1700:
frequency burst. On the other hand, as the STFT consists of a constant window width, it leads to the result having poorer definition. In the second experiment, two more high frequency bursts are added to crossed chirps. In the result, all four frequencies were detected by the S transform. On the other hand, the two high frequencies bursts are not detected by STFT. The high frequencies bursts cross term caused STFT to have a single frequency at lower frequency.
1649:, meanwhile, the window function for S-Transform is a function of f. With a window function proportional to frequency, S Transform performs well in frequency domain analysis when the input frequency is low. When the input frequency is high, S-Transform has a better clarity in the time domain. As table below.
74:
transform algorithm was invented in 2010. It reduces the computational complexity from O to O and makes the transform one-to-one, where the transform has the same number of points as the source signal or image, compared to storage complexity of N for the original formulation. An implementation is
2101:
Rocco
Ditommaso, Marco Mucciarelli, Felice C. Ponzo (2010). S-Transform based filter applied to the analysis of non-linear dynamic behaviour of soil and buildings. 14th European Conference on Earthquake Engineering. Proceedings Volume. Ohrid, Republic of Macedonia. August 30 – September 3, 2010.
1699:
transform and short-time
Fourier transform (STFT). First, a high frequency signal, a low frequency signal, and a high frequency burst signal are used in the experiment to compare the performance. The S transform characteristic of frequency dependent resolution allows the detection of the high
1193:
851:
2097:
Rocco
Ditommaso, Felice Carlo Ponzo, Gianluca Auletta (2015). Damage detection on framed structures: modal curvature evaluation using Stockwell Transform under seismic excitation. Earthquake Engineering and Engineering Vibration. June 2015, Volume 14, Issue 2, pp
273:
1686:
The main problem with the Wigner
Transform is the cross term, which stems from the auto-correlation function in the Wigner Transform function. This cross term may cause noise and distortions in signal analyses. S-transform analyses avoid this issue.
1921:
Brown, Robert A.; Lauzon, M. Louis; Frayne, Richard (January 2010). "A General
Description of Linear Time-Frequency Transforms and Formulation of a Fast, Invertible Transform That Samples the Continuous S-Transform Spectrum Nonredundantly".
2109:
M. Mucciarelli, M. Bianca, R. Ditommaso, M.R. Gallipoli, A. Masi, C Milkereit, S. Parolai, M. Picozzi, M. Vona (2011). FAR FIELD DAMAGE ON RC BUILDINGS: THE CASE STUDY OF NAVELLI DURING THE L’AQUILA (ITALY) SEISMIC SEQUENCE, 2009.
421:
940:
2128:
Jaya
Bharata Reddy, Dusmanta Kumar Mohanta, and B. M. Karan, "Power system disturbance recognition using wavelet and s-transform techniques," Birla institute of Technology, Mesra, Ranchi-835215, 2004.
1540:
1005:
706:
2067:
Ray, Prakash K.; Mohanty, Soumya R.; Kishor, Nand; Dubey, Harish C. (2010). "Coherency determination in grid-connected distributed generation based hybrid system under islanding scenarios".
1300:
43:
and overcoming some of its disadvantages. For one, modulation sinusoids are fixed with respect to the time axis; this localizes the scalable
Gaussian window dilations and translations in
629:
494:
113:
1647:
698:
563:
1336:
2125:
J. J. Ding, "Time-frequency analysis and wavelet transform course note," the
Department of Electrical Engineering, National Taiwan University (NTU), Taipei, Taiwan, 2007.
994:
967:
82:
A general formulation of the S transform makes clear the relationship to other time frequency transforms such as the
Fourier, short time Fourier, and wavelet transforms.
2131:
B. Boashash, "Notes on the use of the wigner distribution for time frequency signal analysis", IEEE Trans. on Acoust. Speech. and Signal
Processing, vol. 26, no. 9, 1987
1724:
transform has been proven to be able to identify a few types of disturbances, like voltage sag, voltage swell, momentary interruption, and oscillatory transients.
1245:
2057:
Hongmei Zhu, and J. Ross Mitchell, "The S Transform in Medical Imaging," University of Calgary Seaman Family MR Research Centre Foothills Medical Centre, Canada.
1576:
1427:
1376:
1855:-transform analysis of gravity wave activity from a small scale network of airglow imagers. PhD thesis, University of Western Ontario, London, Ontario, Canada.
2231:
F. Hlawatsch and G. F. Boudreuax-Bartels, 1992 "Linear and quadratic timefrequency signal representations", IEEE Signal Processing Magazine, pp. 21–67
286:
1736:
transform generates contours which are suitable for simple visual inspection. However, wavelet transform requires specific tools like standard
64:
867:
2084:
1882:
1678:
This kind of property makes S-Transform a powerful tool to analyze sound because human is sensitive to low frequency part in a sound signal.
1591:
The only difference between the Gabor transform (GT) and the S transform is the window size. For GT, the windows size is a Gaussian function
2301:
2111:
1998:
E. Sejdić, I. Djurović, J. Jiang, "Time-frequency feature representation using energy concentration: An overview of recent advances,"
1433:
1188:{\displaystyle S_{x}(n\Delta _{T}\,,m\Delta _{F})=\sum _{p=0}^{N-1}X\,e^{-\pi {\frac {p^{2}}{m^{2}}}}\,e^{\frac {j2pn}{N}}}
846:{\displaystyle S_{x}(t,f)=\int _{-\infty }^{\infty }X(f+\alpha )\,e^{-\pi \alpha ^{2}/f^{2}}\,e^{j2\pi \alpha t}\,d\alpha }
2291:
1777:
60:
40:
36:
1251:
268:{\displaystyle S_{x}(t,f)=\int _{-\infty }^{\infty }x(\tau )|f|e^{-\pi (t-\tau )^{2}f^{2}}e^{-j2\pi f\tau }\,d\tau }
1712:
2296:
574:
439:
1737:
1594:
634:
499:
2211:
Goupillaud, P.; Grossmann, A.; Morlet, J. (1984). "Cycle-octave and related transforms in seismic analysis".
1730:
transform also be applied for other types of disturbances such as notches, harmonics with sag and swells etc.
2046:
2154:
1813:
2013:
1748:
98:
transform is derived as the phase correction of the continuous wavelet transform with window being the
31:
as a time–frequency distribution was developed in 1994 for analyzing geophysics data. In this way, the
1796:
Stockwell, RG; Mansinha, L; Lowe, RP (1996). "Localization of the complex spectrum: the S transform".
2251:
2159:
2134:
R. N. Bracewell, The Fourier Transform and Its Applications, McGraw Hill Book Company, New York, 1978
2028:
1931:
1805:
1818:
1305:
76:
2267:
2207:
D. Gabor, "Theory of communication", J. Inst. Elect. Eng., vol. 93, no. 3, pp. 429–457, 1946
1955:
1896:
1867:
2008 30th Annual International Conference of the IEEE Engineering in Medicine and Biology Society
1831:
972:
945:
436:
The above definition implies that the s-transform function can be express as the convolution of
16:
This article is about the time–frequency transform. For the mathematical use of this term, see
2236:
2080:
1947:
1888:
1878:
1772:
1767:
568:
99:
17:
1865:
Brown, RA; Frayne, R (2008). "A fast discrete S-transform for biomedical signal processing".
2278:
R. K. Young, Wavelet Theory and its Applications, Kluwer Academic Publishers, Dordrecht,1993
2259:
2220:
2196:
2164:
2115:
2072:
2036:
1939:
1870:
1823:
1973:
52:
2175:
I. Daubechies, "The wavelet transform, time-frequency localization and signal analysis",
1210:
2255:
2032:
1935:
1809:
1546:
1381:
1342:
2285:
2224:
2271:
1959:
1900:
1835:
862:
From the spectrum form of S-transform, we can derive the discrete-time S-transform.
51:
transform doesn't have a cross-term problem and yields a better signal clarity than
2200:
2076:
2014:"Analysis of non-stationary structural systems by using a band-variable filter"
1874:
2119:
2040:
1753:
416:{\displaystyle x(\tau )=\int _{-\infty }^{\infty }\left\,e^{j2\pi f\tau }\,df}
1951:
1943:
1985:
1892:
935:{\displaystyle t=n\Delta _{T}\,\,f=m\Delta _{F}\,\,\alpha =p\Delta _{F}}
1827:
2263:
2184:
2168:
2012:
Ditommaso, Rocco; Mucciarelli, Marco; Ponzo, Felice Carlo (2012).
2103:
59:
transform has its own disadvantages: the clarity is worse than
2145:
Cohen, L. (1989). "Time-frequency distributions—A review".
1972:
Kelly Sansom, "Fast S Transform", University of Calgary,
2185:"Wavelet transforms and their application to turbulence"
2141:, Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1974
1974:
https://www.ucalgary.ca/news/utoday/may31-2011/computing
1535:{\displaystyle B=X\cdot e^{-\pi {\frac {p^{2}}{m^{2}}}}}
998:
The Discrete time S-transform can then be expressed as:
2069:
2010 IEEE International Conference on Power and Energy
1597:
1549:
1436:
1384:
1345:
1308:
1254:
1213:
1008:
975:
948:
870:
709:
637:
577:
502:
442:
289:
116:
90:
There are several ways to represent the idea of the
1582:
Comparison with other time–frequency analysis tools
1641:
1570:
1534:
1421:
1370:
1330:
1294:
1239:
1187:
988:
961:
934:
845:
692:
623:
557:
488:
415:
267:
1691:Comparison with the short-time Fourier transform
1847:
1845:
1295:{\displaystyle e^{-\pi {\frac {p^{2}}{m^{2}}}}}
1203:Below is the Pseudo code of the implementation.
1986:"Fast S-Transform download | SourceForge.net"
75:available to the research community under an
8:
2002:, vol. 19, no. 1, pp. 153-183, January 2009.
1199:Implementation of discrete-time S-transform
624:{\displaystyle (x(\tau )e^{-j2\pi f\tau })}
489:{\displaystyle (x(\tau )e^{-j2\pi f\tau })}
2158:
1817:
1628:
1605:
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1548:
1522:
1512:
1506:
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1111:
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1007:
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953:
947:
926:
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911:
905:
891:
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884:
869:
836:
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813:
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750:
742:
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708:
679:
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183:
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149:
121:
115:
1651:
1642:{\displaystyle (e^{-\pi (t-\tau )^{2}})}
1248:loop over m (voices) Step2.Compute
693:{\displaystyle (|f|e^{-\pi t^{2}f^{2}})}
558:{\displaystyle (|f|e^{-\pi t^{2}f^{2}})}
1788:
1924:IEEE Transactions on Signal Processing
1798:IEEE Transactions on Signal Processing
1718:Power system disturbance recognition
35:transform is a generalization of the
7:
1916:
1914:
1912:
1910:
1869:. Vol. 2008. pp. 2586–9.
65:Cohen's class distribution function
2112:Bulletin of Earthquake Engineering
2021:Bulletin of Earthquake Engineering
1480:
1407:
1356:
1319:
1224:
1108:
1043:
1026:
977:
950:
923:
902:
881:
751:
746:
341:
336:
318:
313:
158:
153:
14:
2177:IEEE Trans. on Information Theory
1661:Good clarity in frequency domain
2237:"Wavelets and signal processing"
2235:Rioul, O.; Vetterli, M. (1991).
2189:Annual Review of Fluid Mechanics
1682:Comparison with Wigner transform
2244:IEEE Signal Processing Magazine
1669:Bad clarity in frequency domain
1587:Comparison with Gabor transform
1430:Step4.Multiply Step2 and Step3
2201:10.1146/annurev.fluid.24.1.395
1636:
1625:
1612:
1598:
1565:
1553:
1489:
1476:
1464:
1461:
1452:
1440:
1416:
1403:
1391:
1388:
1365:
1349:
1331:{\displaystyle f=m\Delta _{F}}
1233:
1217:
1117:
1103:
1091:
1088:
1052:
1019:
771:
759:
732:
720:
687:
650:
642:
638:
618:
590:
584:
578:
552:
515:
507:
503:
483:
455:
449:
443:
368:
356:
299:
293:
212:
199:
184:
176:
172:
166:
139:
127:
1:
2104:http://roccoditommaso.xoom.it
969:is the sampling interval and
2225:10.1016/0016-7142(84)90025-5
2179:, vol. 36, no. 5, Sept. 1990
1778:Short-time Fourier transform
1745:Geophysical signal analysis
1672:Good clarity in time domain
61:Wigner distribution function
41:continuous wavelet transform
37:short-time Fourier transform
989:{\displaystyle \Delta _{F}}
962:{\displaystyle \Delta _{T}}
2318:
2139:The Fast Fourier Transform
2077:10.1109/PECON.2010.5697562
1875:10.1109/IEMBS.2008.4649729
1713:Magnetic resonance imaging
1658:Bad clarity in time domain
996:is the sampling frequency.
15:
2120:10.1007/s10518-010-9201-y
2041:10.1007/s10518-012-9338-y
2000:Digital Signal Processing
858:Discrete-time S-transform
47:transform. Moreover, the
1944:10.1109/tsp.2009.2028972
1738:multiresolution analysis
2302:Time–frequency analysis
1851:Stockwell, RG (1999).
1643:
1572:
1536:
1423:
1372:
1332:
1296:
1241:
1189:
1084:
990:
963:
936:
847:
694:
625:
559:
490:
417:
269:
39:(STFT), extending the
1749:Reflection seismology
1644:
1573:
1537:
1424:
1373:
1333:
1297:
1242:
1190:
1058:
991:
964:
937:
848:
695:
626:
560:
491:
418:
270:
1595:
1547:
1434:
1382:
1343:
1306:
1252:
1211:
1006:
973:
946:
868:
707:
635:
575:
500:
440:
287:
114:
94:transform. In here,
2292:Integral transforms
2256:1991ISPM....8...14R
2102:(downloadable from
2033:2012BuEE...10..895D
1936:2010ITSP...58..281B
1810:1996ITSP...44..998S
1695:We can compare the
1240:{\displaystyle X\,}
755:
345:
322:
279:Inverse S-Transform
162:
77:open source license
2183:Farge, M. (1992).
2071:. pp. 85–88.
1639:
1568:
1532:
1419:
1368:
1328:
1292:
1237:
1185:
986:
959:
932:
843:
738:
690:
621:
555:
486:
413:
328:
305:
265:
145:
2086:978-1-4244-8947-3
1988:. 13 August 2018.
1884:978-1-4244-1814-5
1828:10.1109/78.492555
1773:Wavelet transform
1768:Laplace transform
1754:Global seismology
1709:Signal filterings
1676:
1675:
1571:{\displaystyle B}
1528:
1422:{\displaystyle X}
1371:{\displaystyle X}
1288:
1182:
1154:
569:Fourier transform
100:Gaussian function
18:Laplace transform
2309:
2297:Fourier analysis
2275:
2264:10.1109/79.91217
2241:
2228:
2204:
2172:
2162:
2160:10.1.1.1026.2853
2091:
2090:
2064:
2058:
2055:
2049:
2044:
2018:
2009:
2003:
1996:
1990:
1989:
1982:
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1963:
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1904:
1862:
1856:
1849:
1840:
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1821:
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987:
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2210:
2182:
2169:10.1109/5.30749
2144:
2137:E. O. Brigham,
2094:
2087:
2066:
2065:
2061:
2056:
2052:
2016:
2011:
2010:
2006:
1997:
1993:
1984:
1983:
1979:
1971:
1967:
1920:
1919:
1908:
1885:
1864:
1863:
1859:
1850:
1843:
1819:10.1.1.462.1500
1804:(4): 998–1001.
1795:
1794:
1790:
1786:
1764:
1706:
1693:
1684:
1624:
1601:
1593:
1592:
1589:
1584:
1579:
1545:
1544:
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1518:
1508:
1495:
1479:
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1406:
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1379:
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1303:
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949:
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922:
901:
880:
866:
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863:
814:
801:
786:
775:
710:
705:
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675:
665:
654:
633:
632:
593:
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572:
566:
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519:
498:
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458:
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429:
384:
346:
327:
323:
285:
284:
233:
221:
211:
188:
117:
112:
111:
88:
55:. However, the
53:Gabor transform
21:
12:
11:
5:
2315:
2313:
2305:
2304:
2299:
2294:
2284:
2283:
2280:
2279:
2276:
2232:
2229:
2213:Geoexploration
2208:
2205:
2180:
2173:
2153:(7): 941–981.
2142:
2135:
2132:
2129:
2126:
2123:
2107:
2099:
2093:
2092:
2085:
2059:
2050:
2027:(3): 895–911.
2004:
1991:
1977:
1965:
1930:(1): 281–290.
1906:
1883:
1857:
1841:
1787:
1785:
1782:
1781:
1780:
1775:
1770:
1763:
1760:
1759:
1758:
1757:
1756:
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1741:
1731:
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1716:
1710:
1705:
1702:
1692:
1689:
1683:
1680:
1674:
1673:
1670:
1667:
1666:High-frequency
1663:
1662:
1659:
1656:
1638:
1631:
1627:
1623:
1620:
1617:
1614:
1611:
1608:
1604:
1600:
1588:
1585:
1583:
1580:
1578:). Repeat.}
1567:
1564:
1561:
1558:
1555:
1552:
1525:
1521:
1515:
1511:
1505:
1502:
1498:
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1460:
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1413:
1409:
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1402:
1399:
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1367:
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1325:
1321:
1317:
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1311:
1285:
1281:
1275:
1271:
1265:
1262:
1258:
1235:
1230:
1226:
1222:
1219:
1216:
1207:Step1.Compute
1206:
1200:
1197:
1196:
1195:
1181:
1177:
1174:
1171:
1168:
1162:
1151:
1147:
1141:
1137:
1131:
1128:
1124:
1119:
1114:
1110:
1105:
1102:
1099:
1096:
1093:
1090:
1087:
1082:
1079:
1076:
1071:
1068:
1065:
1061:
1057:
1054:
1049:
1045:
1041:
1038:
1032:
1028:
1024:
1021:
1016:
1012:
983:
979:
956:
952:
929:
925:
921:
918:
915:
908:
904:
900:
897:
894:
887:
883:
879:
876:
873:
860:
859:
855:
854:
842:
839:
833:
830:
827:
824:
821:
817:
808:
804:
799:
793:
789:
785:
782:
778:
773:
770:
767:
764:
761:
758:
753:
748:
745:
741:
737:
734:
731:
728:
725:
722:
717:
713:
689:
682:
678:
672:
668:
664:
661:
657:
652:
648:
644:
640:
620:
615:
612:
609:
606:
603:
600:
596:
592:
589:
586:
583:
580:
554:
547:
543:
537:
533:
529:
526:
522:
517:
513:
509:
505:
485:
480:
477:
474:
471:
468:
465:
461:
457:
454:
451:
448:
445:
434:
433:
428:
425:
424:
423:
412:
409:
403:
400:
397:
394:
391:
387:
381:
377:
374:
370:
367:
364:
361:
358:
353:
349:
343:
338:
335:
331:
326:
320:
315:
312:
308:
304:
301:
298:
295:
292:
281:
280:
276:
275:
264:
261:
255:
252:
249:
246:
243:
240:
236:
228:
224:
218:
214:
210:
207:
204:
201:
198:
195:
191:
186:
182:
178:
174:
171:
168:
165:
160:
155:
152:
148:
144:
141:
138:
135:
132:
129:
124:
120:
108:
107:
87:
84:
13:
10:
9:
6:
4:
3:
2:
2314:
2303:
2300:
2298:
2295:
2293:
2290:
2289:
2287:
2277:
2273:
2269:
2265:
2261:
2257:
2253:
2249:
2245:
2238:
2233:
2230:
2226:
2222:
2218:
2214:
2209:
2206:
2202:
2198:
2194:
2190:
2186:
2181:
2178:
2174:
2170:
2166:
2161:
2156:
2152:
2148:
2143:
2140:
2136:
2133:
2130:
2127:
2124:
2121:
2117:
2113:
2108:
2105:
2100:
2096:
2095:
2088:
2082:
2078:
2074:
2070:
2063:
2060:
2054:
2051:
2048:
2042:
2038:
2034:
2030:
2026:
2022:
2015:
2008:
2005:
2001:
1995:
1992:
1987:
1981:
1978:
1975:
1969:
1966:
1961:
1957:
1953:
1949:
1945:
1941:
1937:
1933:
1929:
1925:
1917:
1915:
1913:
1911:
1907:
1902:
1898:
1894:
1890:
1886:
1880:
1876:
1872:
1868:
1861:
1858:
1854:
1848:
1846:
1842:
1837:
1833:
1829:
1825:
1820:
1815:
1811:
1807:
1803:
1799:
1792:
1789:
1783:
1779:
1776:
1774:
1771:
1769:
1766:
1765:
1761:
1755:
1752:
1750:
1747:
1746:
1744:
1739:
1735:
1732:
1729:
1726:
1723:
1720:
1719:
1717:
1714:
1711:
1708:
1707:
1703:
1701:
1698:
1690:
1688:
1681:
1679:
1671:
1668:
1665:
1664:
1660:
1657:
1655:Low-frequency
1654:
1653:
1650:
1629:
1621:
1618:
1615:
1609:
1606:
1602:
1586:
1581:
1562:
1559:
1556:
1550:
1523:
1519:
1513:
1509:
1503:
1500:
1496:
1492:
1484:
1473:
1470:
1467:
1458:
1455:
1449:
1446:
1443:
1437:
1411:
1400:
1397:
1394:
1385:
1360:
1352:
1346:
1323:
1315:
1312:
1309:
1283:
1279:
1273:
1269:
1263:
1260:
1256:
1228:
1220:
1214:
1205:
1198:
1179:
1175:
1172:
1169:
1166:
1160:
1149:
1145:
1139:
1135:
1129:
1126:
1122:
1112:
1100:
1097:
1094:
1085:
1080:
1077:
1074:
1069:
1066:
1063:
1059:
1055:
1047:
1039:
1036:
1030:
1022:
1014:
1010:
1002:
1001:
1000:
981:
954:
927:
919:
916:
913:
906:
898:
895:
892:
885:
877:
874:
871:
857:
856:
840:
837:
831:
828:
825:
822:
819:
815:
806:
802:
797:
791:
787:
783:
780:
776:
768:
765:
762:
756:
743:
739:
735:
729:
726:
723:
715:
711:
703:
702:
701:
680:
676:
670:
666:
662:
659:
655:
646:
613:
610:
607:
604:
601:
598:
594:
587:
581:
570:
567:Applying the
545:
541:
535:
531:
527:
524:
520:
511:
478:
475:
472:
469:
466:
463:
459:
452:
446:
432:Spectrum Form
431:
430:
427:Modified form
426:
410:
407:
401:
398:
395:
392:
389:
385:
379:
375:
372:
365:
362:
359:
351:
347:
333:
329:
324:
310:
306:
302:
296:
290:
283:
282:
278:
277:
262:
259:
253:
250:
247:
244:
241:
238:
234:
226:
222:
216:
208:
205:
202:
196:
193:
189:
180:
169:
163:
150:
146:
142:
136:
133:
130:
122:
118:
110:
109:
105:
104:
103:
101:
97:
93:
85:
83:
80:
78:
73:
68:
66:
62:
58:
54:
50:
46:
42:
38:
34:
30:
27:
26:
19:
2250:(4): 14–38.
2247:
2243:
2216:
2212:
2192:
2188:
2176:
2150:
2146:
2138:
2068:
2062:
2053:
2024:
2020:
2007:
1999:
1994:
1980:
1968:
1927:
1923:
1866:
1860:
1852:
1801:
1797:
1791:
1733:
1727:
1721:
1704:Applications
1696:
1694:
1685:
1677:
1590:
1202:
861:
435:
95:
91:
89:
81:
71:
69:
56:
48:
44:
32:
28:
24:
23:
22:
2195:: 395–457.
2047:MATLAB file
2045:. See also
1543:Step5.IDFT(
1339:Step3.Move
106:S-Transform
2286:Categories
2219:: 85–102.
2147:Proc. IEEE
1784:References
86:Definition
2155:CiteSeerX
1952:1053-587X
1814:CiteSeerX
1622:τ
1619:−
1610:π
1607:−
1504:π
1501:−
1493:⋅
1481:Δ
1408:Δ
1357:Δ
1320:Δ
1264:π
1261:−
1225:Δ
1130:π
1127:−
1109:Δ
1078:−
1060:∑
1044:Δ
1027:Δ
978:Δ
951:Δ
924:Δ
914:α
903:Δ
882:Δ
841:α
829:α
826:π
788:α
784:π
781:−
769:α
752:∞
747:∞
744:−
740:∫
663:π
660:−
614:τ
608:π
599:−
588:τ
528:π
525:−
479:τ
473:π
464:−
453:τ
402:τ
396:π
342:∞
337:∞
334:−
330:∫
319:∞
314:∞
311:−
307:∫
297:τ
263:τ
254:τ
248:π
239:−
209:τ
206:−
197:π
194:−
170:τ
159:∞
154:∞
151:−
147:∫
29:transform
2272:13266737
2098:265–274.
1960:16074001
1901:29974786
1893:19163232
1836:30202517
1762:See also
942:, where
571:to both
2252:Bibcode
2029:Bibcode
1932:Bibcode
1806:Bibcode
70:A fast
2270:
2157:
2083:
1958:
1950:
1899:
1891:
1881:
1834:
1816:
700:gives
2268:S2CID
2240:(PDF)
2017:(PDF)
1956:S2CID
1897:S2CID
1832:S2CID
1715:(MRI)
2081:ISBN
1948:ISSN
1889:PMID
1879:ISBN
1302:for
864:Let
631:and
496:and
63:and
2260:doi
2221:doi
2197:doi
2165:doi
2116:doi
2073:doi
2037:doi
1940:doi
1871:doi
1824:doi
1378:to
2288::
2266:.
2258:.
2246:.
2242:.
2217:23
2215:.
2193:24
2191:.
2187:.
2163:.
2151:77
2149:.
2114:.
2079:.
2035:.
2025:10
2023:.
2019:.
1954:.
1946:.
1938:.
1928:58
1926:.
1909:^
1895:.
1887:.
1877:.
1844:^
1830:.
1822:.
1812:.
1802:44
1800:.
102:.
79:.
67:.
2274:.
2262::
2254::
2248:8
2227:.
2223::
2203:.
2199::
2171:.
2167::
2122:.
2118::
2106:)
2089:.
2075::
2043:.
2039::
2031::
1962:.
1942::
1934::
1903:.
1873::
1853:S
1838:.
1826::
1808::
1740:.
1734:S
1728:S
1722:S
1697:S
1637:)
1630:2
1626:)
1616:t
1613:(
1603:e
1599:(
1566:]
1563:p
1560:,
1557:m
1554:[
1551:B
1524:2
1520:m
1514:2
1510:p
1497:e
1490:]
1485:F
1477:)
1474:m
1471:+
1468:p
1465:(
1462:[
1459:X
1456:=
1453:]
1450:p
1447:,
1444:m
1441:[
1438:B
1417:]
1412:F
1404:)
1401:m
1398:+
1395:p
1392:(
1389:[
1386:X
1366:]
1361:F
1353:p
1350:[
1347:X
1324:F
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1313:=
1310:f
1284:2
1280:m
1274:2
1270:p
1257:e
1234:]
1229:F
1221:p
1218:[
1215:X
1180:N
1176:n
1173:p
1170:2
1167:j
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1140:2
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1101:m
1098:+
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1089:[
1086:X
1081:1
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1070:0
1067:=
1064:p
1056:=
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1048:F
1040:m
1037:,
1031:T
1023:n
1020:(
1015:x
1011:S
982:F
955:T
928:F
920:p
917:=
907:F
899:m
896:=
893:f
886:T
878:n
875:=
872:t
853:.
838:d
832:t
823:2
820:j
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807:2
803:f
798:/
792:2
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766:+
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760:(
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736:=
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730:f
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721:(
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681:2
677:f
671:2
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639:(
619:)
611:f
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579:(
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553:)
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504:(
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167:(
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