102:
1759:
1209:
1127:
654:
1754:{\displaystyle {\begin{aligned}s_{\mathrm {a} }(t)&\triangleq {\mathcal {F}}^{-1}\\&={\mathcal {F}}^{-1}\\&=\underbrace {{\mathcal {F}}^{-1}\{S(f)\}} _{s(t)}+\overbrace {\underbrace {{\mathcal {F}}^{-1}\{\operatorname {sgn}(f)\}} _{j{\frac {1}{\pi t}}}*\underbrace {{\mathcal {F}}^{-1}\{S(f)\}} _{s(t)}} ^{\text{convolution}}\\&=s(t)+j\underbrace {\left} _{\operatorname {\mathcal {H}} }\\&=s(t)+j{\hat {s}}(t),\end{aligned}}}
3749:
3331:
5739:
808:
359:
2782:
3065:
3733:
1122:{\displaystyle {\begin{aligned}S(f)&={\begin{cases}{\frac {1}{2}}S_{\mathrm {a} }(f),&{\text{for}}\ f>0,\\S_{\mathrm {a} }(f),&{\text{for}}\ f=0,\\{\frac {1}{2}}S_{\mathrm {a} }(-f)^{*},&{\text{for}}\ f<0\ {\text{(Hermitian symmetry)}}\end{cases}}\\&={\frac {1}{2}}.\end{aligned}}}
649:{\displaystyle {\begin{aligned}S_{\mathrm {a} }(f)&\triangleq {\begin{cases}2S(f),&{\text{for}}\ f>0,\\S(f),&{\text{for}}\ f=0,\\0,&{\text{for}}\ f<0\end{cases}}\\&=\underbrace {2\operatorname {u} (f)} _{1+\operatorname {sgn}(f)}S(f)=S(f)+\operatorname {sgn}(f)S(f),\end{aligned}}}
81:
of such a spectrum. These negative frequency components can be discarded with no loss of information, provided one is willing to deal with a complex-valued function instead. That makes certain attributes of the function more accessible and facilitates the derivation of modulation and demodulation
4603:
with real coefficients can excise the portion of interest. Another motive is to reduce the highest frequency, which reduces the minimum rate for alias-free sampling. A frequency shift does not undermine the mathematical tractability of the complex signal representation. So in that sense, the
4435:
2537:
3054:
3326:{\displaystyle s_{\mathrm {a} }(t)={\begin{cases}e^{j(\omega t+\theta )}\ \ =\ e^{j|\omega |t}\cdot e^{j\theta },&{\text{if}}\ \omega >0,\\e^{-j(\omega t+\theta )}=\ e^{j|\omega |t}\cdot e^{-j\theta },&{\text{if}}\ \omega <0.\end{cases}}}
4899:
3527:
4252:
The instantaneous amplitude, and the instantaneous phase and frequency are in some applications used to measure and detect local features of the signal. Another application of the analytic representation of a signal relates to demodulation of
5273:
The concept of analytic signal is well-defined for signals of a single variable which typically is time. For signals of two or more variables, an analytic signal can be defined in different ways, and two approaches are presented below.
2142:
4272:
5383:. The analytic signal is then produced by removing all negative frequencies and multiply the result by 2, in accordance to the procedure described for the case of one-variable signals. However, there is no particular direction for
2898:
5082:
2777:{\displaystyle {\begin{aligned}{\hat {s}}(t)&=\cos \left(\omega t-{\frac {\pi }{2}}\right)=\sin(\omega t),\\s_{\mathrm {a} }(t)&=s(t)+j{\hat {s}}(t)=\cos(\omega t)+j\sin(\omega t)=e^{j\omega t}.\end{aligned}}}
5381:
4742:
has no positive frequencies. In that case, extracting the real component restores them, but in reverse order; the low-frequency components are now high ones and vice versa. This can be used to demodulate a type of
3848:
1834:
3926:
4168:
4246:
5205:
4740:
3532:
2542:
1214:
813:
364:
69:, comprising the original function and its Hilbert transform. This representation facilitates many mathematical manipulations. The basic idea is that the negative frequency components of the
4005:
4792:
2918:
2904:
component, and doubling the positive frequency component. And the analytic representation of a sum of sinusoids is the sum of the analytic representations of the individual sinusoids.
5439:
5410:
5316:
3408:. But it might not be a reversible representation, because the original spectrum is not symmetrical in general. So except for this example, the general discussion assumes real-valued
2382:
2215:
2308:
730:
1985:
692:
5338:
4693:
2500:
4608:. However, restoring the real-valued representation is no longer a simple matter of just extracting the real component. Up-conversion may be required, and if the signal has been
4086:
3489:
3377:
2434:
2265:
1201:
3728:{\displaystyle {\begin{aligned}{\hat {s}}(t)&=je^{-j\omega t},\\s_{\mathrm {a} }(t)&=e^{-j\omega t}+j^{2}e^{-j\omega t}=e^{-j\omega t}-e^{-j\omega t}=0.\end{aligned}}}
279:
5640:
B. Boashash, "Notes on the use of the Wigner distribution for time frequency signal analysis", IEEE Trans. on
Acoustics, Speech, and Signal Processing, vol. 26, no. 9, 1987
2526:
2384:
restores the suppressed positive frequency components. Another viewpoint is that the imaginary component in either case is a term that subtracts frequency components from
5678:
3515:
4928:
4787:
4652:
4597:
4563:
4503:
4462:
5596:
B. Boashash, "Estimating and
Interpreting the Instantaneous Frequency of a Signal-Part I: Fundamentals", Proceedings of the IEEE, Vol. 80, No. 4, pp. 519–538, April 1992
5255:
4961:
318:
5453:, as it is defined for one-variable signals. However, the monogenic signal can be extended to arbitrary number of variables in a straightforward manner, producing an
5104:
5257:
as defined above is interpreted as a time-dependent generalization of the complex amplitude. Their relationship is not unlike that in the real-valued case: varying
2414:
4536:
4048:
3435:
3406:
1996:
1867:
1163:
800:
771:
351:
169:
136:
4966:
219:
1915:
1889:
189:
5563:
Smith, J.O. "Analytic
Signals and Hilbert Transform Filters", in Mathematics of the Discrete Fourier Transform (DFT) with Audio Applications, Second Edition,
2798:
5450:
89:), the conversion from complex back to real is just a matter of discarding the imaginary part. The analytic representation is a generalization of the
4430:{\displaystyle {s_{\mathrm {a} }}_{\downarrow }(t)\triangleq s_{\mathrm {a} }(t)e^{-j\omega _{0}t}=s_{\mathrm {m} }(t)e^{j(\phi (t)-\omega _{0}t)},}
93:
concept: while the phasor is restricted to time-invariant amplitude, phase, and frequency, the analytic signal allows for time-variable parameters.
5343:
5282:
A straightforward generalization of the analytic signal can be done for a multi-dimensional signal once it is established what is meant by
3766:
2900:
In general, the analytic representation of a simple sinusoid is obtained by expressing it in terms of complex-exponentials, discarding the
1770:
5109:
3341:
This is another example of using the
Hilbert transform method to remove negative frequency components. Nothing prevents us from computing
3859:
4269:
Analytic signals are often shifted in frequency (down-converted) toward 0 Hz, possibly creating negative frequency components:
5747:
4109:
4189:
5210:
In the field of time-frequency signal processing, it was shown that the analytic signal was needed in the definition of the
50:
components. The real and imaginary parts of an analytic signal are real-valued functions related to each other by the
5837:
4698:
3049:{\displaystyle s(t)=\cos(\omega t+\theta )={\frac {1}{2}}\left(e^{j(\omega t+\theta )}+e^{-j(\omega t+\theta )}\right)}
5775:
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665:
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4657:
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2419:
2220:
1168:
5449:
The real and imaginary parts of the analytic signal correspond to the two elements of the vector-valued
3752:
A function in blue and the magnitude of its analytic representation in red, showing the envelope effect.
5832:
5543:"Complex envelope is an extended interpretation of complex amplitude as a function of time." p. 85
5496:
4894:{\displaystyle \int _{0}^{+\infty }(\omega -\omega _{0})^{2}|S_{\mathrm {a} }(\omega )|^{2}\,d\omega .}
4744:
1987:
this can also be expressed as a filtering operation that directly removes negative frequency components
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5757:
227:
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As long as the manipulated function has no negative frequency components (that is, it is still
5224:
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287:
5089:
2137:{\displaystyle s_{\mathrm {a} }(t)=s(t)*\underbrace {\left} _{{\mathcal {F}}^{-1}\{2u(f)\}}.}
5412:
which must be chosen unless there are some additional constraints. Therefore, the choice of
4261:
and phase (or frequency) modulation, and effectively demodulates certain kinds of signals.
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8:
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20:
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so that the method can have the desirable properties needed for practical applications.
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78:
47:
2893:{\textstyle \cos(\omega t)={\frac {1}{2}}\left(e^{j\omega t}+e^{j(-\omega )t}\right).}
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70:
51:
35:
24:
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operator removes the subtraction, giving the appearance of adding new components.
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5651:
5569:
https://www.dsprelated.com/freebooks/mdft/Analytic_Signals_Hilbert_Transform.html
5277:
4600:
4599:
is selected to be somewhere in the middle of the desired passband. Then a simple
4506:
5619:
4092:
2217:, restoring the negative frequency components is a simple matter of discarding
1918:
4930:
can be chosen to minimize the mean square error in linearly approximating the
5826:
5627:
5606:
Justice, J. (1979-12-01). "Analytic signal processing in music computation".
5506:
4613:
733:
5565:
https://ccrma.stanford.edu/~jos/r320/Analytic_Signals_Hilbert_Transform.html
5217:
Sometimes the phrase "complex envelope" is given the simpler meaning of the
5077:{\displaystyle \int _{-\infty }^{+\infty }^{2}|s_{\mathrm {a} }(t)|^{2}\,dt}
2912:
Here we use Euler's formula to identify and discard the negative frequency.
5287:
1892:
62:
31:
5805:
Time-Frequency Signal
Analysis and Processing: A Comprehensive Reference
4478:. The complex envelope is not unique; it is determined by the choice of
5376:{\displaystyle {\boldsymbol {\xi }}\cdot {\boldsymbol {\hat {u}}}<0}
4617:
4254:
5262:
3843:{\displaystyle s_{\mathrm {a} }(t)=s_{\mathrm {m} }(t)e^{j\phi (t)},}
2792:
1829:{\displaystyle {\hat {s}}(t)\triangleq \operatorname {\mathcal {H}} }
773:. And the operation is reversible, due to the Hermitian symmetry of
5479:
5221:
of a (constant-frequency) phasor; other times the complex envelope
4621:
4474:
3921:{\displaystyle s_{\mathrm {m} }(t)\triangleq |s_{\mathrm {a} }(t)|}
74:
5269:
Extensions of the analytic signal to signals of multiple variables
4758:
Other choices of reference frequency are sometimes considered:
5278:
Multi-dimensional analytic signal based on an ad hoc direction
191:
is the real value denoting frequency), then the transform has
5608:
IEEE Transactions on
Acoustics, Speech, and Signal Processing
4257:. The polar coordinates conveniently separate the effects of
4180:
4163:{\displaystyle \omega (t)\triangleq {\frac {d\phi }{dt}}(t).}
4241:{\displaystyle f(t)\triangleq {\frac {1}{2\pi }}\omega (t).}
3853:
where the following time-variant quantities are introduced:
5534:"the complex envelope (or complex amplitude)", p. 586
3748:
3319:
1027:
516:
5680:
Encyclopedia of
Optical Engineering: Abe-Las, pages 1-1024
5475:
Practical considerations for computing
Hilbert transforms
2267:
which may seem counter-intuitive. The complex conjugate
77:) of a real-valued function are superfluous, due to the
5800:, vol. II, Cambridge University Press, Cambridge, 2009.
2801:
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5346:
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in the
Fourier domain and label any frequency vector
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5461:-dimensional vector-valued function for the case of
5200:{\displaystyle \int _{-\infty }^{+\infty }^{2}\,dt.}
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In the accompanying diagram, the blue curve depicts
3743:
4735:{\displaystyle {s_{\mathrm {a} }}_{\downarrow }(t)}
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1979:
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213:
183:
163:
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5571:, online book, 2007 edition, accessed 2021-04-29.
5425:
5396:
5361:
5302:
5286:for this case. This can be done by introducing a
3965:
2314:the negative frequency components. And therefore
5824:
5650:Hlawatsch, Franz; Auger, François (2013-03-01).
2147:
4654:is chosen larger than the highest frequency of
4505:. This concept is often used when dealing with
4000:{\displaystyle \phi (t)\triangleq \arg \!\left}
5818:Analytic Signals and Hilbert Transform Filters
5649:
105:Transfer function to create an analytic signal
5525:"the complex envelope (or complex amplitude)"
4467:This function goes by various names, such as
4464:is an arbitrary reference angular frequency.
4264:
4050:and the red curve depicts the corresponding
3756:An analytic signal can also be expressed in
2126:
2108:
1552:
1537:
1483:
1465:
1413:
1398:
5579:
5577:
5793:, Prentice Hall, Upper Saddle River, 1995.
5585:The Fourier Transform and Its Applications
5776:Learn how and when to remove this message
5187:
5086:or another alternative (for some optimum
5067:
4881:
5676:
5574:
5434:{\displaystyle {\boldsymbol {\hat {u}}}}
5405:{\displaystyle {\boldsymbol {\hat {u}}}}
5311:{\displaystyle {\boldsymbol {\hat {u}}}}
3747:
2377:{\displaystyle s(t)=\operatorname {Re} }
2210:{\displaystyle s(t)=\operatorname {Re} }
100:
5703:
5605:
5444:
5422:
5393:
5358:
5348:
5326:
5299:
2303:{\displaystyle s_{\mathrm {a} }^{*}(t)}
5825:
725:{\displaystyle \operatorname {sgn}(f)}
1980:{\displaystyle s(t)=s(t)*\delta (t),}
687:{\displaystyle \operatorname {u} (f)}
82:techniques, such as single-sideband.
16:Particular representation of a signal
5732:
5441:is ad hoc, or application specific.
5333:{\displaystyle {\boldsymbol {\xi }}}
4688:{\displaystyle s_{\mathrm {a} }(t),}
2495:{\displaystyle s(t)=\cos(\omega t),}
1165:is the inverse Fourier transform of
4620:) might also be necessary to avoid
4081:{\displaystyle s_{\mathrm {m} }(t)}
3484:{\displaystyle s(t)=e^{-j\omega t}}
3372:{\displaystyle s_{\mathrm {a} }(t)}
2429:{\displaystyle \operatorname {Re} }
2260:{\displaystyle \operatorname {Im} }
1196:{\displaystyle S_{\mathrm {a} }(f)}
13:
5728:
5677:Driggers, Ronald G. (2003-01-01).
5129:
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14:
5854:
5811:
5807:, Elsevier Science, Oxford, 2003.
5707:Global Environment Remote Sensing
4095:instantaneous phase has units of
3744:Instantaneous amplitude and phase
5737:
5704:Okamoto, KenĘĽichi (2001-01-01).
5587:. McGraw-Hill, 2000. pp. 361-362
142:function with Fourier transform
5697:
5537:
5490:
4604:down-converted signal is still
4101:instantaneous angular frequency
274:{\displaystyle S(-f)=S(f)^{*},}
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5760:and help improve the section.
5550:
3738:
2521:{\displaystyle \omega >0.}
2148:Negative frequency components
96:
3510:{\displaystyle \omega >0}
3336:
2907:
2444:
7:
5468:
4923:{\displaystyle \omega _{0}}
4782:{\displaystyle \omega _{0}}
4647:{\displaystyle \omega _{0}}
4592:{\displaystyle \omega _{0}}
4558:{\displaystyle \omega _{0}}
4498:{\displaystyle \omega _{0}}
4457:{\displaystyle \omega _{0}}
4091:The time derivative of the
2439:
10:
5859:
5620:10.1109/TASSP.1979.1163321
5497:Single-sideband modulation
18:
5656:. John Wiley & Sons.
5212:Wigner–Ville distribution
4265:Complex envelope/baseband
5512:
5250:{\displaystyle s_{m}(t)}
4956:{\displaystyle \phi (t)}
4565:might be equated to its
313:{\displaystyle S(f)^{*}}
19:Not to be confused with
5838:Time–frequency analysis
5791:Time-frequency analysis
5653:Time-Frequency Analysis
5099:{\displaystyle \theta }
4538:is a modulated signal,
4176:instantaneous frequency
3930:instantaneous amplitude
696:Heaviside step function
59:analytic representation
44:complex-valued function
5435:
5406:
5377:
5334:
5312:
5261:generalizing constant
5251:
5201:
5100:
5078:
4957:
4924:
4895:
4789:is chosen to minimize
4783:
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4689:
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1981:
1911:
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767:
742:non-negative frequency
726:
688:
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3379:for a complex-valued
3374:
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2895:
2787:The last equality is
2779:
2523:
2497:
2431:
2411:
2409:{\displaystyle s(t).}
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1982:
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727:
689:
651:
348:
315:
276:
216:
186:
166:
133:
104:
5445:The monogenic signal
5416:
5387:
5344:
5322:
5293:
5284:negative frequencies
5225:
5110:
5090:
4967:
4938:
4934:instantaneous phase
4907:
4793:
4766:
4699:
4658:
4631:
4576:
4542:
4531:{\displaystyle s(t)}
4513:
4482:
4441:
4273:
4259:amplitude modulation
4190:
4110:
4099:, and is called the
4054:
4043:{\displaystyle s(t)}
4025:
3944:
3860:
3767:
3528:
3495:
3444:
3430:{\displaystyle s(t)}
3412:
3401:{\displaystyle s(t)}
3383:
3345:
3066:
2919:
2799:
2538:
2506:
2453:
2420:
2388:
2318:
2271:
2221:
2156:
1997:
1929:
1901:
1875:
1862:{\displaystyle s(t)}
1844:
1771:
1210:
1169:
1158:{\displaystyle s(t)}
1140:
1021:(Hermitian symmetry)
809:
795:{\displaystyle S(f)}
777:
766:{\displaystyle S(f)}
748:
704:
666:
360:
346:{\displaystyle S(f)}
328:
288:
228:
199:
175:
164:{\displaystyle S(f)}
146:
131:{\displaystyle s(t)}
113:
5796:Frederick W. King,
5465:-variable signals.
5133:
4990:
4813:
4010:instantaneous phase
2502: where
2361:
2290:
214:{\displaystyle f=0}
195:symmetry about the
21:analytic expression
5798:Hilbert Transforms
5485:Negative frequency
5431:
5402:
5373:
5330:
5308:
5247:
5197:
5113:
5096:
5074:
4970:
4953:
4920:
4891:
4796:
4779:
4732:
4685:
4644:
4589:
4555:
4528:
4495:
4454:
4427:
4238:
4160:
4078:
4040:
3997:
3918:
3840:
3754:
3725:
3723:
3507:
3481:
3427:
3398:
3369:
3323:
3318:
3046:
2902:negative frequency
2890:
2774:
2772:
2518:
2492:
2426:
2406:
2374:
2345:
2300:
2274:
2257:
2207:
2134:
2130:
2088:
1977:
1907:
1881:
1859:
1826:
1751:
1749:
1694:
1662:
1575:
1559:
1512:
1490:
1436:
1420:
1193:
1155:
1119:
1117:
1026:
792:
763:
740:contains only the
722:
684:
646:
644:
581:
556:
515:
343:
310:
271:
211:
181:
161:
128:
107:
79:Hermitian symmetry
48:negative frequency
5833:Signal processing
5786:
5785:
5778:
5502:Quadrature filter
5428:
5399:
5364:
5305:
5219:complex amplitude
4753:inverted sideband
4612:(discrete-time),
4567:carrier frequency
4255:modulated signals
4221:
4146:
3758:polar coordinates
3544:
3306:
3302:
3248:
3195:
3191:
3140:
3134:
3131:
2972:
2831:
2691:
2601:
2554:
2078:
2041:
2039:
1910:{\displaystyle j}
1884:{\displaystyle *}
1838:Hilbert transform
1783:
1732:
1637:
1618:
1616:
1587:
1585:
1580:
1518:
1516:
1509:
1446:
1444:
1379:
1377:
1049:
1022:
1018:
1006:
1002:
961:
936:
932:
886:
882:
851:
532:
530:
503:
499:
471:
467:
430:
426:
322:complex conjugate
184:{\displaystyle f}
71:Fourier transform
52:Hilbert transform
36:signal processing
25:analytic function
5850:
5843:Fourier analysis
5781:
5774:
5770:
5767:
5761:
5756:Please read the
5752:may need cleanup
5741:
5740:
5733:
5722:
5721:
5701:
5695:
5694:
5674:
5668:
5667:
5647:
5641:
5638:
5632:
5631:
5603:
5597:
5594:
5588:
5583:Bracewell, Ron.
5581:
5572:
5561:
5544:
5541:
5535:
5532:
5526:
5523:
5460:
5451:monogenic signal
5440:
5438:
5437:
5432:
5430:
5429:
5421:
5411:
5409:
5408:
5403:
5401:
5400:
5392:
5382:
5380:
5379:
5374:
5366:
5365:
5357:
5351:
5339:
5337:
5336:
5331:
5329:
5317:
5315:
5314:
5309:
5307:
5306:
5298:
5256:
5254:
5253:
5248:
5237:
5236:
5206:
5204:
5203:
5198:
5186:
5185:
5164:
5163:
5132:
5124:
5105:
5103:
5102:
5097:
5083:
5081:
5080:
5075:
5066:
5065:
5060:
5045:
5044:
5043:
5033:
5028:
5027:
5018:
5017:
4989:
4981:
4962:
4960:
4959:
4954:
4929:
4927:
4926:
4921:
4919:
4918:
4900:
4898:
4897:
4892:
4880:
4879:
4874:
4859:
4858:
4857:
4847:
4842:
4841:
4832:
4831:
4812:
4804:
4788:
4786:
4785:
4780:
4778:
4777:
4741:
4739:
4738:
4733:
4722:
4721:
4716:
4715:
4714:
4713:
4694:
4692:
4691:
4686:
4672:
4671:
4670:
4653:
4651:
4650:
4645:
4643:
4642:
4598:
4596:
4595:
4590:
4588:
4587:
4572:In other cases,
4564:
4562:
4561:
4556:
4554:
4553:
4537:
4535:
4534:
4529:
4507:passband signals
4504:
4502:
4501:
4496:
4494:
4493:
4469:complex envelope
4463:
4461:
4460:
4455:
4453:
4452:
4436:
4434:
4433:
4428:
4423:
4422:
4415:
4414:
4370:
4369:
4368:
4355:
4354:
4350:
4349:
4320:
4319:
4318:
4296:
4295:
4290:
4289:
4288:
4287:
4247:
4245:
4244:
4239:
4222:
4220:
4209:
4183:) is therefore:
4169:
4167:
4166:
4161:
4147:
4145:
4137:
4129:
4087:
4085:
4084:
4079:
4068:
4067:
4066:
4049:
4047:
4046:
4041:
4006:
4004:
4003:
3998:
3996:
3992:
3982:
3981:
3980:
3927:
3925:
3924:
3919:
3917:
3903:
3902:
3901:
3891:
3874:
3873:
3872:
3849:
3847:
3846:
3841:
3836:
3835:
3805:
3804:
3803:
3781:
3780:
3779:
3734:
3732:
3731:
3726:
3724:
3714:
3713:
3692:
3691:
3670:
3669:
3651:
3650:
3638:
3637:
3603:
3602:
3601:
3584:
3583:
3546:
3545:
3537:
3516:
3514:
3513:
3508:
3490:
3488:
3487:
3482:
3480:
3479:
3436:
3434:
3433:
3428:
3407:
3405:
3404:
3399:
3378:
3376:
3375:
3370:
3359:
3358:
3357:
3332:
3330:
3329:
3324:
3322:
3321:
3304:
3303:
3300:
3293:
3292:
3274:
3273:
3269:
3261:
3246:
3242:
3241:
3193:
3192:
3189:
3182:
3181:
3166:
3165:
3161:
3153:
3138:
3132:
3129:
3128:
3127:
3080:
3079:
3078:
3055:
3053:
3052:
3047:
3045:
3041:
3040:
3039:
3006:
3005:
2973:
2965:
2899:
2897:
2896:
2891:
2886:
2882:
2881:
2880:
2853:
2852:
2832:
2824:
2783:
2781:
2780:
2775:
2773:
2766:
2765:
2693:
2692:
2684:
2647:
2646:
2645:
2607:
2603:
2602:
2594:
2556:
2555:
2547:
2527:
2525:
2524:
2519:
2501:
2499:
2498:
2493:
2435:
2433:
2432:
2427:
2415:
2413:
2412:
2407:
2383:
2381:
2380:
2375:
2360:
2355:
2354:
2309:
2307:
2306:
2301:
2289:
2284:
2283:
2266:
2264:
2263:
2258:
2244:
2243:
2242:
2216:
2214:
2213:
2208:
2194:
2193:
2192:
2143:
2141:
2140:
2135:
2129:
2107:
2106:
2098:
2097:
2089:
2084:
2080:
2079:
2077:
2066:
2011:
2010:
2009:
1986:
1984:
1983:
1978:
1916:
1914:
1913:
1908:
1890:
1888:
1887:
1882:
1868:
1866:
1865:
1860:
1835:
1833:
1832:
1827:
1804:
1803:
1785:
1784:
1776:
1760:
1758:
1757:
1752:
1750:
1734:
1733:
1725:
1698:
1693:
1671:
1670:
1663:
1658:
1654:
1638:
1636:
1625:
1591:
1586:
1583:
1581:
1576:
1574:
1560:
1555:
1536:
1535:
1527:
1526:
1511:
1510:
1508:
1497:
1491:
1486:
1464:
1463:
1455:
1454:
1442:
1440:
1435:
1421:
1416:
1397:
1396:
1388:
1387:
1370:
1315:
1314:
1306:
1305:
1292:
1276:
1275:
1274:
1261:
1260:
1252:
1251:
1228:
1227:
1226:
1202:
1200:
1199:
1194:
1183:
1182:
1181:
1164:
1162:
1161:
1156:
1128:
1126:
1125:
1120:
1118:
1108:
1107:
1089:
1088:
1087:
1065:
1064:
1063:
1050:
1042:
1034:
1030:
1029:
1023:
1020:
1016:
1004:
1003:
1000:
993:
992:
974:
973:
972:
962:
954:
934:
933:
930:
914:
913:
912:
884:
883:
880:
864:
863:
862:
852:
844:
801:
799:
798:
793:
772:
770:
769:
764:
731:
729:
728:
723:
693:
691:
690:
685:
655:
653:
652:
647:
645:
580:
557:
552:
523:
519:
518:
501:
500:
497:
469:
468:
465:
428:
427:
424:
378:
377:
376:
353:. The function:
352:
350:
349:
344:
319:
317:
316:
311:
309:
308:
280:
278:
277:
272:
267:
266:
220:
218:
217:
212:
190:
188:
187:
182:
170:
168:
167:
162:
137:
135:
134:
129:
5858:
5857:
5853:
5852:
5851:
5849:
5848:
5847:
5823:
5822:
5814:
5782:
5771:
5765:
5762:
5755:
5748:Further reading
5742:
5738:
5731:
5729:Further reading
5726:
5725:
5718:
5702:
5698:
5691:
5675:
5671:
5664:
5648:
5644:
5639:
5635:
5604:
5600:
5595:
5591:
5582:
5575:
5562:
5558:
5553:
5548:
5547:
5542:
5538:
5533:
5529:
5524:
5520:
5515:
5493:
5471:
5454:
5447:
5420:
5419:
5417:
5414:
5413:
5391:
5390:
5388:
5385:
5384:
5356:
5355:
5347:
5345:
5342:
5341:
5340:as negative if
5325:
5323:
5320:
5319:
5297:
5296:
5294:
5291:
5290:
5280:
5271:
5232:
5228:
5226:
5223:
5222:
5181:
5177:
5159:
5155:
5125:
5117:
5111:
5108:
5107:
5091:
5088:
5087:
5061:
5056:
5055:
5039:
5038:
5034:
5029:
5023:
5019:
5013:
5009:
4982:
4974:
4968:
4965:
4964:
4939:
4936:
4935:
4914:
4910:
4908:
4905:
4904:
4903:Alternatively,
4875:
4870:
4869:
4853:
4852:
4848:
4843:
4837:
4833:
4827:
4823:
4805:
4800:
4794:
4791:
4790:
4773:
4769:
4767:
4764:
4763:
4745:single-sideband
4717:
4709:
4708:
4704:
4703:
4702:
4700:
4697:
4696:
4666:
4665:
4661:
4659:
4656:
4655:
4638:
4634:
4632:
4629:
4628:
4601:low-pass filter
4583:
4579:
4577:
4574:
4573:
4549:
4545:
4543:
4540:
4539:
4514:
4511:
4510:
4489:
4485:
4483:
4480:
4479:
4448:
4444:
4442:
4439:
4438:
4410:
4406:
4384:
4380:
4364:
4363:
4359:
4345:
4341:
4334:
4330:
4314:
4313:
4309:
4291:
4283:
4282:
4278:
4277:
4276:
4274:
4271:
4270:
4267:
4213:
4208:
4191:
4188:
4187:
4138:
4130:
4128:
4111:
4108:
4107:
4062:
4061:
4057:
4055:
4052:
4051:
4026:
4023:
4022:
3976:
3975:
3971:
3970:
3966:
3945:
3942:
3941:
3913:
3897:
3896:
3892:
3887:
3868:
3867:
3863:
3861:
3858:
3857:
3819:
3815:
3799:
3798:
3794:
3775:
3774:
3770:
3768:
3765:
3764:
3746:
3741:
3722:
3721:
3700:
3696:
3678:
3674:
3656:
3652:
3646:
3642:
3624:
3620:
3613:
3597:
3596:
3592:
3589:
3588:
3570:
3566:
3556:
3536:
3535:
3531:
3529:
3526:
3525:
3496:
3493:
3492:
3466:
3462:
3445:
3442:
3441:
3413:
3410:
3409:
3384:
3381:
3380:
3353:
3352:
3348:
3346:
3343:
3342:
3339:
3317:
3316:
3299:
3297:
3282:
3278:
3265:
3257:
3253:
3249:
3216:
3212:
3209:
3208:
3188:
3186:
3174:
3170:
3157:
3149:
3145:
3141:
3105:
3101:
3094:
3093:
3074:
3073:
3069:
3067:
3064:
3063:
3014:
3010:
2983:
2979:
2978:
2974:
2964:
2920:
2917:
2916:
2910:
2861:
2857:
2842:
2838:
2837:
2833:
2823:
2800:
2797:
2796:
2789:Euler's formula
2771:
2770:
2755:
2751:
2683:
2682:
2657:
2641:
2640:
2636:
2633:
2632:
2593:
2583:
2579:
2566:
2546:
2545:
2541:
2539:
2536:
2535:
2507:
2504:
2503:
2454:
2451:
2450:
2447:
2442:
2421:
2418:
2417:
2389:
2386:
2385:
2356:
2350:
2349:
2319:
2316:
2315:
2285:
2279:
2278:
2272:
2269:
2268:
2238:
2237:
2233:
2222:
2219:
2218:
2188:
2187:
2183:
2157:
2154:
2153:
2150:
2099:
2093:
2092:
2091:
2090:
2070:
2065:
2046:
2042:
2040:
2005:
2004:
2000:
1998:
1995:
1994:
1930:
1927:
1926:
1902:
1899:
1898:
1876:
1873:
1872:
1845:
1842:
1841:
1799:
1798:
1775:
1774:
1772:
1769:
1768:
1748:
1747:
1724:
1723:
1696:
1695:
1666:
1665:
1664:
1629:
1624:
1623:
1619:
1617:
1589:
1588:
1582:
1561:
1528:
1522:
1521:
1520:
1519:
1517:
1501:
1496:
1492:
1456:
1450:
1449:
1448:
1447:
1445:
1443:
1441:
1422:
1389:
1383:
1382:
1381:
1380:
1378:
1368:
1367:
1307:
1301:
1300:
1299:
1290:
1289:
1270:
1269:
1265:
1253:
1247:
1246:
1245:
1238:
1222:
1221:
1217:
1213:
1211:
1208:
1207:
1177:
1176:
1172:
1170:
1167:
1166:
1141:
1138:
1137:
1134:analytic signal
1116:
1115:
1103:
1099:
1083:
1082:
1078:
1059:
1058:
1054:
1041:
1032:
1031:
1025:
1024:
1019:
999:
997:
988:
984:
968:
967:
963:
953:
950:
949:
929:
927:
908:
907:
903:
900:
899:
879:
877:
858:
857:
853:
843:
836:
835:
828:
812:
810:
807:
806:
778:
775:
774:
749:
746:
745:
705:
702:
701:
667:
664:
663:
643:
642:
558:
533:
531:
521:
520:
514:
513:
496:
494:
485:
484:
464:
462:
444:
443:
423:
421:
396:
395:
388:
372:
371:
367:
363:
361:
358:
357:
329:
326:
325:
304:
300:
289:
286:
285:
262:
258:
229:
226:
225:
200:
197:
196:
176:
173:
172:
147:
144:
143:
114:
111:
110:
99:
67:analytic signal
65:function is an
40:analytic signal
28:
17:
12:
11:
5:
5856:
5846:
5845:
5840:
5835:
5821:
5820:
5813:
5812:External links
5810:
5809:
5808:
5801:
5794:
5784:
5783:
5745:
5743:
5736:
5730:
5727:
5724:
5723:
5716:
5696:
5689:
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5662:
5642:
5633:
5614:(6): 670–684.
5598:
5589:
5573:
5555:
5554:
5552:
5549:
5546:
5545:
5536:
5527:
5517:
5516:
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4856:
4851:
4846:
4840:
4836:
4830:
4826:
4822:
4819:
4816:
4811:
4808:
4803:
4799:
4776:
4772:
4749:lower sideband
4747:signal called
4731:
4728:
4725:
4720:
4712:
4707:
4684:
4681:
4678:
4675:
4669:
4664:
4641:
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4344:
4340:
4337:
4333:
4329:
4326:
4323:
4317:
4312:
4308:
4305:
4302:
4299:
4294:
4286:
4281:
4266:
4263:
4250:
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4237:
4234:
4231:
4228:
4225:
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4212:
4207:
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4198:
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4170:
4159:
4156:
4153:
4150:
4144:
4141:
4136:
4133:
4127:
4124:
4121:
4118:
4115:
4097:radians/second
4077:
4074:
4071:
4065:
4060:
4039:
4036:
4033:
4030:
4019:
4018:
4007:is called the
3995:
3991:
3988:
3985:
3979:
3974:
3969:
3964:
3961:
3958:
3955:
3952:
3949:
3939:
3928:is called the
3916:
3912:
3909:
3906:
3900:
3895:
3890:
3886:
3883:
3880:
3877:
3871:
3866:
3851:
3850:
3839:
3834:
3831:
3828:
3825:
3822:
3818:
3814:
3811:
3808:
3802:
3797:
3793:
3790:
3787:
3784:
3778:
3773:
3745:
3742:
3740:
3737:
3736:
3735:
3720:
3717:
3712:
3709:
3706:
3703:
3699:
3695:
3690:
3687:
3684:
3681:
3677:
3673:
3668:
3665:
3662:
3659:
3655:
3649:
3645:
3641:
3636:
3633:
3630:
3627:
3623:
3619:
3616:
3614:
3612:
3609:
3606:
3600:
3595:
3591:
3590:
3587:
3582:
3579:
3576:
3573:
3569:
3565:
3562:
3559:
3557:
3555:
3552:
3549:
3543:
3540:
3534:
3533:
3519:
3518:
3506:
3503:
3500:
3478:
3475:
3472:
3469:
3465:
3461:
3458:
3455:
3452:
3449:
3426:
3423:
3420:
3417:
3397:
3394:
3391:
3388:
3368:
3365:
3362:
3356:
3351:
3338:
3335:
3334:
3333:
3320:
3315:
3312:
3309:
3298:
3296:
3291:
3288:
3285:
3281:
3277:
3272:
3268:
3264:
3260:
3256:
3252:
3245:
3240:
3237:
3234:
3231:
3228:
3225:
3222:
3219:
3215:
3211:
3210:
3207:
3204:
3201:
3198:
3187:
3185:
3180:
3177:
3173:
3169:
3164:
3160:
3156:
3152:
3148:
3144:
3137:
3126:
3123:
3120:
3117:
3114:
3111:
3108:
3104:
3100:
3099:
3097:
3092:
3089:
3086:
3083:
3077:
3072:
3057:
3056:
3044:
3038:
3035:
3032:
3029:
3026:
3023:
3020:
3017:
3013:
3009:
3004:
3001:
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2879:
2876:
2873:
2870:
2867:
2864:
2860:
2856:
2851:
2848:
2845:
2841:
2836:
2830:
2827:
2822:
2819:
2816:
2813:
2810:
2807:
2804:
2785:
2784:
2769:
2764:
2761:
2758:
2754:
2750:
2747:
2744:
2741:
2738:
2735:
2732:
2729:
2726:
2723:
2720:
2717:
2714:
2711:
2708:
2705:
2702:
2699:
2696:
2690:
2687:
2681:
2678:
2675:
2672:
2669:
2666:
2663:
2660:
2658:
2656:
2653:
2650:
2644:
2639:
2635:
2634:
2631:
2628:
2625:
2622:
2619:
2616:
2613:
2610:
2606:
2600:
2597:
2592:
2589:
2586:
2582:
2578:
2575:
2572:
2569:
2567:
2565:
2562:
2559:
2553:
2550:
2544:
2543:
2529:
2528:
2517:
2514:
2511:
2491:
2488:
2485:
2482:
2479:
2476:
2473:
2470:
2467:
2464:
2461:
2458:
2446:
2443:
2441:
2438:
2425:
2405:
2402:
2399:
2396:
2393:
2373:
2370:
2367:
2364:
2359:
2353:
2348:
2344:
2341:
2338:
2335:
2332:
2329:
2326:
2323:
2299:
2296:
2293:
2288:
2282:
2277:
2256:
2253:
2250:
2247:
2241:
2236:
2232:
2229:
2226:
2206:
2203:
2200:
2197:
2191:
2186:
2182:
2179:
2176:
2173:
2170:
2167:
2164:
2161:
2149:
2146:
2145:
2144:
2133:
2128:
2125:
2122:
2119:
2116:
2113:
2110:
2105:
2102:
2096:
2087:
2083:
2076:
2073:
2069:
2064:
2061:
2058:
2055:
2052:
2049:
2045:
2038:
2035:
2032:
2029:
2026:
2023:
2020:
2017:
2014:
2008:
2003:
1976:
1973:
1970:
1967:
1964:
1961:
1958:
1955:
1952:
1949:
1946:
1943:
1940:
1937:
1934:
1923:
1922:
1919:imaginary unit
1906:
1896:
1891:is the binary
1880:
1870:
1858:
1855:
1852:
1849:
1825:
1822:
1819:
1816:
1813:
1810:
1807:
1802:
1797:
1794:
1791:
1788:
1782:
1779:
1762:
1761:
1746:
1743:
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1731:
1728:
1722:
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1716:
1713:
1710:
1707:
1704:
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1699:
1697:
1692:
1689:
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1683:
1680:
1677:
1674:
1669:
1661:
1657:
1653:
1650:
1647:
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1641:
1635:
1632:
1628:
1622:
1615:
1612:
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1606:
1603:
1600:
1597:
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1590:
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1573:
1570:
1567:
1564:
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1551:
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1534:
1531:
1525:
1515:
1507:
1504:
1500:
1495:
1489:
1485:
1482:
1479:
1476:
1473:
1470:
1467:
1462:
1459:
1453:
1439:
1434:
1431:
1428:
1425:
1419:
1415:
1412:
1409:
1406:
1403:
1400:
1395:
1392:
1386:
1376:
1373:
1371:
1369:
1366:
1363:
1360:
1357:
1354:
1351:
1348:
1345:
1342:
1339:
1336:
1333:
1330:
1327:
1324:
1321:
1318:
1313:
1310:
1304:
1298:
1295:
1293:
1291:
1288:
1285:
1282:
1279:
1273:
1268:
1264:
1259:
1256:
1250:
1244:
1241:
1239:
1237:
1234:
1231:
1225:
1220:
1216:
1215:
1192:
1189:
1186:
1180:
1175:
1154:
1151:
1148:
1145:
1130:
1129:
1114:
1111:
1106:
1102:
1098:
1095:
1092:
1086:
1081:
1077:
1074:
1071:
1068:
1062:
1057:
1053:
1048:
1045:
1040:
1037:
1035:
1033:
1028:
1015:
1012:
1009:
998:
996:
991:
987:
983:
980:
977:
971:
966:
960:
957:
952:
951:
948:
945:
942:
939:
928:
926:
923:
920:
917:
911:
906:
902:
901:
898:
895:
892:
889:
878:
876:
873:
870:
867:
861:
856:
850:
847:
842:
841:
839:
834:
831:
829:
827:
824:
821:
818:
815:
814:
791:
788:
785:
782:
762:
759:
756:
753:
744:components of
738:
737:
721:
718:
715:
712:
709:
699:
683:
680:
677:
674:
671:
657:
656:
641:
638:
635:
632:
629:
626:
623:
620:
617:
614:
611:
608:
605:
602:
599:
596:
593:
590:
587:
584:
579:
576:
573:
570:
567:
564:
561:
555:
551:
548:
545:
542:
539:
536:
529:
526:
524:
522:
517:
512:
509:
506:
495:
493:
490:
487:
486:
483:
480:
477:
474:
463:
461:
458:
455:
452:
449:
446:
445:
442:
439:
436:
433:
422:
420:
417:
414:
411:
408:
405:
402:
401:
399:
394:
391:
389:
387:
384:
381:
375:
370:
366:
365:
342:
339:
336:
333:
307:
303:
299:
296:
293:
282:
281:
270:
265:
261:
257:
254:
251:
248:
245:
242:
239:
236:
233:
210:
207:
204:
180:
160:
157:
154:
151:
127:
124:
121:
118:
98:
95:
15:
9:
6:
4:
3:
2:
5855:
5844:
5841:
5839:
5836:
5834:
5831:
5830:
5828:
5819:
5816:
5815:
5806:
5803:B. Boashash,
5802:
5799:
5795:
5792:
5788:
5787:
5780:
5777:
5769:
5759:
5758:editing guide
5753:
5749:
5744:
5735:
5734:
5719:
5717:9781586031015
5713:
5710:. IOS Press.
5709:
5708:
5700:
5692:
5690:9780824742508
5686:
5683:. CRC Press.
5682:
5681:
5673:
5665:
5663:9781118623831
5659:
5655:
5654:
5646:
5637:
5629:
5625:
5621:
5617:
5613:
5609:
5602:
5593:
5586:
5580:
5578:
5570:
5566:
5560:
5556:
5540:
5531:
5522:
5518:
5508:
5507:Causal filter
5505:
5503:
5500:
5498:
5495:
5494:
5486:
5483:
5481:
5478:
5476:
5473:
5472:
5466:
5464:
5458:
5452:
5442:
5370:
5367:
5352:
5289:
5285:
5275:
5266:
5264:
5260:
5241:
5233:
5229:
5220:
5215:
5213:
5194:
5191:
5188:
5182:
5171:
5168:
5165:
5160:
5156:
5149:
5143:
5137:
5126:
5118:
5114:
5093:
5085:
5071:
5068:
5062:
5049:
5035:
5024:
5014:
5010:
5006:
5000:
4994:
4983:
4975:
4971:
4947:
4941:
4933:
4915:
4911:
4902:
4888:
4885:
4882:
4876:
4863:
4849:
4838:
4828:
4824:
4820:
4817:
4806:
4801:
4797:
4774:
4770:
4761:
4760:
4759:
4756:
4754:
4750:
4746:
4726:
4705:
4682:
4676:
4662:
4639:
4635:
4625:
4623:
4619:
4615:
4614:interpolation
4611:
4607:
4602:
4584:
4580:
4570:
4568:
4550:
4546:
4522:
4516:
4508:
4490:
4486:
4477:
4476:
4470:
4465:
4449:
4445:
4424:
4416:
4411:
4407:
4403:
4397:
4391:
4385:
4381:
4374:
4360:
4356:
4351:
4346:
4342:
4338:
4335:
4331:
4324:
4310:
4306:
4300:
4279:
4262:
4260:
4256:
4235:
4229:
4223:
4217:
4214:
4210:
4205:
4199:
4193:
4186:
4185:
4184:
4182:
4178:
4177:
4157:
4151:
4142:
4139:
4134:
4131:
4125:
4119:
4113:
4106:
4105:
4104:
4102:
4098:
4094:
4089:
4072:
4058:
4034:
4028:
4016:
4012:
4011:
3993:
3986:
3972:
3967:
3962:
3959:
3953:
3947:
3940:
3937:
3936:
3931:
3907:
3893:
3884:
3878:
3864:
3856:
3855:
3854:
3837:
3829:
3823:
3820:
3816:
3809:
3795:
3791:
3785:
3771:
3763:
3762:
3761:
3759:
3750:
3718:
3715:
3710:
3707:
3704:
3701:
3697:
3693:
3688:
3685:
3682:
3679:
3675:
3671:
3666:
3663:
3660:
3657:
3653:
3647:
3643:
3639:
3634:
3631:
3628:
3625:
3621:
3617:
3615:
3607:
3593:
3585:
3580:
3577:
3574:
3571:
3567:
3563:
3560:
3558:
3550:
3538:
3524:
3523:
3522:
3504:
3501:
3498:
3476:
3473:
3470:
3467:
3463:
3459:
3453:
3447:
3440:
3439:
3438:
3421:
3415:
3392:
3386:
3363:
3349:
3313:
3310:
3307:
3294:
3289:
3286:
3283:
3279:
3275:
3270:
3262:
3254:
3250:
3243:
3235:
3232:
3229:
3226:
3220:
3217:
3213:
3205:
3202:
3199:
3196:
3183:
3178:
3175:
3171:
3167:
3162:
3154:
3146:
3142:
3135:
3121:
3118:
3115:
3112:
3106:
3102:
3095:
3090:
3084:
3070:
3062:
3061:
3060:
3042:
3033:
3030:
3027:
3024:
3018:
3015:
3011:
3007:
2999:
2996:
2993:
2990:
2984:
2980:
2975:
2969:
2966:
2961:
2955:
2952:
2949:
2946:
2940:
2937:
2934:
2928:
2922:
2915:
2914:
2913:
2905:
2903:
2887:
2883:
2877:
2871:
2868:
2862:
2858:
2854:
2849:
2846:
2843:
2839:
2834:
2828:
2825:
2820:
2814:
2811:
2805:
2802:
2794:
2791:, of which a
2790:
2767:
2762:
2759:
2756:
2752:
2748:
2742:
2739:
2733:
2730:
2727:
2724:
2718:
2715:
2709:
2706:
2703:
2697:
2685:
2679:
2676:
2670:
2664:
2661:
2659:
2651:
2637:
2629:
2623:
2620:
2614:
2611:
2608:
2604:
2598:
2595:
2590:
2587:
2584:
2580:
2576:
2573:
2570:
2568:
2560:
2548:
2534:
2533:
2532:
2515:
2512:
2509:
2489:
2483:
2480:
2474:
2471:
2468:
2462:
2456:
2449:
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2437:
2423:
2403:
2397:
2391:
2365:
2357:
2346:
2339:
2336:
2333:
2327:
2321:
2313:
2294:
2286:
2275:
2248:
2234:
2227:
2224:
2198:
2184:
2177:
2174:
2171:
2165:
2159:
2131:
2120:
2114:
2111:
2103:
2100:
2085:
2081:
2074:
2071:
2067:
2062:
2059:
2053:
2047:
2043:
2036:
2030:
2024:
2021:
2015:
2001:
1993:
1992:
1991:
1990:
1974:
1968:
1962:
1959:
1953:
1947:
1944:
1938:
1932:
1920:
1904:
1897:
1894:
1878:
1871:
1853:
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1700:
1684:
1678:
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1598:
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1540:
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1529:
1513:
1505:
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1498:
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1487:
1477:
1471:
1468:
1460:
1457:
1437:
1429:
1423:
1417:
1407:
1401:
1393:
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1358:
1352:
1349:
1343:
1337:
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1331:
1325:
1319:
1311:
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1296:
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1280:
1266:
1257:
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1242:
1240:
1232:
1218:
1206:
1205:
1204:
1187:
1173:
1149:
1143:
1135:
1112:
1104:
1096:
1093:
1079:
1075:
1069:
1055:
1046:
1043:
1038:
1036:
1013:
1010:
1007:
994:
989:
981:
978:
964:
958:
955:
946:
943:
940:
937:
924:
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893:
890:
887:
874:
868:
854:
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845:
837:
832:
830:
822:
816:
805:
804:
803:
786:
780:
757:
751:
743:
735:
734:sign function
716:
710:
707:
700:
697:
678:
672:
662:
661:
660:
639:
633:
627:
621:
615:
612:
609:
603:
597:
594:
588:
582:
574:
568:
565:
562:
559:
553:
546:
540:
534:
527:
525:
510:
507:
504:
491:
488:
481:
478:
475:
472:
459:
453:
447:
440:
437:
434:
431:
418:
412:
406:
403:
397:
392:
390:
382:
368:
356:
355:
354:
337:
331:
323:
305:
297:
291:
268:
263:
255:
249:
246:
240:
237:
231:
224:
223:
222:
208:
205:
202:
194:
178:
155:
149:
141:
122:
116:
103:
94:
92:
88:
83:
80:
76:
72:
68:
64:
60:
55:
53:
49:
45:
41:
37:
33:
26:
22:
5804:
5797:
5790:
5789:Leon Cohen,
5772:
5766:October 2014
5763:
5751:
5706:
5699:
5679:
5672:
5652:
5645:
5636:
5611:
5607:
5601:
5592:
5584:
5559:
5539:
5530:
5521:
5491:Applications
5462:
5456:
5448:
5283:
5281:
5272:
5216:
5209:
4931:
4757:
4752:
4748:
4626:
4605:
4571:
4472:
4468:
4466:
4268:
4251:
4174:
4172:
4100:
4096:
4090:
4020:
4014:
4008:
3933:
3929:
3852:
3755:
3520:
3340:
3058:
2911:
2786:
2530:
2311:
2151:
1988:
1925:Noting that
1924:
1763:
1133:
1131:
741:
739:
658:
283:
139:
108:
86:
84:
66:
58:
56:
46:that has no
39:
29:
5288:unit vector
4015:phase angle
1893:convolution
1584:convolution
140:real-valued
63:real-valued
32:mathematics
5827:Categories
5750:" section
5551:References
4762:Sometimes
4618:upsampling
3739:Properties
2310:comprises
97:Definition
5628:0096-3518
5426:^
5397:^
5362:^
5353:⋅
5349:ξ
5327:ξ
5303:^
5263:amplitude
5172:θ
5157:ω
5150:−
5138:ϕ
5130:∞
5122:∞
5119:−
5115:∫
5094:θ
5011:ω
5007:−
4995:ω
4987:∞
4979:∞
4976:−
4972:∫
4942:ϕ
4932:unwrapped
4912:ω
4886:ω
4864:ω
4825:ω
4821:−
4818:ω
4810:∞
4798:∫
4771:ω
4719:↓
4636:ω
4581:ω
4547:ω
4487:ω
4446:ω
4408:ω
4404:−
4392:ϕ
4343:ω
4336:−
4307:≜
4293:↓
4224:ω
4218:π
4206:≜
4135:ϕ
4126:≜
4114:ω
4093:unwrapped
3960:≜
3948:ϕ
3885:≜
3824:ϕ
3708:ω
3702:−
3694:−
3686:ω
3680:−
3664:ω
3658:−
3632:ω
3626:−
3578:ω
3572:−
3542:^
3499:ω
3474:ω
3468:−
3337:Example 3
3308:ω
3290:θ
3284:−
3276:⋅
3263:ω
3236:θ
3227:ω
3218:−
3197:ω
3179:θ
3168:⋅
3155:ω
3122:θ
3113:ω
3034:θ
3025:ω
3016:−
3000:θ
2991:ω
2956:θ
2947:ω
2941:
2908:Example 2
2872:ω
2869:−
2847:ω
2812:ω
2806:
2793:corollary
2760:ω
2740:ω
2734:
2716:ω
2710:
2689:^
2621:ω
2615:
2596:π
2591:−
2585:ω
2577:
2552:^
2510:ω
2481:ω
2475:
2445:Example 1
2358:∗
2340:
2287:∗
2228:
2178:
2101:−
2086:⏟
2072:π
2048:δ
2037:∗
1963:δ
1960:∗
1895:operator;
1879:∗
1806:
1796:≜
1781:^
1730:^
1673:
1660:⏟
1640:∗
1631:π
1578:⏞
1557:⏟
1530:−
1514:∗
1503:π
1488:⏟
1472:
1458:−
1418:⏟
1391:−
1350:⋅
1338:
1309:−
1255:−
1243:≜
1105:∗
1094:−
990:∗
979:−
711:
673:
616:
569:
554:⏟
541:
393:≜
306:∗
264:∗
238:−
193:Hermitian
5480:I/Q data
5469:See also
5259:envelope
4622:aliasing
4606:analytic
4475:baseband
4473:complex
3935:envelope
3491:, where
2440:Examples
87:analytic
75:spectrum
4610:sampled
3932:or the
1917:is the
1836:is the
732:is the
694:is the
320:is the
171:(where
5746:This "
5714:
5687:
5660:
5626:
4437:where
4248:
3521:Then:
3305:
3247:
3194:
3139:
3133:
3130:
3059:Then:
2531:Then:
2152:Since
1764:where
1017:
1005:
935:
885:
659:where
502:
470:
429:
284:where
221:axis:
91:phasor
5567:, or
5513:Notes
4695:then
4509:. If
4181:hertz
138:is a
61:of a
42:is a
38:, an
5712:ISBN
5685:ISBN
5658:ISBN
5624:ISSN
5459:+ 1)
5368:<
4471:and
4179:(in
4173:The
3502:>
3311:<
3200:>
2513:>
2416:The
2312:only
1132:The
1011:<
891:>
508:<
435:>
73:(or
57:The
34:and
5616:doi
5106:):
4751:or
4627:If
4013:or
3963:arg
2938:cos
2803:cos
2795:is
2731:sin
2707:cos
2612:sin
2574:cos
2472:cos
1840:of
1469:sgn
1335:sgn
1136:of
1001:for
931:for
881:for
708:sgn
613:sgn
566:sgn
498:for
466:for
425:for
324:of
109:If
30:In
23:or
5829::
5622:.
5612:27
5610:.
5576:^
5265:.
4963::
4755:.
4624:.
4569:.
4103::
4088:.
3760::
3719:0.
3437:.
3314:0.
3301:if
3190:if
2516:0.
2424:Re
2337:Re
2225:Im
2175:Re
1203::
802::
54:.
5779:)
5773:(
5768:)
5764:(
5754:.
5720:.
5693:.
5666:.
5630:.
5618::
5463:n
5457:n
5455:(
5423:u
5394:u
5371:0
5359:u
5300:u
5245:)
5242:t
5239:(
5234:m
5230:s
5195:.
5192:t
5189:d
5183:2
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5169:+
5166:t
5161:0
5153:(
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5144:t
5141:(
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4998:(
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4984:+
4951:)
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4889:.
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2016:t
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2007:a
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