4707:
36:
4049:
48:
4702:{\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}}}
9137:
8190:
3985:
2346:
7174:
7641:
3449:
61:
6758:
5191:
6153:) 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
2193:
8185:{\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}}}
3980:{\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}}}
7455:
9075:
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
243:
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
2370:
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
9111:) 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
4870:
5734:
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
6111:
9400:
2357:
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
7169:{\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}}}
1970:
493:
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
852:
2736:
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
5380:
2954:
8811:
2732:
3151:
7181:
8956:
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
957:
3273:
6397:
706:
396:
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
5947:
5656:
1262:
8657:
9123:
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
9281:
6578:
5186:{\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 }
717:
5565:, 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):
2956:
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
7646:
7186:
6763:
4054:
3454:
2846:
9044:
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
8682:
2595:
3037:
1078:
216:, 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
9440:
5894:
1965:
2188:{\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}},}
9493:
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
7602:
3356:
470:, 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
248:
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
2805:
6270:
5563:
5237:
4760:
5496:
5942:
4044:
1599:
1327:
2238:
5835:
2324:
857:
8566:
8528:
8330:
8229:
4832:
3408:
3028:
1852:
1735:
7632:
6313:
1779:
1551:
1507:
8432:
7533:
7494:
6737:
6698:
6446:
151:
3444:
2841:
2482:
2295:
1646:
282:
9826:
2590:
629:
9533:
9491:
2272:
9185:(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.
5800:
1906:
9042:
8490:
8461:
8388:
8359:
8291:
8262:
6659:
6630:
6482:
5766:
5568:
5222:
4861:
4793:
2990:
2537:
2439:
2402:
1808:
1675:
1400:
1360:
1167:
1131:
990:
624:
379:
318:
186:
5732:
5704:
6175:
6151:
6131:
5676:
4734:
2557:
2508:
1872:
1695:
1619:
1463:
1435:
1102:
1013:
595:
338:
210:
8579:
7450:{\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}}}
5450:
9724:
8842:
approaches, and may be based on time-domain or frequency-domain analysis. For example, a common parametric technique involves fitting the observations to an
7608:. 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
2404:(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
3164:
517:
Mathematically, it is not necessary to assign physical dimensions to the signal or to the independent variable. In the following discussion the meaning of
9196:, 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.
9091:
Any signal that can be represented as a variable that varies in time has a corresponding frequency spectrum. This includes familiar entities such as
393:, 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.
79:
8974:
A "spectrum" generally means the power spectral density, as discussed above, which depicts the distribution of signal content over frequency. For
1178:
1019:). 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
9012:
of a transfer function, which also includes a phase (or equivalently, a real and imaginary part) as a function of frequency. The time-domain
2326: But in the mathematical sciences the interval is often set to 1, which simplifies the results at the expense of generality. (also see
227:
When the energy of the signal is concentrated around a finite time interval, especially if its total energy is finite, one may compute the
6491:
6113:
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
385:, and generalizations based on Fourier analysis. In many cases the time domain is not specifically employed in practice, such as when a
1854:
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
19:
This article is about signal processing and relation of spectra to time-series. For further applications in the physical sciences, see
9622:
6106:{\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}}
5678:
is finite rather than approaching infinity. This results in decreased spectral coverage and resolution since frequencies of less than
9278:, p. 30) still use the non-normalized Fourier transform in a formal way to formulate a definition of the power spectral density
710:
The energy spectral density is most suitable for transientsâthat is, pulse-like signalsâhaving a finite total energy. Finite or not,
413:. Typically the process is a function of time, but one can similarly discuss data in the spatial domain being decomposed in terms of
8887:
of a signal is the midpoint of its spectral density function, i.e. the frequency that divides the distribution into two equal parts.
7178:
Using the same notation and methods as used for the power spectral density derivation, we exploit
Parseval's theorem and obtain
10328:
10110:
10091:
10072:
10053:
10026:
10007:
9894:
9875:
9856:
9734:
2350:
9543:, whose distribution function is F (but not necessarily as a function and not necessarily possessing a probability density).
6184:
can similarly be calculated; as the PSD is related to the autocorrelation, so is the cross-spectral density related to the
5658:
More generally, similar techniques may be used to estimate a time-varying spectral density. In this case the time interval
9395:{\displaystyle \langle {\hat {x}}(\omega ){\hat {x}}^{\ast }(\omega ')\rangle =2\pi f(\omega )\delta (\omega -\omega '),}
8874:
8862:
10293:
1084:
multiplied by an infinitesimally small frequency interval, describing the energy contained in the signal at frequency
1022:
847:{\displaystyle \int _{-\infty }^{\infty }|x(t)|^{2}\,dt=\int _{-\infty }^{\infty }\left|{\hat {x}}(f)\right|^{2}\,df,}
451:
present in the signal as a function of frequency, per unit frequency. Power spectral density is commonly expressed in
10153:
97:
9446:. Such formal statements may sometimes be useful to guide the intuition, but should always be used with utmost care.
9071:), which is the square root of the PSD; the ASD of a voltage signal has units of V Hz. This is useful when the
9502:
381:
as it fluctuates at an extremely high frequency. Obtaining a spectrum from time series such as these involves the
9220:
8568:
are usually restricted to positive frequencies by convention. Therefore, in typical signal processing, the full
2241:
8834:
from a sequence of time samples. Depending on what is known about the signal, estimation techniques can involve
9405:
6157:. This periodogram converges to the true PSD as the number of estimates as well as the averaging time interval
5840:
1911:
545:
but with only half the amplitude. Noise PSDs are generally one-sided in engineering and two-sided in physics.
9460:
7538:
6598:
5771:
Just as with the energy spectral density, the definition of the power spectral density can be generalized to
5375:{\displaystyle S_{xx}(f)=\int _{-\infty }^{\infty }R_{xx}(\tau )e^{-i2\pi f\tau }\,d\tau ={\hat {R}}_{xx}(f)}
3299:
398:
2949:{\displaystyle P=\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }\left|x_{T}(t)\right|^{2}\,dt.}
2353:
temperature anisotropy in terms of the angular scale. The solid line is a theoretical model, for comparison.
2297:
is needed to keep the correct physical units and to ensure that we recover the continuous case in the limit
9182:
8964:
7604:, so the full contribution to the cross power is, generally, from twice the real part of either individual
2740:
20:
8806:{\displaystyle S_{xy}(f)=\sum _{n=-\infty }^{\infty }R_{xy}(\tau _{n})e^{-i2\pi f\tau _{n}}\,\Delta \tau }
6208:
5501:
4739:
2727:{\displaystyle P=\lim _{T\to \infty }{\frac {1}{T}}\int _{t_{0}-T/2}^{t_{0}+T/2}\left|x(t)\right|^{2}\,dt}
10205:
9456:
9245:
9230:
8821:
6201:
The power spectrum is always real and non-negative, and the spectrum of a real valued process is also an
5455:
5225:
5224:, that the power spectral density can be found as the Fourier transform of the autocorrelation function (
3146:{\displaystyle P=\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }|{\hat {x}}_{T}(f)|^{2}\,df}
2339:
2327:
1363:
5899:
3990:
1556:
1284:
43:
as a function of optical wavelength shows peaks at atomic transitions, indicated by the numbered arrows.
10298:
9494:
2198:
1441:
resistor (so that all of the pulse energy is delivered to the resistor and none is reflected back). By
574:
566:
10126:
1553:
with respect to time over the duration of the pulse. To find the value of the energy spectral density
10318:
9948:
8937:
of the spectrum density. It describes one point in time (one window, to be precise). For example, in
8839:
6297:
5805:
2300:
9003:
8946:
8533:
8495:
8296:
8195:
4798:
3369:
3030:; however, for many signals of interest the Fourier transform does not formally exist. Regardless,
2995:
1817:
1700:
9049:
7611:
10323:
9193:
6580:
Note that the previous expression for total power (signal variance) is a special case where
1740:
1512:
1468:
1373:
As a physical example of how one might measure the energy spectral density of a signal, suppose
9868:
Introduction to Random
Signals and Applied Kalman Filtering with Matlab Exercises and Solutions
9225:
9210:
9149:
8393:
7499:
7460:
6703:
6664:
6412:
117:
35:
9144:
signal with frequency on the horizontal axis and time increasing upwards on the vertical axis.
3413:
2810:
2451:
2277:
1628:
251:
10308:
10146:
9540:
9153:
8835:
6303:
3031:
2562:
711:
352:
of a light source is determined by the spectrum of the electromagnetic wave's electric field
285:
9508:
9466:
2247:
525:) will remain unspecified, but the independent variable will be assumed to be that of time.
9630:
9536:
9443:
8983:
8843:
5778:
5736:
1884:
1415:
475:
436:
426:
9787:
9018:
8986:) the complete frequency response may be graphed in two parts: power versus frequency and
8466:
8437:
8364:
8335:
8267:
8238:
6635:
6606:
6458:
5742:
5198:
4837:
4769:
2966:
2513:
2415:
2378:
1784:
1651:
1376:
1336:
1143:
1107:
966:
600:
355:
294:
162:
75:
8:
10313:
9235:
9157:
5709:
5681:
4710:
3293:
714:(or Plancherel's theorem) gives us an alternate expression for the energy of the signal:
217:
10272:
9986:
9930:
9905:
9250:
9008:
6276:
6160:
6136:
6116:
5661:
4719:
2542:
2493:
2363:
1878:
1857:
1680:
1604:
1448:
1438:
1420:
1403:
1087:
998:
952:{\displaystyle {\hat {x}}(f)\triangleq \int _{-\infty }^{\infty }e^{-i2\pi ft}x(t)\ dt}
580:
542:
402:
323:
195:
10226:
5409:
10210:
10106:
10087:
10068:
10049:
10032:
10022:
10003:
9990:
9978:
9935:
9906:"A Low Computational Cost Algorithm for REM Sleep Detection Using Single Channel EEG"
9890:
9871:
9852:
9730:
9169:
8975:
8883:
8854:
8827:
8232:
7635:
6185:
3359:
3268:{\displaystyle S_{xx}(f)=\lim _{T\to \infty }{\frac {1}{T}}|{\hat {x}}_{T}(f)|^{2}\,}
1411:
960:
414:
382:
111:
40:
10241:
2448:, then indeed the instantaneous power dissipated in that resistor would be given by
1877:
This definition generalizes in a straightforward manner to a discrete signal with a
1367:
10257:
10139:
9968:
9960:
9925:
9917:
9498:
9240:
9013:
6392:{\displaystyle P=\operatorname {Var} (x)=\int _{-\infty }^{\infty }\!S_{xx}(f)\,df}
6310:(average power) of a process by integrating the power spectrum over all frequency:
1081:
483:
386:
213:
8679:, the relationship between the cross-spectral density and the cross-covariance is
9255:
9215:
9104:
9045:
5706:
are not sampled, and results at frequencies which are not an integer multiple of
4763:
2539:
over all time is therefore given by the following time average, where the period
2367:
1622:
1330:
559:
448:
348:
of a musical instrument are immediately determined from a spectral analysis. The
189:
154:
2408:
even when there is no physical power involved. If one were to create a physical
701:{\displaystyle E\triangleq \int _{-\infty }^{\infty }\left|x(t)\right|^{2}\ dt.}
10303:
9964:
9165:
8950:
8938:
8934:
8918:
monitoring, in which case SEF has variously been used to estimate the depth of
29:
9921:
47:
10287:
10036:
9100:
9092:
8987:
8831:
6202:
5772:
5651:{\displaystyle P_{\textsf {bandlimited}}=2\int _{f_{1}}^{f_{2}}S_{xx}(f)\,df}
341:
51:
The voice waveform over time (left) has a broad audio power spectrum (right).
8963:. This is the basis of a number of spectral analysis techniques such as the
8914:
is in the range 75 to 95. It is more particularly a popular measure used in
1677:
dissipated across the resistor. The value of the energy spectral density at
10267:
10183:
9982:
9973:
9939:
9564:
9125:
8942:
8858:
6755:). To begin, let us consider the average power of such a combined signal.
1442:
1257:{\displaystyle {\bar {S}}_{xx}(f)\triangleq \left|{\hat {x}}(f)\right|^{2}}
577:
is used in the generalized sense of signal processing; that is, the energy
503:
390:
10165:
3155:
Then the power spectral density is simply defined as the integrand above.
10262:
10188:
9057:
9053:
8959:
8847:
8652:{\displaystyle \operatorname {CPSD} _{\text{Full}}=2S_{xy}(f)=2S_{yx}(f)}
8390:. In light of this, the PSD is seen to be a special case of the CSD for
6154:
3363:
570:
410:
10131:
9794:
9148:
The concept and use of the power spectrum of a signal is fundamental in
8949:
properties, as defined by the range of brightness levels in each of the
1648:, say) near the frequency of interest and then measure the total energy
10216:
9568:
9205:
9136:
9108:
8919:
440:
9539:
can be used to prove that its
Fourier transform exists as a positive
9116:
9112:
8979:
8910:
percent of the total power of a given signal are located; typically,
8903:
8492:
are real signals (e.g. voltage or current), their
Fourier transforms
3446:, where * represents the complex conjugate. Taking into account that
993:
495:
487:
245:
9951:(2017). "An Ultralow-Power Sleep Spindle Detection System on Chip".
9806:
9680:
1874:
so that the energy spectral density instead has units of V Hz.
1621:, one could insert between the transmission line and the resistor a
9141:
9086:
8968:
6307:
2445:
2372:
2345:
452:
221:
558:"Energy spectral density" redirects here. Not to be confused with
10162:
9851:. Berlin ; New York: Springer Science & Business Media.
9692:
4864:
2409:
511:
467:
432:
406:
28:"Spectral power density" redirects here. Not to be confused with
9115:
component. And additionally there may be peaks corresponding to
16:
Relative importance of certain frequencies in a composite signal
9827:"The Fundamentals of FFT-Based Signal Analysis and Measurement"
6180:
If two signals both possess power spectral densities, then the
345:
220:) as analyzed in terms of its frequency content, is called its
157:
9668:
9644:
9006:
of the transfer function. This is not to be confused with the
9824:
9755:
9704:
9161:
9119:
of a fundamental peak, indicating a periodic signal which is
9096:
8923:
1811:
460:
349:
9946:
9800:
8945:, the spectral envelope of a feature is the boundary of its
3034:
tells us that we can re-write the average power as follows.
10236:
10231:
6191:
4716:
Now, if we divide the time convolution above by the period
2992:, one might like to compute the ordinary Fourier transform
2484:
1407:
456:
10046:
Fundamentals of Noise and
Vibration Analysis for Engineers
1362:
form a
Fourier transform pair, a result also known as the
474:
to the actual power delivered by that signal into a given
320:
often contains essential information about the nature of
10221:
10177:
9887:
9767:
8915:
5498:, can be calculated by integrating over frequency. Since
4713:
has been used when passing from the 3rd to the 4th line.
2843:
is unity within the arbitrary period and zero elsewhere.
2442:
9903:
9812:
9656:
9590:
9048:) forcing the Fourier transform to be real-valued. See
4867:, which is true in most, but not all, practical cases.
1016:
9904:
Imtiaz, Syed Anas; Rodriguez-Villegas, Esther (2014).
9743:
9602:
9511:
9469:
9408:
9284:
9021:
8685:
8582:
8536:
8498:
8469:
8440:
8396:
8367:
8338:
8299:
8270:
8241:
8198:
7644:
7614:
7541:
7502:
7463:
7184:
6761:
6706:
6667:
6638:
6609:
6494:
6461:
6415:
6316:
6211:
6163:
6139:
6119:
5950:
5902:
5843:
5808:
5781:
5745:
5712:
5684:
5664:
5571:
5504:
5458:
5412:
5240:
5201:
4873:
4840:
4801:
4772:
4742:
4722:
4052:
3993:
3452:
3416:
3372:
3302:
3167:
3040:
2998:
2969:
2849:
2813:
2743:
2598:
2565:
2545:
2516:
2496:
2454:
2418:
2381:
2303:
2280:
2250:
2201:
1973:
1914:
1887:
1860:
1820:
1787:
1743:
1703:
1683:
1654:
1631:
1607:
1559:
1515:
1471:
1451:
1423:
1379:
1339:
1287:
1181:
1146:
1110:
1090:
1025:
1001:
969:
860:
720:
632:
603:
583:
358:
326:
297:
254:
198:
165:
120:
9953:
IEEE Transactions on
Biomedical Circuits and Systems
9885:
Davenport, Wilbur B. (Jr); Root, William L. (1987).
9181:
The spectrum analyzer measures the magnitude of the
10086:. New York: Springer Science & Business Media.
9866:Brown, Robert Grover; Hwang, Patrick Y. C. (1997).
6573:{\displaystyle F(f)=2\int _{0}^{f}S_{xx}(f')\,df'.}
6488:power contained in frequencies from DC to f using:
5195:From here we see, again assuming the ergodicity of
478:. So one might use units of V Hz for the PSD.
70:
may be too technical for most readers to understand
10002:. West Sussex, England ; Hoboken, NJ: Wiley.
9722:
9527:
9485:
9434:
9394:
9036:
8805:
8651:
8560:
8522:
8484:
8455:
8426:
8382:
8353:
8324:
8285:
8256:
8223:
8184:
7626:
7596:
7527:
7488:
7449:
7168:
6731:
6692:
6653:
6624:
6572:
6476:
6440:
6391:
6264:
6169:
6145:
6125:
6105:
5936:
5888:
5829:
5794:
5760:
5726:
5698:
5670:
5650:
5557:
5490:
5444:
5406:The power of the signal in a given frequency band
5374:
5216:
5185:
4855:
4826:
4787:
4754:
4728:
4701:
4038:
3979:
3438:
3402:
3350:
3267:
3145:
3022:
2984:
2948:
2835:
2799:
2726:
2584:
2551:
2531:
2502:
2476:
2433:
2396:
2318:
2289:
2266:
2232:
2187:
1959:
1900:
1866:
1846:
1802:
1773:
1729:
1689:
1669:
1640:
1613:
1593:
1545:
1501:
1457:
1429:
1394:
1354:
1321:
1256:
1161:
1125:
1096:
1072:
1007:
984:
951:
846:
700:
618:
589:
482:(ESD) would have units of V s Hz, since
373:
332:
312:
276:
204:
180:
145:
10062:
9686:
9674:
9650:
6661:, each of which possess power spectral densities
6359:
2963:In analyzing the frequency content of the signal
1625:which passes only a narrow range of frequencies (
10285:
10063:Oppenheim, Alan V.; Verghese, George C. (2016).
9578:
8853:The spectral density is usually estimated using
7967:
7702:
7347:
7219:
6946:
6777:
6292:) can be reconstructed from its power spectrum S
5977:
4972:
4875:
3194:
3048:
2857:
2606:
2009:
1073:{\displaystyle \left|{\hat {x}}(f)\right|^{2}df}
9785:
9563:
6592:
10016:
9761:
9710:
9698:
8826:The goal of spectral density estimation is to
1509:, so the total energy is found by integrating
1445:, the power delivered to the resistor at time
573:is distributed with frequency. Here, the term
10147:
10043:
9884:
9825:Michael Cerna & Audrey F. Harvey (2000).
9773:
9596:
1410:) of an electrical pulse propagating along a
486:has units of power multiplied by time (e.g.,
9342:
9285:
9107:(specified by their frequency, or sometimes
1437:, and suppose the line is terminated with a
502: Hz are frequently used for the PSD of
466:When a signal is defined in terms only of a
10198:
10081:
9275:
8846:. A common non-parametric technique is the
6181:
5399:Many authors use this equality to actually
1908:such as a signal sampled at discrete times
537:function of only positive frequencies or a
389:is used to obtain a spectrum of light in a
10154:
10140:
9865:
9749:
9559:
9557:
9555:
9168:technology. Electronic instruments called
9002:. Less commonly, the two parts may be the
5837:with the signal sampled at discrete times
5768:evaluated over the specified time window.
565:Energy spectral density describes how the
553:
239:), which applies to signals existing over
10161:
10105:. New York Weinheim: Wiley-Interscience.
10048:. Cambridge: Cambridge University Press.
9972:
9929:
9435:{\displaystyle \delta (\omega -\omega ')}
9131:
8796:
6555:
6382:
6083:
5927:
5889:{\displaystyle t_{n}=t_{0}+(n\,\Delta t)}
5876:
5802:. As before, we can consider a window of
5641:
5578:
5331:
3264:
3136:
2936:
2717:
2441:and applied it to the terminals of a one
2333:
2109:
1960:{\displaystyle t_{n}=t_{0}+(n\,\Delta t)}
1947:
834:
768:
528:
435:, the signal might be a wave, such as an
98:Learn how and when to remove this message
82:, without removing the technical details.
10017:Miller, Scott; Childers, Donald (2012).
9846:
9801:Iranmanesh & Rodriguez-Villegas 2017
9608:
9135:
6192:Properties of the power spectral density
2960:that diverging integral, in such cases.
2344:
46:
34:
9552:
7597:{\displaystyle S_{xy}^{*}(f)=S_{yx}(f)}
3351:{\displaystyle |{\hat {x}}_{T}(f)|^{2}}
443:, or the vibration of a mechanism. The
405:, as well as in many other branches of
10286:
10044:Norton, M. P.; Karczub, D. G. (2003).
2559:is centered about some arbitrary time
244:all time would generally be infinite.
10135:
10127:Power Spectral Density Matlab scripts
10100:
9997:
9788:"Signal Processing Course, chapter 7"
9662:
9620:
9584:
2800:{\displaystyle x_{T}(t)=x(t)w_{T}(t)}
2351:cosmic microwave background radiation
284:over the time domain, as dictated by
212:composing that signal. According to
80:make it understandable to non-experts
10082:Risken, Hannes; Frank, Till (1996).
9813:Imtiaz & Rodriguez-Villegas 2014
9501:, not in the sense of a statistical
9459:makes sense of this formula for any
9172:are used to observe and measure the
9164:, and related systems, plus passive
8861:), but other techniques such as the
6265:{\displaystyle S_{xx}(-f)=S_{xx}(f)}
6196:Some properties of the PSD include:
5558:{\displaystyle S_{xx}(-f)=S_{xx}(f)}
5231:
4766:function of the non-windowed signal
4755:{\displaystyle T\rightarrow \infty }
3158:
1810:has units of V s Ί =
1172:
54:
8875:spectral density (physical science)
8868:
7634:becomes the Fourier transform of a
7535:are already understood. Note that
7457:where, again, the contributions of
6279:x(t), the autocorrelation function
5491:{\displaystyle 0<f_{1}<f_{2}}
2349:The power spectrum of the measured
1737:. In this example, since the power
291:The spectrum of a physical process
13:
8797:
8730:
8725:
8117:
8112:
8005:
8000:
7977:
7956:
7951:
7852:
7847:
7740:
7735:
7712:
7691:
7686:
7621:
7357:
7229:
6984:
6979:
6956:
6815:
6810:
6787:
6354:
6349:
6084:
5998:
5987:
5937:{\displaystyle T=(2N+1)\,\Delta t}
5928:
5877:
5279:
5274:
5125:
5120:
5010:
5005:
4982:
4961:
4956:
4885:
4749:
4593:
4588:
4570:
4565:
4463:
4458:
4440:
4435:
4364:
4318:
4286:
4240:
4193:
4039:{\displaystyle u(t)=x_{T}^{*}(-t)}
3873:
3789:
3784:
3687:
3682:
3607:
3602:
3521:
3516:
3459:
3204:
3086:
3081:
3058:
2895:
2890:
2867:
2616:
2304:
2281:
2110:
2027:
2019:
1948:
1838:
1721:
1632:
1594:{\displaystyle {\bar {S}}_{xx}(f)}
1322:{\displaystyle {\bar {S}}_{xx}(f)}
898:
893:
791:
786:
734:
729:
652:
647:
447:(PSD) of the signal describes the
14:
10340:
10120:
2233:{\displaystyle {\hat {x}}_{d}(f)}
10193:
10065:Signals, Systems & Inference
10019:Probability and Random Processes
9910:Annals of Biomedical Engineering
9503:Cumulative distribution function
9154:electronic communication systems
4388:
401:and in the statistical study of
59:
9818:
9779:
9716:
9449:
9221:Least-squares spectral analysis
9099:), musical notes (perceived as
9080:
9050:Ultrashort pulse#Spectral phase
6409:) with power spectral density
5896:for a total measurement period
2242:discrete-time Fourier transform
10021:. Boston, MA: Academic Press.
9889:. New York: Wiley-IEEE Press.
9614:
9569:"Spectral Analysis of Signals"
9429:
9412:
9386:
9369:
9363:
9357:
9339:
9328:
9316:
9306:
9300:
9294:
9268:
9076:or over a specified bandwidth.
9031:
9025:
8761:
8748:
8705:
8699:
8646:
8640:
8618:
8612:
8555:
8549:
8543:
8517:
8511:
8505:
8479:
8473:
8450:
8444:
8421:
8415:
8406:
8400:
8377:
8371:
8348:
8342:
8319:
8313:
8280:
8274:
8251:
8245:
8218:
8212:
8141:
8135:
8056:
8050:
8037:
8025:
7974:
7933:
7927:
7876:
7870:
7791:
7785:
7772:
7760:
7709:
7668:
7662:
7618:
7591:
7585:
7566:
7560:
7522:
7516:
7483:
7477:
7435:
7429:
7417:
7407:
7401:
7384:
7354:
7336:
7330:
7307:
7301:
7289:
7279:
7273:
7256:
7226:
7208:
7202:
7146:
7141:
7135:
7121:
7114:
7108:
7095:
7089:
7068:
7062:
7049:
7043:
7015:
7010:
7004:
6990:
6953:
6921:
6915:
6899:
6893:
6864:
6858:
6842:
6836:
6784:
6726:
6720:
6687:
6681:
6648:
6642:
6619:
6613:
6552:
6541:
6504:
6498:
6484:, which specifies the average
6471:
6465:
6435:
6429:
6379:
6373:
6335:
6329:
6259:
6253:
6234:
6225:
6005:
5995:
5984:
5970:
5964:
5924:
5909:
5883:
5870:
5830:{\displaystyle -N\leq n\leq N}
5755:
5749:
5638:
5632:
5552:
5546:
5527:
5518:
5439:
5413:
5388:
5369:
5363:
5348:
5303:
5297:
5260:
5254:
5211:
5205:
5149:
5143:
5061:
5055:
5042:
5030:
4979:
4931:
4925:
4913:
4882:
4850:
4844:
4821:
4815:
4782:
4776:
4746:
4644:
4638:
4625:
4613:
4502:
4496:
4483:
4471:
4409:
4403:
4383:
4377:
4344:
4338:
4305:
4299:
4266:
4260:
4227:
4218:
4178:
4172:
4160:
4141:
4137:
4131:
4119:
4109:
4088:
4082:
4070:
4033:
4024:
4003:
3997:
3959:
3953:
3941:
3899:
3893:
3810:
3804:
3745:
3716:
3713:
3707:
3633:
3627:
3550:
3541:
3493:
3484:
3433:
3427:
3397:
3388:
3338:
3333:
3327:
3315:
3304:
3281:
3254:
3249:
3243:
3231:
3220:
3201:
3187:
3181:
3126:
3121:
3115:
3103:
3092:
3055:
3017:
3011:
3005:
2979:
2973:
2922:
2916:
2864:
2830:
2824:
2794:
2788:
2775:
2769:
2760:
2754:
2703:
2697:
2613:
2526:
2520:
2471:
2465:
2428:
2422:
2391:
2385:
2319:{\displaystyle \Delta t\to 0.}
2310:
2227:
2221:
2209:
2166:
2160:
2148:
2034:
2024:
2016:
2002:
1996:
1981:
1954:
1941:
1830:
1824:
1797:
1791:
1754:
1747:
1713:
1707:
1664:
1658:
1588:
1582:
1567:
1526:
1519:
1482:
1475:
1389:
1383:
1349:
1343:
1316:
1310:
1295:
1270:
1240:
1234:
1228:
1210:
1204:
1189:
1156:
1150:
1050:
1044:
1038:
979:
973:
937:
931:
879:
873:
867:
820:
814:
808:
758:
753:
747:
740:
672:
666:
613:
607:
541:function of both positive and
427:Fourier transform § Units
368:
362:
307:
301:
271:
265:
188:describes the distribution of
175:
169:
140:
134:
1:
9847:Birolini, Alessandro (2007).
9840:
9687:Oppenheim & Verghese 2016
9675:Oppenheim & Verghese 2016
9651:Oppenheim & Verghese 2016
9461:wide-sense stationary process
8898:), usually expressed as "SEF
8815:
8576:s scaled by a factor of two.
8561:{\displaystyle {\hat {y}}(f)}
8523:{\displaystyle {\hat {x}}(f)}
8325:{\displaystyle R_{yx}(\tau )}
8224:{\displaystyle R_{xy}(\tau )}
6739:, it is possible to define a
6599:Coherence (signal processing)
6593:Cross power spectral density
4827:{\displaystyle R_{xx}(\tau )}
3403:{\displaystyle x_{T}^{*}(-t)}
3023:{\displaystyle {\hat {x}}(f)}
2274: The sampling interval
1847:{\displaystyle E(f)/\Delta f}
1781:has units of V Ί, the energy
1730:{\displaystyle E(f)/\Delta f}
548:
399:statistical signal processing
10329:Spectrum (physical sciences)
9505:) instead of functions. If
9188:
9183:short-time Fourier transform
9063:Sometimes one encounters an
8965:short-time Fourier transform
8332:is the cross-correlation of
7627:{\displaystyle T\to \infty }
6741:cross power spectral density
5403:the power spectral density.
231:. More commonly used is the
21:Spectrum (physical sciences)
7:
10101:Stein, Jonathan Y. (2000).
9723:Dennis Ward Ricker (2003).
9246:Spectral power distribution
9231:Spectral density estimation
9199:
8822:Spectral density estimation
6454:power spectral distribution
2340:spectral power distribution
10:
10345:
10084:The Fokker-Planck Equation
9965:10.1109/TBCAS.2017.2690908
9949:Rodriguez-Villegas, Esther
9762:Miller & Childers 2012
9711:Miller & Childers 2012
9699:Miller & Childers 2012
9084:
9065:amplitude spectral density
8872:
8830:the spectral density of a
8819:
6596:
2337:
1774:{\displaystyle V(t)^{2}/Z}
1546:{\displaystyle V(t)^{2}/Z}
1502:{\displaystyle V(t)^{2}/Z}
1104:in the frequency interval
557:
424:
192:into frequency components
39:The spectral density of a
27:
18:
10294:Frequency-domain analysis
10250:
10173:
10103:Digital Signal Processing
9922:10.1007/s10439-014-1085-6
9774:Davenport & Root 1987
9597:Norton & Karczub 2003
9463:under weaker hypotheses:
8865:method can also be used.
8427:{\displaystyle x(t)=y(t)}
7528:{\displaystyle S_{yy}(f)}
7489:{\displaystyle S_{xx}(f)}
6732:{\displaystyle S_{yy}(f)}
6693:{\displaystyle S_{xx}(f)}
6441:{\displaystyle S_{xx}(f)}
6298:inverse Fourier transform
2366:; this describes how the
1814:, and hence the estimate
146:{\displaystyle S_{xx}(f)}
9870:. New York: Wiley-Liss.
9786:William D Penny (2009).
9623:"Power Spectral Density"
9261:
9004:real and imaginary parts
8873:Not to be confused with
3439:{\displaystyle x_{T}(t)}
2836:{\displaystyle w_{T}(t)}
2477:{\displaystyle x^{2}(t)}
2375:of a function over time
2338:Not to be confused with
2290:{\displaystyle \Delta t}
1697:is then estimated to be
1641:{\displaystyle \Delta f}
1080:can be interpreted as a
420:
277:{\displaystyle x^{2}(t)}
9849:Reliability Engineering
9457:WienerâKhinchin theorem
9276:Risken & Frank 1996
9194:Primordial fluctuations
8892:spectral edge frequency
5226:WienerâKhinchin theorem
2585:{\displaystyle t=t_{0}}
2362:(PSD) which exists for
1364:WienerâKhinchin theorem
1138:energy spectral density
554:Energy spectral density
480:Energy spectral density
463:(abbreviated as W/Hz).
229:energy spectral density
9998:Maral, Gerard (2004).
9750:Brown & Hwang 1997
9726:Echo Signal Processing
9567:& R Moses (2005).
9529:
9528:{\displaystyle R_{xx}}
9487:
9486:{\displaystyle R_{xx}}
9436:
9396:
9226:Noise spectral density
9211:Brightness temperature
9150:electrical engineering
9145:
9132:Electrical engineering
9038:
8992:phase spectral density
8807:
8734:
8653:
8562:
8524:
8486:
8457:
8428:
8384:
8355:
8326:
8287:
8258:
8225:
8186:
7628:
7598:
7529:
7490:
7451:
7170:
6749:cross spectral density
6733:
6694:
6655:
6626:
6574:
6478:
6448:, one can compute the
6442:
6393:
6306:, one can compute the
6266:
6182:cross-spectral density
6171:
6147:
6127:
6107:
6049:
5938:
5890:
5831:
5796:
5762:
5728:
5700:
5672:
5652:
5559:
5492:
5446:
5376:
5218:
5187:
4857:
4828:
4795:, which is denoted as
4789:
4756:
4736:and take the limit as
4730:
4703:
4040:
3981:
3440:
3404:
3352:
3292:From here, due to the
3269:
3147:
3024:
2986:
2950:
2837:
2801:
2728:
2586:
2553:
2533:
2504:
2478:
2435:
2412:source which followed
2398:
2360:power spectral density
2354:
2334:Power spectral density
2320:
2291:
2268:
2267:{\displaystyle x_{n}.}
2234:
2189:
2075:
1961:
1902:
1868:
1848:
1804:
1775:
1731:
1691:
1671:
1642:
1615:
1595:
1547:
1503:
1459:
1431:
1396:
1356:
1323:
1258:
1163:
1127:
1098:
1074:
1009:
986:
953:
848:
702:
620:
591:
533:A PSD can be either a
529:One-sided vs two-sided
445:power spectral density
375:
334:
314:
278:
233:power spectral density
206:
182:
147:
52:
44:
9530:
9488:
9437:
9397:
9274:Some authors, e.g., (
9139:
9085:Further information:
9039:
8990:versus frequencyâthe
8857:methods (such as the
8808:
8711:
8661:For discrete signals
8654:
8563:
8525:
8487:
8458:
8429:
8385:
8356:
8327:
8288:
8259:
8226:
8187:
7629:
7599:
7530:
7491:
7452:
7171:
6734:
6695:
6656:
6627:
6575:
6479:
6443:
6394:
6267:
6172:
6148:
6128:
6108:
6026:
5939:
5891:
5832:
5797:
5795:{\displaystyle x_{n}}
5763:
5737:statistical ensembles
5729:
5701:
5673:
5653:
5560:
5493:
5447:
5377:
5219:
5188:
4858:
4829:
4790:
4757:
4731:
4704:
4041:
3982:
3441:
3405:
3353:
3270:
3148:
3025:
2987:
2951:
2838:
2802:
2729:
2587:
2554:
2534:
2505:
2479:
2436:
2399:
2348:
2321:
2292:
2269:
2235:
2190:
2052:
1962:
1903:
1901:{\displaystyle x_{n}}
1869:
1849:
1805:
1776:
1732:
1692:
1672:
1643:
1616:
1596:
1548:
1504:
1460:
1432:
1397:
1357:
1324:
1259:
1164:
1128:
1099:
1075:
1010:
987:
954:
849:
703:
621:
592:
376:
335:
315:
279:
207:
183:
148:
114:, the power spectrum
50:
38:
9665:, pp. 108, 115.
9621:Paschotta, RĂźdiger.
9509:
9467:
9444:Dirac delta function
9406:
9282:
9158:radio communications
9037:{\displaystyle h(t)}
9019:
8844:autoregressive model
8683:
8580:
8534:
8496:
8485:{\displaystyle y(t)}
8467:
8456:{\displaystyle x(t)}
8438:
8394:
8383:{\displaystyle x(t)}
8365:
8354:{\displaystyle y(t)}
8336:
8297:
8286:{\displaystyle y(t)}
8268:
8257:{\displaystyle x(t)}
8239:
8196:
7642:
7612:
7539:
7500:
7461:
7182:
6759:
6704:
6665:
6654:{\displaystyle y(t)}
6636:
6625:{\displaystyle x(t)}
6607:
6492:
6477:{\displaystyle F(f)}
6459:
6413:
6314:
6209:
6161:
6137:
6117:
5948:
5900:
5841:
5806:
5779:
5761:{\displaystyle x(t)}
5743:
5710:
5682:
5662:
5569:
5502:
5456:
5410:
5238:
5217:{\displaystyle x(t)}
5199:
4871:
4856:{\displaystyle x(t)}
4838:
4799:
4788:{\displaystyle x(t)}
4770:
4740:
4720:
4050:
3991:
3450:
3414:
3370:
3300:
3165:
3038:
2996:
2985:{\displaystyle x(t)}
2967:
2847:
2811:
2741:
2596:
2563:
2543:
2532:{\displaystyle x(t)}
2514:
2494:
2452:
2434:{\displaystyle x(t)}
2416:
2397:{\displaystyle x(t)}
2379:
2364:stationary processes
2328:normalized frequency
2301:
2278:
2248:
2199:
1971:
1912:
1885:
1858:
1818:
1803:{\displaystyle E(f)}
1785:
1741:
1701:
1681:
1670:{\displaystyle E(f)}
1652:
1629:
1605:
1557:
1513:
1469:
1449:
1421:
1395:{\displaystyle V(t)}
1377:
1355:{\displaystyle x(t)}
1337:
1285:
1179:
1162:{\displaystyle x(t)}
1144:
1126:{\displaystyle f+df}
1108:
1088:
1023:
999:
985:{\displaystyle x(t)}
967:
959:is the value of the
858:
718:
630:
619:{\displaystyle x(t)}
601:
581:
543:negative frequencies
437:electromagnetic wave
403:stochastic processes
374:{\displaystyle E(t)}
356:
340:. For instance, the
324:
313:{\displaystyle x(t)}
295:
252:
196:
181:{\displaystyle x(t)}
163:
118:
10067:. Boston: Pearson.
9701:, pp. 429â431.
9689:, pp. 422â423.
9236:Spectral efficiency
8572:is just one of the
8121:
8024:
8009:
7960:
7856:
7759:
7744:
7695:
7559:
7400:
7272:
7088:
7042:
6988:
6819:
6527:
6450:integrated spectrum
6401:For a real process
6358:
6177:approach infinity.
5739:of realizations of
5727:{\displaystyle 1/T}
5699:{\displaystyle 1/T}
5618:
5283:
5129:
5029:
5014:
4965:
4711:convolution theorem
4612:
4597:
4574:
4467:
4444:
4217:
4023:
3793:
3706:
3691:
3626:
3611:
3540:
3525:
3483:
3387:
3296:, we can also view
3294:convolution theorem
3090:
2899:
2687:
902:
795:
738:
656:
10273:telecommunications
9947:Iranmanesh, Saam;
9525:
9483:
9432:
9392:
9251:Whittle likelihood
9170:spectrum analyzers
9146:
9140:Spectrogram of an
9034:
9009:frequency response
8976:transfer functions
8902:", represents the
8803:
8649:
8558:
8520:
8482:
8453:
8424:
8380:
8351:
8322:
8283:
8254:
8221:
8182:
8180:
8104:
8010:
7992:
7981:
7943:
7839:
7745:
7727:
7716:
7678:
7624:
7594:
7542:
7525:
7486:
7447:
7445:
7377:
7361:
7249:
7233:
7166:
7164:
7074:
7028:
6971:
6960:
6802:
6791:
6729:
6690:
6651:
6622:
6603:Given two signals
6570:
6513:
6474:
6438:
6389:
6341:
6304:Parseval's theorem
6277:stochastic process
6262:
6167:
6143:
6123:
6103:
5991:
5934:
5886:
5827:
5792:
5758:
5724:
5696:
5668:
5648:
5590:
5555:
5488:
5442:
5372:
5266:
5214:
5183:
5112:
5015:
4997:
4986:
4948:
4889:
4853:
4824:
4785:
4752:
4726:
4699:
4697:
4598:
4580:
4557:
4450:
4427:
4203:
4036:
4009:
3977:
3975:
3776:
3692:
3674:
3612:
3594:
3526:
3508:
3469:
3436:
3400:
3373:
3348:
3265:
3208:
3143:
3073:
3062:
3032:Parseval's theorem
3020:
2982:
2946:
2882:
2871:
2833:
2797:
2724:
2631:
2620:
2582:
2549:
2529:
2500:
2490:The average power
2474:
2431:
2394:
2355:
2316:
2287:
2264:
2230:
2185:
2181:
2132:
2023:
1957:
1898:
1879:countably infinite
1864:
1844:
1800:
1771:
1727:
1687:
1667:
1638:
1611:
1591:
1543:
1499:
1455:
1427:
1392:
1352:
1319:
1254:
1159:
1123:
1094:
1070:
1005:
982:
949:
885:
844:
778:
721:
712:Parseval's theorem
698:
639:
616:
587:
371:
330:
310:
286:Parseval's theorem
274:
202:
178:
143:
53:
45:
10299:Signal processing
10281:
10280:
10112:978-0-471-29546-4
10093:978-3-540-61530-9
10074:978-0-13-394328-3
10055:978-0-521-49913-2
10028:978-0-12-386981-4
10009:978-0-470-86684-9
9896:978-0-87942-235-6
9877:978-0-471-12839-7
9858:978-3-540-49388-4
9736:978-1-4020-7395-3
9537:Bochner's theorem
9497:(in the sense of
9319:
9297:
8931:spectral envelope
8884:spectral centroid
8855:Fourier transform
8590:
8546:
8508:
8233:cross-correlation
7990:
7966:
7725:
7701:
7636:cross-correlation
7420:
7387:
7370:
7346:
7292:
7259:
7242:
7218:
6969:
6945:
6800:
6776:
6296:(f) by using the
6275:For a continuous
6186:cross-correlation
6170:{\displaystyle T}
6146:{\displaystyle T}
6126:{\displaystyle N}
6018:
5976:
5671:{\displaystyle T}
5580:
5396:
5395:
5351:
5102:
4995:
4971:
4916:
4898:
4874:
4762:, it becomes the
4729:{\displaystyle T}
4685:
4163:
4122:
4073:
3944:
3360:Fourier transform
3318:
3289:
3288:
3234:
3217:
3193:
3106:
3071:
3047:
3008:
2880:
2856:
2629:
2605:
2552:{\displaystyle T}
2503:{\displaystyle P}
2212:
2151:
2045:
2043:
2008:
1984:
1881:number of values
1867:{\displaystyle Z}
1690:{\displaystyle f}
1614:{\displaystyle f}
1570:
1458:{\displaystyle t}
1430:{\displaystyle Z}
1412:transmission line
1298:
1278:
1277:
1231:
1192:
1097:{\displaystyle f}
1041:
1008:{\displaystyle f}
961:Fourier transform
942:
870:
811:
688:
590:{\displaystyle E}
569:of a signal or a
415:spatial frequency
383:Fourier transform
333:{\displaystyle x}
205:{\displaystyle f}
112:signal processing
108:
107:
100:
41:fluorescent light
10336:
10319:Fourier analysis
10258:logarithmic unit
10156:
10149:
10142:
10133:
10132:
10116:
10097:
10078:
10059:
10040:
10013:
9994:
9976:
9943:
9933:
9900:
9881:
9862:
9834:
9833:
9831:
9822:
9816:
9810:
9804:
9798:
9792:
9791:
9783:
9777:
9771:
9765:
9759:
9753:
9747:
9741:
9740:
9720:
9714:
9708:
9702:
9696:
9690:
9684:
9678:
9672:
9666:
9660:
9654:
9648:
9642:
9641:
9639:
9638:
9629:. Archived from
9627:rp-photonics.com
9618:
9612:
9606:
9600:
9594:
9588:
9582:
9576:
9575:
9573:
9561:
9544:
9534:
9532:
9531:
9526:
9524:
9523:
9499:Laurent Schwartz
9492:
9490:
9489:
9484:
9482:
9481:
9453:
9447:
9441:
9439:
9438:
9433:
9428:
9401:
9399:
9398:
9393:
9385:
9338:
9327:
9326:
9321:
9320:
9312:
9299:
9298:
9290:
9272:
9241:Spectral leakage
9152:, especially in
9043:
9041:
9040:
9035:
9014:impulse response
8869:Related concepts
8812:
8810:
8809:
8804:
8795:
8794:
8793:
8792:
8760:
8759:
8747:
8746:
8733:
8728:
8698:
8697:
8678:
8669:
8658:
8656:
8655:
8650:
8639:
8638:
8611:
8610:
8592:
8591:
8588:
8567:
8565:
8564:
8559:
8548:
8547:
8539:
8529:
8527:
8526:
8521:
8510:
8509:
8501:
8491:
8489:
8488:
8483:
8462:
8460:
8459:
8454:
8433:
8431:
8430:
8425:
8389:
8387:
8386:
8381:
8360:
8358:
8357:
8352:
8331:
8329:
8328:
8323:
8312:
8311:
8292:
8290:
8289:
8284:
8263:
8261:
8260:
8255:
8230:
8228:
8227:
8222:
8211:
8210:
8191:
8189:
8188:
8183:
8181:
8168:
8167:
8134:
8133:
8120:
8115:
8094:
8093:
8069:
8065:
8049:
8048:
8023:
8018:
8008:
8003:
7991:
7983:
7980:
7959:
7954:
7926:
7925:
7903:
7902:
7869:
7868:
7855:
7850:
7829:
7828:
7804:
7800:
7784:
7783:
7758:
7753:
7743:
7738:
7726:
7718:
7715:
7694:
7689:
7661:
7660:
7633:
7631:
7630:
7625:
7603:
7601:
7600:
7595:
7584:
7583:
7558:
7553:
7534:
7532:
7531:
7526:
7515:
7514:
7495:
7493:
7492:
7487:
7476:
7475:
7456:
7454:
7453:
7448:
7446:
7442:
7438:
7428:
7427:
7422:
7421:
7413:
7399:
7394:
7389:
7388:
7380:
7371:
7363:
7360:
7329:
7328:
7314:
7310:
7300:
7299:
7294:
7293:
7285:
7271:
7266:
7261:
7260:
7252:
7243:
7235:
7232:
7201:
7200:
7175:
7173:
7172:
7167:
7165:
7155:
7154:
7149:
7134:
7133:
7124:
7107:
7106:
7087:
7082:
7061:
7060:
7041:
7036:
7024:
7023:
7018:
7003:
7002:
6993:
6987:
6982:
6970:
6962:
6959:
6938:
6928:
6924:
6914:
6913:
6892:
6891:
6877:
6876:
6871:
6867:
6857:
6856:
6835:
6834:
6818:
6813:
6801:
6793:
6790:
6738:
6736:
6735:
6730:
6719:
6718:
6699:
6697:
6696:
6691:
6680:
6679:
6660:
6658:
6657:
6652:
6631:
6629:
6628:
6623:
6586:
6579:
6577:
6576:
6571:
6566:
6551:
6540:
6539:
6526:
6521:
6483:
6481:
6480:
6475:
6447:
6445:
6444:
6439:
6428:
6427:
6398:
6396:
6395:
6390:
6372:
6371:
6357:
6352:
6271:
6269:
6268:
6263:
6252:
6251:
6224:
6223:
6176:
6174:
6173:
6168:
6152:
6150:
6149:
6144:
6132:
6130:
6129:
6124:
6112:
6110:
6109:
6104:
6102:
6101:
6096:
6092:
6091:
6090:
6059:
6058:
6048:
6043:
6019:
6014:
6013:
6012:
5993:
5990:
5963:
5962:
5943:
5941:
5940:
5935:
5895:
5893:
5892:
5887:
5866:
5865:
5853:
5852:
5836:
5834:
5833:
5828:
5801:
5799:
5798:
5793:
5791:
5790:
5767:
5765:
5764:
5759:
5733:
5731:
5730:
5725:
5720:
5705:
5703:
5702:
5697:
5692:
5677:
5675:
5674:
5669:
5657:
5655:
5654:
5649:
5631:
5630:
5617:
5616:
5615:
5605:
5604:
5603:
5583:
5582:
5581:
5564:
5562:
5561:
5556:
5545:
5544:
5517:
5516:
5497:
5495:
5494:
5489:
5487:
5486:
5474:
5473:
5451:
5449:
5448:
5445:{\displaystyle }
5443:
5438:
5437:
5425:
5424:
5390:
5381:
5379:
5378:
5373:
5362:
5361:
5353:
5352:
5344:
5330:
5329:
5296:
5295:
5282:
5277:
5253:
5252:
5232:
5223:
5221:
5220:
5215:
5192:
5190:
5189:
5184:
5176:
5175:
5142:
5141:
5128:
5123:
5100:
5099:
5098:
5074:
5070:
5054:
5053:
5028:
5023:
5013:
5008:
4996:
4988:
4985:
4964:
4959:
4944:
4943:
4938:
4934:
4924:
4923:
4918:
4917:
4909:
4899:
4891:
4888:
4862:
4860:
4859:
4854:
4834:, provided that
4833:
4831:
4830:
4825:
4814:
4813:
4794:
4792:
4791:
4786:
4761:
4759:
4758:
4753:
4735:
4733:
4732:
4727:
4708:
4706:
4705:
4700:
4698:
4683:
4682:
4681:
4657:
4653:
4637:
4636:
4611:
4606:
4596:
4591:
4573:
4568:
4550:
4540:
4539:
4515:
4511:
4495:
4494:
4466:
4461:
4443:
4438:
4420:
4416:
4412:
4402:
4401:
4392:
4391:
4368:
4367:
4355:
4351:
4347:
4337:
4336:
4322:
4321:
4312:
4308:
4290:
4289:
4277:
4273:
4269:
4259:
4258:
4244:
4243:
4234:
4230:
4216:
4211:
4197:
4196:
4184:
4171:
4170:
4165:
4164:
4156:
4149:
4148:
4130:
4129:
4124:
4123:
4115:
4101:
4100:
4095:
4091:
4081:
4080:
4075:
4074:
4066:
4045:
4043:
4042:
4037:
4022:
4017:
3986:
3984:
3983:
3978:
3976:
3972:
3971:
3966:
3962:
3952:
3951:
3946:
3945:
3937:
3921:
3917:
3916:
3911:
3907:
3906:
3902:
3892:
3891:
3877:
3876:
3858:
3854:
3853:
3848:
3844:
3837:
3836:
3803:
3802:
3792:
3787:
3763:
3753:
3752:
3743:
3742:
3705:
3700:
3690:
3685:
3667:
3657:
3656:
3625:
3620:
3610:
3605:
3587:
3577:
3576:
3539:
3534:
3524:
3519:
3500:
3496:
3482:
3477:
3463:
3462:
3445:
3443:
3442:
3437:
3426:
3425:
3409:
3407:
3406:
3401:
3386:
3381:
3357:
3355:
3354:
3349:
3347:
3346:
3341:
3326:
3325:
3320:
3319:
3311:
3307:
3283:
3274:
3272:
3271:
3266:
3263:
3262:
3257:
3242:
3241:
3236:
3235:
3227:
3223:
3218:
3210:
3207:
3180:
3179:
3159:
3152:
3150:
3149:
3144:
3135:
3134:
3129:
3114:
3113:
3108:
3107:
3099:
3095:
3089:
3084:
3072:
3064:
3061:
3029:
3027:
3026:
3021:
3010:
3009:
3001:
2991:
2989:
2988:
2983:
2955:
2953:
2952:
2947:
2935:
2934:
2929:
2925:
2915:
2914:
2898:
2893:
2881:
2873:
2870:
2842:
2840:
2839:
2834:
2823:
2822:
2806:
2804:
2803:
2798:
2787:
2786:
2753:
2752:
2733:
2731:
2730:
2725:
2716:
2715:
2710:
2706:
2686:
2682:
2671:
2670:
2660:
2656:
2645:
2644:
2630:
2622:
2619:
2591:
2589:
2588:
2583:
2581:
2580:
2558:
2556:
2555:
2550:
2538:
2536:
2535:
2530:
2509:
2507:
2506:
2501:
2483:
2481:
2480:
2475:
2464:
2463:
2440:
2438:
2437:
2432:
2403:
2401:
2400:
2395:
2325:
2323:
2322:
2317:
2296:
2294:
2293:
2288:
2273:
2271:
2270:
2265:
2260:
2259:
2239:
2237:
2236:
2231:
2220:
2219:
2214:
2213:
2205:
2194:
2192:
2191:
2186:
2180:
2179:
2178:
2173:
2169:
2159:
2158:
2153:
2152:
2144:
2133:
2128:
2127:
2122:
2118:
2117:
2116:
2085:
2084:
2074:
2069:
2042:
2041:
2022:
1995:
1994:
1986:
1985:
1977:
1966:
1964:
1963:
1958:
1937:
1936:
1924:
1923:
1907:
1905:
1904:
1899:
1897:
1896:
1873:
1871:
1870:
1865:
1853:
1851:
1850:
1845:
1837:
1809:
1807:
1806:
1801:
1780:
1778:
1777:
1772:
1767:
1762:
1761:
1736:
1734:
1733:
1728:
1720:
1696:
1694:
1693:
1688:
1676:
1674:
1673:
1668:
1647:
1645:
1644:
1639:
1620:
1618:
1617:
1612:
1600:
1598:
1597:
1592:
1581:
1580:
1572:
1571:
1563:
1552:
1550:
1549:
1544:
1539:
1534:
1533:
1508:
1506:
1505:
1500:
1495:
1490:
1489:
1464:
1462:
1461:
1456:
1436:
1434:
1433:
1428:
1401:
1399:
1398:
1393:
1361:
1359:
1358:
1353:
1328:
1326:
1325:
1320:
1309:
1308:
1300:
1299:
1291:
1272:
1263:
1261:
1260:
1255:
1253:
1252:
1247:
1243:
1233:
1232:
1224:
1203:
1202:
1194:
1193:
1185:
1173:
1168:
1166:
1165:
1160:
1132:
1130:
1129:
1124:
1103:
1101:
1100:
1095:
1082:density function
1079:
1077:
1076:
1071:
1063:
1062:
1057:
1053:
1043:
1042:
1034:
1014:
1012:
1011:
1006:
991:
989:
988:
983:
958:
956:
955:
950:
940:
927:
926:
901:
896:
872:
871:
863:
853:
851:
850:
845:
833:
832:
827:
823:
813:
812:
804:
794:
789:
767:
766:
761:
743:
737:
732:
707:
705:
704:
699:
686:
685:
684:
679:
675:
655:
650:
625:
623:
622:
617:
596:
594:
593:
588:
387:dispersive prism
380:
378:
377:
372:
339:
337:
336:
331:
319:
317:
316:
311:
283:
281:
280:
275:
264:
263:
235:(PSD, or simply
214:Fourier analysis
211:
209:
208:
203:
187:
185:
184:
179:
152:
150:
149:
144:
133:
132:
103:
96:
92:
89:
83:
63:
62:
55:
10344:
10343:
10339:
10338:
10337:
10335:
10334:
10333:
10284:
10283:
10282:
10277:
10246:
10169:
10160:
10123:
10113:
10094:
10075:
10056:
10029:
10010:
9916:(11): 2344â59.
9897:
9878:
9859:
9843:
9838:
9837:
9829:
9823:
9819:
9811:
9807:
9799:
9795:
9784:
9780:
9772:
9768:
9760:
9756:
9748:
9744:
9737:
9721:
9717:
9709:
9705:
9697:
9693:
9685:
9681:
9673:
9669:
9661:
9657:
9649:
9645:
9636:
9634:
9619:
9615:
9607:
9603:
9595:
9591:
9583:
9579:
9571:
9562:
9553:
9548:
9547:
9535:is continuous,
9516:
9512:
9510:
9507:
9506:
9474:
9470:
9468:
9465:
9464:
9454:
9450:
9421:
9407:
9404:
9403:
9378:
9331:
9322:
9311:
9310:
9309:
9289:
9288:
9283:
9280:
9279:
9273:
9269:
9264:
9256:Window function
9216:Colors of noise
9202:
9191:
9134:
9089:
9083:
9046:autocorrelation
9020:
9017:
9016:
8878:
8871:
8863:maximum entropy
8824:
8818:
8788:
8784:
8768:
8764:
8755:
8751:
8739:
8735:
8729:
8715:
8690:
8686:
8684:
8681:
8680:
8676:
8671:
8667:
8662:
8631:
8627:
8603:
8599:
8587:
8583:
8581:
8578:
8577:
8538:
8537:
8535:
8532:
8531:
8500:
8499:
8497:
8494:
8493:
8468:
8465:
8464:
8439:
8436:
8435:
8395:
8392:
8391:
8366:
8363:
8362:
8337:
8334:
8333:
8304:
8300:
8298:
8295:
8294:
8269:
8266:
8265:
8240:
8237:
8236:
8203:
8199:
8197:
8194:
8193:
8179:
8178:
8148:
8144:
8126:
8122:
8116:
8108:
8074:
8070:
8044:
8040:
8019:
8014:
8004:
7996:
7982:
7970:
7965:
7961:
7955:
7947:
7936:
7918:
7914:
7911:
7910:
7883:
7879:
7861:
7857:
7851:
7843:
7809:
7805:
7779:
7775:
7754:
7749:
7739:
7731:
7717:
7705:
7700:
7696:
7690:
7682:
7671:
7653:
7649:
7645:
7643:
7640:
7639:
7613:
7610:
7609:
7576:
7572:
7554:
7546:
7540:
7537:
7536:
7507:
7503:
7501:
7498:
7497:
7468:
7464:
7462:
7459:
7458:
7444:
7443:
7423:
7412:
7411:
7410:
7395:
7390:
7379:
7378:
7376:
7372:
7362:
7350:
7339:
7321:
7317:
7315:
7295:
7284:
7283:
7282:
7267:
7262:
7251:
7250:
7248:
7244:
7234:
7222:
7211:
7193:
7189:
7185:
7183:
7180:
7179:
7163:
7162:
7150:
7145:
7144:
7129:
7125:
7120:
7102:
7098:
7083:
7078:
7056:
7052:
7037:
7032:
7019:
7014:
7013:
6998:
6994:
6989:
6983:
6975:
6961:
6949:
6936:
6935:
6909:
6905:
6887:
6883:
6882:
6878:
6872:
6852:
6848:
6830:
6826:
6825:
6821:
6820:
6814:
6806:
6792:
6780:
6769:
6762:
6760:
6757:
6756:
6711:
6707:
6705:
6702:
6701:
6672:
6668:
6666:
6663:
6662:
6637:
6634:
6633:
6608:
6605:
6604:
6601:
6595:
6590:
6581:
6559:
6544:
6532:
6528:
6522:
6517:
6493:
6490:
6489:
6460:
6457:
6456:
6420:
6416:
6414:
6411:
6410:
6364:
6360:
6353:
6345:
6315:
6312:
6311:
6295:
6287:
6244:
6240:
6216:
6212:
6210:
6207:
6206:
6194:
6162:
6159:
6158:
6138:
6135:
6134:
6118:
6115:
6114:
6097:
6064:
6060:
6054:
6050:
6044:
6030:
6025:
6021:
6020:
6008:
6004:
5994:
5992:
5980:
5955:
5951:
5949:
5946:
5945:
5901:
5898:
5897:
5861:
5857:
5848:
5844:
5842:
5839:
5838:
5807:
5804:
5803:
5786:
5782:
5780:
5777:
5776:
5744:
5741:
5740:
5716:
5711:
5708:
5707:
5688:
5683:
5680:
5679:
5663:
5660:
5659:
5623:
5619:
5611:
5607:
5606:
5599:
5595:
5594:
5577:
5576:
5572:
5570:
5567:
5566:
5537:
5533:
5509:
5505:
5503:
5500:
5499:
5482:
5478:
5469:
5465:
5457:
5454:
5453:
5433:
5429:
5420:
5416:
5411:
5408:
5407:
5397:
5354:
5343:
5342:
5341:
5310:
5306:
5288:
5284:
5278:
5270:
5245:
5241:
5239:
5236:
5235:
5200:
5197:
5196:
5156:
5152:
5134:
5130:
5124:
5116:
5079:
5075:
5049:
5045:
5024:
5019:
5009:
5001:
4987:
4975:
4970:
4966:
4960:
4952:
4939:
4919:
4908:
4907:
4906:
4905:
4901:
4900:
4890:
4878:
4872:
4869:
4868:
4839:
4836:
4835:
4806:
4802:
4800:
4797:
4796:
4771:
4768:
4767:
4764:autocorrelation
4741:
4738:
4737:
4721:
4718:
4717:
4696:
4695:
4662:
4658:
4632:
4628:
4607:
4602:
4592:
4584:
4579:
4575:
4569:
4561:
4548:
4547:
4520:
4516:
4490:
4486:
4462:
4454:
4449:
4445:
4439:
4431:
4418:
4417:
4397:
4393:
4387:
4386:
4373:
4369:
4363:
4362:
4353:
4352:
4332:
4328:
4327:
4323:
4317:
4316:
4295:
4291:
4285:
4284:
4275:
4274:
4254:
4250:
4249:
4245:
4239:
4238:
4212:
4207:
4202:
4198:
4192:
4191:
4182:
4181:
4166:
4155:
4154:
4153:
4144:
4140:
4125:
4114:
4113:
4112:
4102:
4096:
4076:
4065:
4064:
4063:
4062:
4058:
4057:
4053:
4051:
4048:
4047:
4018:
4013:
3992:
3989:
3988:
3974:
3973:
3967:
3947:
3936:
3935:
3934:
3933:
3929:
3928:
3919:
3918:
3912:
3887:
3883:
3882:
3878:
3872:
3871:
3870:
3866:
3865:
3856:
3855:
3849:
3817:
3813:
3798:
3794:
3788:
3780:
3775:
3771:
3770:
3761:
3760:
3748:
3744:
3723:
3719:
3701:
3696:
3686:
3678:
3665:
3664:
3640:
3636:
3621:
3616:
3606:
3598:
3585:
3584:
3557:
3553:
3535:
3530:
3520:
3512:
3501:
3478:
3473:
3468:
3464:
3458:
3457:
3453:
3451:
3448:
3447:
3421:
3417:
3415:
3412:
3411:
3382:
3377:
3371:
3368:
3367:
3342:
3337:
3336:
3321:
3310:
3309:
3308:
3303:
3301:
3298:
3297:
3290:
3258:
3253:
3252:
3237:
3226:
3225:
3224:
3219:
3209:
3197:
3172:
3168:
3166:
3163:
3162:
3130:
3125:
3124:
3109:
3098:
3097:
3096:
3091:
3085:
3077:
3063:
3051:
3039:
3036:
3035:
3000:
2999:
2997:
2994:
2993:
2968:
2965:
2964:
2930:
2910:
2906:
2905:
2901:
2900:
2894:
2886:
2872:
2860:
2848:
2845:
2844:
2818:
2814:
2812:
2809:
2808:
2782:
2778:
2748:
2744:
2742:
2739:
2738:
2711:
2693:
2689:
2688:
2678:
2666:
2662:
2661:
2652:
2640:
2636:
2635:
2621:
2609:
2597:
2594:
2593:
2576:
2572:
2564:
2561:
2560:
2544:
2541:
2540:
2515:
2512:
2511:
2495:
2492:
2491:
2459:
2455:
2453:
2450:
2449:
2417:
2414:
2413:
2380:
2377:
2376:
2343:
2336:
2302:
2299:
2298:
2279:
2276:
2275:
2255:
2251:
2249:
2246:
2245:
2215:
2204:
2203:
2202:
2200:
2197:
2196:
2174:
2154:
2143:
2142:
2141:
2140:
2136:
2135:
2134:
2123:
2090:
2086:
2080:
2076:
2070:
2056:
2051:
2047:
2046:
2044:
2037:
2033:
2012:
1987:
1976:
1975:
1974:
1972:
1969:
1968:
1932:
1928:
1919:
1915:
1913:
1910:
1909:
1892:
1888:
1886:
1883:
1882:
1859:
1856:
1855:
1833:
1819:
1816:
1815:
1786:
1783:
1782:
1763:
1757:
1753:
1742:
1739:
1738:
1716:
1702:
1699:
1698:
1682:
1679:
1678:
1653:
1650:
1649:
1630:
1627:
1626:
1623:bandpass filter
1606:
1603:
1602:
1573:
1562:
1561:
1560:
1558:
1555:
1554:
1535:
1529:
1525:
1514:
1511:
1510:
1491:
1485:
1481:
1470:
1467:
1466:
1450:
1447:
1446:
1422:
1419:
1418:
1402:represents the
1378:
1375:
1374:
1338:
1335:
1334:
1331:autocorrelation
1301:
1290:
1289:
1288:
1286:
1283:
1282:
1279:
1248:
1223:
1222:
1221:
1217:
1216:
1195:
1184:
1183:
1182:
1180:
1177:
1176:
1169:is defined as:
1145:
1142:
1141:
1136:Therefore, the
1109:
1106:
1105:
1089:
1086:
1085:
1058:
1033:
1032:
1031:
1027:
1026:
1024:
1021:
1020:
1000:
997:
996:
968:
965:
964:
907:
903:
897:
889:
862:
861:
859:
856:
855:
828:
803:
802:
801:
797:
796:
790:
782:
762:
757:
756:
739:
733:
725:
719:
716:
715:
680:
662:
658:
657:
651:
643:
631:
628:
627:
602:
599:
598:
582:
579:
578:
563:
560:energy spectrum
556:
551:
531:
429:
423:
357:
354:
353:
325:
322:
321:
296:
293:
292:
259:
255:
253:
250:
249:
197:
194:
193:
164:
161:
160:
155:continuous time
125:
121:
119:
116:
115:
104:
93:
87:
84:
76:help improve it
73:
64:
60:
33:
24:
17:
12:
11:
5:
10342:
10332:
10331:
10326:
10324:Radio spectrum
10321:
10316:
10311:
10306:
10301:
10296:
10279:
10278:
10276:
10275:
10270:
10265:
10260:
10255:
10251:
10248:
10247:
10245:
10244:
10239:
10234:
10229:
10224:
10219:
10214:
10208:
10203:
10202:
10201:
10191:
10186:
10181:
10174:
10171:
10170:
10159:
10158:
10151:
10144:
10136:
10130:
10129:
10122:
10121:External links
10119:
10118:
10117:
10111:
10098:
10092:
10079:
10073:
10060:
10054:
10041:
10027:
10014:
10008:
9995:
9959:(4): 858â866.
9944:
9901:
9895:
9882:
9876:
9863:
9857:
9842:
9839:
9836:
9835:
9817:
9805:
9793:
9778:
9766:
9764:, p. 431.
9754:
9742:
9735:
9715:
9713:, p. 433.
9703:
9691:
9679:
9667:
9655:
9643:
9613:
9601:
9589:
9577:
9550:
9549:
9546:
9545:
9522:
9519:
9515:
9480:
9477:
9473:
9448:
9431:
9427:
9424:
9420:
9417:
9414:
9411:
9391:
9388:
9384:
9381:
9377:
9374:
9371:
9368:
9365:
9362:
9359:
9356:
9353:
9350:
9347:
9344:
9341:
9337:
9334:
9330:
9325:
9318:
9315:
9308:
9305:
9302:
9296:
9293:
9287:
9266:
9265:
9263:
9260:
9259:
9258:
9253:
9248:
9243:
9238:
9233:
9228:
9223:
9218:
9213:
9208:
9201:
9198:
9190:
9187:
9166:remote sensing
9133:
9130:
9095:(perceived as
9082:
9079:
9078:
9077:
9061:
9033:
9030:
9027:
9024:
9000:spectral phase
8996:phase spectrum
8972:
8954:
8951:spectral bands
8939:remote sensing
8935:envelope curve
8927:
8922:and stages of
8888:
8870:
8867:
8840:non-parametric
8820:Main article:
8817:
8814:
8802:
8799:
8791:
8787:
8783:
8780:
8777:
8774:
8771:
8767:
8763:
8758:
8754:
8750:
8745:
8742:
8738:
8732:
8727:
8724:
8721:
8718:
8714:
8710:
8707:
8704:
8701:
8696:
8693:
8689:
8674:
8665:
8648:
8645:
8642:
8637:
8634:
8630:
8626:
8623:
8620:
8617:
8614:
8609:
8606:
8602:
8598:
8595:
8586:
8557:
8554:
8551:
8545:
8542:
8519:
8516:
8513:
8507:
8504:
8481:
8478:
8475:
8472:
8452:
8449:
8446:
8443:
8423:
8420:
8417:
8414:
8411:
8408:
8405:
8402:
8399:
8379:
8376:
8373:
8370:
8350:
8347:
8344:
8341:
8321:
8318:
8315:
8310:
8307:
8303:
8282:
8279:
8276:
8273:
8253:
8250:
8247:
8244:
8220:
8217:
8214:
8209:
8206:
8202:
8177:
8174:
8171:
8166:
8163:
8160:
8157:
8154:
8151:
8147:
8143:
8140:
8137:
8132:
8129:
8125:
8119:
8114:
8111:
8107:
8103:
8100:
8097:
8092:
8089:
8086:
8083:
8080:
8077:
8073:
8068:
8064:
8061:
8058:
8055:
8052:
8047:
8043:
8039:
8036:
8033:
8030:
8027:
8022:
8017:
8013:
8007:
8002:
7999:
7995:
7989:
7986:
7979:
7976:
7973:
7969:
7964:
7958:
7953:
7950:
7946:
7942:
7939:
7937:
7935:
7932:
7929:
7924:
7921:
7917:
7913:
7912:
7909:
7906:
7901:
7898:
7895:
7892:
7889:
7886:
7882:
7878:
7875:
7872:
7867:
7864:
7860:
7854:
7849:
7846:
7842:
7838:
7835:
7832:
7827:
7824:
7821:
7818:
7815:
7812:
7808:
7803:
7799:
7796:
7793:
7790:
7787:
7782:
7778:
7774:
7771:
7768:
7765:
7762:
7757:
7752:
7748:
7742:
7737:
7734:
7730:
7724:
7721:
7714:
7711:
7708:
7704:
7699:
7693:
7688:
7685:
7681:
7677:
7674:
7672:
7670:
7667:
7664:
7659:
7656:
7652:
7648:
7647:
7623:
7620:
7617:
7593:
7590:
7587:
7582:
7579:
7575:
7571:
7568:
7565:
7562:
7557:
7552:
7549:
7545:
7524:
7521:
7518:
7513:
7510:
7506:
7485:
7482:
7479:
7474:
7471:
7467:
7441:
7437:
7434:
7431:
7426:
7419:
7416:
7409:
7406:
7403:
7398:
7393:
7386:
7383:
7375:
7369:
7366:
7359:
7356:
7353:
7349:
7345:
7342:
7340:
7338:
7335:
7332:
7327:
7324:
7320:
7316:
7313:
7309:
7306:
7303:
7298:
7291:
7288:
7281:
7278:
7275:
7270:
7265:
7258:
7255:
7247:
7241:
7238:
7231:
7228:
7225:
7221:
7217:
7214:
7212:
7210:
7207:
7204:
7199:
7196:
7192:
7188:
7187:
7161:
7158:
7153:
7148:
7143:
7140:
7137:
7132:
7128:
7123:
7119:
7116:
7113:
7110:
7105:
7101:
7097:
7094:
7091:
7086:
7081:
7077:
7073:
7070:
7067:
7064:
7059:
7055:
7051:
7048:
7045:
7040:
7035:
7031:
7027:
7022:
7017:
7012:
7009:
7006:
7001:
6997:
6992:
6986:
6981:
6978:
6974:
6968:
6965:
6958:
6955:
6952:
6948:
6944:
6941:
6939:
6937:
6934:
6931:
6927:
6923:
6920:
6917:
6912:
6908:
6904:
6901:
6898:
6895:
6890:
6886:
6881:
6875:
6870:
6866:
6863:
6860:
6855:
6851:
6847:
6844:
6841:
6838:
6833:
6829:
6824:
6817:
6812:
6809:
6805:
6799:
6796:
6789:
6786:
6783:
6779:
6775:
6772:
6770:
6768:
6765:
6764:
6728:
6725:
6722:
6717:
6714:
6710:
6689:
6686:
6683:
6678:
6675:
6671:
6650:
6647:
6644:
6641:
6621:
6618:
6615:
6612:
6594:
6591:
6589:
6588:
6569:
6565:
6562:
6558:
6554:
6550:
6547:
6543:
6538:
6535:
6531:
6525:
6520:
6516:
6512:
6509:
6506:
6503:
6500:
6497:
6473:
6470:
6467:
6464:
6437:
6434:
6431:
6426:
6423:
6419:
6399:
6388:
6385:
6381:
6378:
6375:
6370:
6367:
6363:
6356:
6351:
6348:
6344:
6340:
6337:
6334:
6331:
6328:
6325:
6322:
6319:
6300:
6293:
6283:
6273:
6261:
6258:
6255:
6250:
6247:
6243:
6239:
6236:
6233:
6230:
6227:
6222:
6219:
6215:
6205:of frequency:
6198:
6193:
6190:
6166:
6142:
6122:
6100:
6095:
6089:
6086:
6082:
6079:
6076:
6073:
6070:
6067:
6063:
6057:
6053:
6047:
6042:
6039:
6036:
6033:
6029:
6024:
6017:
6011:
6007:
6003:
6000:
5997:
5989:
5986:
5983:
5979:
5975:
5972:
5969:
5966:
5961:
5958:
5954:
5933:
5930:
5926:
5923:
5920:
5917:
5914:
5911:
5908:
5905:
5885:
5882:
5879:
5875:
5872:
5869:
5864:
5860:
5856:
5851:
5847:
5826:
5823:
5820:
5817:
5814:
5811:
5789:
5785:
5757:
5754:
5751:
5748:
5723:
5719:
5715:
5695:
5691:
5687:
5667:
5647:
5644:
5640:
5637:
5634:
5629:
5626:
5622:
5614:
5610:
5602:
5598:
5593:
5589:
5586:
5575:
5554:
5551:
5548:
5543:
5540:
5536:
5532:
5529:
5526:
5523:
5520:
5515:
5512:
5508:
5485:
5481:
5477:
5472:
5468:
5464:
5461:
5441:
5436:
5432:
5428:
5423:
5419:
5415:
5394:
5393:
5384:
5382:
5371:
5368:
5365:
5360:
5357:
5350:
5347:
5340:
5337:
5334:
5328:
5325:
5322:
5319:
5316:
5313:
5309:
5305:
5302:
5299:
5294:
5291:
5287:
5281:
5276:
5273:
5269:
5265:
5262:
5259:
5256:
5251:
5248:
5244:
5230:
5213:
5210:
5207:
5204:
5182:
5179:
5174:
5171:
5168:
5165:
5162:
5159:
5155:
5151:
5148:
5145:
5140:
5137:
5133:
5127:
5122:
5119:
5115:
5111:
5108:
5105:
5097:
5094:
5091:
5088:
5085:
5082:
5078:
5073:
5069:
5066:
5063:
5060:
5057:
5052:
5048:
5044:
5041:
5038:
5035:
5032:
5027:
5022:
5018:
5012:
5007:
5004:
5000:
4994:
4991:
4984:
4981:
4978:
4974:
4969:
4963:
4958:
4955:
4951:
4947:
4942:
4937:
4933:
4930:
4927:
4922:
4915:
4912:
4904:
4897:
4894:
4887:
4884:
4881:
4877:
4852:
4849:
4846:
4843:
4823:
4820:
4817:
4812:
4809:
4805:
4784:
4781:
4778:
4775:
4751:
4748:
4745:
4725:
4694:
4691:
4688:
4680:
4677:
4674:
4671:
4668:
4665:
4661:
4656:
4652:
4649:
4646:
4643:
4640:
4635:
4631:
4627:
4624:
4621:
4618:
4615:
4610:
4605:
4601:
4595:
4590:
4587:
4583:
4578:
4572:
4567:
4564:
4560:
4556:
4553:
4551:
4549:
4546:
4543:
4538:
4535:
4532:
4529:
4526:
4523:
4519:
4514:
4510:
4507:
4504:
4501:
4498:
4493:
4489:
4485:
4482:
4479:
4476:
4473:
4470:
4465:
4460:
4457:
4453:
4448:
4442:
4437:
4434:
4430:
4426:
4423:
4421:
4419:
4415:
4411:
4408:
4405:
4400:
4396:
4390:
4385:
4382:
4379:
4376:
4372:
4366:
4361:
4358:
4356:
4354:
4350:
4346:
4343:
4340:
4335:
4331:
4326:
4320:
4315:
4311:
4307:
4304:
4301:
4298:
4294:
4288:
4283:
4280:
4278:
4276:
4272:
4268:
4265:
4262:
4257:
4253:
4248:
4242:
4237:
4233:
4229:
4226:
4223:
4220:
4215:
4210:
4206:
4201:
4195:
4190:
4187:
4185:
4183:
4180:
4177:
4174:
4169:
4162:
4159:
4152:
4147:
4143:
4139:
4136:
4133:
4128:
4121:
4118:
4111:
4108:
4105:
4103:
4099:
4094:
4090:
4087:
4084:
4079:
4072:
4069:
4061:
4056:
4055:
4035:
4032:
4029:
4026:
4021:
4016:
4012:
4008:
4005:
4002:
3999:
3996:
3970:
3965:
3961:
3958:
3955:
3950:
3943:
3940:
3932:
3927:
3924:
3922:
3920:
3915:
3910:
3905:
3901:
3898:
3895:
3890:
3886:
3881:
3875:
3869:
3864:
3861:
3859:
3857:
3852:
3847:
3843:
3840:
3835:
3832:
3829:
3826:
3823:
3820:
3816:
3812:
3809:
3806:
3801:
3797:
3791:
3786:
3783:
3779:
3774:
3769:
3766:
3764:
3762:
3759:
3756:
3751:
3747:
3741:
3738:
3735:
3732:
3729:
3726:
3722:
3718:
3715:
3712:
3709:
3704:
3699:
3695:
3689:
3684:
3681:
3677:
3673:
3670:
3668:
3666:
3663:
3660:
3655:
3652:
3649:
3646:
3643:
3639:
3635:
3632:
3629:
3624:
3619:
3615:
3609:
3604:
3601:
3597:
3593:
3590:
3588:
3586:
3583:
3580:
3575:
3572:
3569:
3566:
3563:
3560:
3556:
3552:
3549:
3546:
3543:
3538:
3533:
3529:
3523:
3518:
3515:
3511:
3507:
3504:
3502:
3499:
3495:
3492:
3489:
3486:
3481:
3476:
3472:
3467:
3461:
3456:
3455:
3435:
3432:
3429:
3424:
3420:
3399:
3396:
3393:
3390:
3385:
3380:
3376:
3345:
3340:
3335:
3332:
3329:
3324:
3317:
3314:
3306:
3287:
3286:
3277:
3275:
3261:
3256:
3251:
3248:
3245:
3240:
3233:
3230:
3222:
3216:
3213:
3206:
3203:
3200:
3196:
3192:
3189:
3186:
3183:
3178:
3175:
3171:
3157:
3142:
3139:
3133:
3128:
3123:
3120:
3117:
3112:
3105:
3102:
3094:
3088:
3083:
3080:
3076:
3070:
3067:
3060:
3057:
3054:
3050:
3046:
3043:
3019:
3016:
3013:
3007:
3004:
2981:
2978:
2975:
2972:
2945:
2942:
2939:
2933:
2928:
2924:
2921:
2918:
2913:
2909:
2904:
2897:
2892:
2889:
2885:
2879:
2876:
2869:
2866:
2863:
2859:
2855:
2852:
2832:
2829:
2826:
2821:
2817:
2796:
2793:
2790:
2785:
2781:
2777:
2774:
2771:
2768:
2765:
2762:
2759:
2756:
2751:
2747:
2723:
2720:
2714:
2709:
2705:
2702:
2699:
2696:
2692:
2685:
2681:
2677:
2674:
2669:
2665:
2659:
2655:
2651:
2648:
2643:
2639:
2634:
2628:
2625:
2618:
2615:
2612:
2608:
2604:
2601:
2579:
2575:
2571:
2568:
2548:
2528:
2525:
2522:
2519:
2499:
2473:
2470:
2467:
2462:
2458:
2430:
2427:
2424:
2421:
2406:power spectrum
2393:
2390:
2387:
2384:
2335:
2332:
2315:
2312:
2309:
2306:
2286:
2283:
2263:
2258:
2254:
2229:
2226:
2223:
2218:
2211:
2208:
2184:
2177:
2172:
2168:
2165:
2162:
2157:
2150:
2147:
2139:
2131:
2126:
2121:
2115:
2112:
2108:
2105:
2102:
2099:
2096:
2093:
2089:
2083:
2079:
2073:
2068:
2065:
2062:
2059:
2055:
2050:
2040:
2036:
2032:
2029:
2026:
2021:
2018:
2015:
2011:
2007:
2004:
2001:
1998:
1993:
1990:
1983:
1980:
1956:
1953:
1950:
1946:
1943:
1940:
1935:
1931:
1927:
1922:
1918:
1895:
1891:
1863:
1843:
1840:
1836:
1832:
1829:
1826:
1823:
1799:
1796:
1793:
1790:
1770:
1766:
1760:
1756:
1752:
1749:
1746:
1726:
1723:
1719:
1715:
1712:
1709:
1706:
1686:
1666:
1663:
1660:
1657:
1637:
1634:
1610:
1590:
1587:
1584:
1579:
1576:
1569:
1566:
1542:
1538:
1532:
1528:
1524:
1521:
1518:
1498:
1494:
1488:
1484:
1480:
1477:
1474:
1454:
1426:
1391:
1388:
1385:
1382:
1351:
1348:
1345:
1342:
1318:
1315:
1312:
1307:
1304:
1297:
1294:
1276:
1275:
1266:
1264:
1251:
1246:
1242:
1239:
1236:
1230:
1227:
1220:
1215:
1212:
1209:
1206:
1201:
1198:
1191:
1188:
1171:
1158:
1155:
1152:
1149:
1122:
1119:
1116:
1113:
1093:
1069:
1066:
1061:
1056:
1052:
1049:
1046:
1040:
1037:
1030:
1004:
981:
978:
975:
972:
948:
945:
939:
936:
933:
930:
925:
922:
919:
916:
913:
910:
906:
900:
895:
892:
888:
884:
881:
878:
875:
869:
866:
843:
840:
837:
831:
826:
822:
819:
816:
810:
807:
800:
793:
788:
785:
781:
777:
774:
771:
765:
760:
755:
752:
749:
746:
742:
736:
731:
728:
724:
697:
694:
691:
683:
678:
674:
671:
668:
665:
661:
654:
649:
646:
642:
638:
635:
615:
612:
609:
606:
586:
555:
552:
550:
547:
530:
527:
422:
419:
370:
367:
364:
361:
329:
309:
306:
303:
300:
273:
270:
267:
262:
258:
237:power spectrum
201:
177:
174:
171:
168:
142:
139:
136:
131:
128:
124:
106:
105:
67:
65:
58:
30:Spectral power
15:
9:
6:
4:
3:
2:
10341:
10330:
10327:
10325:
10322:
10320:
10317:
10315:
10312:
10310:
10307:
10305:
10302:
10300:
10297:
10295:
10292:
10291:
10289:
10274:
10271:
10269:
10266:
10264:
10261:
10259:
10256:
10253:
10252:
10249:
10243:
10240:
10238:
10235:
10233:
10230:
10228:
10225:
10223:
10220:
10218:
10215:
10212:
10209:
10207:
10204:
10200:
10197:
10196:
10195:
10192:
10190:
10187:
10185:
10182:
10179:
10176:
10175:
10172:
10167:
10164:
10157:
10152:
10150:
10145:
10143:
10138:
10137:
10134:
10128:
10125:
10124:
10114:
10108:
10104:
10099:
10095:
10089:
10085:
10080:
10076:
10070:
10066:
10061:
10057:
10051:
10047:
10042:
10038:
10034:
10030:
10024:
10020:
10015:
10011:
10005:
10001:
10000:VSAT Networks
9996:
9992:
9988:
9984:
9980:
9975:
9974:10044/1/46059
9970:
9966:
9962:
9958:
9954:
9950:
9945:
9941:
9937:
9932:
9927:
9923:
9919:
9915:
9911:
9907:
9902:
9898:
9892:
9888:
9883:
9879:
9873:
9869:
9864:
9860:
9854:
9850:
9845:
9844:
9828:
9821:
9814:
9809:
9802:
9797:
9789:
9782:
9775:
9770:
9763:
9758:
9751:
9746:
9738:
9732:
9728:
9727:
9719:
9712:
9707:
9700:
9695:
9688:
9683:
9677:, p. 14.
9676:
9671:
9664:
9659:
9653:, p. 12.
9652:
9647:
9633:on 2024-04-15
9632:
9628:
9624:
9617:
9611:, p. 83.
9610:
9609:Birolini 2007
9605:
9598:
9593:
9586:
9581:
9570:
9566:
9560:
9558:
9556:
9551:
9542:
9538:
9520:
9517:
9513:
9504:
9500:
9496:
9495:distributions
9478:
9475:
9471:
9462:
9458:
9452:
9445:
9425:
9422:
9418:
9415:
9409:
9389:
9382:
9379:
9375:
9372:
9366:
9360:
9354:
9351:
9348:
9345:
9335:
9332:
9323:
9313:
9303:
9291:
9277:
9271:
9267:
9257:
9254:
9252:
9249:
9247:
9244:
9242:
9239:
9237:
9234:
9232:
9229:
9227:
9224:
9222:
9219:
9217:
9214:
9212:
9209:
9207:
9204:
9203:
9197:
9195:
9186:
9184:
9179:
9177:
9176:
9175:power spectra
9171:
9167:
9163:
9159:
9155:
9151:
9143:
9138:
9129:
9127:
9122:
9118:
9114:
9110:
9106:
9102:
9098:
9094:
9093:visible light
9088:
9074:
9070:
9066:
9062:
9059:
9055:
9051:
9047:
9028:
9022:
9015:
9011:
9010:
9005:
9001:
8997:
8993:
8989:
8985:
8981:
8977:
8973:
8970:
8966:
8962:
8961:
8955:
8952:
8948:
8944:
8940:
8936:
8932:
8928:
8925:
8921:
8917:
8913:
8909:
8905:
8901:
8897:
8893:
8889:
8886:
8885:
8880:
8879:
8876:
8866:
8864:
8860:
8856:
8851:
8849:
8845:
8841:
8837:
8833:
8832:random signal
8829:
8823:
8813:
8800:
8789:
8785:
8781:
8778:
8775:
8772:
8769:
8765:
8756:
8752:
8743:
8740:
8736:
8722:
8719:
8716:
8712:
8708:
8702:
8694:
8691:
8687:
8677:
8668:
8659:
8643:
8635:
8632:
8628:
8624:
8621:
8615:
8607:
8604:
8600:
8596:
8593:
8584:
8575:
8571:
8552:
8540:
8514:
8502:
8476:
8470:
8447:
8441:
8418:
8412:
8409:
8403:
8397:
8374:
8368:
8345:
8339:
8316:
8308:
8305:
8301:
8277:
8271:
8248:
8242:
8234:
8215:
8207:
8204:
8200:
8175:
8172:
8169:
8164:
8161:
8158:
8155:
8152:
8149:
8145:
8138:
8130:
8127:
8123:
8109:
8105:
8101:
8098:
8095:
8090:
8087:
8084:
8081:
8078:
8075:
8071:
8066:
8062:
8059:
8053:
8045:
8041:
8034:
8031:
8028:
8020:
8015:
8011:
7997:
7993:
7987:
7984:
7971:
7962:
7948:
7944:
7940:
7938:
7930:
7922:
7919:
7915:
7907:
7904:
7899:
7896:
7893:
7890:
7887:
7884:
7880:
7873:
7865:
7862:
7858:
7844:
7840:
7836:
7833:
7830:
7825:
7822:
7819:
7816:
7813:
7810:
7806:
7801:
7797:
7794:
7788:
7780:
7776:
7769:
7766:
7763:
7755:
7750:
7746:
7732:
7728:
7722:
7719:
7706:
7697:
7683:
7679:
7675:
7673:
7665:
7657:
7654:
7650:
7637:
7615:
7607:
7588:
7580:
7577:
7573:
7569:
7563:
7555:
7550:
7547:
7543:
7519:
7511:
7508:
7504:
7480:
7472:
7469:
7465:
7439:
7432:
7424:
7414:
7404:
7396:
7391:
7381:
7373:
7367:
7364:
7351:
7343:
7341:
7333:
7325:
7322:
7318:
7311:
7304:
7296:
7286:
7276:
7268:
7263:
7253:
7245:
7239:
7236:
7223:
7215:
7213:
7205:
7197:
7194:
7190:
7176:
7159:
7156:
7151:
7138:
7130:
7126:
7117:
7111:
7103:
7099:
7092:
7084:
7079:
7075:
7071:
7065:
7057:
7053:
7046:
7038:
7033:
7029:
7025:
7020:
7007:
6999:
6995:
6976:
6972:
6966:
6963:
6950:
6942:
6940:
6932:
6929:
6925:
6918:
6910:
6906:
6902:
6896:
6888:
6884:
6879:
6873:
6868:
6861:
6853:
6849:
6845:
6839:
6831:
6827:
6822:
6807:
6803:
6797:
6794:
6781:
6773:
6771:
6766:
6754:
6750:
6746:
6742:
6723:
6715:
6712:
6708:
6684:
6676:
6673:
6669:
6645:
6639:
6616:
6610:
6600:
6584:
6567:
6563:
6560:
6556:
6548:
6545:
6536:
6533:
6529:
6523:
6518:
6514:
6510:
6507:
6501:
6495:
6487:
6468:
6462:
6455:
6451:
6432:
6424:
6421:
6417:
6408:
6404:
6400:
6386:
6383:
6376:
6368:
6365:
6361:
6346:
6342:
6338:
6332:
6326:
6323:
6320:
6317:
6309:
6305:
6301:
6299:
6291:
6286:
6282:
6278:
6274:
6256:
6248:
6245:
6241:
6237:
6231:
6228:
6220:
6217:
6213:
6204:
6203:even function
6200:
6199:
6197:
6189:
6187:
6183:
6178:
6164:
6156:
6140:
6120:
6098:
6093:
6087:
6080:
6077:
6074:
6071:
6068:
6065:
6061:
6055:
6051:
6045:
6040:
6037:
6034:
6031:
6027:
6022:
6015:
6009:
6001:
5981:
5973:
5967:
5959:
5956:
5952:
5931:
5921:
5918:
5915:
5912:
5906:
5903:
5880:
5873:
5867:
5862:
5858:
5854:
5849:
5845:
5824:
5821:
5818:
5815:
5812:
5809:
5787:
5783:
5774:
5773:discrete time
5769:
5752:
5746:
5738:
5721:
5717:
5713:
5693:
5689:
5685:
5665:
5645:
5642:
5635:
5627:
5624:
5620:
5612:
5608:
5600:
5596:
5591:
5587:
5584:
5573:
5549:
5541:
5538:
5534:
5530:
5524:
5521:
5513:
5510:
5506:
5483:
5479:
5475:
5470:
5466:
5462:
5459:
5434:
5430:
5426:
5421:
5417:
5404:
5402:
5392:
5385:
5383:
5366:
5358:
5355:
5345:
5338:
5335:
5332:
5326:
5323:
5320:
5317:
5314:
5311:
5307:
5300:
5292:
5289:
5285:
5271:
5267:
5263:
5257:
5249:
5246:
5242:
5234:
5233:
5229:
5227:
5208:
5202:
5193:
5180:
5177:
5172:
5169:
5166:
5163:
5160:
5157:
5153:
5146:
5138:
5135:
5131:
5117:
5113:
5109:
5106:
5103:
5095:
5092:
5089:
5086:
5083:
5080:
5076:
5071:
5067:
5064:
5058:
5050:
5046:
5039:
5036:
5033:
5025:
5020:
5016:
5002:
4998:
4992:
4989:
4976:
4967:
4953:
4949:
4945:
4940:
4935:
4928:
4920:
4910:
4902:
4895:
4892:
4879:
4866:
4847:
4841:
4818:
4810:
4807:
4803:
4779:
4773:
4765:
4743:
4723:
4714:
4712:
4692:
4689:
4686:
4678:
4675:
4672:
4669:
4666:
4663:
4659:
4654:
4650:
4647:
4641:
4633:
4629:
4622:
4619:
4616:
4608:
4603:
4599:
4585:
4581:
4576:
4562:
4558:
4554:
4552:
4544:
4541:
4536:
4533:
4530:
4527:
4524:
4521:
4517:
4512:
4508:
4505:
4499:
4491:
4487:
4480:
4477:
4474:
4468:
4455:
4451:
4446:
4432:
4428:
4424:
4422:
4413:
4406:
4398:
4394:
4380:
4374:
4370:
4359:
4357:
4348:
4341:
4333:
4329:
4324:
4313:
4309:
4302:
4296:
4292:
4281:
4279:
4270:
4263:
4255:
4251:
4246:
4235:
4231:
4224:
4221:
4213:
4208:
4204:
4199:
4188:
4186:
4175:
4167:
4157:
4150:
4145:
4134:
4126:
4116:
4106:
4104:
4097:
4092:
4085:
4077:
4067:
4059:
4030:
4027:
4019:
4014:
4010:
4006:
4000:
3994:
3968:
3963:
3956:
3948:
3938:
3930:
3925:
3923:
3913:
3908:
3903:
3896:
3888:
3884:
3879:
3867:
3862:
3860:
3850:
3845:
3841:
3838:
3833:
3830:
3827:
3824:
3821:
3818:
3814:
3807:
3799:
3795:
3781:
3777:
3772:
3767:
3765:
3757:
3754:
3749:
3739:
3736:
3733:
3730:
3727:
3724:
3720:
3710:
3702:
3697:
3693:
3679:
3675:
3671:
3669:
3661:
3658:
3653:
3650:
3647:
3644:
3641:
3637:
3630:
3622:
3617:
3613:
3599:
3595:
3591:
3589:
3581:
3578:
3573:
3570:
3567:
3564:
3561:
3558:
3554:
3547:
3544:
3536:
3531:
3527:
3513:
3509:
3505:
3503:
3497:
3490:
3487:
3479:
3474:
3470:
3465:
3430:
3422:
3418:
3394:
3391:
3383:
3378:
3374:
3365:
3361:
3343:
3330:
3322:
3312:
3295:
3285:
3278:
3276:
3259:
3246:
3238:
3228:
3214:
3211:
3198:
3190:
3184:
3176:
3173:
3169:
3161:
3160:
3156:
3153:
3140:
3137:
3131:
3118:
3110:
3100:
3078:
3074:
3068:
3065:
3052:
3044:
3041:
3033:
3014:
3002:
2976:
2970:
2961:
2959:
2943:
2940:
2937:
2931:
2926:
2919:
2911:
2907:
2902:
2887:
2883:
2877:
2874:
2861:
2853:
2850:
2827:
2819:
2815:
2791:
2783:
2779:
2772:
2766:
2763:
2757:
2749:
2745:
2734:
2721:
2718:
2712:
2707:
2700:
2694:
2690:
2683:
2679:
2675:
2672:
2667:
2663:
2657:
2653:
2649:
2646:
2641:
2637:
2632:
2626:
2623:
2610:
2602:
2599:
2577:
2573:
2569:
2566:
2546:
2523:
2517:
2497:
2488:
2486:
2468:
2460:
2456:
2447:
2444:
2425:
2419:
2411:
2407:
2388:
2382:
2374:
2369:
2365:
2361:
2352:
2347:
2341:
2331:
2329:
2313:
2307:
2284:
2261:
2256:
2252:
2243:
2224:
2216:
2206:
2182:
2175:
2170:
2163:
2155:
2145:
2137:
2129:
2124:
2119:
2113:
2106:
2103:
2100:
2097:
2094:
2091:
2087:
2081:
2077:
2071:
2066:
2063:
2060:
2057:
2053:
2048:
2038:
2030:
2013:
2005:
1999:
1991:
1988:
1978:
1951:
1944:
1938:
1933:
1929:
1925:
1920:
1916:
1893:
1889:
1880:
1875:
1861:
1841:
1834:
1827:
1821:
1813:
1794:
1788:
1768:
1764:
1758:
1750:
1744:
1724:
1717:
1710:
1704:
1684:
1661:
1655:
1635:
1624:
1608:
1601:at frequency
1585:
1577:
1574:
1564:
1540:
1536:
1530:
1522:
1516:
1496:
1492:
1486:
1478:
1472:
1452:
1444:
1440:
1424:
1417:
1413:
1409:
1405:
1386:
1380:
1371:
1369:
1365:
1346:
1340:
1332:
1313:
1305:
1302:
1292:
1281:The function
1274:
1267:
1265:
1249:
1244:
1237:
1225:
1218:
1213:
1207:
1199:
1196:
1186:
1175:
1174:
1170:
1153:
1147:
1139:
1134:
1120:
1117:
1114:
1111:
1091:
1083:
1067:
1064:
1059:
1054:
1047:
1035:
1028:
1018:
1002:
995:
976:
970:
962:
946:
943:
934:
928:
923:
920:
917:
914:
911:
908:
904:
890:
886:
882:
876:
864:
841:
838:
835:
829:
824:
817:
805:
798:
783:
779:
775:
772:
769:
763:
750:
744:
726:
722:
713:
708:
695:
692:
689:
681:
676:
669:
663:
659:
644:
640:
636:
633:
610:
604:
584:
576:
572:
568:
561:
546:
544:
540:
536:
526:
524:
520:
515:
513:
509:
505:
501:
497:
491:
489:
485:
481:
477:
473:
469:
464:
462:
458:
454:
450:
446:
442:
441:acoustic wave
438:
434:
428:
418:
416:
412:
408:
404:
400:
394:
392:
388:
384:
365:
359:
351:
347:
343:
327:
304:
298:
289:
287:
268:
260:
256:
247:
242:
238:
234:
230:
225:
223:
219:
215:
199:
191:
172:
166:
159:
156:
137:
129:
126:
122:
113:
102:
99:
91:
81:
77:
71:
68:This article
66:
57:
56:
49:
42:
37:
31:
26:
22:
10309:Spectroscopy
10268:signal noise
10102:
10083:
10064:
10045:
10018:
9999:
9956:
9952:
9913:
9909:
9886:
9867:
9848:
9820:
9808:
9796:
9781:
9769:
9757:
9745:
9729:. Springer.
9725:
9718:
9706:
9694:
9682:
9670:
9658:
9646:
9635:. Retrieved
9631:the original
9626:
9616:
9604:
9592:
9580:
9451:
9270:
9192:
9180:
9178:of signals.
9174:
9173:
9156:, including
9147:
9126:notch filter
9120:
9090:
9081:Applications
9072:
9068:
9064:
9007:
8999:
8995:
8991:
8958:
8953:of interest.
8943:spectrometer
8930:
8911:
8907:
8906:below which
8899:
8895:
8891:
8882:
8859:Welch method
8852:
8825:
8672:
8663:
8660:
8573:
8569:
7605:
7177:
6752:
6748:
6744:
6740:
6602:
6582:
6485:
6453:
6449:
6406:
6402:
6289:
6284:
6280:
6195:
6179:
5770:
5405:
5400:
5398:
5386:
5194:
4715:
3987:and making,
3362:of the time
3291:
3279:
3154:
2962:
2957:
2735:
2510:of a signal
2489:
2405:
2359:
2356:
1876:
1465:is equal to
1372:
1280:
1268:
1137:
1135:
709:
597:of a signal
564:
538:
534:
532:
522:
518:
516:
510:denotes the
507:
504:acceleration
499:
492:
479:
472:proportional
471:
465:
444:
430:
395:
391:spectrograph
290:
240:
236:
232:
228:
226:
109:
94:
85:
69:
25:
10263:link budget
9058:group delay
9054:phase noise
8960:spectrogram
8848:periodogram
6486:bandlimited
6155:periodogram
5579:bandlimited
4046:, we have:
3364:convolution
1368:Periodogram
571:time series
498:, units of
411:engineering
10314:Scattering
10288:Categories
9841:References
9663:Stein 2000
9637:2024-06-26
9585:Maral 2004
9206:Bispectrum
9109:wavelength
8920:anesthesia
8836:parametric
8816:Estimation
7638:function.
6597:See also:
6133:(and thus
5775:variables
4709:where the
1366:(see also
549:Definition
496:vibrations
425:See also:
10180:(or dBmW)
10037:696092052
9991:206608057
9423:ω
9419:−
9416:ω
9410:δ
9380:ω
9376:−
9373:ω
9367:δ
9361:ω
9352:π
9343:⟩
9333:ω
9324:∗
9317:^
9304:ω
9295:^
9286:⟨
9189:Cosmology
9117:harmonics
9113:sine wave
8980:Bode plot
8904:frequency
8801:τ
8798:Δ
8786:τ
8779:π
8770:−
8753:τ
8731:∞
8726:∞
8723:−
8713:∑
8544:^
8506:^
8317:τ
8216:τ
8173:τ
8165:τ
8159:π
8150:−
8139:τ
8118:∞
8113:∞
8110:−
8106:∫
8099:τ
8091:τ
8085:π
8076:−
8035:τ
8032:−
8021:∗
8006:∞
8001:∞
7998:−
7994:∫
7978:∞
7975:→
7957:∞
7952:∞
7949:−
7945:∫
7908:τ
7900:τ
7894:π
7885:−
7874:τ
7853:∞
7848:∞
7845:−
7841:∫
7834:τ
7826:τ
7820:π
7811:−
7770:τ
7767:−
7756:∗
7741:∞
7736:∞
7733:−
7729:∫
7713:∞
7710:→
7692:∞
7687:∞
7684:−
7680:∫
7622:∞
7619:→
7556:∗
7418:^
7397:∗
7385:^
7358:∞
7355:→
7290:^
7269:∗
7257:^
7230:∞
7227:→
7085:∗
7039:∗
6985:∞
6980:∞
6977:−
6973:∫
6957:∞
6954:→
6874:∗
6816:∞
6811:∞
6808:−
6804:∫
6788:∞
6785:→
6515:∫
6355:∞
6350:∞
6347:−
6343:∫
6327:
6229:−
6085:Δ
6075:π
6066:−
6038:−
6028:∑
5999:Δ
5988:∞
5985:→
5929:Δ
5878:Δ
5822:≤
5816:≤
5810:−
5592:∫
5522:−
5349:^
5336:τ
5327:τ
5321:π
5312:−
5301:τ
5280:∞
5275:∞
5272:−
5268:∫
5181:τ
5173:τ
5167:π
5158:−
5147:τ
5126:∞
5121:∞
5118:−
5114:∫
5107:τ
5096:τ
5090:π
5081:−
5040:τ
5037:−
5026:∗
5011:∞
5006:∞
5003:−
4999:∫
4983:∞
4980:→
4962:∞
4957:∞
4954:−
4950:∫
4914:^
4886:∞
4883:→
4819:τ
4750:∞
4747:→
4690:τ
4679:τ
4673:π
4664:−
4623:τ
4620:−
4609:∗
4594:∞
4589:∞
4586:−
4582:∫
4571:∞
4566:∞
4563:−
4559:∫
4545:τ
4537:τ
4531:π
4522:−
4478:−
4475:τ
4464:∞
4459:∞
4456:−
4452:∫
4441:∞
4436:∞
4433:−
4429:∫
4389:∗
4314:⋅
4236:⋅
4222:−
4214:∗
4161:^
4151:⋅
4146:∗
4120:^
4071:^
4028:−
4020:∗
3969:∗
3942:^
3914:∗
3851:∗
3828:π
3819:−
3790:∞
3785:∞
3782:−
3778:∫
3750:∗
3734:π
3725:−
3703:∗
3688:∞
3683:∞
3680:−
3676:∫
3648:π
3623:∗
3608:∞
3603:∞
3600:−
3596:∫
3568:π
3559:−
3545:−
3537:∗
3522:∞
3517:∞
3514:−
3510:∫
3488:−
3480:∗
3392:−
3384:∗
3316:^
3232:^
3205:∞
3202:→
3104:^
3087:∞
3082:∞
3079:−
3075:∫
3059:∞
3056:→
3006:^
2896:∞
2891:∞
2888:−
2884:∫
2868:∞
2865:→
2647:−
2633:∫
2617:∞
2614:→
2311:→
2305:Δ
2282:Δ
2210:^
2149:^
2130:⏟
2111:Δ
2101:π
2092:−
2064:−
2054:∑
2028:Δ
2020:∞
2017:→
1982:¯
1949:Δ
1839:Δ
1722:Δ
1633:Δ
1568:¯
1443:Ohm's law
1416:impedance
1404:potential
1296:¯
1229:^
1214:≜
1190:¯
1039:^
994:frequency
918:π
909:−
899:∞
894:∞
891:−
887:∫
883:≜
868:^
809:^
792:∞
787:∞
784:−
780:∫
735:∞
730:∞
727:−
723:∫
653:∞
648:∞
645:−
641:∫
637:≜
539:two-sided
535:one-sided
488:watt-hour
476:impedance
246:Summation
88:June 2024
10254:See also
10166:suffixes
9983:28541914
9940:25113231
9565:P Stoica
9426:′
9383:′
9336:′
9200:See also
9142:FM radio
9105:radio/TV
9087:Spectrum
8969:wavelets
8947:spectral
8941:using a
8828:estimate
6564:′
6549:′
6308:variance
5452:, where
2446:resistor
2373:variance
1329:and the
506:, where
453:SI units
222:spectrum
10213:(radar)
10163:Decibel
9931:4204008
9541:measure
9442:is the
8978:(e.g.,
8933:is the
8231:is the
4865:ergodic
3358:as the
2410:voltage
2240:is the
1439:matched
854:where:
512:g-force
468:voltage
433:physics
407:physics
74:Please
10194:dBm/Hz
10109:
10090:
10071:
10052:
10035:
10025:
10006:
9989:
9981:
9938:
9928:
9893:
9874:
9855:
9733:
9402:where
9162:radars
8434:. If
8192:where
6302:Using
5401:define
5101:
4684:
2195:where
941:
687:
575:energy
567:energy
484:energy
346:timbre
158:signal
10304:Waves
10242:dB-Hz
9987:S2CID
9830:(PDF)
9572:(PDF)
9262:Notes
9101:pitch
9097:color
9073:shape
8998:, or
8988:phase
8984:chirp
8924:sleep
8361:with
8264:with
6747:) or
2485:watts
2368:power
1408:volts
461:hertz
457:watts
449:power
439:, an
421:Units
350:color
342:pitch
218:noise
190:power
153:of a
10237:dBrn
10232:dBFS
10217:dBsm
10168:(dB)
10107:ISBN
10088:ISBN
10069:ISBN
10050:ISBN
10033:OCLC
10023:ISBN
10004:ISBN
9979:PMID
9936:PMID
9891:ISBN
9872:ISBN
9853:ISBN
9731:ISBN
9455:The
8967:and
8890:The
8881:The
8670:and
8589:Full
8585:CPSD
8574:CPSD
8570:CPSD
8530:and
8463:and
8293:and
7606:CPSD
7496:and
6745:CPSD
6700:and
6632:and
5476:<
5463:<
5389:Eq.3
3410:and
3282:Eq.2
2807:and
1406:(in
1271:Eq.1
1015:(in
626:is:
459:per
409:and
344:and
10227:dBi
10222:dBc
10211:dBZ
10206:dBA
10199:PSD
10189:dBV
10184:dBW
10178:dBm
9969:hdl
9961:doi
9926:PMC
9918:doi
9121:not
9103:),
9069:ASD
8916:EEG
8896:SEF
8838:or
8235:of
7968:lim
7703:lim
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7220:lim
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6778:lim
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6452:or
6324:Var
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3366:of
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2607:lim
2443:ohm
2244:of
2010:lim
1414:of
1370:).
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1140:of
1133:.
992:at
963:of
490:).
455:of
431:In
241:all
110:In
78:to
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9912:.
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9625:.
9554:^
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9128:.
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8982:,
8929:A
8850:.
6294:xx
6285:xx
6188:.
5944:.
2958:is
2592::
2487:.
2330:)
2314:0.
1967::
1017:Hz
514:.
417:.
288:.
224:.
10155:e
10148:t
10141:v
10115:.
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8877:.
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5968:f
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5904:T
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3331:f
3328:(
3323:T
3313:x
3305:|
3284:)
3280:(
3260:2
3255:|
3250:)
3247:f
3244:(
3239:T
3229:x
3221:|
3215:T
3212:1
3199:T
3191:=
3188:)
3185:f
3182:(
3177:x
3174:x
3170:S
3141:f
3138:d
3132:2
3127:|
3122:)
3119:f
3116:(
3111:T
3101:x
3093:|
3069:T
3066:1
3053:T
3045:=
3042:P
3018:)
3015:f
3012:(
3003:x
2980:)
2977:t
2974:(
2971:x
2944:.
2941:t
2938:d
2932:2
2927:|
2923:)
2920:t
2917:(
2912:T
2908:x
2903:|
2878:T
2875:1
2862:T
2854:=
2851:P
2831:)
2828:t
2825:(
2820:T
2816:w
2795:)
2792:t
2789:(
2784:T
2780:w
2776:)
2773:t
2770:(
2767:x
2764:=
2761:)
2758:t
2755:(
2750:T
2746:x
2722:t
2719:d
2713:2
2708:|
2704:)
2701:t
2698:(
2695:x
2691:|
2684:2
2680:/
2676:T
2673:+
2668:0
2664:t
2658:2
2654:/
2650:T
2642:0
2638:t
2627:T
2624:1
2611:T
2603:=
2600:P
2578:0
2574:t
2570:=
2567:t
2547:T
2527:)
2524:t
2521:(
2518:x
2498:P
2472:)
2469:t
2466:(
2461:2
2457:x
2429:)
2426:t
2423:(
2420:x
2392:)
2389:t
2386:(
2383:x
2342:.
2308:t
2285:t
2262:.
2257:n
2253:x
2228:)
2225:f
2222:(
2217:d
2207:x
2183:,
2176:2
2171:|
2167:)
2164:f
2161:(
2156:d
2146:x
2138:|
2125:2
2120:|
2114:t
2107:n
2104:f
2098:2
2095:i
2088:e
2082:n
2078:x
2072:N
2067:N
2061:=
2058:n
2049:|
2039:2
2035:)
2031:t
2025:(
2014:N
2006:=
2003:)
2000:f
1997:(
1992:x
1989:x
1979:S
1955:)
1952:t
1945:n
1942:(
1939:+
1934:0
1930:t
1926:=
1921:n
1917:t
1894:n
1890:x
1862:Z
1842:f
1835:/
1831:)
1828:f
1825:(
1822:E
1812:J
1798:)
1795:f
1792:(
1789:E
1769:Z
1765:/
1759:2
1755:)
1751:t
1748:(
1745:V
1725:f
1718:/
1714:)
1711:f
1708:(
1705:E
1685:f
1665:)
1662:f
1659:(
1656:E
1636:f
1609:f
1589:)
1586:f
1583:(
1578:x
1575:x
1565:S
1541:Z
1537:/
1531:2
1527:)
1523:t
1520:(
1517:V
1497:Z
1493:/
1487:2
1483:)
1479:t
1476:(
1473:V
1453:t
1425:Z
1390:)
1387:t
1384:(
1381:V
1350:)
1347:t
1344:(
1341:x
1317:)
1314:f
1311:(
1306:x
1303:x
1293:S
1273:)
1269:(
1250:2
1245:|
1241:)
1238:f
1235:(
1226:x
1219:|
1211:)
1208:f
1205:(
1200:x
1197:x
1187:S
1157:)
1154:t
1151:(
1148:x
1121:f
1118:d
1115:+
1112:f
1092:f
1068:f
1065:d
1060:2
1055:|
1051:)
1048:f
1045:(
1036:x
1029:|
1003:f
980:)
977:t
974:(
971:x
947:t
944:d
938:)
935:t
932:(
929:x
924:t
921:f
915:2
912:i
905:e
880:)
877:f
874:(
865:x
842:,
839:f
836:d
830:2
825:|
821:)
818:f
815:(
806:x
799:|
776:=
773:t
770:d
764:2
759:|
754:)
751:t
748:(
745:x
741:|
696:.
693:t
690:d
682:2
677:|
673:)
670:t
667:(
664:x
660:|
634:E
614:)
611:t
608:(
605:x
585:E
562:.
523:t
521:(
519:x
508:g
500:g
369:)
366:t
363:(
360:E
328:x
308:)
305:t
302:(
299:x
272:)
269:t
266:(
261:2
257:x
200:f
176:)
173:t
170:(
167:x
141:)
138:f
135:(
130:x
127:x
123:S
101:)
95:(
90:)
86:(
72:.
32:.
23:.
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