In natural language and physical science , a physical object or material object (or simply an object or body ) is a contiguous collection of matter , within a defined boundary (or surface ), that exists in space and time . Usually contrasted with abstract objects and mental objects .
113-543: A black body or blackbody is an idealized physical body that absorbs all incident electromagnetic radiation , regardless of frequency or angle of incidence . The radiation emitted by a black body in thermal equilibrium with its environment is called black-body radiation . The name "black body" is given because it absorbs all colors of light. In contrast, a white body is one with a "rough surface that reflects all incident rays completely and uniformly in all directions." A black body in thermal equilibrium (that is, at
226-405: A cloud , a human body , a banana , a billiard ball, a table, or a proton . This is contrasted with abstract objects such as mental objects , which exist in the mental world , and mathematical objects . Other examples that are not physical bodies are emotions , the concept of " justice ", a feeling of hatred, or the number "3". In some philosophies, like the idealism of George Berkeley ,
339-542: A density function multiplied by an infinitesimally small frequency interval, describing the energy contained in the signal at frequency f {\displaystyle f} in the frequency interval f + d f {\displaystyle f+df} . Therefore, the energy spectral density of x ( t ) {\displaystyle x(t)} is defined as: The function S ¯ x x ( f ) {\displaystyle {\bar {S}}_{xx}(f)} and
452-436: A one-sided function of only positive frequencies or a two-sided function of both positive and negative frequencies but with only half the amplitude. Noise PSDs are generally one-sided in engineering and two-sided in physics. Energy spectral density describes how the energy of a signal or a time series is distributed with frequency. Here, the term energy is used in the generalized sense of signal processing; that is,
565-402: A particle , several interacting smaller bodies ( particulate or otherwise). Discrete objects are in contrast to continuous media . The common conception of physical objects includes that they have extension in the physical world , although there do exist theories of quantum physics and cosmology which arguably challenge this. In modern physics, "extension" is understood in terms of
678-454: A transmission line of impedance Z {\displaystyle Z} , and suppose the line is terminated with a matched resistor (so that all of the pulse energy is delivered to the resistor and none is reflected back). By Ohm's law , the power delivered to the resistor at time t {\displaystyle t} is equal to V ( t ) 2 / Z {\displaystyle V(t)^{2}/Z} , so
791-400: A black body in thermal equilibrium has an emissivity ε = 1 . A source with a lower emissivity, independent of frequency, is often referred to as a gray body. Constructing black bodies with an emissivity as close to 1 as possible remains a topic of current interest. In astronomy , the radiation from stars and planets is sometimes characterized in terms of an effective temperature ,
904-413: A black surface is a small hole in a cavity with walls that are opaque to radiation. Radiation incident on the hole will pass into the cavity, and is very unlikely to be re-emitted if the cavity is large. Lack of any re-emission, means that the hole is behaving like a perfect black surface. The hole is not quite a perfect black surface—in particular, if the wavelength of the incident radiation is greater than
1017-517: A body forms an interface with its surroundings, and this interface may be rough or smooth. A nonreflecting interface separating regions with different refractive indices must be rough, because the laws of reflection and refraction governed by the Fresnel equations for a smooth interface require a reflected ray when the refractive indices of the material and its surroundings differ. A few idealized types of behavior are given particular names: An opaque body
1130-425: A collection of sub objects, down to an infinitesimal division, which interact with each other by forces that may be described internally by pressure and mechanical stress . In quantum mechanics an object is a particle or collection of particles. Until measured, a particle does not have a physical position. A particle is defined by a probability distribution of finding the particle at a particular position. There
1243-531: A consequence, Kirchhoff's perfect black bodies that absorb all the radiation that falls on them cannot be realized in an infinitely thin surface layer, and impose conditions upon scattering of the light within the black body that are difficult to satisfy. A realization of a black body refers to a real world, physical embodiment. Here are a few. In 1898, Otto Lummer and Ferdinand Kurlbaum published an account of their cavity radiation source. Their design has been used largely unchanged for radiation measurements to
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#17328687872241356-444: A constant temperature) emits electromagnetic black-body radiation. The radiation is emitted according to Planck's law , meaning that it has a spectrum that is determined by the temperature alone (see figure at right), not by the body's shape or composition. An ideal black body in thermal equilibrium has two main properties: Real materials emit energy at a fraction—called the emissivity —of black-body energy levels. By definition,
1469-535: A dilute gas. At temperatures below billions of Kelvin, direct photon–photon interactions are usually negligible compared to interactions with matter. Photons are an example of an interacting boson gas, and as described by the H-theorem , under very general conditions any interacting boson gas will approach thermal equilibrium. A body's behavior with regard to thermal radiation is characterized by its transmission τ , absorption α , and reflection ρ . The boundary of
1582-1098: A discrete signal with a countably infinite number of values x n {\displaystyle x_{n}} such as a signal sampled at discrete times t n = t 0 + ( n Δ t ) {\displaystyle t_{n}=t_{0}+(n\,\Delta t)} : S ¯ x x ( f ) = lim N → ∞ ( Δ t ) 2 | ∑ n = − N N x n e − i 2 π f n Δ t | 2 ⏟ | x ^ d ( f ) | 2 , {\displaystyle {\bar {S}}_{xx}(f)=\lim _{N\to \infty }(\Delta t)^{2}\underbrace {\left|\sum _{n=-N}^{N}x_{n}e^{-i2\pi fn\,\Delta t}\right|^{2}} _{\left|{\hat {x}}_{d}(f)\right|^{2}},} where x ^ d ( f ) {\displaystyle {\hat {x}}_{d}(f)}
1695-426: A highly schematic cross-section to illustrate the idea. The photosphere of the star, where the emitted light is generated, is idealized as a layer within which the photons of light interact with the material in the photosphere and achieve a common temperature T that is maintained over a long period of time. Some photons escape and are emitted into space, but the energy they carry away is replaced by energy from within
1808-403: A more accurate estimate of the theoretical PSD of the physical process underlying the individual measurements. This computed PSD is sometimes called a periodogram . This periodogram converges to the true PSD as the number of estimates as well as the averaging time interval T {\displaystyle T} approach infinity. If two signals both possess power spectral densities, then
1921-409: A nearly perfect black-body spectrum, ultimately evaporating . The mechanism for this emission is related to vacuum fluctuations in which a virtual pair of particles is separated by the gravity of the hole, one member being sucked into the hole, and the other being emitted. The energy distribution of emission is described by Planck's law with a temperature T : where c is the speed of light , ℏ
2034-413: A number of discrete frequencies, or a spectrum of frequencies over a continuous range. The statistical average of any 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
2147-410: A particular car might have all its wheels changed, and still be regarded as the same car. The identity of an object may not split. If an object is broken into two pieces at most one of the pieces has the same identity. An object's identity may also be destroyed if the simplest description of the system at a point in time changes from identifying the object to not identifying it. Also an object's identity
2260-410: 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
2373-480: A perfect emitter of radiation, a hot material with black body behavior would create an efficient infrared heater, particularly in space or in a vacuum where convective heating is unavailable. They are also useful in telescopes and cameras as anti-reflection surfaces to reduce stray light, and to gather information about objects in high-contrast areas (for example, observation of planets in orbit around their stars), where blackbody-like materials absorb light that comes from
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#17328687872242486-603: A periodic signal which is not simply sinusoidal. Or a continuous spectrum may show narrow frequency intervals which are strongly enhanced corresponding to resonances, or frequency intervals containing almost zero power as would be produced by a notch filter . The concept and use of the power spectrum of a signal is fundamental in electrical engineering , especially in electronic communication systems , including radio communications , radars , and related systems, plus passive remote sensing technology. Electronic instruments called spectrum analyzers are used to observe and measure
2599-565: A physical body is a mental object , but still has extension in the space of a visual field. Frequency spectrum In signal processing , the power spectrum S x x ( f ) {\displaystyle S_{xx}(f)} of a continuous time signal x ( t ) {\displaystyle x(t)} describes the distribution of power into frequency components f {\displaystyle f} composing that signal. According to Fourier analysis , any physical signal can be decomposed into
2712-569: A physical object has physical properties , as compared to mental objects . In ( reductionistic ) behaviorism , objects and their properties are the (only) meaningful objects of study. While in the modern day behavioral psychotherapy it is still only the means for goal oriented behavior modifications, in Body Psychotherapy it is not a means only anymore, but its felt sense is a goal of its own. In cognitive psychology , physical bodies as they occur in biology are studied in order to understand
2825-462: A second after its formation was a near-ideal black body in thermal equilibrium at a temperature above 10 K. The temperature decreased as the Universe expanded and the matter and radiation in it cooled. The cosmic microwave background radiation observed today is "the most perfect black body ever measured in nature". It has a nearly ideal Planck spectrum at a temperature of about 2.7 K. It departs from
2938-448: A single estimate of the PSD can be obtained through a finite number of samplings. As before, the actual PSD is achieved when N {\displaystyle N} (and thus T {\displaystyle T} ) approaches infinity and the expected value is formally applied. In a real-world application, one would typically average a finite-measurement PSD over many trials to obtain
3051-419: A single such time series, the estimated power spectrum will be very "noisy"; however this can be alleviated if it is possible to evaluate the expected value (in the above equation) using a large (or infinite) number of short-term spectra corresponding to statistical ensembles of realizations of x ( t ) {\displaystyle x(t)} evaluated over the specified time window. Just as with
3164-458: 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
3277-480: A time-varying spectral density. In this case the time interval T {\displaystyle T} is finite rather than approaching infinity. This results in decreased spectral coverage and resolution since frequencies of less than 1 / T {\displaystyle 1/T} are not sampled, and results at frequencies which are not an integer multiple of 1 / T {\displaystyle 1/T} are not independent. Just using
3390-645: A total measurement period T = ( 2 N + 1 ) Δ t {\displaystyle T=(2N+1)\,\Delta t} . S x x ( f ) = lim N → ∞ ( Δ t ) 2 T | ∑ n = − N N x n e − i 2 π f n Δ t | 2 {\displaystyle S_{xx}(f)=\lim _{N\to \infty }{\frac {(\Delta t)^{2}}{T}}\left|\sum _{n=-N}^{N}x_{n}e^{-i2\pi fn\,\Delta t}\right|^{2}} Note that
3503-446: A unit by translation or rotation, in 3-dimensional space . Each object has a unique identity, independent of any other properties. Two objects may be identical, in all properties except position, but still remain distinguishable. In most cases the boundaries of two objects may not overlap at any point in time. The property of identity allows objects to be counted. Examples of models of physical bodies include, but are not limited to
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3616-457: Is S x y ( f ) = ∑ n = − ∞ ∞ R x y ( τ n ) e − i 2 π f τ n Δ τ {\displaystyle S_{xy}(f)=\sum _{n=-\infty }^{\infty }R_{xy}(\tau _{n})e^{-i2\pi f\tau _{n}}\,\Delta \tau } The goal of spectral density estimation
3729-427: Is +0.12. The two indices for two types of most common star sequences are compared in the figure (diagram) with the effective surface temperature of the stars if they were perfect black bodies. There is a rough correlation. For example, for a given B-V index measurement, the curves of both most common sequences of star (the main sequence and the supergiants) lie below the corresponding black-body U-B index that includes
3842-575: Is a limit to the accuracy with which the position and velocity may be measured . A particle or collection of particles is described by a quantum state . These ideas vary from the common usage understanding of what an object is. In particle physics , there is a debate as to whether some elementary particles are not bodies, but are points without extension in physical space within spacetime , or are always extended in at least one dimension of space as in string theory or M theory . In some branches of psychology , depending on school of thought ,
3955-432: Is a contiguous surface which may be used to determine what is inside, and what is outside an object. An object is a single piece of material, whose extent is determined by a description based on the properties of the material. An imaginary sphere of granite within a larger block of granite would not be considered an identifiable object, in common usage. A fossilized skull encased in a rock may be considered an object because it
4068-454: Is an example of physical system . An object is known by the application of senses . The properties of an object are inferred by learning and reasoning based on the information perceived. Abstractly, an object is a construction of our mind consistent with the information provided by our senses, using Occam's razor . In common usage an object is the material inside the boundary of an object, in three-dimensional space. The boundary of an object
4181-411: Is called "black" because it absorbs all the light that hits the horizon, reflecting nothing, making it almost an ideal black body (radiation with a wavelength equal to or larger than the diameter of the hole may not be absorbed, so black holes are not perfect black bodies). Physicists believe that to an outside observer, black holes have a non-zero temperature and emit black-body radiation , radiation with
4294-490: Is commonly expressed in SI units of watts per hertz (abbreviated as W/Hz). When a signal is defined in terms only of a voltage , for instance, there is no unique power associated with the stated amplitude. In this case "power" is simply reckoned in terms of the square of the signal, as this would always be proportional to the actual power delivered by that signal into a given impedance . So one might use units of V Hz for
4407-409: Is created at the first point in time that the simplest model of the system consistent with perception identifies it. An object may be composed of components. A component is an object completely within the boundary of a containing object. A living thing may be an object, and is distinguished from non-living things by the designation of the latter as inanimate objects . Inanimate objects generally lack
4520-1399: Is denoted as R x x ( τ ) {\displaystyle R_{xx}(\tau )} , provided that x ( t ) {\displaystyle x(t)} is ergodic , which is true in most, but not all, practical cases. lim T → ∞ 1 T | x ^ T ( f ) | 2 = ∫ − ∞ ∞ [ lim T → ∞ 1 T ∫ − ∞ ∞ x T ∗ ( t − τ ) x T ( t ) d t ] e − i 2 π f τ d τ = ∫ − ∞ ∞ R x x ( τ ) e − i 2 π f τ d τ {\displaystyle \lim _{T\to \infty }{\frac {1}{T}}\left|{\hat {x}}_{T}(f)\right|^{2}=\int _{-\infty }^{\infty }\left[\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }x_{T}^{*}(t-\tau )x_{T}(t)dt\right]e^{-i2\pi f\tau }\ d\tau =\int _{-\infty }^{\infty }R_{xx}(\tau )e^{-i2\pi f\tau }d\tau } From here we see, again assuming
4633-975: Is most suitable for transients—that is, pulse-like signals—having a finite total energy. Finite or not, Parseval's theorem (or Plancherel's theorem) gives us an alternate expression for the energy of the signal: ∫ − ∞ ∞ | x ( t ) | 2 d t = ∫ − ∞ ∞ | x ^ ( f ) | 2 d f , {\displaystyle \int _{-\infty }^{\infty }|x(t)|^{2}\,dt=\int _{-\infty }^{\infty }\left|{\hat {x}}(f)\right|^{2}\,df,} where: x ^ ( f ) ≜ ∫ − ∞ ∞ e − i 2 π f t x ( t ) d t {\displaystyle {\hat {x}}(f)\triangleq \int _{-\infty }^{\infty }e^{-i2\pi ft}x(t)\ dt}
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4746-446: Is often reflected and refracted within them, until it be stifled and lost? The idea of a black body originally was introduced by Gustav Kirchhoff in 1860 as follows: ...the supposition that bodies can be imagined which, for infinitely small thicknesses, completely absorb all incident rays, and neither reflect nor transmit any. I shall call such bodies perfectly black , or, more briefly, black bodies . A more modern definition drops
4859-399: Is one for which all incident radiation is reflected uniformly in all directions: τ = 0, α = 0, and ρ = 1. For a black body, τ = 0, α = 1, and ρ = 0. Planck offers a theoretical model for perfectly black bodies, which he noted do not exist in nature: besides their opaque interior, they have interfaces that are perfectly transmitting and non-reflective. Kirchhoff in 1860 introduced
4972-440: Is one that transmits none of the radiation that reaches it, although some may be reflected. That is, τ = 0 and α + ρ = 1. A transparent body is one that transmits all the radiation that reaches it. That is, τ = 1 and α = ρ = 0. A grey body is one where α , ρ and τ are constant for all wavelengths; this term also is used to mean a body for which α is temperature- and wavelength-independent. A white body
5085-1362: Is possible to define a cross power spectral density ( CPSD ) or cross spectral density ( CSD ). To begin, let us consider the average power of such a combined signal. P = lim T → ∞ 1 T ∫ − ∞ ∞ [ x T ( t ) + y T ( t ) ] ∗ [ x T ( t ) + y T ( t ) ] d t = lim T → ∞ 1 T ∫ − ∞ ∞ | x T ( t ) | 2 + x T ∗ ( t ) y T ( t ) + y T ∗ ( t ) x T ( t ) + | y T ( t ) | 2 d t {\displaystyle {\begin{aligned}P&=\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }\left[x_{T}(t)+y_{T}(t)\right]^{*}\left[x_{T}(t)+y_{T}(t)\right]dt\\&=\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }|x_{T}(t)|^{2}+x_{T}^{*}(t)y_{T}(t)+y_{T}^{*}(t)x_{T}(t)+|y_{T}(t)|^{2}dt\\\end{aligned}}} Using
5198-482: Is possible to determine the extent of the skull based on the properties of the material. For a rigid body , the boundary of an object may change over time by continuous translation and rotation . For a deformable body the boundary may also be continuously deformed over time in other ways. An object has an identity . In general two objects with identical properties, other than position at an instance in time, may be distinguished as two objects and may not occupy
5311-446: Is the cross-correlation of x ( t ) {\displaystyle x(t)} with y ( t ) {\displaystyle y(t)} and R y x ( τ ) {\displaystyle R_{yx}(\tau )} is the cross-correlation of y ( t ) {\displaystyle y(t)} with x ( t ) {\displaystyle x(t)} . In light of this,
5424-418: Is the discrete-time Fourier transform of x n . {\displaystyle x_{n}.} The sampling interval Δ t {\displaystyle \Delta t} is needed to keep the correct physical units and to ensure that we recover the continuous case in the limit Δ t → 0. {\displaystyle \Delta t\to 0.} But in
5537-559: Is the periodogram . The spectral density is usually estimated using Fourier transform methods (such as the Welch method ), but other techniques such as the maximum entropy method can also be used. Any signal that can be represented as a variable that varies in time has a corresponding frequency spectrum. This includes familiar entities such as visible light (perceived as color ), musical notes (perceived as pitch ), radio/TV (specified by their frequency, or sometimes wavelength ) and even
5650-478: Is the power spectral density (PSD, 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 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
5763-474: Is the reduced Planck constant , k B is the Boltzmann constant , G is the gravitational constant and M is the mass of the black hole. These predictions have not yet been tested either observationally or experimentally. The Big Bang theory is based upon the cosmological principle , which states that on large scales the Universe is homogeneous and isotropic. According to theory, the Universe approximately
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#17328687872245876-541: Is the value of the Fourier transform of x ( t ) {\displaystyle x(t)} at frequency f {\displaystyle f} (in Hz ). The theorem also holds true in the discrete-time cases. Since the integral on the left-hand side is the energy of the signal, the value of | x ^ ( f ) | 2 d f {\displaystyle \left|{\hat {x}}(f)\right|^{2}df} can be interpreted as
5989-402: Is then estimated to be E ( f ) / Δ f {\displaystyle E(f)/\Delta f} . In this example, since the power V ( t ) 2 / Z {\displaystyle V(t)^{2}/Z} has units of V Ω , the energy E ( f ) {\displaystyle E(f)} has units of V s Ω = J , and hence
6102-418: Is to estimate the spectral density of a random signal from a sequence of time samples. Depending on what is known about the signal, estimation techniques can involve parametric or non-parametric approaches, and may be based on time-domain or frequency-domain analysis. For example, a common parametric technique involves fitting the observations to an autoregressive model . A common non-parametric technique
6215-430: Is unity within the arbitrary period and zero elsewhere. P = lim T → ∞ 1 T ∫ − ∞ ∞ | x T ( t ) | 2 d t . {\displaystyle P=\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }\left|x_{T}(t)\right|^{2}\,dt.} Clearly, in cases where
6328-447: The power spectra of signals. The spectrum analyzer measures the magnitude of the short-time Fourier transform (STFT) of an input signal. If the signal being analyzed can be considered a stationary process, the STFT is a good smoothed estimate of its power spectral density. Primordial fluctuations , density variations in the early universe, are quantified by a power spectrum which gives
6441-498: The autocorrelation of x ( t ) {\displaystyle x(t)} form a Fourier transform pair, a result also known as the Wiener–Khinchin theorem (see also Periodogram ). As a physical example of how one might measure the energy spectral density of a signal, suppose V ( t ) {\displaystyle V(t)} represents the potential (in volts ) of an electrical pulse propagating along
6554-443: The convolution theorem has been used when passing from the 3rd to the 4th line. Now, if we divide the time convolution above by the period T {\displaystyle T} and take the limit as T → ∞ {\displaystyle T\rightarrow \infty } , it becomes the autocorrelation function of the non-windowed signal x ( t ) {\displaystyle x(t)} , which
6667-593: The cross-spectral density can similarly be calculated; as the PSD is related to the autocorrelation, so is the cross-spectral density related to the cross-correlation . Some properties of the PSD include: Given two signals x ( t ) {\displaystyle x(t)} and y ( t ) {\displaystyle y(t)} , each of which possess power spectral densities S x x ( f ) {\displaystyle S_{xx}(f)} and S y y ( f ) {\displaystyle S_{yy}(f)} , it
6780-467: The mind , which may not be a physical body, as in functionalist schools of thought. A physical body is an enduring object that exists throughout a particular trajectory of space and orientation over a particular duration of time , and which is located in the world of physical space (i.e., as studied by physics ). This contrasts with abstract objects such as mathematical objects which do not exist at any particular time or place. Examples are
6893-400: The power spectral density (PSD) which exists for stationary processes ; this describes how the power of a signal or time series is distributed over frequency, as in the simple example given previously. Here, power can be the actual physical power, or more often, for convenience with abstract signals, is simply identified with the squared value of the signal. For example, statisticians study
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#17328687872247006-467: The spacetime : roughly speaking, it means that for a given moment of time the body has some location in the space (although not necessarily amounting to the abstraction of a point in space and time ). A physical body as a whole is assumed to have such quantitative properties as mass , momentum , electric charge , other conserved quantities , and possibly other quantities. An object with known composition and described in an adequate physical theory
7119-470: The variance of a function over time x ( t ) {\displaystyle x(t)} (or over another independent variable), and using an analogy with electrical signals (among other physical processes), it is customary to refer to it as the power spectrum even when there is no physical power involved. If one were to create a physical voltage source which followed x ( t ) {\displaystyle x(t)} and applied it to
7232-564: The Fourier transform does not formally exist. Regardless, Parseval's theorem tells us that we can re-write the average power as follows. P = lim T → ∞ 1 T ∫ − ∞ ∞ | x ^ T ( f ) | 2 d f {\displaystyle P=\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }|{\hat {x}}_{T}(f)|^{2}\,df} Then
7345-701: The PSD is seen to be a special case of the CSD for x ( t ) = y ( t ) {\displaystyle x(t)=y(t)} . If x ( t ) {\displaystyle x(t)} and y ( t ) {\displaystyle y(t)} are real signals (e.g. voltage or current), their Fourier transforms x ^ ( f ) {\displaystyle {\hat {x}}(f)} and y ^ ( f ) {\displaystyle {\hat {y}}(f)} are usually restricted to positive frequencies by convention. Therefore, in typical signal processing,
7458-403: The PSD. Energy spectral density (ESD) would have units of V s Hz , since energy has units of power multiplied by time (e.g., watt-hour ). In the general case, the units of PSD will be the ratio of units of variance per unit of frequency; so, for example, a series of displacement values (in meters) over time (in seconds) will have PSD in units of meters squared per hertz, m /Hz. In
7571-526: The above expression for P is non-zero, the integral must grow without bound as T grows without bound. That is the reason why we cannot use the energy of the signal, which is that diverging integral, in such cases. In analyzing the frequency content of the signal x ( t ) {\displaystyle x(t)} , one might like to compute the ordinary Fourier transform x ^ ( f ) {\displaystyle {\hat {x}}(f)} ; however, for many signals of interest
7684-410: The addition or removal of material, if the system may be more simply described with the continued existence of the object, than in any other way. The addition or removal of material may discontinuously change the boundary of the object. The continuation of the object's identity is then based on the description of the system by continued identity being simpler than without continued identity. For example,
7797-422: The analysis of random vibrations , units of g Hz are frequently used for the PSD of acceleration , where g denotes the g-force . Mathematically, it is not necessary to assign physical dimensions to the signal or to the independent variable. In the following discussion the meaning of x ( t ) will remain unspecified, but the independent variable will be assumed to be that of time. A PSD can be either
7910-460: The black body must absorb or internally generate this amount of power P over the given area A . The cooling of a body due to thermal radiation is often approximated using the Stefan–Boltzmann law supplemented with a "gray body" emissivity ε ≤ 1 ( P / A = εσT ). The rate of decrease of the temperature of the emitting body can be estimated from the power radiated and
8023-417: The body's heat capacity . This approach is a simplification that ignores details of the mechanisms behind heat redistribution (which may include changing composition, phase transitions or restructuring of the body) that occur within the body while it cools, and assumes that at each moment in time the body is characterized by a single temperature. It also ignores other possible complications, such as changes in
8136-518: The capacity or desire to undertake actions, although humans in some cultures may tend to attribute such characteristics to non-living things. In classical mechanics a physical body is collection of matter having properties including mass , velocity , momentum and energy . The matter exists in a volume of three-dimensional space . This space is its extension . Interactions between objects are partly described by orientation and external shape. In continuum mechanics an object may be described as
8249-4010: The complex conjugate. Taking into account that F { x T ∗ ( − t ) } = ∫ − ∞ ∞ x T ∗ ( − t ) e − i 2 π f t d t = ∫ − ∞ ∞ x T ∗ ( t ) e i 2 π f t d t = ∫ − ∞ ∞ x T ∗ ( t ) [ e − i 2 π f t ] ∗ d t = [ ∫ − ∞ ∞ x T ( t ) e − i 2 π f t d t ] ∗ = [ F { x T ( t ) } ] ∗ = [ x ^ T ( f ) ] ∗ {\displaystyle {\begin{aligned}{\mathcal {F}}\left\{x_{T}^{*}(-t)\right\}&=\int _{-\infty }^{\infty }x_{T}^{*}(-t)e^{-i2\pi ft}dt\\&=\int _{-\infty }^{\infty }x_{T}^{*}(t)e^{i2\pi ft}dt\\&=\int _{-\infty }^{\infty }x_{T}^{*}(t)[e^{-i2\pi ft}]^{*}dt\\&=\left[\int _{-\infty }^{\infty }x_{T}(t)e^{-i2\pi ft}dt\right]^{*}\\&=\left[{\mathcal {F}}\left\{x_{T}(t)\right\}\right]^{*}\\&=\left[{\hat {x}}_{T}(f)\right]^{*}\end{aligned}}} and making, u ( t ) = x T ∗ ( − t ) {\displaystyle u(t)=x_{T}^{*}(-t)} , we have: | x ^ T ( f ) | 2 = [ x ^ T ( f ) ] ∗ ⋅ x ^ T ( f ) = F { x T ∗ ( − t ) } ⋅ F { x T ( t ) } = F { u ( t ) } ⋅ F { x T ( t ) } = F { u ( t ) ∗ x T ( t ) } = ∫ − ∞ ∞ [ ∫ − ∞ ∞ u ( τ − t ) x T ( t ) d t ] e − i 2 π f τ d τ = ∫ − ∞ ∞ [ ∫ − ∞ ∞ x T ∗ ( t − τ ) x T ( t ) d t ] e − i 2 π f τ d τ , {\displaystyle {\begin{aligned}\left|{\hat {x}}_{T}(f)\right|^{2}&=[{\hat {x}}_{T}(f)]^{*}\cdot {\hat {x}}_{T}(f)\\&={\mathcal {F}}\left\{x_{T}^{*}(-t)\right\}\cdot {\mathcal {F}}\left\{x_{T}(t)\right\}\\&={\mathcal {F}}\left\{u(t)\right\}\cdot {\mathcal {F}}\left\{x_{T}(t)\right\}\\&={\mathcal {F}}\left\{u(t)\mathbin {\mathbf {*} } x_{T}(t)\right\}\\&=\int _{-\infty }^{\infty }\left[\int _{-\infty }^{\infty }u(\tau -t)x_{T}(t)dt\right]e^{-i2\pi f\tau }d\tau \\&=\int _{-\infty }^{\infty }\left[\int _{-\infty }^{\infty }x_{T}^{*}(t-\tau )x_{T}(t)dt\right]e^{-i2\pi f\tau }\ d\tau ,\end{aligned}}} where
8362-423: The contributions of S x x ( f ) {\displaystyle S_{xx}(f)} and S y y ( f ) {\displaystyle S_{yy}(f)} are already understood. Note that S x y ∗ ( f ) = S y x ( f ) {\displaystyle S_{xy}^{*}(f)=S_{yx}(f)} , so the full contribution to
8475-2372: The cross power is, generally, from twice the real part of either individual CPSD . Just as before, from here we recast these products as the Fourier transform of a time convolution, which when divided by the period and taken to the limit T → ∞ {\displaystyle T\to \infty } becomes the Fourier transform of a cross-correlation function. S x y ( f ) = ∫ − ∞ ∞ [ lim T → ∞ 1 T ∫ − ∞ ∞ x T ∗ ( t − τ ) y T ( t ) d t ] e − i 2 π f τ d τ = ∫ − ∞ ∞ R x y ( τ ) e − i 2 π f τ d τ S y x ( f ) = ∫ − ∞ ∞ [ lim T → ∞ 1 T ∫ − ∞ ∞ y T ∗ ( t − τ ) x T ( t ) d t ] e − i 2 π f τ d τ = ∫ − ∞ ∞ R y x ( τ ) e − i 2 π f τ d τ , {\displaystyle {\begin{aligned}S_{xy}(f)&=\int _{-\infty }^{\infty }\left[\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }x_{T}^{*}(t-\tau )y_{T}(t)dt\right]e^{-i2\pi f\tau }d\tau =\int _{-\infty }^{\infty }R_{xy}(\tau )e^{-i2\pi f\tau }d\tau \\S_{yx}(f)&=\int _{-\infty }^{\infty }\left[\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }y_{T}^{*}(t-\tau )x_{T}(t)dt\right]e^{-i2\pi f\tau }d\tau =\int _{-\infty }^{\infty }R_{yx}(\tau )e^{-i2\pi f\tau }d\tau ,\end{aligned}}} where R x y ( τ ) {\displaystyle R_{xy}(\tau )}
8588-412: The diameter of the hole, part will be reflected. Similarly, even in perfect thermal equilibrium, the radiation inside a finite-sized cavity will not have an ideal Planck spectrum for wavelengths comparable to or larger than the size of the cavity. Suppose the cavity is held at a fixed temperature T and the radiation trapped inside the enclosure is at thermal equilibrium with the enclosure. The hole in
8701-408: The emissivity with temperature, and the role of other accompanying forms of energy emission, for example, emission of particles like neutrinos. If a hot emitting body is assumed to follow the Stefan–Boltzmann law and its power emission P and temperature T are known, this law can be used to estimate the dimensions of the emitting object, because the total emitted power is proportional to the area of
8814-417: The emitting surface. In this way it was found that X-ray bursts observed by astronomers originated in neutron stars with a radius of about 10 km, rather than black holes as originally conjectured. An accurate estimate of size requires some knowledge of the emissivity, particularly its spectral and angular dependence. Physical object Also in common usage, an object is not constrained to consist of
8927-438: The enclosure will allow some radiation to escape. If the hole is small, radiation passing in and out of the hole has negligible effect upon the equilibrium of the radiation inside the cavity. This escaping radiation will approximate black-body radiation that exhibits a distribution in energy characteristic of the temperature T and does not depend upon the properties of the cavity or the hole, at least for wavelengths smaller than
9040-427: The energy E {\displaystyle E} of a signal x ( t ) {\displaystyle x(t)} is: E ≜ ∫ − ∞ ∞ | x ( t ) | 2 d t . {\displaystyle E\triangleq \int _{-\infty }^{\infty }\left|x(t)\right|^{2}\ dt.} The energy spectral density
9153-524: The energy spectral density, the definition of the power spectral density can be generalized to discrete time variables x n {\displaystyle x_{n}} . As before, we can consider a window of − N ≤ n ≤ N {\displaystyle -N\leq n\leq N} with the signal sampled at discrete times t n = t 0 + ( n Δ t ) {\displaystyle t_{n}=t_{0}+(n\,\Delta t)} for
9266-891: The ergodicity of x ( t ) {\displaystyle x(t)} , that the power spectral density can be found as the Fourier transform of the autocorrelation function ( Wiener–Khinchin theorem ). Many authors use this equality to actually define the power spectral density. The power of the signal in a given frequency band [ f 1 , f 2 ] {\displaystyle [f_{1},f_{2}]} , where 0 < f 1 < f 2 {\displaystyle 0<f_{1}<f_{2}} , can be calculated by integrating over frequency. Since S x x ( − f ) = S x x ( f ) {\displaystyle S_{xx}(-f)=S_{xx}(f)} , an equal amount of power can be attributed to positive and negative frequency bands, which accounts for
9379-425: The estimate E ( f ) / Δ f {\displaystyle E(f)/\Delta f} of the energy spectral density has units of J Hz , as required. In many situations, it is common to forget the step of dividing by Z {\displaystyle Z} so that the energy spectral density instead has units of V Hz . This definition generalizes in a straightforward manner to
9492-404: The factor of 2 in the following form (such trivial factors depend on the conventions used): P bandlimited = 2 ∫ f 1 f 2 S x x ( f ) d f {\displaystyle P_{\textsf {bandlimited}}=2\int _{f_{1}}^{f_{2}}S_{xx}(f)\,df} More generally, similar techniques may be used to estimate
9605-458: The full CPSD is just one of the CPSD s scaled by a factor of two. CPSD Full = 2 S x y ( f ) = 2 S y x ( f ) {\displaystyle \operatorname {CPSD} _{\text{Full}}=2S_{xy}(f)=2S_{yx}(f)} For discrete signals x n and y n , the relationship between the cross-spectral density and the cross-covariance
9718-511: The incoming light in the spectral range from the ultra-violet to the far-infrared regions. Other examples of nearly perfect black materials are super black , prepared by chemically etching a nickel – phosphorus alloy , vertically aligned carbon nanotube arrays (like Vantablack ) and flower carbon nanostructures; all absorb 99.9% of light or more. A star or planet often is modeled as a black body, and electromagnetic radiation emitted from these bodies as black-body radiation . The figure shows
9831-399: The light), which ranges from about 5000 K at its outer boundary with the chromosphere to about 9500 K at its inner boundary with the convection zone approximately 500 km (310 mi) deep. A black hole is a region of spacetime from which nothing escapes. Around a black hole there is a mathematically defined surface called an event horizon that marks the point of no return . It
9944-422: The mathematical sciences the interval is often set to 1, which simplifies the results at the expense of generality. (also see normalized frequency ) The above definition of energy spectral density is suitable for transients (pulse-like signals) whose energy is concentrated around one time window; then the Fourier transforms of the signals generally exist. For continuous signals over all time, one must rather define
10057-404: 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
10170-405: The outside of the photosphere. With all these assumptions in place, the star emits black-body radiation at the temperature of the photosphere. Using this model the effective temperature of stars is estimated, defined as the temperature of a black body that yields the same surface flux of energy as the star. If a star were a black body, the same effective temperature would result from any region of
10283-597: The perfect isotropy of true black-body radiation by an observed anisotropy that varies with angle on the sky only to about one part in 100,000. The integration of Planck's law over all frequencies provides the total energy per unit of time per unit of surface area radiated by a black body maintained at a temperature T , and is known as the Stefan–Boltzmann law : where σ is the Stefan–Boltzmann constant , σ ≈ 5.67 × 10 W⋅m⋅K To remain in thermal equilibrium at constant temperature T ,
10396-568: The period T {\displaystyle T} is centered about some arbitrary time t = t 0 {\displaystyle t=t_{0}} : P = lim T → ∞ 1 T ∫ t 0 − T / 2 t 0 + T / 2 | x ( t ) | 2 d t {\displaystyle P=\lim _{T\to \infty }{\frac {1}{T}}\int _{t_{0}-T/2}^{t_{0}+T/2}\left|x(t)\right|^{2}\,dt} However, for
10509-563: The power spectral density is simply defined as the integrand above. From here, due to the convolution theorem , we can also view | x ^ T ( f ) | 2 {\displaystyle |{\hat {x}}_{T}(f)|^{2}} as the Fourier transform of the time convolution of x T ∗ ( − t ) {\displaystyle x_{T}^{*}(-t)} and x T ( t ) {\displaystyle x_{T}(t)} , where * represents
10622-589: The present day. It was a hole in the wall of a platinum box, divided by diaphragms, with its interior blackened with iron oxide. It was an important ingredient for the progressively improved measurements that led to the discovery of Planck's law. A version described in 1901 had its interior blackened with a mixture of chromium, nickel, and cobalt oxides. See also Hohlraum . There is interest in blackbody-like materials for camouflage and radar-absorbent materials for radar invisibility. They also have application as solar energy collectors, and infrared thermal detectors. As
10735-438: The process is a function of time, but one can similarly discuss data in the spatial domain being decomposed in terms of spatial frequency . In physics , the signal might be a wave, such as an electromagnetic wave , an acoustic wave , or the vibration of a mechanism. The power spectral density (PSD) of the signal describes the power present in the signal as a function of frequency, per unit frequency. Power spectral density
10848-537: The reference to "infinitely small thicknesses": An ideal body is now defined, called a blackbody . A blackbody allows all incident radiation to pass into it (no reflected energy) and internally absorbs all the incident radiation (no energy transmitted through the body). This is true for radiation of all wavelengths and for all angles of incidence. Hence the blackbody is a perfect absorber for all incident radiation. This section describes some concepts developed in connection with black bodies. A widely used model of
10961-404: The regular rotation of the earth. When these signals are viewed in the form of a frequency spectrum, certain aspects of the received signals or the underlying processes producing them are revealed. In some cases the frequency spectrum may include a distinct peak corresponding to a sine wave component. And additionally there may be peaks corresponding to harmonics of a fundamental peak, indicating
11074-475: The sake of dealing with the math that follows, it is more convenient to deal with time limits in the signal itself rather than time limits in the bounds of the integral. As such, we have an alternative representation of the average power, where x T ( t ) = x ( t ) w T ( t ) {\displaystyle x_{T}(t)=x(t)w_{T}(t)} and w T ( t ) {\displaystyle w_{T}(t)}
11187-451: The same collection of matter . Atoms or parts of an object may change over time. An object is usually meant to be defined by the simplest representation of the boundary consistent with the observations. However the laws of physics only apply directly to objects that consist of the same collection of matter. In physics , an object is an identifiable collection of matter , which may be constrained by an identifiable boundary, and may move as
11300-961: The same notation and methods as used for the power spectral density derivation, we exploit Parseval's theorem and obtain S x y ( f ) = lim T → ∞ 1 T [ x ^ T ∗ ( f ) y ^ T ( f ) ] S y x ( f ) = lim T → ∞ 1 T [ y ^ T ∗ ( f ) x ^ T ( f ) ] {\displaystyle {\begin{aligned}S_{xy}(f)&=\lim _{T\to \infty }{\frac {1}{T}}\left[{\hat {x}}_{T}^{*}(f){\hat {y}}_{T}(f)\right]&S_{yx}(f)&=\lim _{T\to \infty }{\frac {1}{T}}\left[{\hat {y}}_{T}^{*}(f){\hat {x}}_{T}(f)\right]\end{aligned}}} where, again,
11413-406: The same space at the same time (excluding component objects). An object's identity may be tracked using the continuity of the change in its boundary over time. The identity of objects allows objects to be arranged in sets and counted . The material in an object may change over time. For example, a rock may wear away or have pieces broken off it. The object will be regarded as the same object after
11526-519: The size of the hole. See the figure in the Introduction for the spectrum as a function of the frequency of the radiation, which is related to the energy of the radiation by the equation E = hf , with E = energy, h = Planck constant , f = frequency. At any given time the radiation in the cavity may not be in thermal equilibrium, but the second law of thermodynamics states that if left undisturbed it will eventually reach equilibrium, although
11639-448: 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
11752-554: The spectrum. For example, comparisons in the B (blue) or V (visible) range lead to the so-called B-V color index , which increases the redder the star, with the Sun having an index of +0.648 ± 0.006. Combining the U (ultraviolet) and the B indices leads to the U-B index, which becomes more negative the hotter the star and the more the UV radiation. Assuming the Sun is a type G2 V star, its U-B index
11865-429: The star, so that the temperature of the photosphere is nearly steady. Changes in the core lead to changes in the supply of energy to the photosphere, but such changes are slow on the time scale of interest here. Assuming these circumstances can be realized, the outer layer of the star is somewhat analogous to the example of an enclosure with a small hole in it, with the hole replaced by the limited transmission into space at
11978-501: The temperature of a black body that would emit the same total flux of electromagnetic energy. Isaac Newton introduced the notion of a black body in his 1704 book Opticks , with query 6 of the book stating: Do not black Bodies conceive heat more easily from Light than those of other Colours do, by reason that the Light falling on them is not reflected outwards, but enters into the Bodies, and
12091-424: The terminals of a one ohm resistor , then indeed the instantaneous power dissipated in that resistor would be given by x 2 ( t ) {\displaystyle x^{2}(t)} watts . The average power P {\displaystyle P} of a signal x ( t ) {\displaystyle x(t)} over all time is therefore given by the following time average, where
12204-505: The theoretical concept of a perfect black body with a completely absorbing surface layer of infinitely small thickness, but Planck noted some severe restrictions upon this idea. Planck noted three requirements upon a black body: the body must (i) allow radiation to enter but not reflect; (ii) possess a minimum thickness adequate to absorb the incident radiation and prevent its re-emission; (iii) satisfy severe limitations upon scattering to prevent radiation from entering and bouncing back out. As
12317-451: The time it takes to do so may be very long. Typically, equilibrium is reached by continual absorption and emission of radiation by material in the cavity or its walls. Radiation entering the cavity will be " thermalized " by this mechanism: the energy will be redistributed until the ensemble of photons achieves a Planck distribution . The time taken for thermalization is much faster with condensed matter present than with rarefied matter such as
12430-445: The total energy is found by integrating V ( t ) 2 / Z {\displaystyle V(t)^{2}/Z} with respect to time over the duration of the pulse. To find the value of the energy spectral density S ¯ x x ( f ) {\displaystyle {\bar {S}}_{xx}(f)} at frequency f {\displaystyle f} , one could insert between
12543-422: The transmission line and the resistor a bandpass filter which passes only a narrow range of frequencies ( Δ f {\displaystyle \Delta f} , say) near the frequency of interest and then measure the total energy E ( f ) {\displaystyle E(f)} dissipated across the resistor. The value of the energy spectral density at f {\displaystyle f}
12656-485: The ultraviolet spectrum, showing that both groupings of star emit less ultraviolet light than a black body with the same B-V index. It is perhaps surprising that they fit a black body curve as well as they do, considering that stars have greatly different temperatures at different depths. For example, the Sun has an effective temperature of 5780 K, which can be compared to the temperature of its photosphere (the region generating
12769-519: The wrong sources. It has long been known that a lamp-black coating will make a body nearly black. An improvement on lamp-black is found in manufactured carbon nanotubes . Nano-porous materials can achieve refractive indices nearly that of vacuum, in one case obtaining average reflectance of 0.045%. In 2009, a team of Japanese scientists created a material called nanoblack which is close to an ideal black body, based on vertically aligned single-walled carbon nanotubes . This absorbs between 98% and 99% of
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