Talk:Spectral density

Other fields

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)

middle c

" then the pressure variations making up the sound wave would be the signal and "middle C and A" are in a sense the spectral density of the sound signal."

What does that mean? - Omegatron 13:39, May 23, 2005 (UTC)

My idea is to write an article such that the first paragraph would be useful to the newcomer to the subject. I may not have succeeded here, so please fix it if you have a better idea. I just don't want to start the article with "In the space of Lebesgue integrable functions..." PAR 14:58, 23 May 2005 (UTC)

yeah i hate that! i see what you're trying to say now... hmm... - Omegatron 16:57, May 23, 2005 (UTC)

Stationarity

"If the signal is not stationary then the same methods used to calculate the spectral density can still be used, but the result cannot be called the spectral density."

Are you sure of that? - Omegatron 13:39, May 23, 2005 (UTC)

Well, no. It was a line taken from the "power spectrum" article which I merged with this one. Let's check the definition. If you find it to be untrue, please delete it.

I'll try to find something. -Omegatron 16:57, May 23, 2005 (UTC)

It seems that this is true. They are considering the power spectrum to only apply to a stationary signal, and when you take a measurement of a non-stationary signal you are approximating. I know to take the spectrum of an audio clip with an FFT, you window the clip and either pad to infinity with zeros or loop to infinity, which sort of turns it into a stationary signal. - Omegatron 17:56, May 23, 2005 (UTC)

I'm starting to wonder about this. I think of a stationary process as a noise signal, with a constant average value, and a constant degree of correlation from one point to the next (with white noise having no correlation). I don't understand the meaning of stationary if its not with respect to noise, so I don't know whether the top statement is true or not. I'm not sure I understand what you mean by stationary in your example either. PAR 21:18, 23 May 2005 (UTC)

Yeah, my concept of "stationarity" is not terribly well-defined, either. I'm pretty sure a sinusoid is stationary, and a square or triangle wave would be. As far as the power spectrum is concerned, stationary means that the spectrum will be the same no matter what section of the signal you window and measure. An audio signal would not be stationary, for instance. But I am thinking in terms of spectrograms and I'm not really sure of the mathematical foundations behind this. - Omegatron 21:25, May 23, 2005 (UTC)

Omegatron, are you sure that you're not thinking of "cyclostationarity"? I'm not sure if a sinusoid is stationary. I think what IS stationary is a process that looks like sin( t + x ) where t is a deterministic variable and x is a random phase, picked uniformly from [0,2*pi). Without the random variable x, it's not stationary. Lavaka (talk) 16:55, 19 November 2009 (UTC)

Here is the definition of stationary random process:

All of these are NOT functions of time:
1. Its average value, its variance, and all of its statistical moments.
2. Its autocorrelation function, and all of its power spectra and energy spectra.
98.81.0.222 (talk) 23:34, 8 July 2012 (UTC)

Here is the definition of a wide-sense stationary random process:

All of these are NOT functions of time:
1. Its average value and its variance.
2. Its autocorrelation function, its power spectrum, and its energy spectrum, whichever of the later two exists.
98.81.0.222 (talk) 23:34, 8 July 2012 (UTC)

Change Definition

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)

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)

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)

finite time intervall

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)

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)

Units of Phi(omega) for continuous and discrete transforms

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)

Yes, that makes sense. I'll add something. Dicklyon 19:24, 26 March 2007 (UTC)

QFT

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)

Difference between power spectral density and energy spectral density

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)

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)

Spectral Analyzer

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)

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)

Spectral intensity

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)

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)

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)

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)

Survey: bit/s/Hz, (bit/s)/Hz or bit·s⁻¹·Hz⁻¹ as Spectral efficiency unit?

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)

Energy of a signal - doubled wiki entry

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)

Recent changes that confuse power and energy spectral density

A raft of changes introduced a ton of confusion and the ambiguous acronym "SD", with no supporting sources. That's why I reverted them and will revert them again. I realize it was well intentioned, but since I can't discern what the point is, it's hard to see how to fix it. What kind of SD is not already covered in the distinction between power and energy spectral density? Why is variant terminology being introduced? Is there some source that this change is based on? The one that was cited about the W-K theorem was not at all in support of the text it was attached to. Dicklyon (talk) 05:19, 28 November 2008 (UTC)

Be that as it may, it would have been more friendly to discuss the matter and give whoever it was a chance to provide sources, rather than revert what is clearly several hours of work. Particularly since it was probably done by a newbie who may not be aware of the possibility of reversions, or of the need for RS. --Zvika (talk) 13:35, 28 November 2008 (UTC)

OK, it was me (admitedly very newbie in wiki modification) who introduced these reverted changes. Let me explain my point of view.

Spectral density is a widely used concept, not necessarily related to energy or power. Spectral density is therefore defined for physical quantities which are not linked by any mean to a power or energy. OK, I agree that a voltage spectral density is most often called a power spectral density, for the reason that it relates nicely and conventionnaly to a power by the relation V^2/R with R=1 Ohm. However, a lot of other physical quantities do not relates to a power in any conventional way. One crucial exemple of that, which is of prime importance in my own personnal field of work, is the spectral density of phase fluctuations and of frequency fluctuations. Refere for exemple to IEEE Std 1193-1994 for more detail (I recommand draft revision in proc. of IEEE IFCS 1997 p338-357). They do not relate to a power at all. Yes, I know that some author/community conventionnaly use the term "power spectral density" for anything which is the square of a given physical quantity per Hz. This usage is however not followed by everyone (by far) (see again IEEE IFCS 1997 p338-357 and IEEE Std 1193), and I was actually trying to clarify that a little more than what was already made.

The actual form of the article is IMHO not totally adapted to all these fields where SD is being used. The title of the article being "spectral density" and not "spectral density of power (and energy)", It seems to me that it needs to be much less specific and define SD for every physical quantities. Again, this is most specially true when there is a large community who is using SD for physical quantities other than power (or energy) which cannot be turned into a power by any cannonical means.

One possible way to adress all these different communities would be to give the specific explanations for every single SD concept we can find in the literature (PSD, ESD, Frequency fluctuation SD, phase fluctuations SD, Timing fluctuations SD etc...), with the risk of forgetting a lot of them we don't know about (because those working in these fields don't bother correcting Wikipedia). The other possibiliy is to give a very general (mathematical) definition of SD for any physical quantity, and then give some specific exemples of use. This second solution is what I would very strongly recommend.

An other thing which I have strong problem with is the "snake bitting it's tail" situation between WK and spectral density which is very confusing in Wikipedia:en right know : basically the WK theorem states that the spectral density of a signal is the FT of the autocorrelation of the signal, while the PSD is defined by the current article by use of the WK theorem. This doesn't make sense at all and will confuse anyone not already familiar with the concepts (either the WK theorem is a theorem or a definition...). The WK and SD article together should give a better clarification on this topic IMHO. In practice, nowadays, the use of autocorrelation for extracting SD from a signal is not so frequent, and fast fourier transform of the sampled signal is very often prefered, the definition of SD in wikipedia should therefore reflect this reality. Besides, the easy confusion I find in students mind between one sided and two sided spectral densities was also a topic which, IMHO needed clarification. I added some comment on that with this purpose.

Hope this clarifies my admitedly clumsy and too quick attempt at giving the article more generality and clarity.

On the other hand, I admit introducing the Acronym "SD" by facility, while it's not particularly standard (except as slang in the research labs I know...). I agree that SD should therefore be replaced by "spectral density" everywhere it appears.

I'm refrening from restoring for a while, but, seriously, it looks to me that my modifications were, at least, a step in the right direction. —Preceding unsigned comment added by 145.238.204.158 (talk) 16:13, 18 December 2008 (UTC)

Thanks for responding. A step in the right direction is a step that is backed up by sources (see WP:V and WP:RS). What sources do it your way? Without knowing, it's hard to help integrate your viewpoint. As for "the square of a given physical quantity per Hz", that applies to both energy and power spectral density; the difference is in whether it's per unit time or not. An energy spectral density is the the squared magnitude of the fourier transform of a signal with finite integral of its square (finite energy); a power spectral density, on the other hand, is mean square per Hz, as opposed to integral square per Hz, and applies to a signal with a finite mean square, and infinite integral square. It really doesn't matter whether the values related to physical energy. If this is not clear in the article, consult the cited sources and think about clarifying it. As for calculating with an FFT or DFT, that's a detail of how to estimate an SD, not a definition of what it is. So if you'd like to add a section on estimating it, go ahead, as long as you cite a good source. Dicklyon (talk) 05:03, 19 December 2008 (UTC)

Also, if you'd like me to look at those IEEE standards, post us a link, or email me a copy. Dicklyon (talk) 05:04, 19 December 2008 (UTC)

terminology, scaling

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)

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)

Suggestion for better definition of spectral density (June 2010)

The current definition of (Power) Spectral Density (PSD) in this article obviously has dimension "... per squared Hz". A proper definition should have the intended dimension, i.e. ".. per Hz"). Moreover the PSD should never diverge, as will happen when using the present definition in the case of stationary noise for example. Therefore I suggest to change the integral into a time average: integrate over a time interval ΔT and divide the (squared) integral by ΔT. Then take the limit for ΔT → ∞.

• This comment was written after reading a lecture of E. Losery (www.ee.nmt.edu/~elosery/lectures/power_spectral_density.pdf). In his turn E. Losery refers to the definition of Stremmler (F.G. Stremler, Introduction to Communication Systems, 2nd Ed., Addison-Wesley, Massachusetts, 1982).

Eric Hennes June 3, 2010 —Preceding undated comment added 16:06, 2 June 2010 (UTC).

What you say is obvious remains unclear to me. Can you point which formula obviously has these wrong units, and what you would do to fix it? Or did you already fix it? Dicklyon (talk) 05:55, 12 October 2010 (UTC)

Well, suppose the signal s is unit-less, then the autocorrelation integral R(tau) of the signal has dimension "time". Its Fourier integral adds another "time", so we end up with time^2, i.e. "inverse frequency"^2, 1/Hz^2. Eric Hennes December 21, 2010

"S" and "s"

In the section on power spectral density, the discussion is confusing, in part because of confusing 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)

True. Any expectations that the case shift signifies a relationship between the two is not correct here. Otherwise, it looks right. See if you can find a better or more conventional set of names. Dicklyon (talk) 05:53, 12 October 2010 (UTC)'

Adding to the confusion described above, the definition of the term $f_{T}(t)$ is not entirely clear? Yitping (talk) 13:46, 30 March 2011 (UTC)

What you need to understand is that the s(t) in s(t)**2 means "signal" as a function of time, where the signal is in volts. Then s(t)**2 divided by a resistance works out to have the unit of watts. In other words, it is a power. Then, S(f) is a power spectrum in the unit of watts per hertz. There is NOT any kind of a Fourier Transform relationship between s(t) and S(f). Sorry if you are confused, but this is just the way that it is. Scientists and engineers only have 26 letters of the alphabet to work with, after all.98.81.0.222 (talk) 22:41, 8 July 2012 (UTC)

Also, some textbooks, pamphlets, etc., have the bad way of stating that S(f) gives the power measured in a "one-hertz bandwidth". One hertz does not have ANYTHING to do with it, and "one hertz" is just a confusion factor. S(f) actually gives the power measured in an infinitesimal bandwidth divided by that infinitesimal bandwidth. This is a limiting process, and THIS is where all the calculus comes from, including the Wiener-Khinchine Theorem. Do not pretend to understand spectral densities withough understanding differential and integral calculus first.98.81.0.222 (talk) 22:41, 8 July 2012 (UTC)

A good deal of knowledge of stochastic processes is also a necessity to truly understand spectral densities. The subject cannot really be explained on an elementary level.98.81.0.222 (talk) 22:48, 8 July 2012 (UTC)

Stationary vs Cyclostationary

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)

I think you're right; this book agrees. I'll take out "and only if". Dicklyon (talk) 04:20, 31 January 2011 (UTC)

No, cyclostationary processes are a really different "ball game" from wide-sense stationary processes and strictly-stationary processes. There is also something mathematically questionable in allowing Dirac delta functions in spectral densities. If done, this has to be done with great care.98.81.0.222 (talk) 22:53, 8 July 2012 (UTC)

PSD relation to periodogram

If we are to follow the definition of the periodogram given in the peridogram wiki page, which is linked here, the periodogram has units of voltage (given that the signal has units of voltage). The PSD, which according to the definition here is the squared periodogram divided by time, must then have units of voltage squared times frequency, or voltage squared divided by time. If either the formula for the PSD here or the definition of the periodogram omits the division by T, we get the correct units. Thus something must be changed, or are my dimensional analysis skills severely lacking?

Definition section was confusing: Major Rewrite

Why?
A good and useful article should bridge the gap from beautiful abstract mathematics to real-world applications.
It should help engineers to understand the language of mathematicians and vice versa. This was not the case right?
Please let me outline a possible route for reorganizing the article (that builds on suggestions made by others on this page, thanks!):

1.Explain power spectrum in layman's language.

The correct expression in English is "layman's language", instead of what was here before. Also, "explain power spectrum in layman's language" cannot be done. Please believe me. D.A.W.-MSEE-Georgia Tech.
98.81.0.222 (talk) 23:04, 8 July 2012 (UTC)

2.Abstract mathematical definition of power spectrum using infinite-time Fourier transform.

3. Introduce normalized Fourier transform for stationary processes and define power spectrum with respect to those.

4. Discuss the case of discrete-time series and show pseudocode.

Foolish98.81.0.222 (talk) 23:04, 8 July 2012 (UTC)

5. State the Wiener-Khinchine theorem and remark that it is sometimes used even as a definition of the power spectrum.
— Preceding unsigned comment added by Benjamin.friedrich (talk • contribs) 09:24, 25 May 2012 (UTC)

Dear all, I started to rewrite the definition section.
I do very much appreciate the contribution of the former authors.
Still, I think a good Wikipedia article should bridge the gap between elegant textbook definitions and recipes for real world tasks (like analyzing time-series data).
In the interest of all, please do not just undo my changes.
If you feel like adding a little bit of more mathematical rigor, please feel free to do so.
But I honestly believe the link to physics and engineering should be kept. In this spirit, all quantities should have correct physical units.
Let's move this article to a higher quality level together.<br? Best wishes, Ben
Benjamin.friedrich (talk)

These recent redefinitions are a lot to digest. It would not be unreasonable to back them out and take a more incremental approach. The attempt to start a discussion, misplaced at the top, needs time for reactions. I can't tell without a lot more work whether there is an improvement there. Anyone else willing to study it? Dicklyon (talk) 16:53, 25 May 2012 (UTC)

Please note that the recent rewrite takes up suggestion made before. In particular, defining the power spectral density using a finite time equivalent of the Fourier transform was suggested several times before (see sections "finite time interval" and "Suggestion for better definition of spectral density (June 2010)"). I think this suggestion is more transparent than the alternative definition as the Fourier transform of the autocorrelation function (which, of course, should still be mentioned for completeness.) A second point that came up before is how all the different definitions of the spectral density (energy/ power/ time-continuous signals /discrete signals) go together, and which normalization factors should be used, so that they get the right physical units. Correct physical units is not so important in pure math but get crucial in physics and engineering applications. So, I would say the new changes are useful and actually combine previous suggestions by others. But I agree that all this needs discussion, so let's start it here ... Benjamin.friedrich (talk)

The new definition of PSD in terms of the limit of the Fourier transform of a random signal doesn't make sense, as this book explains. It at least needs an expectation over an ensemble average. You mention a source, but there's no citation. And the Weiner–Khintchine theorem isn't linked, and there are lots of errors. Very hard to review this big a change. Dicklyon (talk) 16:12, 28 May 2012 (UTC)

I just added some sources. I very much like the reference you give: The book by Miller very nicely explains how using the truncated Fourier transform ensures the existence of Fourier transformed stochastic signals. I added this as a source too. I agree on your point that for stochastic signals, one is interested not so much in the PSD of a particular realization (although it is perfectly fine to apply the definition to that case), but rather in an ensemble average. I added a sentence on that. Surely, there will be more errors. My personal view is that any larger improvement implies leaving a local maximum of text quality and will transiently introduce errors, before a new (hopefully even better) maximum is found. Thanks a lot for spotting some of these errors. Benjamin.friedrich (talk)

Thanks, I appreciate the support. I did a bit more on it, and changed to definition to be the one in this source. I can't see the relevant page on the other source, but I have a hard time imagining that it said what was in the article. I'm unsure of the rest of that section, as I haven't digested it all yet. I think it needs work. Dicklyon (talk) 18:38, 3 June 2012 (UTC)

Typo? The PSD is now defined as a limit T->0. I wonder if it shouldn't read T->infty as in reference [5]? Otherwise, writing the PSD as an expectation value is fine with me. Benjamin.friedrich (talk) —Preceding undated comment added 13:15, 5 June 2012 (UTC)

Yes, sorry, my mistake. Thanks for fixing the limit. Dicklyon (talk) 14:20, 5 June 2012 (UTC)

Properties section

Could please someone have a look at the properties section? What is so special about the interval [-1/2,1/2]? I wonder if the statement on the variance is true? Benjamin.friedrich (talk) —Preceding undated comment added 14:52, 25 May 2012 (UTC).

The insistance on that interval [-1/2,1/2] is quite strange and doubtless not true. It sounds like someone copied this without understanding why, wherefore, or under what circumstances this holds.

D.A.W., master of science in electrical engineering, Georgia Tech.98.81.0.222 (talk) 22:01, 8 July 2012 (UTC)

Here is the diff where the problem came it. The ref he's using is probably about discrete time series, and this property is in units of cycles per sample, I'd guess. But it needs to be fixed. We can ask User:Tomaschwutz, but he seems to be long dormant. Dicklyon (talk) 22:26, 8 July 2012 (UTC)

The derivative of S(f) at f = 0

The derivative of S(f) at f = 0
This was badly mistated that this derivative MUST be zero.
This derivative might not even exist at f = 0.

1. Imagine an S(f) that contains an isoscoles triangle with its point sticking up at f = 0. The derivative does not exist at f = 0, and you should be able to see this by drawing a simple diagram.

2. Any S(f) that has a Dirac delta function at f = 0 does not have a derivative there. In fact, the value of S(f) is undefined there, unless we wish to define it to be infinity there.
98.81.0.222 (talk) 23:19, 8 July 2012 (UTC)