literature and texts vary on whether the FT of the auto-correlation function is the definition of the power spectral density, 
or the result of a *theorem* because they have already defined the power spectrum.  It is untrue to say that only one of these
alternatives can be mathematically rigorous.  This article has a definition of power spectrum which is not far from being 
rigorous, although it is not very physically motivated either.  The idea of defining the power spectrum by the distribution
function appearing in the Stieltjes integral that gives the spectral decomposition of the auto-correlation function is perfectly
rigorous even for auto-correlation functions that are not square-integrable or absolutely integrable, is the definition used by 
e.g. Chatfield in his undergrad text---without proofs, and it does require a bit of work to prove, which is why it deserves to
be called a theorem. I myself disapprove of using the Wiener--Khintchine formula as a definition since it obscures the physical
content98.109.227.161 (talk) 00:38, 8 November 2012 (UTC)
thread_title: Change Definition

message: It is nice to define the spectral density
by the infinite time Fourier integral of the signal.
However, often one has a signal s(t) defined for 0<t<T only.
One can then define S(w) at the Frequencies w_n=2 pi n/T
as the modulus-squared of the respective Fourier coefficients normalized by 1/T.
The limit T->infinity recovers the infinite time Fourier integral definition.
If found the normalization by 1/T especially tricky. However it is needed to get a
constant power spectrum for white noise independent of T.
This reflects somehow decay of Fourier modes of white noise due to phase diffusion.
reference Fred Rieken, ... : Spikes: Exploring the Neural Code.
:I'm not sure I understand your point.  First of all, a Fourier series (discrete frequencies of a finite or periodic signal) is not a spectral density.  More of a periodogram.  For stationary random process, where PSD is the right concept, periodogram does not approach a limit as T goes to infinity.  And for a signal only defined on 0 to T, taking such a limit makes no sense.
:I just noticed the comment above yours, where in July 2005 the correct definition of PSD based on Fourier transform of autocorrelation function was removed.  That explains why I had to put it back early this year.  I think I didn't go far enough in fixing the article, though.  Dicklyon 19:44, 13 July 2006 (UTC)
replies: text: I take back what I said about the limit as T goes to infinity.  As long as you have the modulus squared inside the limit, it will exist as you said, assuming the signal is defined for all time and is a sample function of a weak-sense-stationary ergodic random process; that is, the time average of the square converges on the expected value of the square, i.e. the variance.  Blackman and Tukey use something like that definition in their book The Measurement of Power Spectra. Dicklyon 05:23, 14 July 2006 (UTC)
thread_title:  finite time intervall 

message: Hi!
I may be wrong, but the letter phi is used for definition of both continuous (Eq. 1) and discrete Fourier transforms (Eq. 2), which is confusing since phi doesn't have the same dimensions (units) from one to the other. Let's say f(t) is the displacement of a mass attached to a spring, then f(t) is in meters and phi(omega) will be in meters squared times seconds squared in the continous definition, and phi(omega) will be in meters squared in the discrete definition.
Using the same letter (phi) for two quantities that do not have the same meaning seems unappropriate. Would it be possible to add a small sentence saying that units differ in the continuous and discrete transforms?
What do you people think of this suggestion?
142.169.53.185 12:11, 26 March 2007 (UTC)
replies: text: Yes, that makes sense.  I'll add something. Dicklyon 19:24, 26 March 2007 (UTC)
thread_title:  Units of Phi(omega) for continuous and discrete transforms 

message: The concept of a spectral mass function appears in quantum field theory, but that isn't mentioned here.  I'm don't feel confident enough in my knowledge to write a summary of it, however.  --Starwed 13:44, 12 September 2007 (UTC)
thread_title:  QFT 

message: The difference between power spectral density and energy spectral density is very unclear. Take a look at Signals Sounds and Sensations from William Hartmann, chapter 14. You can find this book almost entirely at http://books.google.com/.
It is not really my field unfortunately, so I would rather not change the text myself.  —Preceding unsigned comment added by 62.131.137.4 (talk) 13:52, 29 December 2007 (UTC)
replies: text: Hartmann talks about power spectral density, as is typical in the sound field, since they're speaking of signals that are presumed to be stationary, that is, with same statistics at all times.  Such signals do not have a finite energy, but do have a finite energy per time, or power.  The alterative, energy spectral density, applies to signals with finite energy, that is, things you can take a Fourier transform of.  I tried to make this clear in the article a while back, but maybe it needs some help. Dicklyon (talk) 16:05, 29 December 2007 (UTC)
thread_title:  Difference between power spectral density and energy spectral density  

message: If I am not mistaken, spectral analyzers (SAs) do not always measure the magnitude of the short-time Fourier transform (STFT) of an input signal, as suggested by the text. Indeed, I understand that (as indicated by the link to SAs) this kind of measurement is not performed by analog SAs. Digital SAs do perform some kind of Fourier Transform on the input signal but then the spectrum becomes susceptible to aliasing. This could be discussed in the text. 
What do you guys know about it? Anyway, I don't feel confident enough for changing the text.  —Preceding unsigned comment added by Abbade (talk • contribs) 05:02, 6 January 2008 (UTC)
replies: text: Analog spectrum analyzers use a heterodyne/filter technique, sort of like an AM radio.  The result is not so much different from using an FFT of a windowed segment; both give you an estimate of spectral density, with ways to control bandwidth, resolution, and leakage; and each way can be re-expressed, at least approximately, in terms of the other. Read all about it. Dicklyon (talk) 05:31, 6 January 2008 (UTC)
thread_title:  Spectral Analyzer 

message: Some sources call F(w) the spectral density and Phi(w) the spectral intensity (cf. Palmer & Rogalski). Is this a British vs. American thing or did I misunderstand?--Adoniscik (talk) 20:28, 23 January 2008 (UTC)
replies: text: Neither the math nor the terminology in that section connects to anything I can understand.  Do those Fourier-like integrals make any sense to you?  How did they get them to be one-sided?  Anyway, I'd be surprised to find that usage of spectral density in other places; let us know what you find.  In general, it appears to me that "density" means on a per-frequency basis, while "intensity" can mean just about anything.   I don't see any other sources that would present a non-squared Fourier spectrum as a "density"; it doesn't make sense.  Dicklyon (talk) 23:30, 23 January 2008 (UTC)

text: I didn't dwell on it, but the one-sidedness of the Fourier transform follows because it assumes the source to be real (opening paragraph, second sentence, and also after equation 20.6). Im[f(t)]=0 implies F(w)=F(-w) (Fourier transform#Functional relationships). I'm thinking the density here is akin to the density in "probability density function" … aka the "probability distribution function". See also Special:Whatlinkshere/Spectral_intensity… --Adoniscik (talk) 00:53, 24 January 2008 (UTC)
replies: text: No, there's no such relationship, and nothing that looks like that on the page you cite.  Read it again.  Now if he had said the wave was even symmetric about zero, that would be different; but he didn't.  If he had said the power spectrum was symmetric about zero, that would be OK; but didn't, he implied the Fourier amplitude spectrum is symmetric, and it's not, due to phase effects (it's Hermitian).  Your interpretation of "density" is correct; that's why it has to be in an additive domain, such as power. Dicklyon (talk) 01:26, 24 January 2008 (UTC)
thread_title:  Spectral intensity

message: Please vote at Talk:Eb/N0#Survey on which unit that should be used at Wikipedia for measuring  Spectral efficiency. For a background discussion, see Talk:Spectral_efficiency#Bit/s/Hz and Talk:Eb/N0#Bit/s/Hz.
Mange01 (talk) 07:21, 16 April 2008 (UTC)
thread_title:  Survey: bit/s/Hz, (bit/s)/Hz or bit·s<sup>−1</sup>·Hz<sup>−1</sup> as [[Spectral efficiency]] unit? 

message: There is another wiki page on signal energy. Most of its content is better explained in the spectral density page. As far as appropriate, the two pages should be merged. I suggest keeping only the explanation of why energy of a signal is called energy in the other page. I found this page navigating from energy disambigation where I created a link to the spectral density entry.Sigmout (talk) 09:16, 18 August 2008 (UTC)
thread_title:  Energy of a signal - doubled wiki entry 

message: Is there any difference in the terms 'power spectral density', 'power spectrum', 'power density spectrum', 'spectral power distribution', and similarly for energy? Also, 'spectral density' and 'spectrum level' ? I'm looking through some of my textbooks and I don't see any attempt to distinguish between these terms, so I assume there are simply two concepts, the power spectrum and the energy spectrum. Also, doesn't the definition of the power spectrum require a scale factor to ensure that the total signal power is equal to the integral of the power spectrum (to account for the arbitrary scale factor in front of the Fourier transform)?. Since several different definitions of the Fourier transform are commonly used, wouldn't it be appropriate to elaborate on how these definitions depend on each other? 146.6.200.213 (talk) 17:27, 11 May 2009 (UTC)
replies: text: The statement at the end of the "energy spectral density" section notes that the definition depends on the scale factor, but it does not specify how exactly. The page on Fourier transforms might elaborate on this; its hard to say because that page is longer than it should be. It is also one of many Wikipedia pages where intuitive explanations have been replaced with abstract mathematics definitions (more than would be necessary to ensure the article is strictly correct) 146.6.200.213 (talk) 19:09, 11 May 2009 (UTC)
replies: text: An encyclopedia has to make correct statements and refrain from making incorrect ones.  Many times, this looks like unnecessarily abstract mathematics, but in this case, it is the minimum necessary to avoid the howlers that pervade the first chapters of so many engineering textbooks on the subject.  A lecture can focus on intuitive explanations that are full of "little white lies", but not an encyclopedia article. The current state of this article is very bad in this respect, for example talking about the expectation of x(t) when x is deterministic and X is the stochastic process of which x is one sample.66.167.204.242 (talk) 06:27, 18 April 2014 (UTC)
thread_title:  terminology, scaling 

message: In the section on power spectral density, the discussion is confusing, in part because of bad notation. At one point s(t)**2 is called 'power' at at another point S(f) (without a square) is called 'power'. For someone trying to understand, this confuses things. One might expect that S(f) = F(s(t)), where 'F' denotes a Fourier transform, etc.  —Preceding unsigned comment added by 136.177.20.13 (talk) 15:09, 11 October 2010 (UTC)
replies: text: True.  Any expectations that the case shift signifies a relationship between the two is not correct here.  Otherwise, it looks rihgt.  See if you can find a better or more conventional set of names. Dicklyon (talk) 05:53, 12 October 2010 (UTC)
thread_title:  s and S 

message: The article says "The power spectral density of a signal exists if and only if the signal is a wide-sense stationary process." Is this correct?  I would think that it also exists if the signal is wide-sense cyclostationary.  Lavaka (talk) 00:31, 31 January 2011 (UTC)
replies: text: I think you're right; this book agrees.  I'll take out "and only if".  Dicklyon (talk) 04:20, 31 January 2011 (UTC)

text: This is one of the many inaccuracies in the article.  The integrated spectrum of a signal exists if the signal comes from a stationary process almost surely, but there is no «only if».  But even so, the spectral density need not exist, since the integrated spectrum might not be absolutely continuous.  If the stationary process is also purely indeterministic, then the spectral density will exist almost everywhere, but even then might not be differentiable itself (it is the integrated spectrum which is then differentiable almost everywhere, and its derivative gives the spectral density). I fear that a cyclostationary process will not have a continuous integrated spectrum, and so its spectral density will not exist, and I would rather omit the topic of cyclostationarity from this article altogether, as being too specialised and too advanced. 98.109.227.161 (talk) 15:49, 9 November 2012 (UTC)
thread_title:  stationary vs cyclostationary 

message: I have added a section to motivate the concepts and theorems: the example of a finite sum of sines and cosines.  No complex numbers, no negative frequencies, no difficult convergence issues.  
But even this simple example is an example of where a commonly offered definition of the power spectrum breaks down: take f(t) to be just simply cosine t.  The average power is finite.  The truncated Fourier transform, normalised as one of the editors has suggested here,
is always zero or infinity.... that is not very useful.  This is why mathematically careful writers define the spectral distribution function first, and then, *if* it is differentiable, define the spectral density, as is usual in probability and statistics.  
I will make the notation consistent throughout the article, later, consistent with standard time-series and signal processing texts.  (Not all are completely consistent with each other, but those square brackets have to go... 98.109.240.7 (talk) 02:47, 13 November 2012 (UTC)
replies: text: It's kind of long, and doesn't end up with something that has a spectral density, as you note.  So maybe it's not the best way to motivate this stuff?  I did some copyedits for WP style (WP:MOS).  Dicklyon (talk) 03:04, 13 November 2012 (UTC)
replies: text: Well, the example of the autocorrelation function could be left out if people thought it was not worth the space.  One important point is that the definitions of power density given later don't make sense for the general wide-sense stationary process.  For example, the comment that the expectation (ensemble average) of the truncated Fourier transform will have a limit even if the typical sample path, or signal, does not, is just a mistake, as seen by the simple example of  \cos (t + p)  where p is a random variable uniformly distributed on the interval from zero to  2\pi.  So it is crucial to talk about the power spectrum even when the density is always zero or a delta function.  That is why careful writers first define the integrated spectrum, or spectral distribution function, first. 98.109.240.7 (talk) 03:33, 13 November 2012 (UTC)

text: It would also be shorter if the summation notation were removed and one simply did the example of sin(t).98.109.240.7 (talk) 13:15, 13 November 2012 (UTC)
thread_title:  New section, simple example for motivation 

message: using phi was just my habit, Wiener always uses it.  Now I have been consulting the few (less than a dozen) time-series of signal processing texts which I happen to have with me to pick notation.  I am thinking as follows: as always in stats, caps for random variables, lower case for the sample.  greek for theoretical statistics, roman for the sample statistic.  So, the stochastic process will be  X_t and the signal will be  x(t).  Frequency will be f, angular frequency omega.  Discrete time will be n, sampled signal will be  x_n.  Correlation will be rho for theoretical, r for sample.  Covariance will be gamma for theoretical, c for sample.  But as usual, what most statisticians call the autocovariance function will be called the autocorrelation function in accordance with Wiener and many signal processing engineers, so it will be c.  When cross-correlations are involved, r and c will be supplied with subscripts, but not in the basic discussion of the univariate case.  STochastic integrals and processes with orthogonal increments will only be mentioned at the tail end, as dessert...as is usual in statistics or engineering texts.  So far, the consensus on this page seems to be not to bring up Laurent Schwartz's theory of distributions as a key building block in the definitions.  I actually don't know of a reliabel source for the off-hand statement often made, that that theory makes the use of all these non-convergent integrals rigorous....  and such a reliance on Schwartz is not normal in careful statistics texts.  Some mention should be made of it at the tail end.  98.109.240.7 (talk) 17:26, 13 November 2012 (UTC)
thread_title:  notational consistency 

message: For real?  This is an relatively long article on PSD and there is no sample graph?!
—DIV (137.111.13.36 (talk) 23:29, 4 December 2013 (UTC)
replies: text: Thank you for your suggestion. When you believe an article needs improvement, please feel free to make those changes.  Wikipedia is a wiki, so anyone can edit almost any article by simply following the  link at the top.</br> The Wikipedia community encourages you to be bold in updating pages. Don't worry too much about making honest mistakes—they're likely to be found and corrected quickly. If you're not sure how editing works, check out how to edit a page, or use the sandbox to try out your editing skills.  New contributors are always welcome. You don't even need to log in (although there are many reasons why you might want to). Zueignung (talk) 04:18, 5 December 2013 (UTC)
thread_title:  Illustrative image(s) 

message: The concept of spectrum is more fundamental than that of spectral density. For the same reasons, the concept of power spectrum is more fundamental than that of power spectral density. When analysing real data, one first looks at the power spectrum as a whole. Then filters out the discontinuous part. Only the remainder, the continuous part, has a power spectral density. So it is a pity that the title "Power Spectrum" redirects to "spectral density". I think that including the word "density" in this title is disorienting to the whole article, and its baleful influence disorients both the readers and the writers.66.167.204.242 (talk) 07:13, 18 April 2014 (UTC)
thread_title: Title

message: The definition.
The formula given for the definition is nonsensical, i.e., it makes no sense.  It is
:" S_{xx}(\omega) = \lim_{T \rightarrow \infty} \mathbf{E} \left[ | \hat{x}_T(\omega) | ^ 2 \right]. 
:"Here E denotes the expected value; explicitly, we have
: \mathbf{E} \left[ | \hat{x}_T(\omega) |^2 \right] = \mathbf{E} \left[ \frac{1}{T} \int\limits_0^T x^*(t) e^{i\omega t}\, dt \int\limits_0^T x(t') e^{-i\omega t'}\, dt' \right] = \frac{1}{T} \int\limits_0^T \int\limits_0^T \mathbf{E}\left[x^*(t) x(t')\right] e^{i\omega (t-t')}\, dt\, dt'."
But in the notation of this article, x is a deterministic signal, so it does not have a probabilistic expectation, so the formula is nonsensical.
But even if this were fixed, the formula would be a mistake.  Because the process is wide-sense stationary, its expectation is a constant.  And even if this were fixed, one would run into convergence problems.66.167.204.242 (talk) 07:44, 18 April 2014 (UTC)
replies: text: I totally agree that this needs to be changed. I got so confused because S_{xx}(\omega) is defined as both the energy spectral density AND the power spectral density in the article. Please someone fix this!Nscozzaro (talk) 05:48, 21 September 2016 (UTC)
thread_title: The most serious mistake at present

message: The Amazon user reviews of an introductory text in one of the Wiley series points out mathematical "howlers" in the foundational chapter on Fourier analysis.  Yet this is a printed text, published by a reputable publisher.  
http://www.amazon.com/Principles-Random-Signal-Analysis-Design/dp/0471226173
Ah ha...but reputable in what precise sense? This is an engineering series.  Engineers are professionally trained, and the reviewers for an engineering series are known for, getting the right answers and making things that work.  They are not professionally trained to make mathematical statements that are not mistaken.  The foundational chapter in question, then, does not fall within the sphere of the good repute of this series of texts, nor of its authors, nor of its reviewers.  The conclusion, fortified by my comparison of how statistics textbooks treat this topic with how engineering textbooks treat this topic, is that an engineering text is not a reliable source for any mathematical statement on this topic.  
So I then looked at my little brother's old textbook on this topic, written by the head of AT&T's French division of research.  He makes the mistake of saying that the operational properties of the Fourier transform, of taking products into convolutions and vice versa, are still true when the Fourier transform in the sense of distribution is used.  But everyone knows that you cannot multiply together two distributions!  Digital Processing of Signals, M. Bellanger, Wiley, 1984. 66.167.204.242 (talk) 17:56, 18 April 2014 (UTC)
thread_title: Another two examples, this time published

message: The Wikipedia article on the Fourier transform puts the 2 pi in the exponent, so 
that the argument is frequency in cycles per secon, i.e., Hertz.  For consistency, we 
should do the same, and not use angular frequency.  In general, many mathematicians, such
as Wiener, and electrical engineering texts do use angular frequency, and then the 
2 pi has to be put into the definition of the transform or else the spectral synthesis.
But then the units aren't quite Hz, so that is just another reason for being consistent
with the other Wikipedia article and not do it in terms of radians, but frequency.
Furthermore...most statisticians prefer to use real frequencies and thus put the 2 pi
in the exponent or inside the argument of the sine and cosine.  Chatfield,
for example.  Also, among signal processing texts, the classic Blackman and Tukey do so, 
as does Laurent Schwartz in his book on mathematics for Physics.  There are many 
computational and conceptual advantages, pointed out by such authors.  
Furthermore, since all data are real, and all measurements are real, many statisticians
prefer not to work with complex exponentials either.  This then has the advantage that 
all frequencies are positive, and the formulas for the power contributed by frequencies
between a and b are simpler.  The concept of a negative frequency is not used in Statistics.
Complex exponentials have advantages when dealing with proofs and the operational properties
of the Fourier transform, and are useful in electrical engineering in order to deal with
phases and phase shifts.  But this article will omit nearly all proofs, and the power 
spectrum ignores all phase relations, so a text such as Chatfield does not use complex
exponentials.68.164.80.96 (talk) 06:22, 3 May 2014 (UTC)
thread_title:  Constant normalising factors in definition of fourier Transform 

message: Signal frequency spectrum here: "distribution over the frequency domain" is synonym with spectral distribution. Fgnievinski (talk) 04:29, 27 June 2015 (UTC)
replies: text: I disagree.  There are many ways to measure a spectral density, but the signal frequency spectrum is measured [obtained] by measuring a time or space domain signal and then computing the Fourier transform.  The signal frequency spectrum is a special case of a much more general idea. Rather than combine the two articles, the spectral density article needs to be fixed so that it is clear to readers that the spectral density represents any effective quantifying [quantification] of the density of some response (output) as a function of frequency or wavelength input.  Fourier1789 (talk) 17:39, 3 July 2015 (UTC)
replies: text:  I hesitate to disagree with Baron Jean Baptiste Joseph Fourier himself, but I don't think signal frequency spectrum necessarily implies the application of a particular spectral density estimation method, your highness; could you provide any sources documenting the distinction you make? I do agree with either (i)  clarifying spectral density so as to remove any assumptions that a time series is a pre-requisite or (ii) rename the article to signal spectral density and refer the reader to Spectrum#Physical science for energy spectrum, mass spectrum, etc.; which one would you prefer? Fgnievinski (talk) 19:11, 3 July 2015 (UTC)
replies: text: The "signal" in signal frequency spectrum implies a subset of signal processing.  In contrast, if you do a google scholar search for things like spectral density in astronomy, spectral density in chemistry, and spectral density in atomic physics, you will find lots of scholarly uses for spectral density unrelated to signal processing or taking a Fourier transform of time series data.  I think the better approach is to keep separate articles, because "signal frequency spectrum" is sufficiently notable to have a separate article independent of the more general "spectral density" article.  I propose the spectral density article be made more general to include other areas as outlined above.Fourier1789 (talk) 20:22, 3 July 2015 (UTC)
replies: text:  So let's create a new spectral density (physical sciences), move the current spectral density to spectral density (signal processing), and leave a disambiguation page in spectral density. Then signal frequency spectrum can be merged into spectral density (signal processing). Fgnievinski (talk) 20:35, 3 July 2015 (UTC)
thread_title:  merge 

message: The page doesn't seem to have a clear explanation on what a spectral density actually is. 
This seems quite... unintuitive? Why have an article for spectral density as a concept but not have the explanation. Reading a few resources has left me with around 20 ideas of what it should be with nothing quite definitive as a concept. Would anyone with a better understanding of the concept be able to provide an explanation in the definition or explanation?  — Preceding unsigned comment added by 65.39.42.160 (talk) 02:18, 19 November 2018 (UTC)
thread_title:  Explanation/Definition 

message: The opening thesis fails logic and learning.
"According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies"
Fourier neither allows or dis-allows any of the above, ie categorizations or partitioning style.
It would be like saying, "addition allows groups of things to be added together"  — Preceding unsigned comment added by 2601:143:400:547B:B5C2:D763:11F8:A71D (talk) 18:45, 28 April 2019 (UTC)
thread_title:  total hog wash 

message: I often see mentioned that the PSD of complex signals are not necessarily symmetric. So I don't know why this statement is here
"The spectrum of a real valued process (or even a complex process using the above definition[dubious – discuss]) is real and an even function of frequency:"
174.62.124.25 (talk) 01:59, 12 December 2020 (UTC)
thread_title:  Symmetry of complex PSD 

message: I don't know which came first, Spectral_density#Energy_spectral_density or Spectral_density#Properties, but now they are redundant.  If there is any new information, e.g. the name Wiener–Khinchin theorem, it could probably be incorporated into Energy_spectral_density.  Another problem with Properties, as currently written, is the mixture of g(t) and R_{xx}(0).  g(t) should be changed to x(t) for consistency with the other sections.
--Bob K (talk) 13:36, 30 December 2020 (UTC)
thread_title:  section "Properties" is redundant 

message: Just below Eq.2, there is an unnumbered equation, whose first part is:
: |\hat{x}_{T}(f)|^2=\mathcal{F}\left \{ x_{T}^*(-t)\mathbf {*}x_{T}(t) \right \}
Is this correct?  How do you get a convolution?  I would expect the complex conjugates to multiply to get an absolute value, not convolve.  The rest of the equation is then used as a proof (without using Wiener–Khinchin theorem) that power spectral density and the autocorrelation of a function are Fourier transform pairs
-- 108.60.43.48
:See Fourier_transform#Cross-correlation_theorem.  For a more convincing argument, see Multi-dimensional derivation of Eq.1, for the dimension n=1, and for the special case of g(x) \triangleq h^*(-x), whose Fourier transform is H^*(f).
:--Bob K (talk) 21:42, 27 May 2021 (UTC)
thread_title:  How does the magnitude squared of a Fourier transform become a time convolution? 

message: This section needs work.  The first sentence should be extended to actually define the csd: "a csd can be defined... as something".  There is actually no definition either in text or in math.  And towards the end there's a "therefore it is one of them with a factor of 2", which kind of comes out of nowhere since it was not defined yet.Moo (talk) 16:11, 17 January 2023 (UTC)
thread_title:  Cross spectral density section