Talk:Spectral density

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Contents 1 Other fields 2 middle c 3 stationarity 4 change definition 5 finite time intervall 6 Units of Phi(omega) for continuous and discrete transforms 7 QFT 8 Difference between power spectral density and energy spectral density 9 Spectral Analyzer 10 Spectral intensity 11 Survey: bit/s/Hz, (bit/s)/Hz or bit·s−1·Hz−1 as Spectral efficiency unit?

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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 abs. 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 setup 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)