3684:
2735:
3679:{\displaystyle {\begin{aligned}R_{x}^{n/T_{0}}(\tau )&={\frac {1}{T_{0}}}\int _{-T_{0}/2}^{T_{0}/2}R_{x}(t,\tau )e^{-j2\pi {\frac {n}{T_{0}}}t}\,\mathrm {d} t\\&={\frac {1}{T_{0}}}\int _{-T_{0}/2}^{T_{0}/2}\sigma _{a}^{2}\sum _{k=-\infty }^{\infty }p(t+\tau -kT_{0})p^{*}(t-kT_{0})e^{-j2\pi {\frac {n}{T_{0}}}t}\mathrm {d} t\\&={\frac {\sigma _{a}^{2}}{T_{0}}}\sum _{k=-\infty }^{\infty }\int _{-T_{0}/2-kT_{0}}^{T_{0}/2-kT_{0}}p(\lambda +\tau )p^{*}(\lambda )e^{-j2\pi {\frac {n}{T_{0}}}(\lambda +kT_{0})}\mathrm {d} \lambda \\&={\frac {\sigma _{a}^{2}}{T_{0}}}\int _{-\infty }^{\infty }p(\lambda +\tau )p^{*}(\lambda )e^{-j2\pi {\frac {n}{T_{0}}}\lambda }\mathrm {d} \lambda \\&={\frac {\sigma _{a}^{2}}{T_{0}}}p(\tau )*\left\{p^{*}(-\tau )e^{j2\pi {\frac {n}{T_{0}}}\tau }\right\}.\end{aligned}}}
45:
the more empirical
Fraction Of Time (FOT) probability for the alternative model. The FOT probability of some event associated with the time series is defined to be the fraction of time that event occurs over the lifetime of the time series. In both approaches, the process or time series is said to be cyclostationary if and only if its associated probability distributions vary periodically with time. However, in the non-stochastic time-series approach, there is an alternative but equivalent definition: A time series that contains no finite-strength additive sine-wave components is said to exhibit cyclostationarity if and only if there exists some nonlinear time-invariant transformation of the time series that produces finite-strength (non-zero) additive sine-wave components.
28:. For example, the maximum daily temperature in New York City can be modeled as a cyclostationary process: the maximum temperature on July 21 is statistically different from the temperature on December 20; however, it is a reasonable approximation that the temperature on December 20 of different years has identical statistics. Thus, we can view the random process composed of daily maximum temperatures as 365 interleaved stationary processes, each of which takes on a new value once per year.
4545:
4672:. On the other hand, if the speed of rotation changes with time, then the signal is no longer cyclostationary (unless the speed varies periodically). Therefore, it is not a model for cyclostationary signals. It is not even a model for time-warped cyclostationarity, although it can be a useful approximation for sufficiently slow changes in speed of rotation.
2656:
1392:
4322:
4009:
periodic modulations). This happens to be the case for noise and vibration produced by gear mechanisms, bearings, internal combustion engines, turbofans, pumps, propellers, etc. The explicit modelling of mechanical signals as cyclostationary processes has been found useful in several applications, such as in
3950:
In practice, signals exhibiting cyclicity with more than one incommensurate period arise and require a generalization of the theory of cyclostationarity. Such signals are called polycyclostationary if they exhibit a finite number of incommensurate periods and almost cyclostationary if they exhibit a
44:
of data--that which has actually been measured in practice and, for some parts of theory, conceptually extended from an observed finite time interval to an infinite interval. Both mathematical models lead to probabilistic theories: abstract stochastic probability for the stochastic process model and
3959:
The wide sense theory of time series exhibiting cyclostationarity, polycyclostationarity and almost cyclostationarity originated and developed by
Gardner was also generalized by Gardner to a theory of higher-order temporal and spectral moments and cumulants and a strict sense theory of cumulative
3951:
countably infinite number. Such signals arise frequently in radio communications due to multiple transmissions with differing sine-wave carrier frequencies and digital symbol rates. The theory was introduced in for stochastic processes and further developed in for non-stochastic time series.
4008:
Mechanical signals produced by rotating or reciprocating machines are remarkably well modelled as cyclostationary processes. The cyclostationary family accepts all signals with hidden periodicities, either of the additive type (presence of tonal components) or multiplicative type (presence of
1904:
For time series, the reason the cyclic spectral density function is called the spectral correlation density function is that it equals the limit, as filter bandwidth approaches zero, of the average over all time of the product of the output of a one-sided bandpass filter with center frequency
2316:
1045:
3920:
This same result can be obtained for the non-stochastic time series model of linearly modulated digital signals in which expectation is replaced with infinite time average, but this requires a somewhat modified mathematical method as originally observed and proved in.
1773:
4965:
2018 W. A. Gardner. STATISTICALLY INFERRED TIME WARPING: EXTENDING THE CYCLOSTATIONARITY PARADIGM FROM REGULAR TO IRREGULAR STATISTICAL CYCLICITY IN SCIENTIFIC DATA. EURASIP Journal on
Advances in Signal Processing volume 2018, Article number: 59. doi:
3864:
776:
4901:
W. A. Gardner. STATISTICALLY INFERRED TIME WARPING: EXTENDING THE CYCLOSTATIONARITY PARADIGM FROM REGULAR TO IRREGULAR STATISTICAL CYCLICITY IN SCIENTIFIC DATA. EURASIP Journal on
Advances in Signal Processing volume 2018, Article number: 59. doi:
1179:
4540:{\displaystyle S_{x}^{\alpha }(f)=\lim _{S\rightarrow +\infty }{\frac {1}{S}}\int _{-S/2}^{S/2}\int _{-\infty }^{+\infty }R_{x}(\theta ,\tau )e^{-j2\pi f\tau }e^{-j2\pi \alpha {\frac {\theta }{\Theta }}}\,\mathrm {d} \tau \,\mathrm {d} \theta }
4152:
4028:
of rotation of a specific component – the “cycle” of the machine. At the same time, a temporal description must be preserved to reflect the nature of dynamical phenomena that are governed by differential equations of time. Therefore, the
1832:
is called the spectral correlation density function is that it equals the limit, as filter bandwidth approaches zero, of the expected value of the product of the output of a one-sided bandpass filter with center frequency
2740:
614:
2651:{\displaystyle {\begin{aligned}R_{x}(t,\tau )&=\operatorname {E} \\&=\sum _{k,n}\operatorname {E} p(t+\tau -kT_{0})p^{*}(t-nT_{0})\\&=\sigma _{a}^{2}\sum _{k}p(t+\tau -kT_{0})p^{*}(t-kT_{0}).\end{aligned}}}
1627:
244:
507:
851:
3937:
in which the autoregression coefficients and residual variance are no longer constant but vary cyclically with time. His work follows a number of other studies of cyclostationary processes within the field of
2321:
2082:
1520:
3960:
probability distributions. The encyclopedic book comprehensively teaches all of this and provides a scholarly treatment of the originating publications by
Gardner and contributions thereafter by others.
2306:
4749:
1983 R. A. Boyles and W. A. Gardner. CYCLOERGODIC PROPERTIES OF DISCRETE-PARAMETER NONSTATIONARY STOCHASTIC PROCESSES. IEEE Transactions on
Information Theory, Vol. IT-29, No. 1, pp. 105-114.
1653:
1103:
2125:
1157:
837:
3723:
2122:
637:
4288:
2232:
1830:
321:
141:
4650:
1387:{\displaystyle {\widehat {R}}_{x}^{n/T_{0}}(\tau )=\lim _{T\rightarrow +\infty }{\frac {1}{T}}\int _{-T/2}^{T/2}x(t+\tau )x^{*}(t)e^{-j2\pi {\frac {n}{T_{0}}}t}\mathrm {d} t.}
363:
1971:
1937:
1899:
1865:
4206:
4041:
3914:
4929:
W. A. Gardner. SIGNAL INTERCEPTION: A UNIFYING THEORETICAL FRAMEWORK FOR FEATURE DETECTION. IEEE Transactions on
Communications, Vol. COM-36, No. 8, pp. 897-906. 1988
4670:
4621:
4570:
4308:
4226:
4177:
413:
4246:
2725:
280:
2698:
2184:
2155:
100:
4598:
3969:
Cyclostationarity has extremely diverse applications in essentially all fields of engineering and science, as thoroughly documented in and. A few examples are:
3709:
383:
36:
There are two differing approaches to the treatment of cyclostationary processes. The stochastic approach is to view measurements as an instance of an abstract
1167:
A signal that is just a function of time and not a sample path of a stochastic process can exhibit cyclostationarity properties in the framework of the
4652:. Consequently, the angle-time autocorrelation is simply a cyclicity-scaled traditional autocorrelation; that is, the cycle frequencies are scaled by
519:
1525:
151:
5000:
423:
65:
processes. The exact definition differs depending on whether the signal is treated as a stochastic process or as a deterministic time series.
1040:{\displaystyle R_{x}^{n/T_{0}}(\tau )={\frac {1}{T_{0}}}\int _{-T_{0}/2}^{T_{0}/2}R_{x}(t,\tau )e^{-j2\pi {\frac {n}{T_{0}}}t}\mathrm {d} t.}
4310:, are called (wide-sense) angle-time cyclostationary. The double Fourier transform of the angle-time autocorrelation function defines the
3930:
1993:
1522:. If the signal is further cycloergodic, all sample paths exhibit the same cyclic time-averages with probability equal to 1 and thus
1402:
53:
An important special case of cyclostationary signals is one that exhibits cyclostationarity in second-order statistics (e.g., the
4759:
Kipnis, Alon; Goldsmith, Andrea; Eldar, Yonina (May 2018). "The
Distortion Rate Function of Cyclostationary Gaussian Processes".
4803:
W. A. Gardner. INTRODUCTION TO RANDOM PROCESSES WITH APPLICATIONS TO SIGNALS AND SYSTEMS. Macmillan, New York, 434 pages, 1985
4010:
1973:, with both filter outputs frequency shifted to a common center frequency, such as zero, as originally observed and proved in.
1901:, with both filter outputs frequency shifted to a common center frequency, such as zero, as originally observed and proved in.
4883:
W. A. Gardner. STATISTICAL SPECTRAL ANALYSIS: A NONPROBABILISTIC THEORY. Prentice-Hall, Englewood Cliffs, NJ, 565 pages, 1987.
4874:
W. A. Gardner. STATISTICAL SPECTRAL ANALYSIS: A NONPROBABILISTIC THEORY. Prentice-Hall, Englewood Cliffs, NJ, 565 pages, 1987.
4821:
W. A. Gardner. STATISTICAL SPECTRAL ANALYSIS: A NONPROBABILISTIC THEORY. Prentice-Hall, Englewood Cliffs, NJ, 565 pages, 1987.
4812:
W. A. Gardner. STATISTICAL SPECTRAL ANALYSIS: A NONPROBABILISTIC THEORY. Prentice-Hall, Englewood Cliffs, NJ, 565 pages, 1987.
24:
having statistical properties that vary cyclically with time. A cyclostationary process can be viewed as multiple interleaved
4865:
W. A. Gardner. STATIONARIZABLE RANDOM PROCESSES. IEEE Transactions on
Information Theory, Vol. IT-24, No. 1, pp. 8-22. 1978
4911:
A. Napolitano, Cyclostationary
Processes and Time Series: Theory, Applications, and Generalizations. Academic Press, 2020.
4892:
A. Napolitano, Cyclostationary Processes and Time Series: Theory, Applications, and Generalizations. Academic Press, 2020.
4248:
for time delay. Processes whose angle-time autocorrelation function exhibit a component periodic in angle, i.e. such that
2237:
1768:{\displaystyle S_{x}^{\alpha }(f)=\int _{-\infty }^{+\infty }R_{x}^{\alpha }(\tau )e^{-j2\pi f\tau }\mathrm {d} \tau .}
21:
4920:
W. A. Gardner. CYCLOSTATIONARITY IN COMMUNICATIONS AND SIGNAL PROCESSING. Piscataway, NJ: IEEE Press. 504 pages.1984.
1055:
1785:
4691:
Gardner, William A.; Antonio Napolitano; Luigi Paura (2006). "Cyclostationarity: Half a century of research".
3859:{\displaystyle S_{x}^{n/T_{0}}(f)={\frac {\sigma _{a}^{2}}{T_{0}}}P(f)P^{*}\left(f-{\frac {n}{T_{0}}}\right).}
771:{\displaystyle R_{x}(t,\tau )=\sum _{n=-\infty }^{\infty }R_{x}^{n/T_{0}}(\tau )e^{j2\pi {\frac {n}{T_{0}}}t}}
4723:
Gardner, William A. (1991). "Two alternative philosophies for estimation of the parameters of time-series".
1115:
786:
2092:
4251:
2192:
4024:
One peculiarity of rotating machine signals is that the period of the process is strictly linked to the
3999:
Cyclostationarity is used to analyze mechanical signals produced by rotating and reciprocating machines.
1794:
285:
105:
62:
4626:
4014:
3939:
3934:
3872:
326:
4147:{\displaystyle R_{x}(\theta ,\tau )=\operatorname {E} \{x(t(\theta )+\tau )x^{*}(t(\theta ))\},\,}
4786:
4768:
3973:
2664:
1942:
1908:
1870:
1836:
37:
25:
4182:
3878:
252:
40:
model. As an alternative, the more empirical approach is to view the measurements as a single
4856:
Pagano, M. (1978) "On periodic and multiple autoregressions." Ann. Stat., 6, 1310–1317.
4655:
4606:
4555:
4293:
4211:
4162:
1637:
The Fourier transform of the cyclic autocorrelation function at cyclic frequency α is called
4978:
Noise in mixers, oscillators, samplers, and logic: an introduction to cyclostationary noise
4948:
4778:
4732:
4700:
1781:
388:
4231:
2703:
1939:
and the conjugate of the output of another one-sided bandpass filter with center frequency
1867:
and the conjugate of the output of another one-sided bandpass filter with center frequency
258:
3993:
3977:
2674:
2160:
2131:
1112:
Wide-sense stationary processes are a special case of cyclostationary processes with only
76:
54:
4982:
4979:
4583:
3694:
626:
368:
4985:
4994:
4790:
3987:
3990:, cyclostationarity is used to analyze the periodic behavior of financial-markets;
3980:, transmitter and receiver optimization, and spectrum sensing for cognitive radio;
4704:
4952:
3712:
1171:
point of view. This way, the cyclic autocorrelation function can be defined by:
41:
4844:
4831:
1982:
4782:
3996:
utilizes cyclostationary theory to analyze computer networks and car traffic;
609:{\displaystyle R_{x}(t,\tau )=R_{x}(t+T_{0};\tau ){\text{ for all }}t,\tau .}
4843:
Jones, R.H., Brelsford, W.M. (1967) "Time series with periodic structure."
3983:
In signals intelligence, cyclostationarity is used for signal interception;
4021:, a popular analysis technique used in the diagnostics of bearing faults.
4017:. In the latter field, cyclostationarity has been found to generalize the
1622:{\displaystyle R_{x}^{n/T_{0}}(\tau )={\widehat {R}}_{x}^{n/T_{0}}(\tau )}
239:{\displaystyle R_{x}(t,\tau )=\operatorname {E} \{x(t+\tau )x^{*}(t)\},\,}
3933:
to incorporate cyclostationary behaviour. For example, Troutman treated
502:{\displaystyle \operatorname {E} =\operatorname {E} {\text{ for all }}t}
4736:
1780:
The cyclic spectrum at zero cyclic frequency is also called average
4773:
1399:
If the time-series is a sample path of a stochastic process it is
4830:
Troutman, B.M. (1979) "Some results in periodic autoregression."
2077:{\displaystyle x(t)=\sum _{k=-\infty }^{\infty }a_{k}p(t-kT_{0})}
4290:
has a non-zero Fourier-Bohr coefficient for some angular period
1515:{\displaystyle R_{x}^{n/T_{0}}(\tau )=\operatorname {E} \left}
4939:
Antoni, Jérôme (2009). "Cyclostationarity by examples".
255:, is said to be wide-sense cyclostationary with period
4658:
4629:
4609:
4586:
4558:
4325:
4296:
4254:
4234:
4214:
4185:
4165:
4044:
3881:
3726:
3697:
2738:
2706:
2677:
2319:
2240:
2195:
2163:
2134:
2095:
1996:
1945:
1911:
1873:
1839:
1797:
1656:
1528:
1405:
1182:
1118:
1058:
854:
789:
640:
522:
426:
391:
371:
329:
288:
261:
154:
108:
79:
1788:can be expressed in terms of its cyclic spectrum.
4664:
4644:
4615:
4592:
4564:
4539:
4302:
4282:
4240:
4220:
4200:
4171:
4146:
4004:Angle-time cyclostationarity of mechanical signals
3908:
3858:
3703:
3678:
2719:
2692:
2650:
2301:{\displaystyle \operatorname {E} =\sigma _{a}^{2}}
2300:
2226:
2178:
2149:
2116:
2076:
1965:
1931:
1893:
1859:
1824:
1767:
1621:
1514:
1386:
1151:
1097:
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831:
770:
608:
501:
407:
377:
357:
315:
274:
238:
135:
94:
3875:adopted in digital communications have thus only
621:The autocorrelation function is thus periodic in
4354:
1235:
3955:Higher Order and Strict Sense Cyclostationarity
1784:. For a Gaussian cyclostationary process, its
2186:, is the supporting pulse of the modulation.
8:
4208:for the time instant corresponding to angle
4137:
4079:
1981:An example of cyclostationary signal is the
1098:{\displaystyle n/T_{0},\,n\in \mathbb {Z} ,}
229:
189:
4718:
4716:
4714:
3929:It is possible to generalise the class of
1977:Example: linearly modulated digital signal
4772:
4657:
4628:
4608:
4585:
4557:
4529:
4528:
4520:
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4507:
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4428:
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4406:
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4213:
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4043:
3880:
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3790:
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3749:
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3642:
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3079:
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3011:
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2016:
1995:
1955:
1944:
1921:
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1807:
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1709:
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1428:
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1238:
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1185:
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1117:
1088:
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1080:
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1004:
991:
966:
952:
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926:
918:
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868:
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812:
803:
799:
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788:
755:
746:
736:
715:
706:
702:
697:
687:
673:
645:
639:
589:
574:
555:
527:
521:
491:
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370:
334:
328:
287:
266:
260:
235:
214:
159:
153:
107:
78:
4941:Mechanical Systems and Signal Processing
2700:is a cyclostationary signal with period
4761:IEEE Transactions on Information Theory
4683:
1152:{\displaystyle R_{x}^{0}(\tau )\neq 0}
832:{\displaystyle R_{x}^{n/T_{0}}(\tau )}
2727:and cyclic autocorrelation function:
2117:{\displaystyle a_{k}\in \mathbb {C} }
1643:spectral correlation density function
7:
4312:order-frequency spectral correlation
4283:{\displaystyle R_{x}(\theta ;\tau )}
3931:autoregressive moving average models
2308:, the auto-correlation function is:
2227:{\displaystyle \operatorname {E} =0}
4031:angle-time autocorrelation function
4530:
4521:
4512:
4432:
4424:
4367:
4297:
4073:
3540:
3458:
3453:
3399:
3215:
3210:
3150:
3036:
3031:
2920:
2431:
2356:
2241:
2196:
2031:
2026:
1825:{\displaystyle S_{x}^{\alpha }(f)}
1755:
1700:
1692:
1448:
1374:
1248:
1027:
688:
683:
454:
427:
316:{\displaystyle \operatorname {E} }
289:
183:
136:{\displaystyle \operatorname {E} }
109:
69:Cyclostationary stochastic process
14:
4623:, angle is proportional to time,
1983:linearly modulated digital signal
4645:{\displaystyle \theta =\omega t}
4603:For constant speed of rotation,
4011:noise, vibration, and harshness
2128:random variables. The waveform
841:cyclic autocorrelation function
4459:
4447:
4361:
4347:
4341:
4277:
4265:
4195:
4189:
4134:
4131:
4125:
4119:
4106:
4097:
4091:
4085:
4067:
4055:
3807:
3801:
3763:
3757:
3625:
3616:
3595:
3589:
3497:
3491:
3478:
3466:
3393:
3371:
3334:
3328:
3315:
3303:
3107:
3085:
3072:
3044:
2876:
2864:
2779:
2773:
2687:
2681:
2638:
2616:
2603:
2575:
2534:
2512:
2499:
2471:
2465:
2437:
2402:
2399:
2393:
2380:
2368:
2362:
2346:
2334:
2277:
2267:
2251:
2247:
2215:
2202:
2173:
2167:
2144:
2138:
2071:
2049:
2006:
2000:
1819:
1813:
1726:
1720:
1678:
1672:
1616:
1610:
1565:
1559:
1504:
1498:
1442:
1436:
1331:
1325:
1312:
1300:
1242:
1228:
1222:
1140:
1134:
984:
972:
891:
885:
826:
820:
729:
723:
663:
651:
586:
561:
545:
533:
488:
485:
466:
460:
448:
445:
439:
433:
358:{\displaystyle R_{x}(t,\tau )}
352:
340:
310:
307:
301:
295:
226:
220:
207:
195:
177:
165:
143:and autocorrelation function:
130:
127:
121:
115:
89:
83:
61:signals, and are analogous to
1:
5001:Statistical signal processing
3972:Cyclostationarity is used in
3916:non-zero cyclic frequencies.
2667:, hence a signal periodic in
4705:10.1016/j.sigpro.2005.06.016
57:function). These are called
49:Wide-sense cyclostationarity
4953:10.1016/j.ymssp.2008.10.010
1966:{\displaystyle f-\alpha /2}
1932:{\displaystyle f+\alpha /2}
1894:{\displaystyle f-\alpha /2}
1860:{\displaystyle f+\alpha /2}
1163:Cyclostationary time series
5017:
4600:a frequency (unit in Hz).
4201:{\displaystyle t(\theta )}
3715:. The cyclic spectrum is:
59:wide-sense cyclostationary
4966:10.1186/s13634-018-0564-6
4947:(4). Elsevier: 987–1036.
4902:10.1186/s13634-018-0564-6
2157:, with Fourier transform
1633:Frequency domain behavior
4783:10.1109/TIT.2017.2741978
4699:(4). Elsevier: 639–697.
3909:{\displaystyle n=-1,0,1}
2663:The last summation is a
1786:rate distortion function
4834:, 66 (2), 219–228
4725:IEEE Trans. Inf. Theory
4665:{\displaystyle \omega }
4616:{\displaystyle \omega }
4565:{\displaystyle \alpha }
4303:{\displaystyle \Theta }
4221:{\displaystyle \theta }
4172:{\displaystyle \theta }
625:and can be expanded in
251:where the star denotes
18:cyclostationary process
4983:annotated presentation
4666:
4646:
4617:
4594:
4566:
4541:
4304:
4284:
4242:
4222:
4202:
4173:
4148:
3925:Cyclostationary models
3910:
3860:
3705:
3680:
3219:
3040:
2721:
2694:
2652:
2302:
2228:
2180:
2151:
2118:
2078:
2035:
1967:
1933:
1895:
1861:
1826:
1782:power spectral density
1769:
1623:
1516:
1388:
1153:
1099:
1041:
833:
772:
692:
610:
503:
409:
408:{\displaystyle T_{0},}
379:
359:
317:
276:
240:
137:
96:
4667:
4647:
4618:
4595:
4578:events per revolution
4567:
4542:
4305:
4285:
4243:
4241:{\displaystyle \tau }
4223:
4203:
4174:
4149:
3946:Polycyclostationarity
3911:
3861:
3706:
3681:
3196:
3017:
2722:
2720:{\displaystyle T_{0}}
2695:
2653:
2303:
2229:
2181:
2152:
2119:
2079:
2012:
1968:
1934:
1896:
1862:
1827:
1770:
1624:
1517:
1389:
1154:
1100:
1042:
834:
773:
669:
611:
504:
410:
380:
360:
318:
277:
275:{\displaystyle T_{0}}
241:
138:
97:
73:A stochastic process
63:wide-sense stationary
4656:
4627:
4607:
4584:
4556:
4323:
4294:
4252:
4232:
4212:
4183:
4163:
4042:
4015:condition monitoring
3940:time series analysis
3879:
3873:raised-cosine pulses
3724:
3695:
2736:
2704:
2693:{\displaystyle x(t)}
2675:
2317:
2238:
2193:
2179:{\displaystyle P(f)}
2161:
2150:{\displaystyle p(t)}
2132:
2093:
1994:
1943:
1909:
1871:
1837:
1795:
1654:
1629:with probability 1.
1526:
1403:
1180:
1116:
1056:
852:
787:
638:
520:
424:
389:
369:
327:
286:
259:
152:
106:
95:{\displaystyle x(t)}
77:
26:stationary processes
4847:, 54, 403–410
4436:
4415:
4340:
3785:
3756:
3573:
3462:
3432:
3299:
3183:
3016:
3001:
2853:
2772:
2561:
2464:
2297:
1812:
1719:
1704:
1671:
1609:
1558:
1497:
1435:
1296:
1221:
1133:
961:
884:
819:
722:
591: for all
493: for all
253:complex conjugation
4662:
4642:
4613:
4590:
4562:
4537:
4416:
4382:
4371:
4326:
4300:
4280:
4238:
4218:
4198:
4179:stands for angle,
4169:
4144:
3974:telecommunications
3906:
3856:
3771:
3727:
3701:
3676:
3674:
3559:
3445:
3418:
3220:
3169:
3002:
2954:
2806:
2743:
2717:
2690:
2665:periodic summation
2648:
2646:
2571:
2547:
2450:
2430:
2298:
2283:
2224:
2176:
2147:
2114:
2074:
1963:
1929:
1891:
1857:
1822:
1798:
1765:
1705:
1684:
1657:
1619:
1571:
1529:
1512:
1459:
1406:
1384:
1263:
1252:
1183:
1149:
1119:
1095:
1037:
914:
855:
829:
790:
768:
693:
606:
499:
405:
375:
355:
313:
272:
236:
133:
92:
38:stochastic process
4693:Signal Processing
4593:{\displaystyle f}
4515:
4380:
4353:
4019:envelope spectrum
3846:
3796:
3704:{\displaystyle *}
3657:
3584:
3532:
3443:
3369:
3194:
3142:
2952:
2911:
2804:
2562:
2415:
1645:and is equal to:
1581:
1469:
1366:
1261:
1234:
1193:
1107:cycle frequencies
1019:
912:
761:
592:
494:
378:{\displaystyle t}
5008:
4967:
4963:
4957:
4956:
4936:
4930:
4927:
4921:
4918:
4912:
4909:
4903:
4899:
4893:
4890:
4884:
4881:
4875:
4872:
4866:
4863:
4857:
4854:
4848:
4841:
4835:
4828:
4822:
4819:
4813:
4810:
4804:
4801:
4795:
4794:
4776:
4767:(5): 3810–3824.
4756:
4750:
4747:
4741:
4740:
4737:10.1109/18.61145
4720:
4709:
4708:
4688:
4671:
4669:
4668:
4663:
4651:
4649:
4648:
4643:
4622:
4620:
4619:
4614:
4599:
4597:
4596:
4591:
4571:
4569:
4568:
4563:
4546:
4544:
4543:
4538:
4533:
4524:
4518:
4517:
4516:
4508:
4486:
4485:
4446:
4445:
4435:
4427:
4414:
4410:
4401:
4397:
4381:
4373:
4370:
4339:
4334:
4309:
4307:
4306:
4301:
4289:
4287:
4286:
4281:
4264:
4263:
4247:
4245:
4244:
4239:
4227:
4225:
4224:
4219:
4207:
4205:
4204:
4199:
4178:
4176:
4175:
4170:
4153:
4151:
4150:
4145:
4118:
4117:
4054:
4053:
3915:
3913:
3912:
3907:
3865:
3863:
3862:
3857:
3852:
3848:
3847:
3845:
3844:
3832:
3819:
3818:
3797:
3795:
3794:
3784:
3779:
3770:
3755:
3754:
3753:
3744:
3735:
3710:
3708:
3707:
3702:
3685:
3683:
3682:
3677:
3675:
3668:
3664:
3663:
3662:
3658:
3656:
3655:
3643:
3615:
3614:
3585:
3583:
3582:
3572:
3567:
3558:
3550:
3543:
3538:
3537:
3533:
3531:
3530:
3518:
3490:
3489:
3461:
3456:
3444:
3442:
3441:
3431:
3426:
3417:
3409:
3402:
3397:
3396:
3392:
3391:
3370:
3368:
3367:
3355:
3327:
3326:
3298:
3297:
3296:
3278:
3273:
3272:
3262:
3261:
3260:
3242:
3237:
3236:
3218:
3213:
3195:
3193:
3192:
3182:
3177:
3168:
3160:
3153:
3148:
3147:
3143:
3141:
3140:
3128:
3106:
3105:
3084:
3083:
3071:
3070:
3039:
3034:
3015:
3010:
3000:
2996:
2991:
2990:
2980:
2976:
2971:
2970:
2953:
2951:
2950:
2938:
2930:
2923:
2917:
2916:
2912:
2910:
2909:
2897:
2863:
2862:
2852:
2848:
2843:
2842:
2832:
2828:
2823:
2822:
2805:
2803:
2802:
2790:
2771:
2770:
2769:
2760:
2751:
2726:
2724:
2723:
2718:
2716:
2715:
2699:
2697:
2696:
2691:
2657:
2655:
2654:
2649:
2647:
2637:
2636:
2615:
2614:
2602:
2601:
2570:
2560:
2555:
2540:
2533:
2532:
2511:
2510:
2498:
2497:
2463:
2458:
2449:
2448:
2429:
2408:
2392:
2391:
2333:
2332:
2307:
2305:
2304:
2299:
2296:
2291:
2276:
2275:
2270:
2264:
2263:
2254:
2233:
2231:
2230:
2225:
2214:
2213:
2185:
2183:
2182:
2177:
2156:
2154:
2153:
2148:
2123:
2121:
2120:
2115:
2113:
2105:
2104:
2083:
2081:
2080:
2075:
2070:
2069:
2045:
2044:
2034:
2029:
1972:
1970:
1969:
1964:
1959:
1938:
1936:
1935:
1930:
1925:
1900:
1898:
1897:
1892:
1887:
1866:
1864:
1863:
1858:
1853:
1831:
1829:
1828:
1823:
1811:
1806:
1774:
1772:
1771:
1766:
1758:
1753:
1752:
1718:
1713:
1703:
1695:
1670:
1665:
1628:
1626:
1625:
1620:
1608:
1607:
1606:
1597:
1588:
1583:
1582:
1574:
1557:
1556:
1555:
1546:
1537:
1521:
1519:
1518:
1513:
1511:
1507:
1496:
1495:
1494:
1485:
1476:
1471:
1470:
1462:
1434:
1433:
1432:
1423:
1414:
1393:
1391:
1390:
1385:
1377:
1372:
1371:
1367:
1365:
1364:
1352:
1324:
1323:
1295:
1291:
1282:
1278:
1262:
1254:
1251:
1220:
1219:
1218:
1209:
1200:
1195:
1194:
1186:
1169:fraction-of-time
1158:
1156:
1155:
1150:
1132:
1127:
1104:
1102:
1101:
1096:
1091:
1076:
1075:
1066:
1052:The frequencies
1046:
1044:
1043:
1038:
1030:
1025:
1024:
1020:
1018:
1017:
1005:
971:
970:
960:
956:
951:
950:
940:
936:
931:
930:
913:
911:
910:
898:
883:
882:
881:
872:
863:
838:
836:
835:
830:
818:
817:
816:
807:
798:
777:
775:
774:
769:
767:
766:
762:
760:
759:
747:
721:
720:
719:
710:
701:
691:
686:
650:
649:
615:
613:
612:
607:
593:
590:
579:
578:
560:
559:
532:
531:
508:
506:
505:
500:
495:
492:
484:
483:
414:
412:
411:
406:
401:
400:
384:
382:
381:
376:
364:
362:
361:
356:
339:
338:
322:
320:
319:
314:
281:
279:
278:
273:
271:
270:
245:
243:
242:
237:
219:
218:
164:
163:
142:
140:
139:
134:
101:
99:
98:
93:
5016:
5015:
5011:
5010:
5009:
5007:
5006:
5005:
4991:
4990:
4975:
4970:
4964:
4960:
4938:
4937:
4933:
4928:
4924:
4919:
4915:
4910:
4906:
4900:
4896:
4891:
4887:
4882:
4878:
4873:
4869:
4864:
4860:
4855:
4851:
4842:
4838:
4829:
4825:
4820:
4816:
4811:
4807:
4802:
4798:
4758:
4757:
4753:
4748:
4744:
4722:
4721:
4712:
4690:
4689:
4685:
4681:
4675:
4654:
4653:
4625:
4624:
4605:
4604:
4582:
4581:
4554:
4553:
4487:
4462:
4437:
4321:
4320:
4292:
4291:
4255:
4250:
4249:
4230:
4229:
4210:
4209:
4181:
4180:
4161:
4160:
4109:
4045:
4040:
4039:
4006:
3994:Queueing theory
3978:synchronization
3966:
3957:
3948:
3935:autoregressions
3927:
3919:
3877:
3876:
3836:
3824:
3820:
3810:
3786:
3745:
3722:
3721:
3693:
3692:
3673:
3672:
3647:
3628:
3606:
3605:
3601:
3574:
3548:
3547:
3522:
3500:
3481:
3433:
3407:
3406:
3383:
3359:
3337:
3318:
3288:
3264:
3252:
3228:
3184:
3158:
3157:
3132:
3110:
3097:
3075:
3062:
2982:
2962:
2942:
2928:
2927:
2901:
2879:
2854:
2834:
2814:
2794:
2782:
2761:
2734:
2733:
2707:
2702:
2701:
2673:
2672:
2645:
2644:
2628:
2606:
2593:
2538:
2537:
2524:
2502:
2489:
2440:
2406:
2405:
2383:
2349:
2324:
2315:
2314:
2265:
2255:
2236:
2235:
2205:
2191:
2190:
2159:
2158:
2130:
2129:
2096:
2091:
2090:
2061:
2036:
1992:
1991:
1979:
1941:
1940:
1907:
1906:
1869:
1868:
1835:
1834:
1793:
1792:
1729:
1652:
1651:
1639:cyclic spectrum
1635:
1598:
1547:
1524:
1523:
1486:
1458:
1454:
1424:
1401:
1400:
1356:
1334:
1315:
1210:
1178:
1177:
1165:
1114:
1113:
1067:
1054:
1053:
1009:
987:
962:
942:
922:
902:
873:
850:
849:
808:
785:
784:
751:
732:
711:
641:
636:
635:
570:
551:
523:
518:
517:
475:
422:
421:
392:
387:
386:
367:
366:
330:
325:
324:
284:
283:
262:
257:
256:
210:
155:
150:
149:
104:
103:
75:
74:
71:
55:autocorrelation
51:
34:
12:
11:
5:
5014:
5012:
5004:
5003:
4993:
4992:
4989:
4988:
4974:
4973:External links
4971:
4969:
4968:
4958:
4931:
4922:
4913:
4904:
4894:
4885:
4876:
4867:
4858:
4849:
4836:
4823:
4814:
4805:
4796:
4751:
4742:
4731:(1): 216–218.
4710:
4682:
4680:
4677:
4661:
4641:
4638:
4635:
4632:
4612:
4589:
4561:
4550:
4549:
4548:
4547:
4536:
4532:
4527:
4523:
4514:
4511:
4506:
4503:
4500:
4497:
4494:
4490:
4484:
4481:
4478:
4475:
4472:
4469:
4465:
4461:
4458:
4455:
4452:
4449:
4444:
4440:
4434:
4431:
4426:
4423:
4419:
4413:
4409:
4405:
4400:
4396:
4392:
4389:
4385:
4379:
4376:
4369:
4366:
4363:
4360:
4356:
4352:
4349:
4346:
4343:
4338:
4333:
4329:
4299:
4279:
4276:
4273:
4270:
4267:
4262:
4258:
4237:
4217:
4197:
4194:
4191:
4188:
4168:
4157:
4156:
4155:
4154:
4142:
4139:
4136:
4133:
4130:
4127:
4124:
4121:
4116:
4112:
4108:
4105:
4102:
4099:
4096:
4093:
4090:
4087:
4084:
4081:
4078:
4075:
4072:
4069:
4066:
4063:
4060:
4057:
4052:
4048:
4005:
4002:
4001:
4000:
3997:
3991:
3984:
3981:
3970:
3965:
3962:
3956:
3953:
3947:
3944:
3926:
3923:
3905:
3902:
3899:
3896:
3893:
3890:
3887:
3884:
3869:
3868:
3867:
3866:
3855:
3851:
3843:
3839:
3835:
3830:
3827:
3823:
3817:
3813:
3809:
3806:
3803:
3800:
3793:
3789:
3783:
3778:
3774:
3768:
3765:
3762:
3759:
3752:
3748:
3743:
3739:
3734:
3730:
3700:
3689:
3688:
3687:
3686:
3671:
3667:
3661:
3654:
3650:
3646:
3641:
3638:
3635:
3631:
3627:
3624:
3621:
3618:
3613:
3609:
3604:
3600:
3597:
3594:
3591:
3588:
3581:
3577:
3571:
3566:
3562:
3556:
3553:
3551:
3549:
3546:
3542:
3536:
3529:
3525:
3521:
3516:
3513:
3510:
3507:
3503:
3499:
3496:
3493:
3488:
3484:
3480:
3477:
3474:
3471:
3468:
3465:
3460:
3455:
3452:
3448:
3440:
3436:
3430:
3425:
3421:
3415:
3412:
3410:
3408:
3405:
3401:
3395:
3390:
3386:
3382:
3379:
3376:
3373:
3366:
3362:
3358:
3353:
3350:
3347:
3344:
3340:
3336:
3333:
3330:
3325:
3321:
3317:
3314:
3311:
3308:
3305:
3302:
3295:
3291:
3287:
3284:
3281:
3277:
3271:
3267:
3259:
3255:
3251:
3248:
3245:
3241:
3235:
3231:
3227:
3223:
3217:
3212:
3209:
3206:
3203:
3199:
3191:
3187:
3181:
3176:
3172:
3166:
3163:
3161:
3159:
3156:
3152:
3146:
3139:
3135:
3131:
3126:
3123:
3120:
3117:
3113:
3109:
3104:
3100:
3096:
3093:
3090:
3087:
3082:
3078:
3074:
3069:
3065:
3061:
3058:
3055:
3052:
3049:
3046:
3043:
3038:
3033:
3030:
3027:
3024:
3020:
3014:
3009:
3005:
2999:
2995:
2989:
2985:
2979:
2975:
2969:
2965:
2961:
2957:
2949:
2945:
2941:
2936:
2933:
2931:
2929:
2926:
2922:
2915:
2908:
2904:
2900:
2895:
2892:
2889:
2886:
2882:
2878:
2875:
2872:
2869:
2866:
2861:
2857:
2851:
2847:
2841:
2837:
2831:
2827:
2821:
2817:
2813:
2809:
2801:
2797:
2793:
2788:
2785:
2783:
2781:
2778:
2775:
2768:
2764:
2759:
2755:
2750:
2746:
2742:
2741:
2714:
2710:
2689:
2686:
2683:
2680:
2661:
2660:
2659:
2658:
2643:
2640:
2635:
2631:
2627:
2624:
2621:
2618:
2613:
2609:
2605:
2600:
2596:
2592:
2589:
2586:
2583:
2580:
2577:
2574:
2569:
2565:
2559:
2554:
2550:
2546:
2543:
2541:
2539:
2536:
2531:
2527:
2523:
2520:
2517:
2514:
2509:
2505:
2501:
2496:
2492:
2488:
2485:
2482:
2479:
2476:
2473:
2470:
2467:
2462:
2457:
2453:
2447:
2443:
2439:
2436:
2433:
2428:
2425:
2422:
2418:
2414:
2411:
2409:
2407:
2404:
2401:
2398:
2395:
2390:
2386:
2382:
2379:
2376:
2373:
2370:
2367:
2364:
2361:
2358:
2355:
2352:
2350:
2348:
2345:
2342:
2339:
2336:
2331:
2327:
2323:
2322:
2295:
2290:
2286:
2282:
2279:
2274:
2269:
2262:
2258:
2253:
2249:
2246:
2243:
2223:
2220:
2217:
2212:
2208:
2204:
2201:
2198:
2175:
2172:
2169:
2166:
2146:
2143:
2140:
2137:
2112:
2108:
2103:
2099:
2087:
2086:
2085:
2084:
2073:
2068:
2064:
2060:
2057:
2054:
2051:
2048:
2043:
2039:
2033:
2028:
2025:
2022:
2019:
2015:
2011:
2008:
2005:
2002:
1999:
1978:
1975:
1962:
1958:
1954:
1951:
1948:
1928:
1924:
1920:
1917:
1914:
1890:
1886:
1882:
1879:
1876:
1856:
1852:
1848:
1845:
1842:
1821:
1818:
1815:
1810:
1805:
1801:
1778:
1777:
1776:
1775:
1764:
1761:
1757:
1751:
1748:
1745:
1742:
1739:
1736:
1732:
1728:
1725:
1722:
1717:
1712:
1708:
1702:
1699:
1694:
1691:
1687:
1683:
1680:
1677:
1674:
1669:
1664:
1660:
1634:
1631:
1618:
1615:
1612:
1605:
1601:
1596:
1592:
1587:
1580:
1577:
1570:
1567:
1564:
1561:
1554:
1550:
1545:
1541:
1536:
1532:
1510:
1506:
1503:
1500:
1493:
1489:
1484:
1480:
1475:
1468:
1465:
1457:
1453:
1450:
1447:
1444:
1441:
1438:
1431:
1427:
1422:
1418:
1413:
1409:
1397:
1396:
1395:
1394:
1383:
1380:
1376:
1370:
1363:
1359:
1355:
1350:
1347:
1344:
1341:
1337:
1333:
1330:
1327:
1322:
1318:
1314:
1311:
1308:
1305:
1302:
1299:
1294:
1290:
1286:
1281:
1277:
1273:
1270:
1266:
1260:
1257:
1250:
1247:
1244:
1241:
1237:
1233:
1230:
1227:
1224:
1217:
1213:
1208:
1204:
1199:
1192:
1189:
1164:
1161:
1148:
1145:
1142:
1139:
1136:
1131:
1126:
1122:
1094:
1090:
1086:
1083:
1079:
1074:
1070:
1065:
1061:
1050:
1049:
1048:
1047:
1036:
1033:
1029:
1023:
1016:
1012:
1008:
1003:
1000:
997:
994:
990:
986:
983:
980:
977:
974:
969:
965:
959:
955:
949:
945:
939:
935:
929:
925:
921:
917:
909:
905:
901:
896:
893:
890:
887:
880:
876:
871:
867:
862:
858:
843:and equal to:
828:
825:
822:
815:
811:
806:
802:
797:
793:
781:
780:
779:
778:
765:
758:
754:
750:
745:
742:
739:
735:
731:
728:
725:
718:
714:
709:
705:
700:
696:
690:
685:
682:
679:
676:
672:
668:
665:
662:
659:
656:
653:
648:
644:
627:Fourier series
619:
618:
617:
616:
605:
602:
599:
596:
588:
585:
582:
577:
573:
569:
566:
563:
558:
554:
550:
547:
544:
541:
538:
535:
530:
526:
512:
511:
510:
509:
498:
490:
487:
482:
478:
474:
471:
468:
465:
462:
459:
456:
453:
450:
447:
444:
441:
438:
435:
432:
429:
404:
399:
395:
374:
365:are cyclic in
354:
351:
348:
345:
342:
337:
333:
312:
309:
306:
303:
300:
297:
294:
291:
269:
265:
249:
248:
247:
246:
234:
231:
228:
225:
222:
217:
213:
209:
206:
203:
200:
197:
194:
191:
188:
185:
182:
179:
176:
173:
170:
167:
162:
158:
132:
129:
126:
123:
120:
117:
114:
111:
91:
88:
85:
82:
70:
67:
50:
47:
33:
30:
13:
10:
9:
6:
4:
3:
2:
5013:
5002:
4999:
4998:
4996:
4987:
4984:
4981:
4977:
4976:
4972:
4962:
4959:
4954:
4950:
4946:
4942:
4935:
4932:
4926:
4923:
4917:
4914:
4908:
4905:
4898:
4895:
4889:
4886:
4880:
4877:
4871:
4868:
4862:
4859:
4853:
4850:
4846:
4840:
4837:
4833:
4827:
4824:
4818:
4815:
4809:
4806:
4800:
4797:
4792:
4788:
4784:
4780:
4775:
4770:
4766:
4762:
4755:
4752:
4746:
4743:
4738:
4734:
4730:
4726:
4719:
4717:
4715:
4711:
4706:
4702:
4698:
4694:
4687:
4684:
4678:
4676:
4673:
4659:
4639:
4636:
4633:
4630:
4610:
4601:
4587:
4579:
4575:
4559:
4534:
4525:
4509:
4504:
4501:
4498:
4495:
4492:
4488:
4482:
4479:
4476:
4473:
4470:
4467:
4463:
4456:
4453:
4450:
4442:
4438:
4429:
4421:
4417:
4411:
4407:
4403:
4398:
4394:
4390:
4387:
4383:
4377:
4374:
4364:
4358:
4350:
4344:
4336:
4331:
4327:
4319:
4318:
4317:
4316:
4315:
4313:
4274:
4271:
4268:
4260:
4256:
4235:
4215:
4192:
4186:
4166:
4140:
4128:
4122:
4114:
4110:
4103:
4100:
4094:
4088:
4082:
4076:
4070:
4064:
4061:
4058:
4050:
4046:
4038:
4037:
4036:
4035:
4034:
4032:
4027:
4022:
4020:
4016:
4013:(NVH) and in
4012:
4003:
3998:
3995:
3992:
3989:
3985:
3982:
3979:
3975:
3971:
3968:
3967:
3963:
3961:
3954:
3952:
3945:
3943:
3941:
3936:
3932:
3924:
3922:
3917:
3903:
3900:
3897:
3894:
3891:
3888:
3885:
3882:
3874:
3853:
3849:
3841:
3837:
3833:
3828:
3825:
3821:
3815:
3811:
3804:
3798:
3791:
3787:
3781:
3776:
3772:
3766:
3760:
3750:
3746:
3741:
3737:
3732:
3728:
3720:
3719:
3718:
3717:
3716:
3714:
3698:
3669:
3665:
3659:
3652:
3648:
3644:
3639:
3636:
3633:
3629:
3622:
3619:
3611:
3607:
3602:
3598:
3592:
3586:
3579:
3575:
3569:
3564:
3560:
3554:
3552:
3544:
3534:
3527:
3523:
3519:
3514:
3511:
3508:
3505:
3501:
3494:
3486:
3482:
3475:
3472:
3469:
3463:
3450:
3446:
3438:
3434:
3428:
3423:
3419:
3413:
3411:
3403:
3388:
3384:
3380:
3377:
3374:
3364:
3360:
3356:
3351:
3348:
3345:
3342:
3338:
3331:
3323:
3319:
3312:
3309:
3306:
3300:
3293:
3289:
3285:
3282:
3279:
3275:
3269:
3265:
3257:
3253:
3249:
3246:
3243:
3239:
3233:
3229:
3225:
3221:
3207:
3204:
3201:
3197:
3189:
3185:
3179:
3174:
3170:
3164:
3162:
3154:
3144:
3137:
3133:
3129:
3124:
3121:
3118:
3115:
3111:
3102:
3098:
3094:
3091:
3088:
3080:
3076:
3067:
3063:
3059:
3056:
3053:
3050:
3047:
3041:
3028:
3025:
3022:
3018:
3012:
3007:
3003:
2997:
2993:
2987:
2983:
2977:
2973:
2967:
2963:
2959:
2955:
2947:
2943:
2939:
2934:
2932:
2924:
2913:
2906:
2902:
2898:
2893:
2890:
2887:
2884:
2880:
2873:
2870:
2867:
2859:
2855:
2849:
2845:
2839:
2835:
2829:
2825:
2819:
2815:
2811:
2807:
2799:
2795:
2791:
2786:
2784:
2776:
2766:
2762:
2757:
2753:
2748:
2744:
2732:
2731:
2730:
2729:
2728:
2712:
2708:
2684:
2678:
2670:
2666:
2641:
2633:
2629:
2625:
2622:
2619:
2611:
2607:
2598:
2594:
2590:
2587:
2584:
2581:
2578:
2572:
2567:
2563:
2557:
2552:
2548:
2544:
2542:
2529:
2525:
2521:
2518:
2515:
2507:
2503:
2494:
2490:
2486:
2483:
2480:
2477:
2474:
2468:
2460:
2455:
2451:
2445:
2441:
2434:
2426:
2423:
2420:
2416:
2412:
2410:
2396:
2388:
2384:
2377:
2374:
2371:
2365:
2359:
2353:
2351:
2343:
2340:
2337:
2329:
2325:
2313:
2312:
2311:
2310:
2309:
2293:
2288:
2284:
2280:
2272:
2260:
2256:
2244:
2221:
2218:
2210:
2206:
2199:
2187:
2170:
2164:
2141:
2135:
2127:
2106:
2101:
2097:
2066:
2062:
2058:
2055:
2052:
2046:
2041:
2037:
2023:
2020:
2017:
2013:
2009:
2003:
1997:
1990:
1989:
1988:
1987:
1986:
1984:
1976:
1974:
1960:
1956:
1952:
1949:
1946:
1926:
1922:
1918:
1915:
1912:
1902:
1888:
1884:
1880:
1877:
1874:
1854:
1850:
1846:
1843:
1840:
1816:
1808:
1803:
1799:
1789:
1787:
1783:
1762:
1759:
1749:
1746:
1743:
1740:
1737:
1734:
1730:
1723:
1715:
1710:
1706:
1697:
1689:
1685:
1681:
1675:
1667:
1662:
1658:
1650:
1649:
1648:
1647:
1646:
1644:
1640:
1632:
1630:
1613:
1603:
1599:
1594:
1590:
1585:
1578:
1575:
1568:
1562:
1552:
1548:
1543:
1539:
1534:
1530:
1508:
1501:
1491:
1487:
1482:
1478:
1473:
1466:
1463:
1455:
1451:
1445:
1439:
1429:
1425:
1420:
1416:
1411:
1407:
1381:
1378:
1368:
1361:
1357:
1353:
1348:
1345:
1342:
1339:
1335:
1328:
1320:
1316:
1309:
1306:
1303:
1297:
1292:
1288:
1284:
1279:
1275:
1271:
1268:
1264:
1258:
1255:
1245:
1239:
1231:
1225:
1215:
1211:
1206:
1202:
1197:
1190:
1187:
1176:
1175:
1174:
1173:
1172:
1170:
1162:
1160:
1146:
1143:
1137:
1129:
1124:
1120:
1110:
1108:
1092:
1084:
1081:
1077:
1072:
1068:
1063:
1059:
1034:
1031:
1021:
1014:
1010:
1006:
1001:
998:
995:
992:
988:
981:
978:
975:
967:
963:
957:
953:
947:
943:
937:
933:
927:
923:
919:
915:
907:
903:
899:
894:
888:
878:
874:
869:
865:
860:
856:
848:
847:
846:
845:
844:
842:
823:
813:
809:
804:
800:
795:
791:
763:
756:
752:
748:
743:
740:
737:
733:
726:
716:
712:
707:
703:
698:
694:
680:
677:
674:
670:
666:
660:
657:
654:
646:
642:
634:
633:
632:
631:
630:
628:
624:
603:
600:
597:
594:
583:
580:
575:
571:
567:
564:
556:
552:
548:
542:
539:
536:
528:
524:
516:
515:
514:
513:
496:
480:
476:
472:
469:
463:
457:
451:
442:
436:
430:
420:
419:
418:
417:
416:
402:
397:
393:
372:
349:
346:
343:
335:
331:
304:
298:
292:
267:
263:
254:
232:
223:
215:
211:
204:
201:
198:
192:
186:
180:
174:
171:
168:
160:
156:
148:
147:
146:
145:
144:
124:
118:
112:
86:
80:
68:
66:
64:
60:
56:
48:
46:
43:
39:
31:
29:
27:
23:
19:
4986:presentation
4961:
4944:
4940:
4934:
4925:
4916:
4907:
4897:
4888:
4879:
4870:
4861:
4852:
4839:
4826:
4817:
4808:
4799:
4764:
4760:
4754:
4745:
4728:
4724:
4696:
4692:
4686:
4674:
4602:
4577:
4573:
4551:
4311:
4158:
4030:
4025:
4023:
4018:
4007:
3988:econometrics
3964:Applications
3958:
3949:
3928:
3918:
3870:
3690:
2671:. This way,
2668:
2662:
2189:By assuming
2188:
2088:
1980:
1903:
1790:
1779:
1642:
1638:
1636:
1398:
1168:
1166:
1111:
1106:
1051:
840:
782:
622:
620:
385:with period
250:
72:
58:
52:
35:
17:
15:
3976:for signal
3713:convolution
3711:indicating
1791:The reason
1105:are called
42:time series
4980:manuscript
4845:Biometrika
4832:Biometrika
4774:1505.05586
4679:References
839:is called
32:Definition
4660:ω
4637:ω
4631:θ
4611:ω
4576:(unit in
4560:α
4535:θ
4526:τ
4513:Θ
4510:θ
4505:α
4502:π
4493:−
4483:τ
4477:π
4468:−
4457:τ
4451:θ
4433:∞
4425:∞
4422:−
4418:∫
4388:−
4384:∫
4368:∞
4362:→
4337:α
4298:Θ
4275:τ
4269:θ
4236:τ
4216:θ
4193:θ
4167:θ
4129:θ
4115:∗
4104:τ
4095:θ
4077:
4065:τ
4059:θ
4033:is used,
3889:−
3829:−
3816:∗
3773:σ
3699:∗
3660:τ
3640:π
3623:τ
3620:−
3612:∗
3599:∗
3593:τ
3561:σ
3545:λ
3535:λ
3515:π
3506:−
3495:λ
3487:∗
3476:τ
3470:λ
3459:∞
3454:∞
3451:−
3447:∫
3420:σ
3404:λ
3375:λ
3352:π
3343:−
3332:λ
3324:∗
3313:τ
3307:λ
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