4730:
47:
4072:
59:
4725:{\displaystyle {\begin{aligned}\left|{\hat {x}}_{T}(f)\right|^{2}&=^{*}\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 }\lefte^{-i2\pi f\tau }d\tau \\&=\int _{-\infty }^{\infty }\lefte^{-i2\pi f\tau }\ d\tau ,\end{aligned}}}
9160:
8213:
4008:
2369:
7197:
7664:
3472:
72:
6781:
5214:
6176:) 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 a more accurate estimate of the theoretical PSD of the physical process underlying the individual measurements. This computed PSD is sometimes called a
2216:
8208:{\displaystyle {\begin{aligned}S_{xy}(f)&=\int _{-\infty }^{\infty }\lefte^{-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 }\lefte^{-i2\pi f\tau }d\tau =\int _{-\infty }^{\infty }R_{yx}(\tau )e^{-i2\pi f\tau }d\tau ,\end{aligned}}}
4003:{\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)^{*}dt\\&=\left^{*}\\&=\left^{*}\\&=\left^{*}\end{aligned}}}
7478:
9098:
of the spectrum is rather constant, since variations in the ASD will then be proportional to variations in the signal's voltage level itself. But it is mathematically preferred to use the PSD, since only in that case is the area under the curve meaningful in terms of actual power over all frequency
254:
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
2393:
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 the
9134:) and even 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
4893:
5757:
are not independent. Just using 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
6134:
9423:
2380:
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 the
7192:{\displaystyle {\begin{aligned}P&=\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }\left^{*}\leftdt\\&=\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}}}
1993:
516:
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 the analysis of random
875:
2759:
However, for 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
5403:
2977:
8834:
2755:
3174:
7204:
8979:
The spectral density is a function of frequency, not a function of time. However, the spectral density of a small window of a longer signal may be calculated, and plotted versus time associated with the window. Such a graph is called a
980:
3296:
6420:
729:
407:
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
5970:
5679:
1285:
8680:
9146:
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
9304:
6601:
5209:{\displaystyle \lim _{T\to \infty }{\frac {1}{T}}\left|{\hat {x}}_{T}(f)\right|^{2}=\int _{-\infty }^{\infty }\lefte^{-i2\pi f\tau }\ d\tau =\int _{-\infty }^{\infty }R_{xx}(\tau )e^{-i2\pi f\tau }d\tau }
740:
5588:, an equal amount of power can be attributed to positive and negative frequency bands, which accounts for the factor of 2 in the following form (such trivial factors depend on the conventions used):
2979:
Clearly, in cases where 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
7669:
7209:
6786:
4077:
3477:
2869:
9067:
cannot generally be uniquely recovered from the power spectral density alone without the phase part. Although these are also
Fourier transform pairs, there is no symmetry (as there is for the
8705:
2618:
3060:
1101:
227:, 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 any sort of signal (including
9463:
5917:
1988:
2211:{\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}},}
9516:
does not need to be absolutely integrable, it only needs to exist. But the integral can no longer be interpreted as usual. The formula also makes sense if interpreted as involving
7625:
3379:
493:, 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
259:
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
2828:
6293:
5586:
5260:
4783:
5519:
5965:
4067:
1622:
1350:
2261:
5858:
2347:
880:
8589:
8551:
8353:
8252:
4855:
3431:
3051:
1875:
1758:
7655:
6336:
1802:
1574:
1530:
8455:
7556:
7517:
6760:
6721:
6469:
162:
3467:
2864:
2505:
2318:
1669:
293:
9849:
2613:
652:
9556:
9514:
2295:
9208:(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.
5823:
1929:
9065:
8513:
8484:
8411:
8382:
8314:
8285:
6682:
6653:
6505:
5789:
5591:
5245:
4884:
4816:
3013:
2560:
2462:
2425:
1831:
1698:
1423:
1383:
1190:
1154:
1013:
647:
390:
329:
197:
5755:
5727:
6198:
6174:
6154:
5699:
4757:
2580:
2531:
1895:
1718:
1642:
1486:
1458:
1125:
1036:
618:
349:
221:
8602:
7473:{\displaystyle {\begin{aligned}S_{xy}(f)&=\lim _{T\to \infty }{\frac {1}{T}}\left&S_{yx}(f)&=\lim _{T\to \infty }{\frac {1}{T}}\left\end{aligned}}}
5473:
9747:
8865:
approaches, and may be based on time-domain or frequency-domain analysis. For example, a common parametric technique involves fitting the observations to an
7631:. 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
2427:(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
3187:
540:
Mathematically, it is not necessary to assign physical dimensions to the signal or to the independent variable. In the following discussion the meaning of
9219:, density variations in the early universe, are quantified by a power spectrum which gives the power of the variations as a function of spatial scale.
9114:
Any signal that can be represented as a variable that varies in time has a corresponding frequency spectrum. This includes familiar entities such as
404:, 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.
90:
8997:
A "spectrum" generally means the power spectral density, as discussed above, which depicts the distribution of signal content over frequency. For
1201:
1042:). 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
9035:
of a transfer function, which also includes a phase (or equivalently, a real and imaginary part) as a function of frequency. The time-domain
2349: But in the mathematical sciences the interval is often set to 1, which simplifies the results at the expense of generality. (also see
238:
When the energy of the signal is concentrated around a finite time interval, especially if its total energy is finite, one may compute the
6514:
6136:
Note that a single estimate of the PSD can be obtained through a finite number of samplings. As before, the actual PSD is achieved when
396:, and generalizations based on Fourier analysis. In many cases the time domain is not specifically employed in practice, such as when a
1877:
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
30:
This article is about signal processing and relation of spectra to time-series. For further applications in the physical sciences, see
9645:
6129:{\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}}
5701:
is finite rather than approaching infinity. This results in decreased spectral coverage and resolution since frequencies of less than
9301:, p. 30) still use the non-normalized Fourier transform in a formal way to formulate a definition of the power spectral density
733:
The energy spectral density is most suitable for transientsâthat is, pulse-like signalsâhaving a finite total energy. Finite or not,
424:. Typically the process is a function of time, but one can similarly discuss data in the spatial domain being decomposed in terms of
8910:
of a signal is the midpoint of its spectral density function, i.e. the frequency that divides the distribution into two equal parts.
7201:
Using the same notation and methods as used for the power spectral density derivation, we exploit
Parseval's theorem and obtain
10351:
10133:
10114:
10095:
10076:
10049:
10030:
9917:
9898:
9879:
9757:
2373:
9566:, whose distribution function is F (but not necessarily as a function and not necessarily possessing a probability density).
6207:
can similarly be calculated; as the PSD is related to the autocorrelation, so is the cross-spectral density related to the
5681:
More generally, similar techniques may be used to estimate a time-varying spectral density. In this case the time interval
9418:{\displaystyle \langle {\hat {x}}(\omega ){\hat {x}}^{\ast }(\omega ')\rangle =2\pi f(\omega )\delta (\omega -\omega '),}
8897:
8885:
10316:
1107:
multiplied by an infinitesimally small frequency interval, describing the energy contained in the signal at frequency
1045:
870:{\displaystyle \int _{-\infty }^{\infty }|x(t)|^{2}\,dt=\int _{-\infty }^{\infty }\left|{\hat {x}}(f)\right|^{2}\,df,}
462:
present in the signal as a function of frequency, per unit frequency. Power spectral density is commonly expressed in
10176:
108:
17:
9469:. Such formal statements may sometimes be useful to guide the intuition, but should always be used with utmost care.
9094:), which is the square root of the PSD; the ASD of a voltage signal has units of V Hz. This is useful when the
9525:
392:
as it fluctuates at an extremely high frequency. Obtaining a spectrum from time series such as these involves the
9243:
8591:
are usually restricted to positive frequencies by convention. Therefore, in typical signal processing, the full
2264:
8857:
from a sequence of time samples. Depending on what is known about the signal, estimation techniques can involve
9428:
6180:. This periodogram converges to the true PSD as the number of estimates as well as the averaging time interval
5863:
1934:
568:
but with only half the amplitude. Noise PSDs are generally one-sided in engineering and two-sided in physics.
9483:
7561:
6621:
5794:
Just as with the energy spectral density, the definition of the power spectral density can be generalized to
5398:{\displaystyle S_{xx}(f)=\int _{-\infty }^{\infty }R_{xx}(\tau )e^{-i2\pi f\tau }\,d\tau ={\hat {R}}_{xx}(f)}
3322:
409:
2972:{\displaystyle P=\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }\left|x_{T}(t)\right|^{2}\,dt.}
2376:
temperature anisotropy in terms of the angular scale. The solid line is a theoretical model, for comparison.
2320:
is needed to keep the correct physical units and to ensure that we recover the continuous case in the limit
9205:
8987:
7627:, so the full contribution to the cross power is, generally, from twice the real part of either individual
2763:
31:
8829:{\displaystyle S_{xy}(f)=\sum _{n=-\infty }^{\infty }R_{xy}(\tau _{n})e^{-i2\pi f\tau _{n}}\,\Delta \tau }
6231:
5524:
4762:
2750:{\displaystyle P=\lim _{T\to \infty }{\frac {1}{T}}\int _{t_{0}-T/2}^{t_{0}+T/2}\left|x(t)\right|^{2}\,dt}
10228:
9479:
9268:
9253:
8844:
6224:
The power spectrum is always real and non-negative, and the spectrum of a real valued process is also an
5478:
5248:
5247:, that the power spectral density can be found as the Fourier transform of the autocorrelation function (
3169:{\displaystyle P=\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }|{\hat {x}}_{T}(f)|^{2}\,df}
2362:
2350:
1386:
5922:
4013:
1579:
1307:
54:
as a function of optical wavelength shows peaks at atomic transitions, indicated by the numbered arrows.
10321:
9517:
2221:
1464:
resistor (so that all of the pulse energy is delivered to the resistor and none is reflected back). By
597:
589:
10149:
1576:
with respect to time over the duration of the pulse. To find the value of the energy spectral density
10341:
9971:
8960:
of the spectrum density. It describes one point in time (one window, to be precise). For example, in
8862:
6320:
5828:
2323:
9026:
8969:
8556:
8518:
8319:
8218:
4821:
3392:
3053:; however, for many signals of interest the Fourier transform does not formally exist. Regardless,
3018:
1840:
1723:
9072:
7634:
10346:
9216:
6603:
Note that the previous expression for total power (signal variance) is a special case where
1763:
1535:
1491:
1396:
As a physical example of how one might measure the energy spectral density of a signal, suppose
9891:
Introduction to Random
Signals and Applied Kalman Filtering with Matlab Exercises and Solutions
9248:
9233:
9172:
8416:
7522:
7483:
6726:
6687:
6435:
128:
46:
9167:
signal with frequency on the horizontal axis and time increasing upwards on the vertical axis.
3436:
2833:
2474:
2300:
1651:
262:
10331:
10169:
9563:
9176:
8858:
6326:
3054:
2585:
734:
363:
of a light source is determined by the spectrum of the electromagnetic wave's electric field
296:
9531:
9489:
2270:
548:) will remain unspecified, but the independent variable will be assumed to be that of time.
9653:
9559:
9466:
9006:
8866:
5801:
5759:
1907:
1438:
498:
447:
437:
9810:
9041:
9009:) the complete frequency response may be graphed in two parts: power versus frequency and
8489:
8460:
8387:
8358:
8290:
8261:
6658:
6629:
6481:
5765:
5221:
4860:
4792:
2989:
2536:
2438:
2401:
1807:
1674:
1399:
1359:
1166:
1130:
989:
623:
366:
305:
173:
86:
8:
10336:
9258:
9180:
5732:
5704:
4733:
3316:
737:(or Plancherel's theorem) gives us an alternate expression for the energy of the signal:
228:
10295:
10009:
9953:
9928:
9273:
9031:
6299:
6183:
6159:
6139:
5684:
4742:
2565:
2516:
2386:
1901:
1880:
1703:
1627:
1471:
1461:
1443:
1426:
1110:
1021:
975:{\displaystyle {\hat {x}}(f)\triangleq \int _{-\infty }^{\infty }e^{-i2\pi ft}x(t)\ dt}
603:
565:
413:
334:
206:
10249:
5432:
10233:
10129:
10110:
10091:
10072:
10055:
10045:
10026:
10013:
10001:
9958:
9929:"A Low Computational Cost Algorithm for REM Sleep Detection Using Single Channel EEG"
9913:
9894:
9875:
9753:
9192:
8998:
8906:
8877:
8850:
8255:
7658:
6208:
3382:
3291:{\displaystyle S_{xx}(f)=\lim _{T\to \infty }{\frac {1}{T}}|{\hat {x}}_{T}(f)|^{2}\,}
1434:
983:
425:
393:
122:
51:
10264:
2471:, then indeed the instantaneous power dissipated in that resistor would be given by
1900:
This definition generalizes in a straightforward manner to a discrete signal with a
1390:
10280:
10162:
9991:
9983:
9948:
9940:
9521:
9263:
9036:
6415:{\displaystyle P=\operatorname {Var} (x)=\int _{-\infty }^{\infty }\!S_{xx}(f)\,df}
6333:(average power) of a process by integrating the power spectrum over all frequency:
1104:
506:
397:
224:
8702:, the relationship between the cross-spectral density and the cross-covariance is
9278:
9238:
9127:
9068:
5729:
are not sampled, and results at frequencies which are not an integer multiple of
4786:
2562:
over all time is therefore given by the following time average, where the period
2390:
1645:
1353:
582:
459:
359:
of a musical instrument are immediately determined from a spectral analysis. The
200:
165:
2431:
even when there is no physical power involved. If one were to create a physical
724:{\displaystyle E\triangleq \int _{-\infty }^{\infty }\left|x(t)\right|^{2}\ dt.}
10326:
9987:
9188:
8973:
8961:
8957:
8941:
monitoring, in which case SEF has variously been used to estimate the depth of
40:
9944:
58:
10310:
10059:
9123:
9115:
9010:
8854:
6225:
5795:
5674:{\displaystyle P_{\textsf {bandlimited}}=2\int _{f_{1}}^{f_{2}}S_{xx}(f)\,df}
352:
62:
The voice waveform over time (left) has a broad audio power spectrum (right).
8986:. This is the basis of a number of spectral analysis techniques such as the
8937:
is in the range 75 to 95. It is more particularly a popular measure used in
1700:
dissipated across the resistor. The value of the energy spectral density at
10290:
10206:
10005:
9996:
9962:
9587:
9148:
8965:
8881:
6778:). To begin, let us consider the average power of such a combined signal.
1465:
1280:{\displaystyle {\bar {S}}_{xx}(f)\triangleq \left|{\hat {x}}(f)\right|^{2}}
600:
is used in the generalized sense of signal processing; that is, the energy
526:
401:
10188:
3178:
Then the power spectral density is simply defined as the integrand above.
10285:
10211:
9080:
9076:
8982:
8870:
8675:{\displaystyle \operatorname {CPSD} _{\text{Full}}=2S_{xy}(f)=2S_{yx}(f)}
8413:. In light of this, the PSD is seen to be a special case of the CSD for
6177:
3386:
593:
421:
10154:
9817:
9171:
The concept and use of the power spectrum of a signal is fundamental in
8972:
properties, as defined by the range of brightness levels in each of the
1671:, say) near the frequency of interest and then measure the total energy
10239:
9591:
9228:
9159:
9131:
8942:
451:
9562:
can be used to prove that its
Fourier transform exists as a positive
9139:
9135:
9002:
8933:
percent of the total power of a given signal are located; typically,
8926:
8515:
are real signals (e.g. voltage or current), their
Fourier transforms
3469:, where * represents the complex conjugate. Taking into account that
1016:
518:
510:
256:
9974:(2017). "An Ultralow-Power Sleep Spindle Detection System on Chip".
9829:
9703:
1897:
so that the energy spectral density instead has units of V Hz.
1644:, one could insert between the transmission line and the resistor a
9164:
9109:
8991:
6330:
2468:
2395:
2368:
463:
232:
581:"Energy spectral density" redirects here. Not to be confused with
10185:
9874:. Berlin ; New York: Springer Science & Business Media.
9715:
4887:
2432:
534:
490:
443:
417:
39:"Spectral power density" redirects here. Not to be confused with
9138:
component. And additionally there may be peaks corresponding to
27:
Relative importance of certain frequencies in a composite signal
9850:"The Fundamentals of FFT-Based Signal Analysis and Measurement"
6203:
If two signals both possess power spectral densities, then the
356:
231:) as analyzed in terms of its frequency content, is called its
168:
9691:
9667:
9029:
of the transfer function. This is not to be confused with the
9847:
9778:
9727:
9184:
9142:
of a fundamental peak, indicating a periodic signal which is
9119:
8946:
1834:
479:
360:
9969:
9823:
8968:, the spectral envelope of a feature is the boundary of its
3057:
tells us that we can re-write the average power as follows.
10259:
10254:
6214:
4739:
Now, if we divide the time convolution above by the period
3015:, one might like to compute the ordinary Fourier transform
2507:
1430:
471:
10069:
Fundamentals of Noise and
Vibration Analysis for Engineers
1385:
form a
Fourier transform pair, a result also known as the
497:
to the actual power delivered by that signal into a given
331:
often contains essential information about the nature of
10244:
10200:
9910:
9790:
8938:
5521:, can be calculated by integrating over frequency. Since
4736:
has been used when passing from the 3rd to the 4th line.
2866:
is unity within the arbitrary period and zero elsewhere.
2465:
9926:
9835:
9679:
9613:
9071:) forcing the Fourier transform to be real-valued. See
4890:, which is true in most, but not all, practical cases.
1039:
9927:
Imtiaz, Syed Anas; Rodriguez-Villegas, Esther (2014).
9766:
9625:
9534:
9492:
9431:
9307:
9044:
8708:
8605:
8559:
8521:
8492:
8463:
8419:
8390:
8361:
8322:
8293:
8264:
8221:
7667:
7637:
7564:
7525:
7486:
7207:
6784:
6729:
6690:
6661:
6632:
6517:
6484:
6438:
6339:
6234:
6186:
6162:
6142:
5973:
5925:
5866:
5831:
5804:
5768:
5735:
5707:
5687:
5594:
5527:
5481:
5435:
5263:
5224:
4896:
4863:
4824:
4795:
4765:
4745:
4075:
4016:
3475:
3439:
3395:
3325:
3190:
3063:
3021:
2992:
2872:
2836:
2766:
2621:
2588:
2568:
2539:
2519:
2477:
2441:
2404:
2326:
2303:
2273:
2224:
1996:
1937:
1910:
1883:
1843:
1810:
1766:
1726:
1706:
1677:
1654:
1630:
1582:
1538:
1494:
1474:
1446:
1402:
1362:
1310:
1204:
1169:
1133:
1113:
1048:
1024:
992:
883:
743:
655:
626:
606:
369:
337:
308:
265:
209:
176:
131:
9976:
IEEE Transactions on
Biomedical Circuits and Systems
9908:
Davenport, Wilbur B. (Jr); Root, William L. (1987).
9204:
The spectrum analyzer measures the magnitude of the
10109:. New York: Springer Science & Business Media.
9889:Brown, Robert Grover; Hwang, Patrick Y. C. (1997).
6596:{\displaystyle F(f)=2\int _{0}^{f}S_{xx}(f')\,df'.}
6511:power contained in frequencies from DC to f using:
5218:From here we see, again assuming the ergodicity of
501:. So one might use units of V Hz for the PSD.
81:
may be too technical for most readers to understand
10025:. West Sussex, England ; Hoboken, NJ: Wiley.
9745:
9550:
9508:
9457:
9417:
9059:
8828:
8674:
8583:
8545:
8507:
8478:
8449:
8405:
8376:
8347:
8308:
8279:
8246:
8207:
7649:
7619:
7550:
7511:
7472:
7191:
6754:
6715:
6676:
6647:
6595:
6499:
6463:
6414:
6287:
6192:
6168:
6148:
6128:
5959:
5911:
5852:
5817:
5783:
5749:
5721:
5693:
5673:
5580:
5513:
5467:
5429:The power of the signal in a given frequency band
5397:
5239:
5208:
4878:
4849:
4810:
4777:
4751:
4724:
4061:
4002:
3461:
3425:
3373:
3290:
3168:
3045:
3007:
2971:
2858:
2822:
2749:
2607:
2574:
2554:
2525:
2499:
2456:
2419:
2341:
2312:
2289:
2255:
2210:
1982:
1923:
1889:
1869:
1825:
1796:
1752:
1712:
1692:
1663:
1636:
1616:
1568:
1524:
1480:
1452:
1417:
1377:
1344:
1279:
1184:
1148:
1119:
1095:
1030:
1007:
974:
869:
723:
641:
612:
505:(ESD) would have units of V s Hz, since
384:
343:
323:
287:
215:
191:
156:
10085:
9709:
9697:
9673:
6684:, each of which possess power spectral densities
6382:
2986:In analyzing the frequency content of the signal
1648:which passes only a narrow range of frequencies (
10308:
10086:Oppenheim, Alan V.; Verghese, George C. (2016).
9601:
8876:The spectral density is usually estimated using
7990:
7725:
7370:
7242:
6969:
6800:
6315:) can be reconstructed from its power spectrum S
6000:
4995:
4898:
3217:
3071:
2880:
2629:
2032:
1096:{\displaystyle \left|{\hat {x}}(f)\right|^{2}df}
9808:
9586:
6615:
10039:
9784:
9733:
9721:
8849:The goal of spectral density estimation is to
1532:, so the total energy is found by integrating
1468:, the power delivered to the resistor at time
596:is distributed with frequency. Here, the term
10170:
10066:
9907:
9848:Michael Cerna & Audrey F. Harvey (2000).
9796:
9619:
1433:) of an electrical pulse propagating along a
509:has units of power multiplied by time (e.g.,
9365:
9308:
9130:(specified by their frequency, or sometimes
1460:, and suppose the line is terminated with a
525: Hz are frequently used for the PSD of
489:When a signal is defined in terms only of a
10221:
10104:
9298:
8869:. A common non-parametric technique is the
6204:
5422:Many authors use this equality to actually
1931:such as a signal sampled at discrete times
560:function of only positive frequencies or a
400:is used to obtain a spectrum of light in a
10177:
10163:
9888:
9772:
9582:
9580:
9578:
9191:technology. Electronic instruments called
9025:. Less commonly, the two parts may be the
5860:with the signal sampled at discrete times
5791:evaluated over the specified time window.
588:Energy spectral density describes how the
576:
250:), which applies to signals existing over
10184:
10128:. New York Weinheim: Wiley-Interscience.
10071:. Cambridge: Cambridge University Press.
9995:
9952:
9458:{\displaystyle \delta (\omega -\omega ')}
9154:
8819:
6578:
6405:
6106:
5950:
5912:{\displaystyle t_{n}=t_{0}+(n\,\Delta t)}
5899:
5825:. As before, we can consider a window of
5664:
5601:
5354:
3287:
3159:
2959:
2740:
2464:and applied it to the terminals of a one
2356:
2132:
1983:{\displaystyle t_{n}=t_{0}+(n\,\Delta t)}
1970:
857:
791:
551:
446:, the signal might be a wave, such as an
109:Learn how and when to remove this message
93:, without removing the technical details.
10040:Miller, Scott; Childers, Donald (2012).
9869:
9824:Iranmanesh & Rodriguez-Villegas 2017
9631:
9158:
6215:Properties of the power spectral density
2983:that diverging integral, in such cases.
2367:
57:
45:
9575:
7620:{\displaystyle S_{xy}^{*}(f)=S_{yx}(f)}
3374:{\displaystyle |{\hat {x}}_{T}(f)|^{2}}
454:, or the vibration of a mechanism. The
416:, as well as in many other branches of
14:
10309:
10067:Norton, M. P.; Karczub, D. G. (2003).
2582:is centered about some arbitrary time
255:all time would generally be infinite.
10158:
10150:Power Spectral Density Matlab scripts
10123:
10020:
9811:"Signal Processing Course, chapter 7"
9685:
9643:
9607:
2823:{\displaystyle x_{T}(t)=x(t)w_{T}(t)}
2374:cosmic microwave background radiation
295:over the time domain, as dictated by
223:composing that signal. According to
91:make it understandable to non-experts
10105:Risken, Hannes; Frank, Till (1996).
9836:Imtiaz & Rodriguez-Villegas 2014
9524:, not in the sense of a statistical
9482:makes sense of this formula for any
9195:are used to observe and measure the
9187:, and related systems, plus passive
8884:), but other techniques such as the
6288:{\displaystyle S_{xx}(-f)=S_{xx}(f)}
6219:Some properties of the PSD include:
5581:{\displaystyle S_{xx}(-f)=S_{xx}(f)}
5254:
4789:function of the non-windowed signal
4778:{\displaystyle T\rightarrow \infty }
3181:
1833:has units of V s Ί =
1195:
65:
8898:spectral density (physical science)
8891:
7657:becomes the Fourier transform of a
7558:are already understood. Note that
7480:where, again, the contributions of
6302:x(t), the autocorrelation function
5514:{\displaystyle 0<f_{1}<f_{2}}
2372:The power spectrum of the measured
1760:. In this example, since the power
302:The spectrum of a physical process
24:
8820:
8753:
8748:
8140:
8135:
8028:
8023:
8000:
7979:
7974:
7875:
7870:
7763:
7758:
7735:
7714:
7709:
7644:
7380:
7252:
7007:
7002:
6979:
6838:
6833:
6810:
6377:
6372:
6107:
6021:
6010:
5960:{\displaystyle T=(2N+1)\,\Delta t}
5951:
5900:
5302:
5297:
5148:
5143:
5033:
5028:
5005:
4984:
4979:
4908:
4772:
4616:
4611:
4593:
4588:
4486:
4481:
4463:
4458:
4387:
4341:
4309:
4263:
4216:
4062:{\displaystyle u(t)=x_{T}^{*}(-t)}
3896:
3812:
3807:
3710:
3705:
3630:
3625:
3544:
3539:
3482:
3227:
3109:
3104:
3081:
2918:
2913:
2890:
2639:
2327:
2304:
2133:
2050:
2042:
1971:
1861:
1744:
1655:
1617:{\displaystyle {\bar {S}}_{xx}(f)}
1345:{\displaystyle {\bar {S}}_{xx}(f)}
921:
916:
814:
809:
757:
752:
675:
670:
458:(PSD) of the signal describes the
25:
10363:
10143:
2256:{\displaystyle {\hat {x}}_{d}(f)}
10216:
10088:Signals, Systems & Inference
10042:Probability and Random Processes
9933:Annals of Biomedical Engineering
9526:Cumulative distribution function
9177:electronic communication systems
4411:
412:and in the statistical study of
70:
9841:
9802:
9739:
9472:
9244:Least-squares spectral analysis
9122:), musical notes (perceived as
9103:
9073:Ultrashort pulse#Spectral phase
6432:) with power spectral density
5919:for a total measurement period
2265:discrete-time Fourier transform
10044:. Boston, MA: Academic Press.
9912:. New York: Wiley-IEEE Press.
9637:
9592:"Spectral Analysis of Signals"
9452:
9435:
9409:
9392:
9386:
9380:
9362:
9351:
9339:
9329:
9323:
9317:
9291:
9099:or over a specified bandwidth.
9054:
9048:
8784:
8771:
8728:
8722:
8669:
8663:
8641:
8635:
8578:
8572:
8566:
8540:
8534:
8528:
8502:
8496:
8473:
8467:
8444:
8438:
8429:
8423:
8400:
8394:
8371:
8365:
8342:
8336:
8303:
8297:
8274:
8268:
8241:
8235:
8164:
8158:
8079:
8073:
8060:
8048:
7997:
7956:
7950:
7899:
7893:
7814:
7808:
7795:
7783:
7732:
7691:
7685:
7641:
7614:
7608:
7589:
7583:
7545:
7539:
7506:
7500:
7458:
7452:
7440:
7430:
7424:
7407:
7377:
7359:
7353:
7330:
7324:
7312:
7302:
7296:
7279:
7249:
7231:
7225:
7169:
7164:
7158:
7144:
7137:
7131:
7118:
7112:
7091:
7085:
7072:
7066:
7038:
7033:
7027:
7013:
6976:
6944:
6938:
6922:
6916:
6887:
6881:
6865:
6859:
6807:
6749:
6743:
6710:
6704:
6671:
6665:
6642:
6636:
6575:
6564:
6527:
6521:
6507:, which specifies the average
6494:
6488:
6458:
6452:
6402:
6396:
6358:
6352:
6282:
6276:
6257:
6248:
6028:
6018:
6007:
5993:
5987:
5947:
5932:
5906:
5893:
5853:{\displaystyle -N\leq n\leq N}
5778:
5772:
5661:
5655:
5575:
5569:
5550:
5541:
5462:
5436:
5411:
5392:
5386:
5371:
5326:
5320:
5283:
5277:
5234:
5228:
5172:
5166:
5084:
5078:
5065:
5053:
5002:
4954:
4948:
4936:
4905:
4873:
4867:
4844:
4838:
4805:
4799:
4769:
4667:
4661:
4648:
4636:
4525:
4519:
4506:
4494:
4432:
4426:
4406:
4400:
4367:
4361:
4328:
4322:
4289:
4283:
4250:
4241:
4201:
4195:
4183:
4164:
4160:
4154:
4142:
4132:
4111:
4105:
4093:
4056:
4047:
4026:
4020:
3982:
3976:
3964:
3922:
3916:
3833:
3827:
3768:
3739:
3736:
3730:
3656:
3650:
3573:
3564:
3516:
3507:
3456:
3450:
3420:
3411:
3361:
3356:
3350:
3338:
3327:
3304:
3277:
3272:
3266:
3254:
3243:
3224:
3210:
3204:
3149:
3144:
3138:
3126:
3115:
3078:
3040:
3034:
3028:
3002:
2996:
2945:
2939:
2887:
2853:
2847:
2817:
2811:
2798:
2792:
2783:
2777:
2726:
2720:
2636:
2549:
2543:
2494:
2488:
2451:
2445:
2414:
2408:
2342:{\displaystyle \Delta t\to 0.}
2333:
2250:
2244:
2232:
2189:
2183:
2171:
2057:
2047:
2039:
2025:
2019:
2004:
1977:
1964:
1853:
1847:
1820:
1814:
1777:
1770:
1736:
1730:
1687:
1681:
1611:
1605:
1590:
1549:
1542:
1505:
1498:
1412:
1406:
1372:
1366:
1339:
1333:
1318:
1293:
1263:
1257:
1251:
1233:
1227:
1212:
1179:
1173:
1073:
1067:
1061:
1002:
996:
960:
954:
902:
896:
890:
843:
837:
831:
781:
776:
770:
763:
695:
689:
636:
630:
564:function of both positive and
438:Fourier transform § Units
379:
373:
318:
312:
282:
276:
199:describes the distribution of
186:
180:
151:
145:
13:
1:
9870:Birolini, Alessandro (2007).
9863:
9710:Oppenheim & Verghese 2016
9698:Oppenheim & Verghese 2016
9674:Oppenheim & Verghese 2016
9484:wide-sense stationary process
8921:), usually expressed as "SEF
8838:
8599:s scaled by a factor of two.
8584:{\displaystyle {\hat {y}}(f)}
8546:{\displaystyle {\hat {x}}(f)}
8348:{\displaystyle R_{yx}(\tau )}
8247:{\displaystyle R_{xy}(\tau )}
6762:, it is possible to define a
6622:Coherence (signal processing)
6616:Cross power spectral density
4850:{\displaystyle R_{xx}(\tau )}
3426:{\displaystyle x_{T}^{*}(-t)}
3046:{\displaystyle {\hat {x}}(f)}
2297: The sampling interval
1870:{\displaystyle E(f)/\Delta f}
1804:has units of V Ί, the energy
1753:{\displaystyle E(f)/\Delta f}
571:
410:statistical signal processing
10352:Spectrum (physical sciences)
9528:) instead of functions. If
9211:
9206:short-time Fourier transform
9086:Sometimes one encounters an
8988:short-time Fourier transform
8355:is the cross-correlation of
7650:{\displaystyle T\to \infty }
6764:cross power spectral density
5426:the power spectral density.
242:. More commonly used is the
32:Spectrum (physical sciences)
7:
10124:Stein, Jonathan Y. (2000).
9746:Dennis Ward Ricker (2003).
9269:Spectral power distribution
9254:Spectral density estimation
9222:
8845:Spectral density estimation
6477:power spectral distribution
2363:spectral power distribution
10:
10368:
10107:The Fokker-Planck Equation
9988:10.1109/TBCAS.2017.2690908
9972:Rodriguez-Villegas, Esther
9785:Miller & Childers 2012
9734:Miller & Childers 2012
9722:Miller & Childers 2012
9107:
9088:amplitude spectral density
8895:
8853:the spectral density of a
8842:
6619:
2360:
1797:{\displaystyle V(t)^{2}/Z}
1569:{\displaystyle V(t)^{2}/Z}
1525:{\displaystyle V(t)^{2}/Z}
1127:in the frequency interval
580:
435:
203:into frequency components
50:The spectral density of a
38:
29:
10317:Frequency-domain analysis
10273:
10196:
10126:Digital Signal Processing
9945:10.1007/s10439-014-1085-6
9797:Davenport & Root 1987
9620:Norton & Karczub 2003
9486:under weaker hypotheses:
8888:method can also be used.
8450:{\displaystyle x(t)=y(t)}
7551:{\displaystyle S_{yy}(f)}
7512:{\displaystyle S_{xx}(f)}
6755:{\displaystyle S_{yy}(f)}
6716:{\displaystyle S_{xx}(f)}
6464:{\displaystyle S_{xx}(f)}
6321:inverse Fourier transform
2389:; this describes how the
1837:, and hence the estimate
157:{\displaystyle S_{xx}(f)}
9893:. New York: Wiley-Liss.
9809:William D Penny (2009).
9646:"Power Spectral Density"
9284:
9027:real and imaginary parts
8896:Not to be confused with
3462:{\displaystyle x_{T}(t)}
2859:{\displaystyle w_{T}(t)}
2500:{\displaystyle x^{2}(t)}
2398:of a function over time
2361:Not to be confused with
2313:{\displaystyle \Delta t}
1720:is then estimated to be
1664:{\displaystyle \Delta f}
1103:can be interpreted as a
431:
288:{\displaystyle x^{2}(t)}
9872:Reliability Engineering
9480:WienerâKhinchin theorem
9299:Risken & Frank 1996
9217:Primordial fluctuations
8915:spectral edge frequency
5249:WienerâKhinchin theorem
2608:{\displaystyle t=t_{0}}
2385:(PSD) which exists for
1387:WienerâKhinchin theorem
1161:energy spectral density
577:Energy spectral density
503:Energy spectral density
486:(abbreviated as W/Hz).
240:energy spectral density
10021:Maral, Gerard (2004).
9773:Brown & Hwang 1997
9749:Echo Signal Processing
9590:& R Moses (2005).
9552:
9551:{\displaystyle R_{xx}}
9510:
9509:{\displaystyle R_{xx}}
9459:
9419:
9249:Noise spectral density
9234:Brightness temperature
9173:electrical engineering
9168:
9155:Electrical engineering
9061:
9015:phase spectral density
8830:
8757:
8676:
8585:
8547:
8509:
8480:
8451:
8407:
8378:
8349:
8310:
8281:
8248:
8209:
7651:
7621:
7552:
7513:
7474:
7193:
6772:cross spectral density
6756:
6717:
6678:
6649:
6597:
6501:
6471:, one can compute the
6465:
6416:
6329:, one can compute the
6289:
6205:cross-spectral density
6194:
6170:
6150:
6130:
6072:
5961:
5913:
5854:
5819:
5785:
5751:
5723:
5695:
5675:
5582:
5515:
5469:
5399:
5241:
5210:
4880:
4851:
4818:, which is denoted as
4812:
4779:
4759:and take the limit as
4753:
4726:
4063:
4004:
3463:
3427:
3375:
3315:From here, due to the
3292:
3170:
3047:
3009:
2973:
2860:
2824:
2751:
2609:
2576:
2556:
2527:
2501:
2458:
2435:source which followed
2421:
2383:power spectral density
2377:
2357:Power spectral density
2343:
2314:
2291:
2290:{\displaystyle x_{n}.}
2257:
2212:
2098:
1984:
1925:
1891:
1871:
1827:
1798:
1754:
1714:
1694:
1665:
1638:
1618:
1570:
1526:
1482:
1454:
1419:
1379:
1346:
1281:
1186:
1150:
1121:
1097:
1032:
1009:
976:
871:
725:
643:
614:
556:A PSD can be either a
552:One-sided vs two-sided
456:power spectral density
386:
345:
325:
289:
244:power spectral density
217:
193:
158:
63:
55:
9553:
9511:
9460:
9420:
9297:Some authors, e.g., (
9162:
9108:Further information:
9062:
9013:versus frequencyâthe
8880:methods (such as the
8831:
8734:
8684:For discrete signals
8677:
8586:
8548:
8510:
8481:
8452:
8408:
8379:
8350:
8311:
8282:
8249:
8210:
7652:
7622:
7553:
7514:
7475:
7194:
6757:
6718:
6679:
6650:
6598:
6502:
6466:
6417:
6290:
6195:
6171:
6151:
6131:
6049:
5962:
5914:
5855:
5820:
5818:{\displaystyle x_{n}}
5786:
5760:statistical ensembles
5752:
5724:
5696:
5676:
5583:
5516:
5470:
5400:
5242:
5211:
4881:
4852:
4813:
4780:
4754:
4727:
4064:
4005:
3464:
3428:
3376:
3293:
3171:
3048:
3010:
2974:
2861:
2825:
2752:
2610:
2577:
2557:
2528:
2502:
2459:
2422:
2371:
2344:
2315:
2292:
2258:
2213:
2075:
1985:
1926:
1924:{\displaystyle x_{n}}
1892:
1872:
1828:
1799:
1755:
1715:
1695:
1666:
1639:
1619:
1571:
1527:
1483:
1455:
1420:
1380:
1347:
1282:
1187:
1151:
1122:
1098:
1033:
1010:
977:
872:
726:
644:
615:
387:
346:
326:
290:
218:
194:
159:
125:, the power spectrum
61:
49:
9688:, pp. 108, 115.
9644:Paschotta, RĂźdiger.
9532:
9490:
9467:Dirac delta function
9429:
9305:
9181:radio communications
9060:{\displaystyle h(t)}
9042:
8867:autoregressive model
8706:
8603:
8557:
8519:
8508:{\displaystyle y(t)}
8490:
8479:{\displaystyle x(t)}
8461:
8417:
8406:{\displaystyle x(t)}
8388:
8377:{\displaystyle y(t)}
8359:
8320:
8309:{\displaystyle y(t)}
8291:
8280:{\displaystyle x(t)}
8262:
8219:
7665:
7635:
7562:
7523:
7484:
7205:
6782:
6727:
6688:
6677:{\displaystyle y(t)}
6659:
6648:{\displaystyle x(t)}
6630:
6515:
6500:{\displaystyle F(f)}
6482:
6436:
6337:
6232:
6184:
6160:
6140:
5971:
5923:
5864:
5829:
5802:
5784:{\displaystyle x(t)}
5766:
5733:
5705:
5685:
5592:
5525:
5479:
5433:
5261:
5240:{\displaystyle x(t)}
5222:
4894:
4879:{\displaystyle x(t)}
4861:
4822:
4811:{\displaystyle x(t)}
4793:
4763:
4743:
4073:
4014:
3473:
3437:
3393:
3323:
3188:
3061:
3019:
3008:{\displaystyle x(t)}
2990:
2870:
2834:
2764:
2619:
2586:
2566:
2555:{\displaystyle x(t)}
2537:
2517:
2475:
2457:{\displaystyle x(t)}
2439:
2420:{\displaystyle x(t)}
2402:
2387:stationary processes
2351:normalized frequency
2324:
2301:
2271:
2222:
1994:
1935:
1908:
1881:
1841:
1826:{\displaystyle E(f)}
1808:
1764:
1724:
1704:
1693:{\displaystyle E(f)}
1675:
1652:
1628:
1580:
1536:
1492:
1472:
1444:
1418:{\displaystyle V(t)}
1400:
1378:{\displaystyle x(t)}
1360:
1308:
1202:
1185:{\displaystyle x(t)}
1167:
1149:{\displaystyle f+df}
1131:
1111:
1046:
1022:
1008:{\displaystyle x(t)}
990:
982:is the value of the
881:
741:
653:
642:{\displaystyle x(t)}
624:
604:
566:negative frequencies
448:electromagnetic wave
414:stochastic processes
385:{\displaystyle E(t)}
367:
351:. For instance, the
335:
324:{\displaystyle x(t)}
306:
263:
207:
192:{\displaystyle x(t)}
174:
129:
10090:. Boston: Pearson.
9724:, pp. 429â431.
9712:, pp. 422â423.
9259:Spectral efficiency
8595:is just one of the
8144:
8047:
8032:
7983:
7879:
7782:
7767:
7718:
7582:
7423:
7295:
7111:
7065:
7011:
6842:
6550:
6473:integrated spectrum
6424:For a real process
6381:
6200:approach infinity.
5762:of realizations of
5750:{\displaystyle 1/T}
5722:{\displaystyle 1/T}
5641:
5306:
5152:
5052:
5037:
4988:
4734:convolution theorem
4635:
4620:
4597:
4490:
4467:
4240:
4046:
3816:
3729:
3714:
3649:
3634:
3563:
3548:
3506:
3410:
3319:, we can also view
3317:convolution theorem
3113:
2922:
2710:
925:
818:
761:
679:
10296:telecommunications
9970:Iranmanesh, Saam;
9548:
9506:
9455:
9415:
9274:Whittle likelihood
9193:spectrum analyzers
9169:
9163:Spectrogram of an
9057:
9032:frequency response
8999:transfer functions
8925:", represents the
8826:
8672:
8581:
8543:
8505:
8476:
8447:
8403:
8374:
8345:
8306:
8277:
8244:
8205:
8203:
8127:
8033:
8015:
8004:
7966:
7862:
7768:
7750:
7739:
7701:
7647:
7617:
7565:
7548:
7509:
7470:
7468:
7400:
7384:
7272:
7256:
7189:
7187:
7097:
7051:
6994:
6983:
6825:
6814:
6752:
6713:
6674:
6645:
6626:Given two signals
6593:
6536:
6497:
6461:
6412:
6364:
6327:Parseval's theorem
6300:stochastic process
6285:
6190:
6166:
6146:
6126:
6014:
5957:
5909:
5850:
5815:
5781:
5747:
5719:
5691:
5671:
5613:
5578:
5511:
5465:
5395:
5289:
5237:
5206:
5135:
5038:
5020:
5009:
4971:
4912:
4876:
4847:
4808:
4775:
4749:
4722:
4720:
4621:
4603:
4580:
4473:
4450:
4226:
4059:
4032:
4000:
3998:
3799:
3715:
3697:
3635:
3617:
3549:
3531:
3492:
3459:
3423:
3396:
3371:
3288:
3231:
3166:
3096:
3085:
3055:Parseval's theorem
3043:
3005:
2969:
2905:
2894:
2856:
2820:
2747:
2654:
2643:
2605:
2572:
2552:
2523:
2513:The average power
2497:
2454:
2417:
2378:
2339:
2310:
2287:
2253:
2208:
2204:
2155:
2046:
1980:
1921:
1902:countably infinite
1887:
1867:
1823:
1794:
1750:
1710:
1690:
1661:
1634:
1614:
1566:
1522:
1478:
1450:
1415:
1375:
1342:
1277:
1182:
1146:
1117:
1093:
1028:
1005:
972:
908:
867:
801:
744:
735:Parseval's theorem
721:
662:
639:
610:
382:
341:
321:
297:Parseval's theorem
285:
213:
189:
154:
64:
56:
10322:Signal processing
10304:
10303:
10135:978-0-471-29546-4
10116:978-3-540-61530-9
10097:978-0-13-394328-3
10078:978-0-521-49913-2
10051:978-0-12-386981-4
10032:978-0-470-86684-9
9919:978-0-87942-235-6
9900:978-0-471-12839-7
9881:978-3-540-49388-4
9759:978-1-4020-7395-3
9560:Bochner's theorem
9520:(in the sense of
9342:
9320:
8954:spectral envelope
8907:spectral centroid
8878:Fourier transform
8613:
8569:
8531:
8256:cross-correlation
8013:
7989:
7748:
7724:
7659:cross-correlation
7443:
7410:
7393:
7369:
7315:
7282:
7265:
7241:
6992:
6968:
6823:
6799:
6319:(f) by using the
6298:For a continuous
6209:cross-correlation
6193:{\displaystyle T}
6169:{\displaystyle T}
6149:{\displaystyle N}
6041:
5999:
5694:{\displaystyle T}
5603:
5419:
5418:
5374:
5125:
5018:
4994:
4939:
4921:
4897:
4785:, it becomes the
4752:{\displaystyle T}
4708:
4186:
4145:
4096:
3967:
3383:Fourier transform
3341:
3312:
3311:
3257:
3240:
3216:
3129:
3094:
3070:
3031:
2903:
2879:
2652:
2628:
2575:{\displaystyle T}
2526:{\displaystyle P}
2235:
2174:
2068:
2066:
2031:
2007:
1904:number of values
1890:{\displaystyle Z}
1713:{\displaystyle f}
1637:{\displaystyle f}
1593:
1481:{\displaystyle t}
1453:{\displaystyle Z}
1435:transmission line
1321:
1301:
1300:
1254:
1215:
1120:{\displaystyle f}
1064:
1031:{\displaystyle f}
984:Fourier transform
965:
893:
834:
711:
613:{\displaystyle E}
592:of a signal or a
426:spatial frequency
394:Fourier transform
344:{\displaystyle x}
216:{\displaystyle f}
123:signal processing
119:
118:
111:
52:fluorescent light
18:Spectral function
16:(Redirected from
10359:
10342:Fourier analysis
10281:logarithmic unit
10179:
10172:
10165:
10156:
10155:
10139:
10120:
10101:
10082:
10063:
10036:
10017:
9999:
9966:
9956:
9923:
9904:
9885:
9857:
9856:
9854:
9845:
9839:
9833:
9827:
9821:
9815:
9814:
9806:
9800:
9794:
9788:
9782:
9776:
9770:
9764:
9763:
9743:
9737:
9731:
9725:
9719:
9713:
9707:
9701:
9695:
9689:
9683:
9677:
9671:
9665:
9664:
9662:
9661:
9652:. Archived from
9650:rp-photonics.com
9641:
9635:
9629:
9623:
9617:
9611:
9605:
9599:
9598:
9596:
9584:
9567:
9557:
9555:
9554:
9549:
9547:
9546:
9522:Laurent Schwartz
9515:
9513:
9512:
9507:
9505:
9504:
9476:
9470:
9464:
9462:
9461:
9456:
9451:
9424:
9422:
9421:
9416:
9408:
9361:
9350:
9349:
9344:
9343:
9335:
9322:
9321:
9313:
9295:
9264:Spectral leakage
9175:, especially in
9066:
9064:
9063:
9058:
9037:impulse response
8892:Related concepts
8835:
8833:
8832:
8827:
8818:
8817:
8816:
8815:
8783:
8782:
8770:
8769:
8756:
8751:
8721:
8720:
8701:
8692:
8681:
8679:
8678:
8673:
8662:
8661:
8634:
8633:
8615:
8614:
8611:
8590:
8588:
8587:
8582:
8571:
8570:
8562:
8552:
8550:
8549:
8544:
8533:
8532:
8524:
8514:
8512:
8511:
8506:
8485:
8483:
8482:
8477:
8456:
8454:
8453:
8448:
8412:
8410:
8409:
8404:
8383:
8381:
8380:
8375:
8354:
8352:
8351:
8346:
8335:
8334:
8315:
8313:
8312:
8307:
8286:
8284:
8283:
8278:
8253:
8251:
8250:
8245:
8234:
8233:
8214:
8212:
8211:
8206:
8204:
8191:
8190:
8157:
8156:
8143:
8138:
8117:
8116:
8092:
8088:
8072:
8071:
8046:
8041:
8031:
8026:
8014:
8006:
8003:
7982:
7977:
7949:
7948:
7926:
7925:
7892:
7891:
7878:
7873:
7852:
7851:
7827:
7823:
7807:
7806:
7781:
7776:
7766:
7761:
7749:
7741:
7738:
7717:
7712:
7684:
7683:
7656:
7654:
7653:
7648:
7626:
7624:
7623:
7618:
7607:
7606:
7581:
7576:
7557:
7555:
7554:
7549:
7538:
7537:
7518:
7516:
7515:
7510:
7499:
7498:
7479:
7477:
7476:
7471:
7469:
7465:
7461:
7451:
7450:
7445:
7444:
7436:
7422:
7417:
7412:
7411:
7403:
7394:
7386:
7383:
7352:
7351:
7337:
7333:
7323:
7322:
7317:
7316:
7308:
7294:
7289:
7284:
7283:
7275:
7266:
7258:
7255:
7224:
7223:
7198:
7196:
7195:
7190:
7188:
7178:
7177:
7172:
7157:
7156:
7147:
7130:
7129:
7110:
7105:
7084:
7083:
7064:
7059:
7047:
7046:
7041:
7026:
7025:
7016:
7010:
7005:
6993:
6985:
6982:
6961:
6951:
6947:
6937:
6936:
6915:
6914:
6900:
6899:
6894:
6890:
6880:
6879:
6858:
6857:
6841:
6836:
6824:
6816:
6813:
6761:
6759:
6758:
6753:
6742:
6741:
6722:
6720:
6719:
6714:
6703:
6702:
6683:
6681:
6680:
6675:
6654:
6652:
6651:
6646:
6609:
6602:
6600:
6599:
6594:
6589:
6574:
6563:
6562:
6549:
6544:
6506:
6504:
6503:
6498:
6470:
6468:
6467:
6462:
6451:
6450:
6421:
6419:
6418:
6413:
6395:
6394:
6380:
6375:
6294:
6292:
6291:
6286:
6275:
6274:
6247:
6246:
6199:
6197:
6196:
6191:
6175:
6173:
6172:
6167:
6155:
6153:
6152:
6147:
6135:
6133:
6132:
6127:
6125:
6124:
6119:
6115:
6114:
6113:
6082:
6081:
6071:
6066:
6042:
6037:
6036:
6035:
6016:
6013:
5986:
5985:
5966:
5964:
5963:
5958:
5918:
5916:
5915:
5910:
5889:
5888:
5876:
5875:
5859:
5857:
5856:
5851:
5824:
5822:
5821:
5816:
5814:
5813:
5790:
5788:
5787:
5782:
5756:
5754:
5753:
5748:
5743:
5728:
5726:
5725:
5720:
5715:
5700:
5698:
5697:
5692:
5680:
5678:
5677:
5672:
5654:
5653:
5640:
5639:
5638:
5628:
5627:
5626:
5606:
5605:
5604:
5587:
5585:
5584:
5579:
5568:
5567:
5540:
5539:
5520:
5518:
5517:
5512:
5510:
5509:
5497:
5496:
5474:
5472:
5471:
5468:{\displaystyle }
5466:
5461:
5460:
5448:
5447:
5413:
5404:
5402:
5401:
5396:
5385:
5384:
5376:
5375:
5367:
5353:
5352:
5319:
5318:
5305:
5300:
5276:
5275:
5255:
5246:
5244:
5243:
5238:
5215:
5213:
5212:
5207:
5199:
5198:
5165:
5164:
5151:
5146:
5123:
5122:
5121:
5097:
5093:
5077:
5076:
5051:
5046:
5036:
5031:
5019:
5011:
5008:
4987:
4982:
4967:
4966:
4961:
4957:
4947:
4946:
4941:
4940:
4932:
4922:
4914:
4911:
4885:
4883:
4882:
4877:
4857:, provided that
4856:
4854:
4853:
4848:
4837:
4836:
4817:
4815:
4814:
4809:
4784:
4782:
4781:
4776:
4758:
4756:
4755:
4750:
4731:
4729:
4728:
4723:
4721:
4706:
4705:
4704:
4680:
4676:
4660:
4659:
4634:
4629:
4619:
4614:
4596:
4591:
4573:
4563:
4562:
4538:
4534:
4518:
4517:
4489:
4484:
4466:
4461:
4443:
4439:
4435:
4425:
4424:
4415:
4414:
4391:
4390:
4378:
4374:
4370:
4360:
4359:
4345:
4344:
4335:
4331:
4313:
4312:
4300:
4296:
4292:
4282:
4281:
4267:
4266:
4257:
4253:
4239:
4234:
4220:
4219:
4207:
4194:
4193:
4188:
4187:
4179:
4172:
4171:
4153:
4152:
4147:
4146:
4138:
4124:
4123:
4118:
4114:
4104:
4103:
4098:
4097:
4089:
4068:
4066:
4065:
4060:
4045:
4040:
4009:
4007:
4006:
4001:
3999:
3995:
3994:
3989:
3985:
3975:
3974:
3969:
3968:
3960:
3944:
3940:
3939:
3934:
3930:
3929:
3925:
3915:
3914:
3900:
3899:
3881:
3877:
3876:
3871:
3867:
3860:
3859:
3826:
3825:
3815:
3810:
3786:
3776:
3775:
3766:
3765:
3728:
3723:
3713:
3708:
3690:
3680:
3679:
3648:
3643:
3633:
3628:
3610:
3600:
3599:
3562:
3557:
3547:
3542:
3523:
3519:
3505:
3500:
3486:
3485:
3468:
3466:
3465:
3460:
3449:
3448:
3432:
3430:
3429:
3424:
3409:
3404:
3380:
3378:
3377:
3372:
3370:
3369:
3364:
3349:
3348:
3343:
3342:
3334:
3330:
3306:
3297:
3295:
3294:
3289:
3286:
3285:
3280:
3265:
3264:
3259:
3258:
3250:
3246:
3241:
3233:
3230:
3203:
3202:
3182:
3175:
3173:
3172:
3167:
3158:
3157:
3152:
3137:
3136:
3131:
3130:
3122:
3118:
3112:
3107:
3095:
3087:
3084:
3052:
3050:
3049:
3044:
3033:
3032:
3024:
3014:
3012:
3011:
3006:
2978:
2976:
2975:
2970:
2958:
2957:
2952:
2948:
2938:
2937:
2921:
2916:
2904:
2896:
2893:
2865:
2863:
2862:
2857:
2846:
2845:
2829:
2827:
2826:
2821:
2810:
2809:
2776:
2775:
2756:
2754:
2753:
2748:
2739:
2738:
2733:
2729:
2709:
2705:
2694:
2693:
2683:
2679:
2668:
2667:
2653:
2645:
2642:
2614:
2612:
2611:
2606:
2604:
2603:
2581:
2579:
2578:
2573:
2561:
2559:
2558:
2553:
2532:
2530:
2529:
2524:
2506:
2504:
2503:
2498:
2487:
2486:
2463:
2461:
2460:
2455:
2426:
2424:
2423:
2418:
2348:
2346:
2345:
2340:
2319:
2317:
2316:
2311:
2296:
2294:
2293:
2288:
2283:
2282:
2262:
2260:
2259:
2254:
2243:
2242:
2237:
2236:
2228:
2217:
2215:
2214:
2209:
2203:
2202:
2201:
2196:
2192:
2182:
2181:
2176:
2175:
2167:
2156:
2151:
2150:
2145:
2141:
2140:
2139:
2108:
2107:
2097:
2092:
2065:
2064:
2045:
2018:
2017:
2009:
2008:
2000:
1989:
1987:
1986:
1981:
1960:
1959:
1947:
1946:
1930:
1928:
1927:
1922:
1920:
1919:
1896:
1894:
1893:
1888:
1876:
1874:
1873:
1868:
1860:
1832:
1830:
1829:
1824:
1803:
1801:
1800:
1795:
1790:
1785:
1784:
1759:
1757:
1756:
1751:
1743:
1719:
1717:
1716:
1711:
1699:
1697:
1696:
1691:
1670:
1668:
1667:
1662:
1643:
1641:
1640:
1635:
1623:
1621:
1620:
1615:
1604:
1603:
1595:
1594:
1586:
1575:
1573:
1572:
1567:
1562:
1557:
1556:
1531:
1529:
1528:
1523:
1518:
1513:
1512:
1487:
1485:
1484:
1479:
1459:
1457:
1456:
1451:
1424:
1422:
1421:
1416:
1384:
1382:
1381:
1376:
1351:
1349:
1348:
1343:
1332:
1331:
1323:
1322:
1314:
1295:
1286:
1284:
1283:
1278:
1276:
1275:
1270:
1266:
1256:
1255:
1247:
1226:
1225:
1217:
1216:
1208:
1196:
1191:
1189:
1188:
1183:
1155:
1153:
1152:
1147:
1126:
1124:
1123:
1118:
1105:density function
1102:
1100:
1099:
1094:
1086:
1085:
1080:
1076:
1066:
1065:
1057:
1037:
1035:
1034:
1029:
1014:
1012:
1011:
1006:
981:
979:
978:
973:
963:
950:
949:
924:
919:
895:
894:
886:
876:
874:
873:
868:
856:
855:
850:
846:
836:
835:
827:
817:
812:
790:
789:
784:
766:
760:
755:
730:
728:
727:
722:
709:
708:
707:
702:
698:
678:
673:
648:
646:
645:
640:
619:
617:
616:
611:
485:
483:
482:
477:
474:
398:dispersive prism
391:
389:
388:
383:
350:
348:
347:
342:
330:
328:
327:
322:
294:
292:
291:
286:
275:
274:
246:(PSD, or simply
225:Fourier analysis
222:
220:
219:
214:
198:
196:
195:
190:
163:
161:
160:
155:
144:
143:
114:
107:
103:
100:
94:
74:
73:
66:
21:
10367:
10366:
10362:
10361:
10360:
10358:
10357:
10356:
10307:
10306:
10305:
10300:
10269:
10192:
10183:
10146:
10136:
10117:
10098:
10079:
10052:
10033:
9939:(11): 2344â59.
9920:
9901:
9882:
9866:
9861:
9860:
9852:
9846:
9842:
9834:
9830:
9822:
9818:
9807:
9803:
9795:
9791:
9783:
9779:
9771:
9767:
9760:
9744:
9740:
9732:
9728:
9720:
9716:
9708:
9704:
9696:
9692:
9684:
9680:
9672:
9668:
9659:
9657:
9642:
9638:
9630:
9626:
9618:
9614:
9606:
9602:
9594:
9585:
9576:
9571:
9570:
9558:is continuous,
9539:
9535:
9533:
9530:
9529:
9497:
9493:
9491:
9488:
9487:
9477:
9473:
9444:
9430:
9427:
9426:
9401:
9354:
9345:
9334:
9333:
9332:
9312:
9311:
9306:
9303:
9302:
9296:
9292:
9287:
9279:Window function
9239:Colors of noise
9225:
9214:
9157:
9112:
9106:
9069:autocorrelation
9043:
9040:
9039:
8901:
8894:
8886:maximum entropy
8847:
8841:
8811:
8807:
8791:
8787:
8778:
8774:
8762:
8758:
8752:
8738:
8713:
8709:
8707:
8704:
8703:
8699:
8694:
8690:
8685:
8654:
8650:
8626:
8622:
8610:
8606:
8604:
8601:
8600:
8561:
8560:
8558:
8555:
8554:
8523:
8522:
8520:
8517:
8516:
8491:
8488:
8487:
8462:
8459:
8458:
8418:
8415:
8414:
8389:
8386:
8385:
8360:
8357:
8356:
8327:
8323:
8321:
8318:
8317:
8292:
8289:
8288:
8263:
8260:
8259:
8226:
8222:
8220:
8217:
8216:
8202:
8201:
8171:
8167:
8149:
8145:
8139:
8131:
8097:
8093:
8067:
8063:
8042:
8037:
8027:
8019:
8005:
7993:
7988:
7984:
7978:
7970:
7959:
7941:
7937:
7934:
7933:
7906:
7902:
7884:
7880:
7874:
7866:
7832:
7828:
7802:
7798:
7777:
7772:
7762:
7754:
7740:
7728:
7723:
7719:
7713:
7705:
7694:
7676:
7672:
7668:
7666:
7663:
7662:
7636:
7633:
7632:
7599:
7595:
7577:
7569:
7563:
7560:
7559:
7530:
7526:
7524:
7521:
7520:
7491:
7487:
7485:
7482:
7481:
7467:
7466:
7446:
7435:
7434:
7433:
7418:
7413:
7402:
7401:
7399:
7395:
7385:
7373:
7362:
7344:
7340:
7338:
7318:
7307:
7306:
7305:
7290:
7285:
7274:
7273:
7271:
7267:
7257:
7245:
7234:
7216:
7212:
7208:
7206:
7203:
7202:
7186:
7185:
7173:
7168:
7167:
7152:
7148:
7143:
7125:
7121:
7106:
7101:
7079:
7075:
7060:
7055:
7042:
7037:
7036:
7021:
7017:
7012:
7006:
6998:
6984:
6972:
6959:
6958:
6932:
6928:
6910:
6906:
6905:
6901:
6895:
6875:
6871:
6853:
6849:
6848:
6844:
6843:
6837:
6829:
6815:
6803:
6792:
6785:
6783:
6780:
6779:
6734:
6730:
6728:
6725:
6724:
6695:
6691:
6689:
6686:
6685:
6660:
6657:
6656:
6631:
6628:
6627:
6624:
6618:
6613:
6604:
6582:
6567:
6555:
6551:
6545:
6540:
6516:
6513:
6512:
6483:
6480:
6479:
6443:
6439:
6437:
6434:
6433:
6387:
6383:
6376:
6368:
6338:
6335:
6334:
6318:
6310:
6267:
6263:
6239:
6235:
6233:
6230:
6229:
6217:
6185:
6182:
6181:
6161:
6158:
6157:
6141:
6138:
6137:
6120:
6087:
6083:
6077:
6073:
6067:
6053:
6048:
6044:
6043:
6031:
6027:
6017:
6015:
6003:
5978:
5974:
5972:
5969:
5968:
5924:
5921:
5920:
5884:
5880:
5871:
5867:
5865:
5862:
5861:
5830:
5827:
5826:
5809:
5805:
5803:
5800:
5799:
5767:
5764:
5763:
5739:
5734:
5731:
5730:
5711:
5706:
5703:
5702:
5686:
5683:
5682:
5646:
5642:
5634:
5630:
5629:
5622:
5618:
5617:
5600:
5599:
5595:
5593:
5590:
5589:
5560:
5556:
5532:
5528:
5526:
5523:
5522:
5505:
5501:
5492:
5488:
5480:
5477:
5476:
5456:
5452:
5443:
5439:
5434:
5431:
5430:
5420:
5377:
5366:
5365:
5364:
5333:
5329:
5311:
5307:
5301:
5293:
5268:
5264:
5262:
5259:
5258:
5223:
5220:
5219:
5179:
5175:
5157:
5153:
5147:
5139:
5102:
5098:
5072:
5068:
5047:
5042:
5032:
5024:
5010:
4998:
4993:
4989:
4983:
4975:
4962:
4942:
4931:
4930:
4929:
4928:
4924:
4923:
4913:
4901:
4895:
4892:
4891:
4862:
4859:
4858:
4829:
4825:
4823:
4820:
4819:
4794:
4791:
4790:
4787:autocorrelation
4764:
4761:
4760:
4744:
4741:
4740:
4719:
4718:
4685:
4681:
4655:
4651:
4630:
4625:
4615:
4607:
4602:
4598:
4592:
4584:
4571:
4570:
4543:
4539:
4513:
4509:
4485:
4477:
4472:
4468:
4462:
4454:
4441:
4440:
4420:
4416:
4410:
4409:
4396:
4392:
4386:
4385:
4376:
4375:
4355:
4351:
4350:
4346:
4340:
4339:
4318:
4314:
4308:
4307:
4298:
4297:
4277:
4273:
4272:
4268:
4262:
4261:
4235:
4230:
4225:
4221:
4215:
4214:
4205:
4204:
4189:
4178:
4177:
4176:
4167:
4163:
4148:
4137:
4136:
4135:
4125:
4119:
4099:
4088:
4087:
4086:
4085:
4081:
4080:
4076:
4074:
4071:
4070:
4041:
4036:
4015:
4012:
4011:
3997:
3996:
3990:
3970:
3959:
3958:
3957:
3956:
3952:
3951:
3942:
3941:
3935:
3910:
3906:
3905:
3901:
3895:
3894:
3893:
3889:
3888:
3879:
3878:
3872:
3840:
3836:
3821:
3817:
3811:
3803:
3798:
3794:
3793:
3784:
3783:
3771:
3767:
3746:
3742:
3724:
3719:
3709:
3701:
3688:
3687:
3663:
3659:
3644:
3639:
3629:
3621:
3608:
3607:
3580:
3576:
3558:
3553:
3543:
3535:
3524:
3501:
3496:
3491:
3487:
3481:
3480:
3476:
3474:
3471:
3470:
3444:
3440:
3438:
3435:
3434:
3405:
3400:
3394:
3391:
3390:
3365:
3360:
3359:
3344:
3333:
3332:
3331:
3326:
3324:
3321:
3320:
3313:
3281:
3276:
3275:
3260:
3249:
3248:
3247:
3242:
3232:
3220:
3195:
3191:
3189:
3186:
3185:
3153:
3148:
3147:
3132:
3121:
3120:
3119:
3114:
3108:
3100:
3086:
3074:
3062:
3059:
3058:
3023:
3022:
3020:
3017:
3016:
2991:
2988:
2987:
2953:
2933:
2929:
2928:
2924:
2923:
2917:
2909:
2895:
2883:
2871:
2868:
2867:
2841:
2837:
2835:
2832:
2831:
2805:
2801:
2771:
2767:
2765:
2762:
2761:
2734:
2716:
2712:
2711:
2701:
2689:
2685:
2684:
2675:
2663:
2659:
2658:
2644:
2632:
2620:
2617:
2616:
2599:
2595:
2587:
2584:
2583:
2567:
2564:
2563:
2538:
2535:
2534:
2518:
2515:
2514:
2482:
2478:
2476:
2473:
2472:
2440:
2437:
2436:
2403:
2400:
2399:
2366:
2359:
2325:
2322:
2321:
2302:
2299:
2298:
2278:
2274:
2272:
2269:
2268:
2238:
2227:
2226:
2225:
2223:
2220:
2219:
2197:
2177:
2166:
2165:
2164:
2163:
2159:
2158:
2157:
2146:
2113:
2109:
2103:
2099:
2093:
2079:
2074:
2070:
2069:
2067:
2060:
2056:
2035:
2010:
1999:
1998:
1997:
1995:
1992:
1991:
1955:
1951:
1942:
1938:
1936:
1933:
1932:
1915:
1911:
1909:
1906:
1905:
1882:
1879:
1878:
1856:
1842:
1839:
1838:
1809:
1806:
1805:
1786:
1780:
1776:
1765:
1762:
1761:
1739:
1725:
1722:
1721:
1705:
1702:
1701:
1676:
1673:
1672:
1653:
1650:
1649:
1646:bandpass filter
1629:
1626:
1625:
1596:
1585:
1584:
1583:
1581:
1578:
1577:
1558:
1552:
1548:
1537:
1534:
1533:
1514:
1508:
1504:
1493:
1490:
1489:
1473:
1470:
1469:
1445:
1442:
1441:
1425:represents the
1401:
1398:
1397:
1361:
1358:
1357:
1354:autocorrelation
1324:
1313:
1312:
1311:
1309:
1306:
1305:
1302:
1271:
1246:
1245:
1244:
1240:
1239:
1218:
1207:
1206:
1205:
1203:
1200:
1199:
1192:is defined as:
1168:
1165:
1164:
1159:Therefore, the
1132:
1129:
1128:
1112:
1109:
1108:
1081:
1056:
1055:
1054:
1050:
1049:
1047:
1044:
1043:
1023:
1020:
1019:
991:
988:
987:
930:
926:
920:
912:
885:
884:
882:
879:
878:
851:
826:
825:
824:
820:
819:
813:
805:
785:
780:
779:
762:
756:
748:
742:
739:
738:
703:
685:
681:
680:
674:
666:
654:
651:
650:
625:
622:
621:
605:
602:
601:
586:
583:energy spectrum
579:
574:
554:
478:
475:
470:
469:
467:
440:
434:
368:
365:
364:
336:
333:
332:
307:
304:
303:
270:
266:
264:
261:
260:
208:
205:
204:
175:
172:
171:
166:continuous time
136:
132:
130:
127:
126:
115:
104:
98:
95:
87:help improve it
84:
75:
71:
44:
35:
28:
23:
22:
15:
12:
11:
5:
10365:
10355:
10354:
10349:
10347:Radio spectrum
10344:
10339:
10334:
10329:
10324:
10319:
10302:
10301:
10299:
10298:
10293:
10288:
10283:
10278:
10274:
10271:
10270:
10268:
10267:
10262:
10257:
10252:
10247:
10242:
10237:
10231:
10226:
10225:
10224:
10214:
10209:
10204:
10197:
10194:
10193:
10182:
10181:
10174:
10167:
10159:
10153:
10152:
10145:
10144:External links
10142:
10141:
10140:
10134:
10121:
10115:
10102:
10096:
10083:
10077:
10064:
10050:
10037:
10031:
10018:
9982:(4): 858â866.
9967:
9924:
9918:
9905:
9899:
9886:
9880:
9865:
9862:
9859:
9858:
9840:
9828:
9816:
9801:
9789:
9787:, p. 431.
9777:
9765:
9758:
9738:
9736:, p. 433.
9726:
9714:
9702:
9690:
9678:
9666:
9636:
9624:
9612:
9600:
9573:
9572:
9569:
9568:
9545:
9542:
9538:
9503:
9500:
9496:
9471:
9454:
9450:
9447:
9443:
9440:
9437:
9434:
9414:
9411:
9407:
9404:
9400:
9397:
9394:
9391:
9388:
9385:
9382:
9379:
9376:
9373:
9370:
9367:
9364:
9360:
9357:
9353:
9348:
9341:
9338:
9331:
9328:
9325:
9319:
9316:
9310:
9289:
9288:
9286:
9283:
9282:
9281:
9276:
9271:
9266:
9261:
9256:
9251:
9246:
9241:
9236:
9231:
9224:
9221:
9213:
9210:
9189:remote sensing
9156:
9153:
9118:(perceived as
9105:
9102:
9101:
9100:
9084:
9056:
9053:
9050:
9047:
9023:spectral phase
9019:phase spectrum
8995:
8977:
8974:spectral bands
8962:remote sensing
8958:envelope curve
8950:
8945:and stages of
8911:
8893:
8890:
8863:non-parametric
8843:Main article:
8840:
8837:
8825:
8822:
8814:
8810:
8806:
8803:
8800:
8797:
8794:
8790:
8786:
8781:
8777:
8773:
8768:
8765:
8761:
8755:
8750:
8747:
8744:
8741:
8737:
8733:
8730:
8727:
8724:
8719:
8716:
8712:
8697:
8688:
8671:
8668:
8665:
8660:
8657:
8653:
8649:
8646:
8643:
8640:
8637:
8632:
8629:
8625:
8621:
8618:
8609:
8580:
8577:
8574:
8568:
8565:
8542:
8539:
8536:
8530:
8527:
8504:
8501:
8498:
8495:
8475:
8472:
8469:
8466:
8446:
8443:
8440:
8437:
8434:
8431:
8428:
8425:
8422:
8402:
8399:
8396:
8393:
8373:
8370:
8367:
8364:
8344:
8341:
8338:
8333:
8330:
8326:
8305:
8302:
8299:
8296:
8276:
8273:
8270:
8267:
8243:
8240:
8237:
8232:
8229:
8225:
8200:
8197:
8194:
8189:
8186:
8183:
8180:
8177:
8174:
8170:
8166:
8163:
8160:
8155:
8152:
8148:
8142:
8137:
8134:
8130:
8126:
8123:
8120:
8115:
8112:
8109:
8106:
8103:
8100:
8096:
8091:
8087:
8084:
8081:
8078:
8075:
8070:
8066:
8062:
8059:
8056:
8053:
8050:
8045:
8040:
8036:
8030:
8025:
8022:
8018:
8012:
8009:
8002:
7999:
7996:
7992:
7987:
7981:
7976:
7973:
7969:
7965:
7962:
7960:
7958:
7955:
7952:
7947:
7944:
7940:
7936:
7935:
7932:
7929:
7924:
7921:
7918:
7915:
7912:
7909:
7905:
7901:
7898:
7895:
7890:
7887:
7883:
7877:
7872:
7869:
7865:
7861:
7858:
7855:
7850:
7847:
7844:
7841:
7838:
7835:
7831:
7826:
7822:
7819:
7816:
7813:
7810:
7805:
7801:
7797:
7794:
7791:
7788:
7785:
7780:
7775:
7771:
7765:
7760:
7757:
7753:
7747:
7744:
7737:
7734:
7731:
7727:
7722:
7716:
7711:
7708:
7704:
7700:
7697:
7695:
7693:
7690:
7687:
7682:
7679:
7675:
7671:
7670:
7646:
7643:
7640:
7616:
7613:
7610:
7605:
7602:
7598:
7594:
7591:
7588:
7585:
7580:
7575:
7572:
7568:
7547:
7544:
7541:
7536:
7533:
7529:
7508:
7505:
7502:
7497:
7494:
7490:
7464:
7460:
7457:
7454:
7449:
7442:
7439:
7432:
7429:
7426:
7421:
7416:
7409:
7406:
7398:
7392:
7389:
7382:
7379:
7376:
7372:
7368:
7365:
7363:
7361:
7358:
7355:
7350:
7347:
7343:
7339:
7336:
7332:
7329:
7326:
7321:
7314:
7311:
7304:
7301:
7298:
7293:
7288:
7281:
7278:
7270:
7264:
7261:
7254:
7251:
7248:
7244:
7240:
7237:
7235:
7233:
7230:
7227:
7222:
7219:
7215:
7211:
7210:
7184:
7181:
7176:
7171:
7166:
7163:
7160:
7155:
7151:
7146:
7142:
7139:
7136:
7133:
7128:
7124:
7120:
7117:
7114:
7109:
7104:
7100:
7096:
7093:
7090:
7087:
7082:
7078:
7074:
7071:
7068:
7063:
7058:
7054:
7050:
7045:
7040:
7035:
7032:
7029:
7024:
7020:
7015:
7009:
7004:
7001:
6997:
6991:
6988:
6981:
6978:
6975:
6971:
6967:
6964:
6962:
6960:
6957:
6954:
6950:
6946:
6943:
6940:
6935:
6931:
6927:
6924:
6921:
6918:
6913:
6909:
6904:
6898:
6893:
6889:
6886:
6883:
6878:
6874:
6870:
6867:
6864:
6861:
6856:
6852:
6847:
6840:
6835:
6832:
6828:
6822:
6819:
6812:
6809:
6806:
6802:
6798:
6795:
6793:
6791:
6788:
6787:
6751:
6748:
6745:
6740:
6737:
6733:
6712:
6709:
6706:
6701:
6698:
6694:
6673:
6670:
6667:
6664:
6644:
6641:
6638:
6635:
6617:
6614:
6612:
6611:
6592:
6588:
6585:
6581:
6577:
6573:
6570:
6566:
6561:
6558:
6554:
6548:
6543:
6539:
6535:
6532:
6529:
6526:
6523:
6520:
6496:
6493:
6490:
6487:
6460:
6457:
6454:
6449:
6446:
6442:
6422:
6411:
6408:
6404:
6401:
6398:
6393:
6390:
6386:
6379:
6374:
6371:
6367:
6363:
6360:
6357:
6354:
6351:
6348:
6345:
6342:
6323:
6316:
6306:
6296:
6284:
6281:
6278:
6273:
6270:
6266:
6262:
6259:
6256:
6253:
6250:
6245:
6242:
6238:
6228:of frequency:
6221:
6216:
6213:
6189:
6165:
6145:
6123:
6118:
6112:
6109:
6105:
6102:
6099:
6096:
6093:
6090:
6086:
6080:
6076:
6070:
6065:
6062:
6059:
6056:
6052:
6047:
6040:
6034:
6030:
6026:
6023:
6020:
6012:
6009:
6006:
6002:
5998:
5995:
5992:
5989:
5984:
5981:
5977:
5956:
5953:
5949:
5946:
5943:
5940:
5937:
5934:
5931:
5928:
5908:
5905:
5902:
5898:
5895:
5892:
5887:
5883:
5879:
5874:
5870:
5849:
5846:
5843:
5840:
5837:
5834:
5812:
5808:
5780:
5777:
5774:
5771:
5746:
5742:
5738:
5718:
5714:
5710:
5690:
5670:
5667:
5663:
5660:
5657:
5652:
5649:
5645:
5637:
5633:
5625:
5621:
5616:
5612:
5609:
5598:
5577:
5574:
5571:
5566:
5563:
5559:
5555:
5552:
5549:
5546:
5543:
5538:
5535:
5531:
5508:
5504:
5500:
5495:
5491:
5487:
5484:
5464:
5459:
5455:
5451:
5446:
5442:
5438:
5417:
5416:
5407:
5405:
5394:
5391:
5388:
5383:
5380:
5373:
5370:
5363:
5360:
5357:
5351:
5348:
5345:
5342:
5339:
5336:
5332:
5328:
5325:
5322:
5317:
5314:
5310:
5304:
5299:
5296:
5292:
5288:
5285:
5282:
5279:
5274:
5271:
5267:
5253:
5236:
5233:
5230:
5227:
5205:
5202:
5197:
5194:
5191:
5188:
5185:
5182:
5178:
5174:
5171:
5168:
5163:
5160:
5156:
5150:
5145:
5142:
5138:
5134:
5131:
5128:
5120:
5117:
5114:
5111:
5108:
5105:
5101:
5096:
5092:
5089:
5086:
5083:
5080:
5075:
5071:
5067:
5064:
5061:
5058:
5055:
5050:
5045:
5041:
5035:
5030:
5027:
5023:
5017:
5014:
5007:
5004:
5001:
4997:
4992:
4986:
4981:
4978:
4974:
4970:
4965:
4960:
4956:
4953:
4950:
4945:
4938:
4935:
4927:
4920:
4917:
4910:
4907:
4904:
4900:
4875:
4872:
4869:
4866:
4846:
4843:
4840:
4835:
4832:
4828:
4807:
4804:
4801:
4798:
4774:
4771:
4768:
4748:
4717:
4714:
4711:
4703:
4700:
4697:
4694:
4691:
4688:
4684:
4679:
4675:
4672:
4669:
4666:
4663:
4658:
4654:
4650:
4647:
4644:
4641:
4638:
4633:
4628:
4624:
4618:
4613:
4610:
4606:
4601:
4595:
4590:
4587:
4583:
4579:
4576:
4574:
4572:
4569:
4566:
4561:
4558:
4555:
4552:
4549:
4546:
4542:
4537:
4533:
4530:
4527:
4524:
4521:
4516:
4512:
4508:
4505:
4502:
4499:
4496:
4493:
4488:
4483:
4480:
4476:
4471:
4465:
4460:
4457:
4453:
4449:
4446:
4444:
4442:
4438:
4434:
4431:
4428:
4423:
4419:
4413:
4408:
4405:
4402:
4399:
4395:
4389:
4384:
4381:
4379:
4377:
4373:
4369:
4366:
4363:
4358:
4354:
4349:
4343:
4338:
4334:
4330:
4327:
4324:
4321:
4317:
4311:
4306:
4303:
4301:
4299:
4295:
4291:
4288:
4285:
4280:
4276:
4271:
4265:
4260:
4256:
4252:
4249:
4246:
4243:
4238:
4233:
4229:
4224:
4218:
4213:
4210:
4208:
4206:
4203:
4200:
4197:
4192:
4185:
4182:
4175:
4170:
4166:
4162:
4159:
4156:
4151:
4144:
4141:
4134:
4131:
4128:
4126:
4122:
4117:
4113:
4110:
4107:
4102:
4095:
4092:
4084:
4079:
4078:
4058:
4055:
4052:
4049:
4044:
4039:
4035:
4031:
4028:
4025:
4022:
4019:
3993:
3988:
3984:
3981:
3978:
3973:
3966:
3963:
3955:
3950:
3947:
3945:
3943:
3938:
3933:
3928:
3924:
3921:
3918:
3913:
3909:
3904:
3898:
3892:
3887:
3884:
3882:
3880:
3875:
3870:
3866:
3863:
3858:
3855:
3852:
3849:
3846:
3843:
3839:
3835:
3832:
3829:
3824:
3820:
3814:
3809:
3806:
3802:
3797:
3792:
3789:
3787:
3785:
3782:
3779:
3774:
3770:
3764:
3761:
3758:
3755:
3752:
3749:
3745:
3741:
3738:
3735:
3732:
3727:
3722:
3718:
3712:
3707:
3704:
3700:
3696:
3693:
3691:
3689:
3686:
3683:
3678:
3675:
3672:
3669:
3666:
3662:
3658:
3655:
3652:
3647:
3642:
3638:
3632:
3627:
3624:
3620:
3616:
3613:
3611:
3609:
3606:
3603:
3598:
3595:
3592:
3589:
3586:
3583:
3579:
3575:
3572:
3569:
3566:
3561:
3556:
3552:
3546:
3541:
3538:
3534:
3530:
3527:
3525:
3522:
3518:
3515:
3512:
3509:
3504:
3499:
3495:
3490:
3484:
3479:
3478:
3458:
3455:
3452:
3447:
3443:
3422:
3419:
3416:
3413:
3408:
3403:
3399:
3368:
3363:
3358:
3355:
3352:
3347:
3340:
3337:
3329:
3310:
3309:
3300:
3298:
3284:
3279:
3274:
3271:
3268:
3263:
3256:
3253:
3245:
3239:
3236:
3229:
3226:
3223:
3219:
3215:
3212:
3209:
3206:
3201:
3198:
3194:
3180:
3165:
3162:
3156:
3151:
3146:
3143:
3140:
3135:
3128:
3125:
3117:
3111:
3106:
3103:
3099:
3093:
3090:
3083:
3080:
3077:
3073:
3069:
3066:
3042:
3039:
3036:
3030:
3027:
3004:
3001:
2998:
2995:
2968:
2965:
2962:
2956:
2951:
2947:
2944:
2941:
2936:
2932:
2927:
2920:
2915:
2912:
2908:
2902:
2899:
2892:
2889:
2886:
2882:
2878:
2875:
2855:
2852:
2849:
2844:
2840:
2819:
2816:
2813:
2808:
2804:
2800:
2797:
2794:
2791:
2788:
2785:
2782:
2779:
2774:
2770:
2746:
2743:
2737:
2732:
2728:
2725:
2722:
2719:
2715:
2708:
2704:
2700:
2697:
2692:
2688:
2682:
2678:
2674:
2671:
2666:
2662:
2657:
2651:
2648:
2641:
2638:
2635:
2631:
2627:
2624:
2602:
2598:
2594:
2591:
2571:
2551:
2548:
2545:
2542:
2522:
2496:
2493:
2490:
2485:
2481:
2453:
2450:
2447:
2444:
2429:power spectrum
2416:
2413:
2410:
2407:
2358:
2355:
2338:
2335:
2332:
2329:
2309:
2306:
2286:
2281:
2277:
2252:
2249:
2246:
2241:
2234:
2231:
2207:
2200:
2195:
2191:
2188:
2185:
2180:
2173:
2170:
2162:
2154:
2149:
2144:
2138:
2135:
2131:
2128:
2125:
2122:
2119:
2116:
2112:
2106:
2102:
2096:
2091:
2088:
2085:
2082:
2078:
2073:
2063:
2059:
2055:
2052:
2049:
2044:
2041:
2038:
2034:
2030:
2027:
2024:
2021:
2016:
2013:
2006:
2003:
1979:
1976:
1973:
1969:
1966:
1963:
1958:
1954:
1950:
1945:
1941:
1918:
1914:
1886:
1866:
1863:
1859:
1855:
1852:
1849:
1846:
1822:
1819:
1816:
1813:
1793:
1789:
1783:
1779:
1775:
1772:
1769:
1749:
1746:
1742:
1738:
1735:
1732:
1729:
1709:
1689:
1686:
1683:
1680:
1660:
1657:
1633:
1613:
1610:
1607:
1602:
1599:
1592:
1589:
1565:
1561:
1555:
1551:
1547:
1544:
1541:
1521:
1517:
1511:
1507:
1503:
1500:
1497:
1477:
1449:
1414:
1411:
1408:
1405:
1374:
1371:
1368:
1365:
1341:
1338:
1335:
1330:
1327:
1320:
1317:
1299:
1298:
1289:
1287:
1274:
1269:
1265:
1262:
1259:
1253:
1250:
1243:
1238:
1235:
1232:
1229:
1224:
1221:
1214:
1211:
1194:
1181:
1178:
1175:
1172:
1145:
1142:
1139:
1136:
1116:
1092:
1089:
1084:
1079:
1075:
1072:
1069:
1063:
1060:
1053:
1027:
1004:
1001:
998:
995:
971:
968:
962:
959:
956:
953:
948:
945:
942:
939:
936:
933:
929:
923:
918:
915:
911:
907:
904:
901:
898:
892:
889:
866:
863:
860:
854:
849:
845:
842:
839:
833:
830:
823:
816:
811:
808:
804:
800:
797:
794:
788:
783:
778:
775:
772:
769:
765:
759:
754:
751:
747:
720:
717:
714:
706:
701:
697:
694:
691:
688:
684:
677:
672:
669:
665:
661:
658:
638:
635:
632:
629:
609:
578:
575:
573:
570:
553:
550:
433:
430:
381:
378:
375:
372:
340:
320:
317:
314:
311:
284:
281:
278:
273:
269:
248:power spectrum
212:
188:
185:
182:
179:
153:
150:
147:
142:
139:
135:
117:
116:
78:
76:
69:
41:Spectral power
26:
9:
6:
4:
3:
2:
10364:
10353:
10350:
10348:
10345:
10343:
10340:
10338:
10335:
10333:
10330:
10328:
10325:
10323:
10320:
10318:
10315:
10314:
10312:
10297:
10294:
10292:
10289:
10287:
10284:
10282:
10279:
10276:
10275:
10272:
10266:
10263:
10261:
10258:
10256:
10253:
10251:
10248:
10246:
10243:
10241:
10238:
10235:
10232:
10230:
10227:
10223:
10220:
10219:
10218:
10215:
10213:
10210:
10208:
10205:
10202:
10199:
10198:
10195:
10190:
10187:
10180:
10175:
10173:
10168:
10166:
10161:
10160:
10157:
10151:
10148:
10147:
10137:
10131:
10127:
10122:
10118:
10112:
10108:
10103:
10099:
10093:
10089:
10084:
10080:
10074:
10070:
10065:
10061:
10057:
10053:
10047:
10043:
10038:
10034:
10028:
10024:
10023:VSAT Networks
10019:
10015:
10011:
10007:
10003:
9998:
9997:10044/1/46059
9993:
9989:
9985:
9981:
9977:
9973:
9968:
9964:
9960:
9955:
9950:
9946:
9942:
9938:
9934:
9930:
9925:
9921:
9915:
9911:
9906:
9902:
9896:
9892:
9887:
9883:
9877:
9873:
9868:
9867:
9851:
9844:
9837:
9832:
9825:
9820:
9812:
9805:
9798:
9793:
9786:
9781:
9774:
9769:
9761:
9755:
9751:
9750:
9742:
9735:
9730:
9723:
9718:
9711:
9706:
9700:, p. 14.
9699:
9694:
9687:
9682:
9676:, p. 12.
9675:
9670:
9656:on 2024-04-15
9655:
9651:
9647:
9640:
9634:, p. 83.
9633:
9632:Birolini 2007
9628:
9621:
9616:
9609:
9604:
9593:
9589:
9583:
9581:
9579:
9574:
9565:
9561:
9543:
9540:
9536:
9527:
9523:
9519:
9518:distributions
9501:
9498:
9494:
9485:
9481:
9475:
9468:
9448:
9445:
9441:
9438:
9432:
9412:
9405:
9402:
9398:
9395:
9389:
9383:
9377:
9374:
9371:
9368:
9358:
9355:
9346:
9336:
9326:
9314:
9300:
9294:
9290:
9280:
9277:
9275:
9272:
9270:
9267:
9265:
9262:
9260:
9257:
9255:
9252:
9250:
9247:
9245:
9242:
9240:
9237:
9235:
9232:
9230:
9227:
9226:
9220:
9218:
9209:
9207:
9202:
9200:
9199:
9198:power spectra
9194:
9190:
9186:
9182:
9178:
9174:
9166:
9161:
9152:
9150:
9145:
9141:
9137:
9133:
9129:
9125:
9121:
9117:
9116:visible light
9111:
9097:
9093:
9089:
9085:
9082:
9078:
9074:
9070:
9051:
9045:
9038:
9034:
9033:
9028:
9024:
9020:
9016:
9012:
9008:
9004:
9000:
8996:
8993:
8989:
8985:
8984:
8978:
8975:
8971:
8967:
8963:
8959:
8955:
8951:
8948:
8944:
8940:
8936:
8932:
8928:
8924:
8920:
8916:
8912:
8909:
8908:
8903:
8902:
8899:
8889:
8887:
8883:
8879:
8874:
8872:
8868:
8864:
8860:
8856:
8855:random signal
8852:
8846:
8836:
8823:
8812:
8808:
8804:
8801:
8798:
8795:
8792:
8788:
8779:
8775:
8766:
8763:
8759:
8745:
8742:
8739:
8735:
8731:
8725:
8717:
8714:
8710:
8700:
8691:
8682:
8666:
8658:
8655:
8651:
8647:
8644:
8638:
8630:
8627:
8623:
8619:
8616:
8607:
8598:
8594:
8575:
8563:
8537:
8525:
8499:
8493:
8470:
8464:
8441:
8435:
8432:
8426:
8420:
8397:
8391:
8368:
8362:
8339:
8331:
8328:
8324:
8300:
8294:
8271:
8265:
8257:
8238:
8230:
8227:
8223:
8198:
8195:
8192:
8187:
8184:
8181:
8178:
8175:
8172:
8168:
8161:
8153:
8150:
8146:
8132:
8128:
8124:
8121:
8118:
8113:
8110:
8107:
8104:
8101:
8098:
8094:
8089:
8085:
8082:
8076:
8068:
8064:
8057:
8054:
8051:
8043:
8038:
8034:
8020:
8016:
8010:
8007:
7994:
7985:
7971:
7967:
7963:
7961:
7953:
7945:
7942:
7938:
7930:
7927:
7922:
7919:
7916:
7913:
7910:
7907:
7903:
7896:
7888:
7885:
7881:
7867:
7863:
7859:
7856:
7853:
7848:
7845:
7842:
7839:
7836:
7833:
7829:
7824:
7820:
7817:
7811:
7803:
7799:
7792:
7789:
7786:
7778:
7773:
7769:
7755:
7751:
7745:
7742:
7729:
7720:
7706:
7702:
7698:
7696:
7688:
7680:
7677:
7673:
7660:
7638:
7630:
7611:
7603:
7600:
7596:
7592:
7586:
7578:
7573:
7570:
7566:
7542:
7534:
7531:
7527:
7503:
7495:
7492:
7488:
7462:
7455:
7447:
7437:
7427:
7419:
7414:
7404:
7396:
7390:
7387:
7374:
7366:
7364:
7356:
7348:
7345:
7341:
7334:
7327:
7319:
7309:
7299:
7291:
7286:
7276:
7268:
7262:
7259:
7246:
7238:
7236:
7228:
7220:
7217:
7213:
7199:
7182:
7179:
7174:
7161:
7153:
7149:
7140:
7134:
7126:
7122:
7115:
7107:
7102:
7098:
7094:
7088:
7080:
7076:
7069:
7061:
7056:
7052:
7048:
7043:
7030:
7022:
7018:
6999:
6995:
6989:
6986:
6973:
6965:
6963:
6955:
6952:
6948:
6941:
6933:
6929:
6925:
6919:
6911:
6907:
6902:
6896:
6891:
6884:
6876:
6872:
6868:
6862:
6854:
6850:
6845:
6830:
6826:
6820:
6817:
6804:
6796:
6794:
6789:
6777:
6773:
6769:
6765:
6746:
6738:
6735:
6731:
6707:
6699:
6696:
6692:
6668:
6662:
6639:
6633:
6623:
6607:
6590:
6586:
6583:
6579:
6571:
6568:
6559:
6556:
6552:
6546:
6541:
6537:
6533:
6530:
6524:
6518:
6510:
6491:
6485:
6478:
6474:
6455:
6447:
6444:
6440:
6431:
6427:
6423:
6409:
6406:
6399:
6391:
6388:
6384:
6369:
6365:
6361:
6355:
6349:
6346:
6343:
6340:
6332:
6328:
6324:
6322:
6314:
6309:
6305:
6301:
6297:
6279:
6271:
6268:
6264:
6260:
6254:
6251:
6243:
6240:
6236:
6227:
6226:even function
6223:
6222:
6220:
6212:
6210:
6206:
6201:
6187:
6179:
6163:
6143:
6121:
6116:
6110:
6103:
6100:
6097:
6094:
6091:
6088:
6084:
6078:
6074:
6068:
6063:
6060:
6057:
6054:
6050:
6045:
6038:
6032:
6024:
6004:
5996:
5990:
5982:
5979:
5975:
5954:
5944:
5941:
5938:
5935:
5929:
5926:
5903:
5896:
5890:
5885:
5881:
5877:
5872:
5868:
5847:
5844:
5841:
5838:
5835:
5832:
5810:
5806:
5797:
5796:discrete time
5792:
5775:
5769:
5761:
5744:
5740:
5736:
5716:
5712:
5708:
5688:
5668:
5665:
5658:
5650:
5647:
5643:
5635:
5631:
5623:
5619:
5614:
5610:
5607:
5596:
5572:
5564:
5561:
5557:
5553:
5547:
5544:
5536:
5533:
5529:
5506:
5502:
5498:
5493:
5489:
5485:
5482:
5457:
5453:
5449:
5444:
5440:
5427:
5425:
5415:
5408:
5406:
5389:
5381:
5378:
5368:
5361:
5358:
5355:
5349:
5346:
5343:
5340:
5337:
5334:
5330:
5323:
5315:
5312:
5308:
5294:
5290:
5286:
5280:
5272:
5269:
5265:
5257:
5256:
5252:
5250:
5231:
5225:
5216:
5203:
5200:
5195:
5192:
5189:
5186:
5183:
5180:
5176:
5169:
5161:
5158:
5154:
5140:
5136:
5132:
5129:
5126:
5118:
5115:
5112:
5109:
5106:
5103:
5099:
5094:
5090:
5087:
5081:
5073:
5069:
5062:
5059:
5056:
5048:
5043:
5039:
5025:
5021:
5015:
5012:
4999:
4990:
4976:
4972:
4968:
4963:
4958:
4951:
4943:
4933:
4925:
4918:
4915:
4902:
4889:
4870:
4864:
4841:
4833:
4830:
4826:
4802:
4796:
4788:
4766:
4746:
4737:
4735:
4715:
4712:
4709:
4701:
4698:
4695:
4692:
4689:
4686:
4682:
4677:
4673:
4670:
4664:
4656:
4652:
4645:
4642:
4639:
4631:
4626:
4622:
4608:
4604:
4599:
4585:
4581:
4577:
4575:
4567:
4564:
4559:
4556:
4553:
4550:
4547:
4544:
4540:
4535:
4531:
4528:
4522:
4514:
4510:
4503:
4500:
4497:
4491:
4478:
4474:
4469:
4455:
4451:
4447:
4445:
4436:
4429:
4421:
4417:
4403:
4397:
4393:
4382:
4380:
4371:
4364:
4356:
4352:
4347:
4336:
4332:
4325:
4319:
4315:
4304:
4302:
4293:
4286:
4278:
4274:
4269:
4258:
4254:
4247:
4244:
4236:
4231:
4227:
4222:
4211:
4209:
4198:
4190:
4180:
4173:
4168:
4157:
4149:
4139:
4129:
4127:
4120:
4115:
4108:
4100:
4090:
4082:
4053:
4050:
4042:
4037:
4033:
4029:
4023:
4017:
3991:
3986:
3979:
3971:
3961:
3953:
3948:
3946:
3936:
3931:
3926:
3919:
3911:
3907:
3902:
3890:
3885:
3883:
3873:
3868:
3864:
3861:
3856:
3853:
3850:
3847:
3844:
3841:
3837:
3830:
3822:
3818:
3804:
3800:
3795:
3790:
3788:
3780:
3777:
3772:
3762:
3759:
3756:
3753:
3750:
3747:
3743:
3733:
3725:
3720:
3716:
3702:
3698:
3694:
3692:
3684:
3681:
3676:
3673:
3670:
3667:
3664:
3660:
3653:
3645:
3640:
3636:
3622:
3618:
3614:
3612:
3604:
3601:
3596:
3593:
3590:
3587:
3584:
3581:
3577:
3570:
3567:
3559:
3554:
3550:
3536:
3532:
3528:
3526:
3520:
3513:
3510:
3502:
3497:
3493:
3488:
3453:
3445:
3441:
3417:
3414:
3406:
3401:
3397:
3388:
3384:
3366:
3353:
3345:
3335:
3318:
3308:
3301:
3299:
3282:
3269:
3261:
3251:
3237:
3234:
3221:
3213:
3207:
3199:
3196:
3192:
3184:
3183:
3179:
3176:
3163:
3160:
3154:
3141:
3133:
3123:
3101:
3097:
3091:
3088:
3075:
3067:
3064:
3056:
3037:
3025:
2999:
2993:
2984:
2982:
2966:
2963:
2960:
2954:
2949:
2942:
2934:
2930:
2925:
2910:
2906:
2900:
2897:
2884:
2876:
2873:
2850:
2842:
2838:
2814:
2806:
2802:
2795:
2789:
2786:
2780:
2772:
2768:
2757:
2744:
2741:
2735:
2730:
2723:
2717:
2713:
2706:
2702:
2698:
2695:
2690:
2686:
2680:
2676:
2672:
2669:
2664:
2660:
2655:
2649:
2646:
2633:
2625:
2622:
2600:
2596:
2592:
2589:
2569:
2546:
2540:
2520:
2511:
2509:
2491:
2483:
2479:
2470:
2467:
2448:
2442:
2434:
2430:
2411:
2405:
2397:
2392:
2388:
2384:
2375:
2370:
2364:
2354:
2352:
2336:
2330:
2307:
2284:
2279:
2275:
2266:
2247:
2239:
2229:
2205:
2198:
2193:
2186:
2178:
2168:
2160:
2152:
2147:
2142:
2136:
2129:
2126:
2123:
2120:
2117:
2114:
2110:
2104:
2100:
2094:
2089:
2086:
2083:
2080:
2076:
2071:
2061:
2053:
2036:
2028:
2022:
2014:
2011:
2001:
1974:
1967:
1961:
1956:
1952:
1948:
1943:
1939:
1916:
1912:
1903:
1898:
1884:
1864:
1857:
1850:
1844:
1836:
1817:
1811:
1791:
1787:
1781:
1773:
1767:
1747:
1740:
1733:
1727:
1707:
1684:
1678:
1658:
1647:
1631:
1624:at frequency
1608:
1600:
1597:
1587:
1563:
1559:
1553:
1545:
1539:
1519:
1515:
1509:
1501:
1495:
1475:
1467:
1463:
1447:
1440:
1436:
1432:
1428:
1409:
1403:
1394:
1392:
1388:
1369:
1363:
1355:
1336:
1328:
1325:
1315:
1304:The function
1297:
1290:
1288:
1272:
1267:
1260:
1248:
1241:
1236:
1230:
1222:
1219:
1209:
1198:
1197:
1193:
1176:
1170:
1162:
1157:
1143:
1140:
1137:
1134:
1114:
1106:
1090:
1087:
1082:
1077:
1070:
1058:
1051:
1041:
1025:
1018:
999:
993:
985:
969:
966:
957:
951:
946:
943:
940:
937:
934:
931:
927:
913:
909:
905:
899:
887:
864:
861:
858:
852:
847:
840:
828:
821:
806:
802:
798:
795:
792:
786:
773:
767:
749:
745:
736:
731:
718:
715:
712:
704:
699:
692:
686:
682:
667:
663:
659:
656:
633:
627:
607:
599:
595:
591:
584:
569:
567:
563:
559:
549:
547:
543:
538:
536:
532:
528:
524:
520:
514:
512:
508:
504:
500:
496:
492:
487:
481:
473:
465:
461:
457:
453:
452:acoustic wave
449:
445:
439:
429:
427:
423:
419:
415:
411:
405:
403:
399:
395:
376:
370:
362:
358:
354:
338:
315:
309:
300:
298:
279:
271:
267:
258:
253:
249:
245:
241:
236:
234:
230:
226:
210:
202:
183:
177:
170:
167:
148:
140:
137:
133:
124:
113:
110:
102:
92:
88:
82:
79:This article
77:
68:
67:
60:
53:
48:
42:
37:
33:
19:
10332:Spectroscopy
10291:signal noise
10125:
10106:
10087:
10068:
10041:
10022:
9979:
9975:
9936:
9932:
9909:
9890:
9871:
9843:
9831:
9819:
9804:
9792:
9780:
9768:
9752:. Springer.
9748:
9741:
9729:
9717:
9705:
9693:
9681:
9669:
9658:. Retrieved
9654:the original
9649:
9639:
9627:
9615:
9603:
9474:
9293:
9215:
9203:
9201:of signals.
9197:
9196:
9179:, including
9170:
9149:notch filter
9143:
9113:
9104:Applications
9095:
9091:
9087:
9030:
9022:
9018:
9014:
8981:
8976:of interest.
8966:spectrometer
8953:
8934:
8930:
8929:below which
8922:
8918:
8914:
8905:
8882:Welch method
8875:
8848:
8695:
8686:
8683:
8596:
8592:
7628:
7200:
6775:
6771:
6767:
6763:
6625:
6605:
6508:
6476:
6472:
6429:
6425:
6312:
6307:
6303:
6218:
6202:
5793:
5428:
5423:
5421:
5409:
5217:
4738:
4010:and making,
3385:of the time
3314:
3302:
3177:
2985:
2980:
2758:
2533:of a signal
2512:
2428:
2382:
2379:
1899:
1488:is equal to
1395:
1303:
1291:
1160:
1158:
732:
620:of a signal
587:
561:
557:
555:
545:
541:
539:
533:denotes the
530:
527:acceleration
522:
515:
502:
495:proportional
494:
488:
455:
441:
406:
402:spectrograph
301:
251:
247:
243:
239:
237:
120:
105:
96:
80:
36:
10286:link budget
9081:group delay
9077:phase noise
8983:spectrogram
8871:periodogram
6509:bandlimited
6178:periodogram
5602:bandlimited
4069:, we have:
3387:convolution
1391:Periodogram
594:time series
521:, units of
422:engineering
10337:Scattering
10311:Categories
9864:References
9686:Stein 2000
9660:2024-06-26
9608:Maral 2004
9229:Bispectrum
9132:wavelength
8943:anesthesia
8859:parametric
8839:Estimation
7661:function.
6620:See also:
6156:(and thus
5798:variables
4732:where the
1389:(see also
572:Definition
519:vibrations
436:See also:
10203:(or dBmW)
10060:696092052
10014:206608057
9446:ω
9442:−
9439:ω
9433:δ
9403:ω
9399:−
9396:ω
9390:δ
9384:ω
9375:π
9366:⟩
9356:ω
9347:∗
9340:^
9327:ω
9318:^
9309:⟨
9212:Cosmology
9140:harmonics
9136:sine wave
9003:Bode plot
8927:frequency
8824:τ
8821:Δ
8809:τ
8802:π
8793:−
8776:τ
8754:∞
8749:∞
8746:−
8736:∑
8567:^
8529:^
8340:τ
8239:τ
8196:τ
8188:τ
8182:π
8173:−
8162:τ
8141:∞
8136:∞
8133:−
8129:∫
8122:τ
8114:τ
8108:π
8099:−
8058:τ
8055:−
8044:∗
8029:∞
8024:∞
8021:−
8017:∫
8001:∞
7998:→
7980:∞
7975:∞
7972:−
7968:∫
7931:τ
7923:τ
7917:π
7908:−
7897:τ
7876:∞
7871:∞
7868:−
7864:∫
7857:τ
7849:τ
7843:π
7834:−
7793:τ
7790:−
7779:∗
7764:∞
7759:∞
7756:−
7752:∫
7736:∞
7733:→
7715:∞
7710:∞
7707:−
7703:∫
7645:∞
7642:→
7579:∗
7441:^
7420:∗
7408:^
7381:∞
7378:→
7313:^
7292:∗
7280:^
7253:∞
7250:→
7108:∗
7062:∗
7008:∞
7003:∞
7000:−
6996:∫
6980:∞
6977:→
6897:∗
6839:∞
6834:∞
6831:−
6827:∫
6811:∞
6808:→
6538:∫
6378:∞
6373:∞
6370:−
6366:∫
6350:
6252:−
6108:Δ
6098:π
6089:−
6061:−
6051:∑
6022:Δ
6011:∞
6008:→
5952:Δ
5901:Δ
5845:≤
5839:≤
5833:−
5615:∫
5545:−
5372:^
5359:τ
5350:τ
5344:π
5335:−
5324:τ
5303:∞
5298:∞
5295:−
5291:∫
5204:τ
5196:τ
5190:π
5181:−
5170:τ
5149:∞
5144:∞
5141:−
5137:∫
5130:τ
5119:τ
5113:π
5104:−
5063:τ
5060:−
5049:∗
5034:∞
5029:∞
5026:−
5022:∫
5006:∞
5003:→
4985:∞
4980:∞
4977:−
4973:∫
4937:^
4909:∞
4906:→
4842:τ
4773:∞
4770:→
4713:τ
4702:τ
4696:π
4687:−
4646:τ
4643:−
4632:∗
4617:∞
4612:∞
4609:−
4605:∫
4594:∞
4589:∞
4586:−
4582:∫
4568:τ
4560:τ
4554:π
4545:−
4501:−
4498:τ
4487:∞
4482:∞
4479:−
4475:∫
4464:∞
4459:∞
4456:−
4452:∫
4412:∗
4337:⋅
4259:⋅
4245:−
4237:∗
4184:^
4174:⋅
4169:∗
4143:^
4094:^
4051:−
4043:∗
3992:∗
3965:^
3937:∗
3874:∗
3851:π
3842:−
3813:∞
3808:∞
3805:−
3801:∫
3773:∗
3757:π
3748:−
3726:∗
3711:∞
3706:∞
3703:−
3699:∫
3671:π
3646:∗
3631:∞
3626:∞
3623:−
3619:∫
3591:π
3582:−
3568:−
3560:∗
3545:∞
3540:∞
3537:−
3533:∫
3511:−
3503:∗
3415:−
3407:∗
3339:^
3255:^
3228:∞
3225:→
3127:^
3110:∞
3105:∞
3102:−
3098:∫
3082:∞
3079:→
3029:^
2919:∞
2914:∞
2911:−
2907:∫
2891:∞
2888:→
2670:−
2656:∫
2640:∞
2637:→
2334:→
2328:Δ
2305:Δ
2233:^
2172:^
2153:⏟
2134:Δ
2124:π
2115:−
2087:−
2077:∑
2051:Δ
2043:∞
2040:→
2005:¯
1972:Δ
1862:Δ
1745:Δ
1656:Δ
1591:¯
1466:Ohm's law
1439:impedance
1427:potential
1319:¯
1252:^
1237:≜
1213:¯
1062:^
1017:frequency
941:π
932:−
922:∞
917:∞
914:−
910:∫
906:≜
891:^
832:^
815:∞
810:∞
807:−
803:∫
758:∞
753:∞
750:−
746:∫
676:∞
671:∞
668:−
664:∫
660:≜
562:two-sided
558:one-sided
511:watt-hour
499:impedance
257:Summation
99:June 2024
10277:See also
10189:suffixes
10006:28541914
9963:25113231
9588:P Stoica
9449:′
9406:′
9359:′
9223:See also
9165:FM radio
9128:radio/TV
9110:Spectrum
8992:wavelets
8970:spectral
8964:using a
8851:estimate
6587:′
6572:′
6331:variance
5475:, where
2469:resistor
2396:variance
1352:and the
529:, where
464:SI units
233:spectrum
10236:(radar)
10186:Decibel
9954:4204008
9564:measure
9465:is the
9001:(e.g.,
8956:is the
8254:is the
4888:ergodic
3381:as the
2433:voltage
2263:is the
1462:matched
877:where:
535:g-force
491:voltage
484:
468:
444:physics
418:physics
85:Please
10217:dBm/Hz
10132:
10113:
10094:
10075:
10058:
10048:
10029:
10012:
10004:
9961:
9951:
9916:
9897:
9878:
9756:
9425:where
9185:radars
8457:. If
8215:where
6325:Using
5424:define
5124:
4707:
2218:where
964:
710:
598:energy
590:energy
507:energy
357:timbre
169:signal
10327:Waves
10265:dB-Hz
10010:S2CID
9853:(PDF)
9595:(PDF)
9285:Notes
9124:pitch
9120:color
9096:shape
9021:, or
9011:phase
9007:chirp
8947:sleep
8384:with
8287:with
6770:) or
2508:watts
2391:power
1431:volts
480:hertz
472:watts
460:power
450:, an
432:Units
361:color
353:pitch
229:noise
201:power
164:of a
10260:dBrn
10255:dBFS
10240:dBsm
10191:(dB)
10130:ISBN
10111:ISBN
10092:ISBN
10073:ISBN
10056:OCLC
10046:ISBN
10027:ISBN
10002:PMID
9959:PMID
9914:ISBN
9895:ISBN
9876:ISBN
9754:ISBN
9478:The
8990:and
8913:The
8904:The
8693:and
8612:Full
8608:CPSD
8597:CPSD
8593:CPSD
8553:and
8486:and
8316:and
7629:CPSD
7519:and
6768:CPSD
6723:and
6655:and
5499:<
5486:<
5412:Eq.3
3433:and
3305:Eq.2
2830:and
1429:(in
1294:Eq.1
1038:(in
649:is:
420:and
355:and
10250:dBi
10245:dBc
10234:dBZ
10229:dBA
10222:PSD
10212:dBV
10207:dBW
10201:dBm
9992:hdl
9984:doi
9949:PMC
9941:doi
9144:not
9126:),
9092:ASD
8939:EEG
8919:SEF
8861:or
8258:of
7991:lim
7726:lim
7371:lim
7243:lim
6970:lim
6801:lim
6776:CSD
6608:â â
6475:or
6347:Var
6001:lim
5251:).
4996:lim
4899:lim
4886:is
3389:of
3218:lim
3072:lim
2881:lim
2630:lim
2466:ohm
2267:of
2033:lim
1437:of
1393:).
1356:of
1163:of
1156:.
1015:at
986:of
513:).
466:of
442:In
252:all
121:In
89:to
10313::
10054:.
10008:.
10000:.
9990:.
9980:11
9978:.
9957:.
9947:.
9937:42
9935:.
9931:.
9648:.
9577:^
9183:,
9151:.
9079:,
9075:,
9017:,
9005:,
8952:A
8873:.
6317:xx
6308:xx
6211:.
5967:.
2981:is
2615::
2510:.
2353:)
2337:0.
1990::
1040:Hz
537:.
428:.
299:.
235:.
10178:e
10171:t
10164:v
10138:.
10119:.
10100:.
10081:.
10062:.
10035:.
10016:.
9994::
9986::
9965:.
9943::
9922:.
9903:.
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9838:.
9826:.
9813:.
9799:.
9775:.
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9622:.
9610:.
9597:.
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9495:R
9453:)
9436:(
9413:,
9410:)
9393:(
9387:)
9381:(
9378:f
9372:2
9369:=
9363:)
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9337:x
9330:)
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9315:x
9090:(
9083:.
9055:)
9052:t
9049:(
9046:h
8994:.
8949:.
8935:x
8931:x
8923:x
8917:(
8900:.
8813:n
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8799:2
8796:i
8789:e
8785:)
8780:n
8772:(
8767:y
8764:x
8760:R
8743:=
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8732:=
8729:)
8726:f
8723:(
8718:y
8715:x
8711:S
8698:n
8696:y
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8670:)
8667:f
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8639:f
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8617:=
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8576:f
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8541:)
8538:f
8535:(
8526:x
8503:)
8500:t
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8465:x
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8433:=
8430:)
8427:t
8424:(
8421:x
8401:)
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8083:d
8080:)
8077:t
8074:(
8069:T
8065:x
8061:)
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7964:=
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7900:)
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7860:=
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7840:2
7837:i
7830:e
7825:]
7821:t
7818:d
7815:)
7812:t
7809:(
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7800:y
7796:)
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7699:=
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7431:)
7428:f
7425:(
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7360:)
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7331:)
7328:f
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7310:y
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7300:f
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7239:=
7232:)
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7214:S
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7119:)
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7095:+
7092:)
7089:t
7086:(
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7067:(
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7028:(
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6949:]
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6939:(
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6930:y
6926:+
6923:)
6920:t
6917:(
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6892:]
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6797:=
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6774:(
6766:(
6750:)
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6705:(
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6666:(
6663:y
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6640:t
6637:(
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6610:.
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6591:.
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6547:f
6542:0
6534:2
6531:=
6528:)
6525:f
6522:(
6519:F
6495:)
6492:f
6489:(
6486:F
6459:)
6456:f
6453:(
6448:x
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6428:(
6426:x
6410:f
6407:d
6403:)
6400:f
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6389:x
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6359:)
6356:x
6353:(
6344:=
6341:P
6313:t
6311:(
6304:R
6295:.
6283:)
6280:f
6277:(
6272:x
6269:x
6265:S
6261:=
6258:)
6255:f
6249:(
6244:x
6241:x
6237:S
6188:T
6164:T
6144:N
6122:2
6117:|
6111:t
6104:n
6101:f
6095:2
6092:i
6085:e
6079:n
6075:x
6069:N
6064:N
6058:=
6055:n
6046:|
6039:T
6033:2
6029:)
6025:t
6019:(
6005:N
5997:=
5994:)
5991:f
5988:(
5983:x
5980:x
5976:S
5955:t
5948:)
5945:1
5942:+
5939:N
5936:2
5933:(
5930:=
5927:T
5907:)
5904:t
5897:n
5894:(
5891:+
5886:0
5882:t
5878:=
5873:n
5869:t
5848:N
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5836:N
5811:n
5807:x
5779:)
5776:t
5773:(
5770:x
5745:T
5741:/
5737:1
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5713:/
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5669:f
5666:d
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5659:f
5656:(
5651:x
5648:x
5644:S
5636:2
5632:f
5624:1
5620:f
5611:2
5608:=
5597:P
5576:)
5573:f
5570:(
5565:x
5562:x
5558:S
5554:=
5551:)
5548:f
5542:(
5537:x
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5494:1
5490:f
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5445:1
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5437:[
5414:)
5410:(
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5379:x
5369:R
5362:=
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5327:)
5321:(
5316:x
5313:x
5309:R
5287:=
5284:)
5281:f
5278:(
5273:x
5270:x
5266:S
5235:)
5232:t
5229:(
5226:x
5201:d
5193:f
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5177:e
5173:)
5167:(
5162:x
5159:x
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5133:=
5127:d
5116:f
5110:2
5107:i
5100:e
5095:]
5091:t
5088:d
5085:)
5082:t
5079:(
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5070:x
5066:)
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5054:(
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5000:T
4991:[
4969:=
4964:2
4959:|
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4952:f
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4316:{
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4270:{
4264:F
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4212:=
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3932:]
3927:}
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3897:F
3891:[
3886:=
3869:]
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3791:=
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3737:)
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3731:(
3721:T
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3695:=
3685:t
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3665:i
3661:e
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3654:t
3651:(
3641:T
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3615:=
3605:t
3602:d
3597:t
3594:f
3588:2
3585:i
3578:e
3574:)
3571:t
3565:(
3555:T
3551:x
3529:=
3521:}
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3514:t
3508:(
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3494:x
3489:{
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3214:=
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2839:w
2818:)
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2787:=
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2677:/
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2626:=
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2601:0
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2137:t
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2105:n
2101:x
2095:N
2090:N
2084:=
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2072:|
2062:2
2058:)
2054:t
2048:(
2037:N
2029:=
2026:)
2023:f
2020:(
2015:x
2012:x
2002:S
1978:)
1975:t
1968:n
1965:(
1962:+
1957:0
1953:t
1949:=
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1913:x
1885:Z
1865:f
1858:/
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1851:f
1848:(
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1835:J
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1588:S
1564:Z
1560:/
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1540:V
1520:Z
1516:/
1510:2
1506:)
1502:t
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1476:t
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1404:V
1373:)
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1316:S
1296:)
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1242:|
1234:)
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1068:(
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1052:|
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888:x
865:,
862:f
859:d
853:2
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838:(
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822:|
799:=
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793:d
787:2
782:|
777:)
774:t
771:(
768:x
764:|
719:.
716:t
713:d
705:2
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657:E
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628:x
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585:.
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544:(
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531:g
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339:x
319:)
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313:(
310:x
283:)
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268:x
211:f
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181:(
178:x
152:)
149:f
146:(
141:x
138:x
134:S
112:)
106:(
101:)
97:(
83:.
43:.
34:.
20:)
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