What is Power spectrum: Definition and 58 Discussions

The power spectrum




S

x
x


(
f
)


{\displaystyle S_{xx}(f)}
of a time series



x
(
t
)


{\displaystyle x(t)}
describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, or a spectrum of frequencies over a continuous range. The statistical average of a certain signal or sort of signal (including noise) as analyzed in terms of its frequency content, is called its spectrum.
When the energy of the signal is concentrated around a finite time interval, especially if its total energy is finite, one may compute the energy spectral density. More commonly used is the power spectral density (or simply power spectrum), which applies to signals existing over all time, or over a time period large enough (especially in relation to the duration of a measurement) that it could as well have been over an infinite time interval. The power spectral density (PSD) then refers to the spectral energy distribution that would be found per unit time, since the total energy of such a signal over all time would generally be infinite. Summation or integration of the spectral components yields the total power (for a physical process) or variance (in a statistical process), identical to what would be obtained by integrating




x

2


(
t
)


{\displaystyle x^{2}(t)}
over the time domain, as dictated by Parseval's theorem.The spectrum of a physical process



x
(
t
)


{\displaystyle x(t)}
often contains essential information about the nature of



x


{\displaystyle x}
. For instance, the pitch and timbre of a musical instrument are immediately determined from a spectral analysis. The color of a light source is determined by the spectrum of the electromagnetic wave's electric field



E
(
t
)


{\displaystyle E(t)}
as it fluctuates at an extremely high frequency. Obtaining a spectrum from time series such as these involves the Fourier transform, and generalizations based on Fourier analysis. In many cases the time domain is not specifically employed in practice, such as when a dispersive prism is used to obtain a spectrum of light in a spectrograph, or when a sound is perceived through its effect on the auditory receptors of the inner ear, each of which is sensitive to a particular frequency.
However this article concentrates on situations in which the time series is known (at least in a statistical sense) or directly measured (such as by a microphone sampled by a computer). The power spectrum is important in statistical signal processing and in the statistical study of stochastic processes, as well as in many other branches of physics and engineering. Typically the process is a function of time, but one can similarly discuss data in the spatial domain being decomposed in terms of spatial frequency.

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  1. ergospherical

    I Toward the spectral index in slow roll

    Getting a bit confused over a manipulation; I have a power spectrum ##P_{R}(k)## in terms of ##V## and ##\epsilon_V## (to be evaluated at ##k=aH##) for the curvature perturbation ##R## and from this need to get a spectral index, i.e. apply ##d/d(\ln{k})##. So we need to transform this operator...
  2. Skaiserollz89

    Relationship between Fourier coefficients and power spectral density

    Here, ##\Phi(f_{x_n},f_{y_m})=|\mathscr{F(\phi(x,y))}|^2 ## is the Power Spectral Density of ##\phi(x,y)## and ##\mathscr{F}## is the Fourier transform operator. Parseval's Theorem relates the phase ##\phi(x,y)## to the power spectral density ##\Phi(f_{x_n},f_{y_m})## by...
  3. SherLOCKed

    I Maximizing S/N in Angular Power Spectrum Signals

    The signal-to-noise ratio for angular power spectrum signal Cl under theoretical noise Nl, where Cl and Nl are functions of multipole l, is given as (S/N)^2= \sum (2l+1) (Cl/Nl)^2To increase the S/N we bin the power spectrum signal, if bin width \Delta l, this in principle decreases Nl by a...
  4. arcTomato

    I Power spectrum definition

    Hello PF. I am thinking about the power spectrum when observing X-rays. We are trying to obtain the power spectrum by applying a window function ##w(t)## to a light curve ##a(t)## and then Fourier transforming it. I have seen the following definition of power spectrum ##P(\omega)##. Suppose...
  5. arcTomato

    I The power spectrum of Poisson noise

    I thought that if we Fourier transformed the counts of the sum of the signal from the source and the Poisson noise, and obtained the power spectrum, we would get the following, ##P_{j}=P_{j, \text { signal }}+P_{j, \text { noise }}+\text { cross terms }## but I found the following description...
  6. F

    A Analytical expression of Cosmic Variance - Poisson distribution?

    I have an expression of Matter Angular power spectrum which can be computed numerically by a simple rectangular integration method (see below). I make appear in this expression the spectroscopic bias ##b_{s p}^{2}## and the Cosmic variance ##N^{C}##. ## \begin{aligned}...
  7. F

    A Relation between Matter Power spectrum and Angular power spectrum

    From a previous post about the Relationship between the angular and 3D power spectra , I have got a demonstration making the link between the Angular power spectrum ##C_{\ell}## and the 3D Matter power spectrum ##P(k)## : 1) For example, I have the following demonstration, ## C_{\ell}\left(z...
  8. F

    A Angular power spectrum dependence in redshift z and k

    Hi, I wanted to have a precision about a question that has been post on this relation between P(k) and C_l The author writes the ##C_\ell## like this : $$C_\ell(z,z') = \int_0^\infty dkk^2 j_\ell(kz)j_\ell(kz')P(k)$$ I don't undertstand the meaning of ##z## and ##z'## : these are not...
  9. arcTomato

    A Average of the power spectrum of Poisson noise

    I am learning about noise that follows a Poisson distribution. When I do a Fourier transform of the data with only Poisson noise to get the power spectrum, what is the average value of the power spectrum?
  10. arcTomato

    A Why the Poisson noise level is set to 2 after applying Leahy norm

    I am studying about power spectrum analysis in high energy astrophysics. I cannot understand why the Poisson noise level is set to 2 after applying Leahy normalization. $$P_{j}=2 /_{N \mathrm{ph}}\left|a_{j}\right|^{2}$$ The above is the equation for leahy norm, Can I expand the equation from...
  11. John321

    A Relationship between the angular and 3D power spectra

    I have the following equation, $$ C_\ell(z,z') = \int_0^\infty dkk^2 j_\ell(kz)j_\ell(kz')P(k),$$ where $$j_\ell$$ are the spherical Bessel functions. I would like to invert this relation and write P(k) as a function of C_l. I don't know if this is a well known result, but I couldn't find...
  12. W

    B Nearly Scale Invariant Power Spectrum in inflation

    I recall hearing once a very intuitive explanation as to why inflation is thought to lead to a nearly scale invariant power spectrum but i can't recall it. Can anyone offer an explanation that might help me? Why is it nearly scale invariant and not perfectly scale invariant? many thanks
  13. arcTomato

    Comp Sci Power spectrum when the wave number is not an integer

    Hi all. I made a program of DFT, so I made the power spectrum of a sin wave. This is the sin wave I used. All data number ##N=100## and the frequency of sine wave is 4.5Hz. And the power spectrum is this. The wave number is not integer so the spectrum has the side lobe. But I think this is...
  14. arcTomato

    The power spectrum of Poisson noise

    Dear all. I have made the power spectrum of Poisson noise(expected value ##λ=2##), and it becomes like this I think this is not good. I don't know why the power is so high when Random variable(x axis) is 0. I tried another expected value version ,but the result didn't change. so I would like to...
  15. arcTomato

    I I would like to know the calculation process of this power spectrum

    Summary: Derive the formula of power spectrum from Discrete Fourier Transform. Hi all I don't know where should I post this, so if I am wrong, I apologize.(But this is almost math problem so ) I would like to know the calculation process when derive Eq(6.3) in this paper. Eq 2.4a is...
  16. arcTomato

    Engineering The power spectrum of a sine wave (C language)

    Hi I would like to Derive the power spectrum of sinusoid.I tried like this. But It doesn't work. <Moderator: CODE tags added> #include <stdio.h> #include <math.h> #define pi 3.1415926535 FILE *in_file, *out_file; int main() { dft(); } int dft(int argc, char *argv[]) { char...
  17. F

    I Cross-correlations: what size to select for the matrix?

    Hello, I am working on Fisher's formalism in order to get constraints on cosmological parameters. I am trying to do cross-correlation between 2 types of galaxy populations (LRG/ELG) into a total set of 3 types of population (BGS,LRG,ELG). From the following article...
  18. Rafid Mahbub

    A Numerical solution of the Mukhanov-Sasaki equation

    Hi, I am trying to figure out how to solve the Mukhanov equation numerically in Mathematica, but have some problems dealing with it. In terms of the number of efolds, the Fourier modes satisfy the following ODE in terms of the Hubble slow roll parameters: $$...
  19. Spinnor

    I ....Alternatives to Inflation in Primordial Power Spectrum

    Unique Fingerprints of Alternatives to Inflation in the Primordial Power Spectrum "Massive fields in the primordial universe function as standard clocks and imprint clock signals in the density perturbations that directly records the scale factor of the primordial universe as a function of...
  20. B

    Gaussian Smoothing of the Matter Power Spectrum

    Homework Statement Consider the density perturbation smoothed with a Gaussian of scale ##\sigma##, ##\Delta_{\sigma}(\vec x') = \int d^3 \vec x \frac{e^{- \frac{(\vec x - \vec x')^2}{2 \sigma^2}}}{(2 \pi \sigma)^{3/2}} \Delta (\vec x)## Calculate the power spectrum ##P_{\Delta_{\sigma}}## of...
  21. DoobleD

    I Vertical axis of the CMB power spectrum

    Why is the vertical axis in the CMB power spectrum usually chosen as ##l(l+1)C_l/2\pi## instead of simply ##C_l## ? The only answer I found come from this post on stackexchange, but the answer doesn't seem very complete. Anyone knows ?
  22. C

    A Angular power spectrum, bias from N weighted events

    My general question is: What is the angular power spectrum C_{l,N,ω} of N weighted (weight ω_i for event i) events from a full sky map with distribution C_l? I'm interested in: Mean of C_{l,N,ω}: <C_{l,N,ω}> Variance of C_{l,N,ω}: Var(C_{l,N,ω}) The question is important, since we observe in...
  23. S

    I Primordial Power Spectrum

    Why is the power spectrum defined as ##P(k) = \frac{k^3}{2π^2} |w_k|^2 ## where ##w_k## is the mode function? Cosmology books and papers just states that it is defined that way but there are no details on why.
  24. F

    I Primordial power spectrum

    I have been reading the TASI Lectures on Inflation by William Kinney, (https://arxiv.org/pdf/0902.1529v2.pdf). I came across the mode function eq (128) (which obeys a generalization of the Klein-Gordon equation to an expanding spacetime), as I read through until eq (163), I know that it is the...
  25. W

    I Scale invariance in the power spectrum

    I understand the inflation predicts a nearly scale invariant power spectrum but some have claimed this was predicted before inflation (by Harrison and Zeldovitch?) My understanding is that perfectly scale invariance would predict ns=1 but inflation predicts ns =.96. So did the prior prediction...
  26. sunrah

    CAMB python convergence power spectrum code

    I'm trying to understand this python CAMB code: http://camb.readthedocs.io/en/latest/CAMBdemo.html Scroll down to In[29] and In[30] to see it. It's an integration over chi (comoving distance), yet scipy.integrate.quad is not called. It seems that the fun stuff happens in the last for-loop in...
  27. sunrah

    A Plotting the CS power spectrum

    I'm having difficulty plotting the kernel I(k_{1},k) of the cosmic shear power spectrum which is defined as I(k_{1},k) = k_{1}\int^{\infty}_{0}r j_{l}(k_{1}r)dr \int^{r}_{0}\frac{r-r'}{r'}j_{l}(kr')\sqrt{P^{\Phi\Phi}(k)}dr' where the jl are spherical bessel functions. I'v tried plotting I vs...
  28. G

    Finding the One Degree Peak in a Power Spectrum

    Can someone tell me how i can determine the one degree peak from power spectrum ?
  29. J

    Cosmic acoustics -- why no intermediate waves on CMB map

    I fully understand the representation of the set of waves that are either at full compression or full rarefaction at recombination, thus, yielding a CMB map. But at this time are there no waves that are intermediate, e.g. 50% of the way to full compression or full rarefaction. Why don't these...
  30. C

    Intuition Behind Scale Invariance Power Spectrum

    In the book "Statistical physics for cosmic structures" at p. 171 a read a definition of scale invariance (leading to the so called scale invariant power spectrum) given as the requirement that ##\sigma^2_M(R=R_H(t)) = constant##, where ##R_H(t)## is the horizon, i.e. the maximal distance that...
  31. A

    The peak of the thermal radiation power spectrum (dR/dλ)

    Homework Statement The peak of the thermal radiation power spectrum (dR/dλ) is at a wacelength of about λm=hc/5kT. Why is the peak of the same power spectrum plotted as dR/df not at fm=c/λm= 5kT/h? Homework Equations dR/dλ= 2πhc2/(λ5(e(hc/λkT)-1)) f=c/λ The Attempt at a Solution
  32. ChrisVer

    Power Spectrum of CMB angle-l

    http://pdg.lbl.gov/2013/reviews/rpp2013-rev-cosmic-microwave-background.pdf Here one reads in sec. 26.2.4 that: However it states that a single Y_{lm} corresponds to angular variations of \theta \sim \pi /l. I am not getting these statements. Also I find it difficult to understand, since...
  33. T

    Primodial power spectrum of scalar perturbations

    For scalar modes \mathcal{R}_k originating in the Bunch-Davies vacuum at the onset of inflation, I have the following equation for their primordial power spectrum: P_{\mathcal{R}}(k)=\frac{4\pi}{\epsilon(\eta_k)}\bigg( \frac{H(\eta_k)}{2\pi} \bigg)^2, where: c = G = ħ = 1, k is the...
  34. P

    Fast Fourier Transform (FFT) power spectrum angle

    Dear Physics Buddies, How are well all, okay I hope. I was wondering if I might browse all your infinite intellects and ask you a very simple question. I am working with some medical images in MATLAB and my collaborators would like to know the orientation of the fibres that it contains...
  35. M

    Understanding the Linear Power Spectrum: How Do BAOs Create Wiggles?

    Dear Experts, I try to understand power spectra for large scale structure and CMB analyses for my exams. I constantly find the expression for linear power spectra P(k) = A\cdot k^{n_s}\cdot T^2(k). I understand that this comes from primodial primodial fluctuations and the tranfer function...
  36. M

    Wave numbers of Power Spectrum

    Hi. I read some basic cosmology where it is always said that density fluctuations, pertubations can be described in modes of waves. In particular if you use linearised theory where δ(x,t) is Fourier transformed δ(k,t). What exactly is the reason for this? What do the wave modes describe...
  37. J

    Linear vs non linear matter power spectrum

    Hi, I have troubles understanding the difference between linear and non linear matter power spectrum. These words are commonly used in the litterature, but I have found no definitions yet. My understanding is that there is one definition of the power spectrum for matter distribution...
  38. H

    Power spectrum for real, imaginary and complex functions.

    Homework Statement What can we say about the evenness and oddness of the power spectrum (|F(s)|^{2}) if the input fuction is purely real, purely imaginary or complex? I know that a real function will give an even power spectrum. But I can't prove it! Homework Equations F(s) =...
  39. G

    Best,Jimmy

    Hello! I am doing a thesis in ecology using a two dimensional binary cellular automata. I have sampled a parameter for a number of generations and I would like to calculate the power spectrum for this data. I had been planning to use a software for this, but now it seems I have to do it...
  40. A

    Why is the CMB power spectrum often plotted as l(l+1) C_l and not only C_L ?

    Hello everybody, (sorry for the eventual Engrish) I can't find any convincing answer for the following question : Why do we always (or often) plot the CMB power spectrum in this way : jb.man.ac.uk/research/cosmos/vsa/images/CMB_power_spectrum.gif I mean the y-axis is $$C_\ell \ell...
  41. P

    Power Spectrum vs. Power Spectral Density Spectrum

    Does anyone know the difference between the power spectrum of a signal and the power spectral density (PSD) spectrum of a signal? I've read on the net lots of things ranging from: i) They are identical ii) Power spectrum is units of Watts, power spectral density spectrum units of Watts/Hz...
  42. D

    Understanding Power Spectrum

    Could anybody explain what "power spectrum" in cosmology means. I am trying to understand the WMAP power spectrum graph, in particular. For example, what do the multiple moments physically represent? The graph is here: http://casa.colorado.edu/~ajsh/cosmo_04/wmapcl.html
  43. Y

    Plotting Power Spectrum Density in Matlab

    Homework Statement I have an autocorrelation function given by R(\tau) = J_0(2\pif_d\tau) I have already plotted it in Matlab using the following commands ts=0.0000001; fs=1/ts; fd=100; tau=0:0.0000001:0.05; R=besselj(0,2*pi*fd.*tau); How do I plot its PSD in Matlab and the plotted picture...
  44. P

    MATLAB Matlab-Plot Power Spectrum of time series

    Hi all, and thank you for reading/responding to my thread.I have a problem which involves time series and its power spectra. Some background: I have an 10.000xN matrix(called matr) whose columns are individual time series with real data values and I want to plot their power spectra...
  45. TrickyDicky

    CMB Power Spectrum and Angular Power Spectrum Plot from WMAP

    Anyone knows if the CMB map of anisotropies from WMAP is used to implement the angular power spectrum plot(acoustic peaks)? I'm not sure, but I tend to think it is not.
  46. M

    How to measure the CDM power spectrum?

    Hi all! It's my first post, but I follow this forum since a couple of years. Now I have a big question for you: how to measure the CDM (or \LambdaCDM) power spectrum (better, the variance \propto k^3 P(k)) from the observations? What parameters are necessary?
  47. C

    Why does power spectrum is calculated at the radiation/matter epochs?

    In the standard inflationary scenario, the power spectrum is evaluated at the cosmological time when one assumes an equation of state $ P= \omega \rho$ , that is, one is assuming a particular radiation or matter dominated universe. Why does it has to be in these cosmological epochs? does it...
  48. H

    Can someone explain angular power spectrum

    Hi I'm trying to get my head around this plot of the cosmic microwave background radiation. http://www.astro.ucla.edu/~wright/CMB-LCDM-w-WMAP.gif I've been searching all around to find out what exactly a angular power spectrum is. I know what an ordinary power spectrum is. I'm a second...
  49. X

    Fourier transformation: power spectrum

    Lots of works about the high-order harmonic generation in the intense laser-atom physics obtain the harmonic spectrum by Fourier transformation of the dipole moment d(t) (=\int\varphi\varphi^{*}z): p(\omega)=|\frac{1}{tf-ti}\int d(t)exp(-i\omega)dt|^{2} Here, I want to use the Monte-Carlo...
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