# Power spectrum Definition and 8 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. ### 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...
2. ### 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}...
3. ### 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...
4. ### 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...
5. ### 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...
6. ### 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...
7. ### 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: ...
8. ### 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...