419:. (0,0,0) (1/2,1/2,-1/2) (1/3,1/3,0) (1/2,0,0) (1/4,1/4,1/4). The plane containing the points (1/6,1/6,1/6) (1/4,1/4,0) (1/2,0,0) divides this volume into an inner and an outer region. A stationary signal will have zero strength (statistically) in the outer region. The trispectrum support is divided into regions by the plane identified above and by the (f1,f2) plane. Each region has different requirements in terms of the bandwidth of signal required for non-zero values.
360:
has been proven to be the corresponding correlation coefficient. Just as correlation cannot sufficiently demonstrate the presence of causality, spectrum and bicoherence also cannot sufficiently substantiate the existence of an nonlinear interaction.
355:
The bispectrum reflects the energy budget of interactions, as it can be interpreted as a covariance defined between energy-supplying and energy-receiving parties of waves involved in an nonlinear interaction. On the other hand,
229:
317:
describes observations made at two wavelengths. It is often used by scientists to analyze elemental makeup of a planetary atmosphere by analyzing the amount of light reflected and received through various color
652:
Nemer, Elias J. (1999). "Speech analysis and quality enhancement using higher order cumulants" (Document). Ottawa: Ottawa-Carleton
Institute for Electrical and Computer Engineering.
433:
The trispectrum has been used to investigate the domains of applicability of maximum kurtosis phase estimation used in the deconvolution of seismic data to find layer structure.
276:
249:
377:
and provide supplementary information to the power spectrum. The third order polyspectrum (bispectrum) is the easiest to compute, and hence the most popular.
347:
Biphase (phase of polyspectrum) can be used for detection of phase couplings, noise reduction of polharmonic (particularly, speech ) signal analysis.
726:
Mendel JM (1991). "Tutorial on higher-order statistics (spectra) in signal processing and system theory: theoretical results and some applications".
99:
786:
330:
photographs that, while not particularly useful for scientific analysis, are popular for public display in textbooks and fund raising campaigns.
791:
753:
toolbox for spectral and polyspectral analysis, and time-frequency distributions. The documentation explains polyspectra in great detail.
449:
Greb U, Rusbridge MG (1988). "The interpretation of the bispectrum and bicoherence for non-linear interactions of continuous spectra".
641:
Fackrell, Justin W. A. (September 1996). "Bispectral analysis of speech signals" (Document). Edinburgh: The
University of Edinburgh.
415:
of the trispectrum allow a much reduced support set to be defined, contained within the following vertices, where 1 is the
771:
304:
signals monitoring. It was also shown that bispectra characterize differences between families of musical instruments.
297:
may be applied to the case of non-linear interactions of a continuous spectrum of propagating waves in one dimension.
411:, and provides supplementary information to the power spectrum. The trispectrum is a three-dimensional construct. The
776:
766:
322:. By combining and removing two filters, much can be gleaned from only two filters. Through modern computerized
301:
333:
Bispectral analysis can also be used to analyze interactions between wave patterns and tides on Earth.
781:
581:
458:
311:, signals rarely have adequate duration for making sensible bispectral estimates from time averages.
21:
396:
The
Fourier transform of C4 (t1, t2, t3) (fourth-order cumulant-generating function) is called the
93:
523:
474:
408:
541:
Dubnov S, Tishby N and Cohen D. (1997). "Polyspectra as
Measures of Sound Texture and Timbre".
705:
687:
618:
515:
416:
37:
735:
695:
679:
589:
550:
505:
466:
337:
746:
426:
as a function of frequency triples, the trispectrum identifies contributions to a signal's
254:
319:
287:
45:
570:"Evidence for nonlinear coupling of planetary waves and tides in the Antarctic mesopause"
585:
462:
700:
494:"Development and clinical application of electroencephalographic bispectrum monitoring"
234:
49:
568:
Kamalabadi, F.; Forbes, J. M.; Makarov, N. M.; Portnyagin, Yu. I. (27 February 1997).
760:
667:
510:
493:
478:
470:
323:
527:
407:
The trispectrum T(f1,f2,f3) falls into the category of higher-order spectra, or
357:
327:
294:
17:
683:
610:
554:
308:
691:
224:{\displaystyle B(f_{1},f_{2})=F(f_{1})\cdot F(f_{2})\cdot F^{*}(f_{1}+f_{2})}
709:
622:
519:
422:
In the same way that the bispectrum identifies contributions to a signal's
668:"Rossby wave second harmonic generation observed in the middle atmosphere"
286:"Bispectral analysis" redirects here. For the speckle imaging method, see
427:
423:
412:
77:
41:
609:
Mathur, Surbhi; Patel, Jashvin; Goldstein, Sheldon; Jain, Ankit (2021),
594:
569:
750:
739:
341:
28:
is a statistic used to search for nonlinear interactions.
257:
237:
102:
326:, a third virtual filter can be created to recreate
300:Bispectral measurements have been carried out for
270:
243:
223:
80:-generating function) is called the bispectrum or
251:denotes the Fourier transform of the signal, and
666:He, Maosheng; Forbes, Jeffrey M. (2022-12-07).
617:, Treasure Island (FL): StatPearls Publishing,
747:HOSA - Higher Order Spectral Analysis Toolbox
8:
574:Journal of Geophysical Research: Atmospheres
96:allows fast calculation of the bispectrum:
344:waveforms to monitor depth of anesthesia.
699:
593:
509:
336:A form of bispectral analysis called the
262:
256:
236:
212:
199:
186:
170:
148:
126:
113:
101:
430:as a function of frequency quadruplets.
492:Johansen JW, Sebel PS (November 2000).
441:
380:A statistic defined analogously is the
7:
661:
659:
369:Bispectra fall in the category of
14:
511:10.1097/00000542-200011000-00029
787:Nonlinear time series analysis
218:
192:
176:
163:
154:
141:
132:
106:
1:
792:Statistical signal processing
543:Journal of New Music Research
48:function, is the traditional
451:Plasma Phys. Control. Fusion
808:
684:10.1038/s41467-022-35142-3
471:10.1088/0741-3335/30/5/005
285:
555:10.1080/09298219708570732
351:A physical interpretation
55:The Fourier transform of
272:
245:
225:
672:Nature Communications
273:
271:{\displaystyle F^{*}}
246:
226:
382:bispectral coherency
371:higher-order spectra
255:
235:
100:
40:of the second-order
22:statistical analysis
772:Integral transforms
586:1997JGR...102.4437K
463:1988PPCF...30..537G
402:trispectral density
315:Bispectral analysis
94:convolution theorem
611:"Bispectral Index"
268:
241:
221:
82:bispectral density
595:10.1029/96JD01996
580:(D4): 4437–4446.
417:Nyquist frequency
244:{\displaystyle F}
38:Fourier transform
20:, in the area of
799:
777:Fourier analysis
767:Complex analysis
743:
714:
713:
703:
663:
654:
653:
649:
643:
642:
638:
632:
631:
630:
629:
606:
600:
599:
597:
565:
559:
558:
538:
532:
531:
513:
489:
483:
482:
446:
338:bispectral index
277:
275:
274:
269:
267:
266:
250:
248:
247:
242:
230:
228:
227:
222:
217:
216:
204:
203:
191:
190:
175:
174:
153:
152:
131:
130:
118:
117:
807:
806:
802:
801:
800:
798:
797:
796:
757:
756:
740:10.1109/5.75086
725:
722:
720:Further reading
717:
665:
664:
657:
651:
650:
646:
640:
639:
635:
627:
625:
608:
607:
603:
567:
566:
562:
540:
539:
535:
491:
490:
486:
448:
447:
443:
439:
394:
367:
365:Generalizations
353:
293:Bispectrum and
291:
288:Speckle masking
284:
278:its conjugate.
258:
253:
252:
233:
232:
208:
195:
182:
166:
144:
122:
109:
98:
97:
90:
76:) (third-order
75:
68:
61:
46:autocorrelation
34:
12:
11:
5:
805:
803:
795:
794:
789:
784:
779:
774:
769:
759:
758:
755:
754:
744:
734:(3): 278–305.
721:
718:
716:
715:
655:
644:
633:
601:
560:
549:(4): 277–314.
533:
504:(5): 1336–44.
498:Anesthesiology
484:
440:
438:
435:
393:
390:
366:
363:
352:
349:
340:is applied to
283:
280:
265:
261:
240:
220:
215:
211:
207:
202:
198:
194:
189:
185:
181:
178:
173:
169:
165:
162:
159:
156:
151:
147:
143:
140:
137:
134:
129:
125:
121:
116:
112:
108:
105:
89:
86:
73:
66:
59:
50:power spectrum
33:
30:
13:
10:
9:
6:
4:
3:
2:
804:
793:
790:
788:
785:
783:
780:
778:
775:
773:
770:
768:
765:
764:
762:
752:
748:
745:
741:
737:
733:
729:
724:
723:
719:
711:
707:
702:
697:
693:
689:
685:
681:
677:
673:
669:
662:
660:
656:
648:
645:
637:
634:
624:
620:
616:
612:
605:
602:
596:
591:
587:
583:
579:
575:
571:
564:
561:
556:
552:
548:
544:
537:
534:
529:
525:
521:
517:
512:
507:
503:
499:
495:
488:
485:
480:
476:
472:
468:
464:
460:
457:(5): 537–49.
456:
452:
445:
442:
436:
434:
431:
429:
425:
420:
418:
414:
410:
405:
403:
399:
391:
389:
387:
383:
378:
376:
372:
364:
362:
359:
350:
348:
345:
343:
339:
334:
331:
329:
325:
324:interpolation
321:
316:
312:
310:
305:
303:
298:
296:
289:
281:
279:
263:
259:
238:
213:
209:
205:
200:
196:
187:
183:
179:
171:
167:
160:
157:
149:
145:
138:
135:
127:
123:
119:
114:
110:
103:
95:
92:Applying the
87:
85:
83:
79:
72:
65:
58:
53:
51:
47:
43:
39:
31:
29:
27:
23:
19:
731:
727:
675:
671:
647:
636:
626:, retrieved
614:
604:
577:
573:
563:
546:
542:
536:
501:
497:
487:
454:
450:
444:
432:
421:
406:
401:
397:
395:
385:
381:
379:
374:
370:
368:
354:
346:
335:
332:
314:
313:
306:
299:
292:
282:Applications
91:
81:
70:
63:
56:
54:
44:, i.e., the
35:
25:
15:
782:Time series
678:(1): 7544.
409:polyspectra
398:trispectrum
392:Trispectrum
386:bicoherence
375:polyspectra
358:bicoherence
295:bicoherence
88:Calculation
32:Definitions
18:mathematics
761:Categories
728:Proc. IEEE
628:2021-04-08
615:StatPearls
437:References
413:symmetries
328:true color
309:seismology
26:bispectrum
692:2041-1723
479:250741815
264:∗
188:∗
180:⋅
158:⋅
710:36476614
623:30969631
520:11046224
428:kurtosis
424:skewness
231:, where
78:cumulant
42:cumulant
701:9729661
582:Bibcode
459:Bibcode
320:filters
69:,
751:MATLAB
708:
698:
690:
621:
528:379085
526:
518:
477:
24:, the
524:S2CID
475:S2CID
373:, or
749:: A
706:PMID
688:ISSN
619:PMID
516:PMID
36:The
736:doi
696:PMC
680:doi
590:doi
578:102
551:doi
506:doi
467:doi
400:or
384:or
342:EEG
307:In
302:EEG
84:.
52:.
16:In
763::
732:79
730:.
704:.
694:.
686:.
676:13
674:.
670:.
658:^
613:,
588:.
576:.
572:.
547:26
545:.
522:.
514:.
502:93
500:.
496:.
473:.
465:.
455:30
453:.
404:.
388:.
742:.
738::
712:.
682::
598:.
592::
584::
557:.
553::
530:.
508::
481:.
469::
461::
290:.
260:F
239:F
219:)
214:2
210:f
206:+
201:1
197:f
193:(
184:F
177:)
172:2
168:f
164:(
161:F
155:)
150:1
146:f
142:(
139:F
136:=
133:)
128:2
124:f
120:,
115:1
111:f
107:(
104:B
74:2
71:t
67:1
64:t
62:(
60:3
57:C
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.