172:
2073:. For example, if the sampling rate of an audio signal is 44100 Hz and there is a large peak in the cepstrum whose quefrency is 100 samples, the peak indicates the presence of a fundamental frequency that is 44100/100 = 441 Hz. This peak occurs in the cepstrum because the harmonics in the spectrum are periodic and the period corresponds to the fundamental frequency, since harmonics are integer multiples of the fundamental frequency.
2080:, which stands for "Kolmogorov-equation power-series time response", is similar to the cepstrum and has the same relation to it as expected value has to statistical average, i.e. cepstrum is the empirically measured quantity, while kepstrum is the theoretical quantity. It was in use before the cepstrum.
207:
References to the Bogert paper, in a bibliography, are often edited incorrectly. The terms "quefrency", "alanysis", "cepstrum" and "saphe" were invented by the authors by rearranging the letters in frequency, analysis, spectrum, and phase. The invented terms are defined in analogy to the older terms.
2172:
Note that a pure sine wave can not be used to test the cepstrum for its pitch determination from quefrency as a pure sine wave does not contain any harmonics and does not lead to quefrency peaks. Rather, a test signal containing harmonics should be used (such as the sum of at least two sines where
2137:(such as a source and filter) into sums of their cepstra, for linear separation. In particular, the power cepstrum is often used as a feature vector for representing the human voice and musical signals. For these applications, the spectrum is usually first transformed using the
1810:
734:
863:
2290:
signal, from the power spectrum of the sEMG signal itself. In this way, only information about the motor unit action potential (MUAP) shape and amplitude was maintained, which was then used to estimate the parameters of a time-domain model of the MUAP itself.
1066:
1173:
872:. It can be shown that both formulas are consistent with each other as the frequency spectral distribution remains the same, the only difference being a scaling factor which can be applied afterwards. Some articles prefer the second formula.
2036:
Main application is the modification of the signal in the quefrency domain (liftering) as an analog operation to filtering in the spectral frequency domain. An example is the suppression of echo effects by suppression of certain quefrencies.
1393:
2024:
1918:
1502:
1644:
2619:
G. Biagetti, P. Crippa, S. Orcioni, and C. Turchetti, “Homomorphic deconvolution for muap estimation from surface emg signals,” IEEE Journal of
Biomedical and Health Informatics, vol. 21, no. 2, pp. 328– 338, March
952:
2256:
1650:
609:
748:
2098:. It can be implemented by multiplying by a window in the quefrency domain and then converting back to the frequency domain, resulting in a modified signal, i.e. with signal echo being reduced.
2598:"A kepstrum approach to filtering, smoothing and prediction with application to speech enhancement", T. J. Moir and J. F. Barrett, Proc. Royal Society A, vol. 459, 2003, pp. 2957–2976.
1271:
2493:
963:
410:
2173:
the second sine is some harmonic (multiple) of the first sine, or better, a signal with a square or triangle waveform, as such signals provide many overtones in the spectrum.).
553:
1553:
1529:
501:
355:
1072:
2642:
A. Michael Noll (1964), “Short-Time
Spectrum and Cepstrum Techniques for Vocal-Pitch Detection”, Journal of the Acoustical Society of America, Vol. 36, No. 2, pp. 296–302.
1923:
The complex cepstrum retains the information about the phase. Thus it is always possible to return from the quefrency domain to the time domain by the inverse operation:
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1929:
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Recently, cepstrum-based deconvolution was used on surface electromyography signals, to remove the effect of the stochastic impulse train, which originates an
2106:
The cepstrum can be seen as information about the rate of change in the different spectrum bands. It was originally invented for characterizing the seismic
1413:
2587:"Predictive decomposition of time series with applications to seismic exploration", E. A. Robinson MIT report 1954; Geophysics 1967 vol. 32, pp. 418–484;
875:
Other notations are possible due to the fact that the log of the power spectrum is equal to the log of the spectrum if a scaling factor 2 is applied:
1565:
1555:
as a product of magnitude and phase, and subsequently as a sum. Further simplification is obvious, if log is a natural logarithm with base
2589:"Use of the kepstrum in signal analysis", M. T. Silvia and E. A. Robinson, Geoexploration, volume 16, issues 1–2, April 1978, pages 55–73.
2287:
881:
739:
The power cepstrum has main applications in analysis of sound and vibration signals. It is a complementary tool to spectral analysis.
1805:{\displaystyle \log _{e}({\mathcal {F}})=\log _{e}({\mathcal {|F|}})+\log _{e}(e^{i\varphi })=\log _{e}({\mathcal {|F|}})+i\varphi }
2760:
2610:
A. Michael Noll (1967), “Cepstrum Pitch
Determination”, Journal of the Acoustical Society of America, Vol. 41, No. 2, pp. 293–309.
729:{\displaystyle C_{p}=\left|{\mathcal {F}}^{-1}\left\{\log \left(\left|{\mathcal {F}}\{f(t)\}\right|^{2}\right)\right\}\right|^{2}}
2122:
signal returns. Cepstrum pitch determination is particularly effective because the effects of the vocal excitation (pitch) and
188:. It serves as a tool to investigate periodic structures in frequency spectra. Such effects are related to noticeable echos or
858:{\displaystyle C_{p}=\left|{\mathcal {F}}\left\{\log \left(\left|{\mathcal {F}}\{f(t)\}\right|^{2}\right)\right\}\right|^{2}}
1407:
was defined by
Oppenheim in his development of homomorphic system theory. The formula is provided also in other literature.
2633:, "Short-Time 'Cepstrum' Pitch Detection," (abstract) Journal of the Acoustical Society of America, Vol. 36, No. 5, p. 1030
2145:
or MFC (its coefficients are called mel-frequency cepstral coefficients, or MFCCs). It is used for voice identification,
2062:
2776:
2186:
2522:
2379:
2149:
and much more. The cepstrum is useful in these applications because the low-frequency periodic excitation from the
2474:
A. V. Oppenheim, "Superposition in a class of nonlinear systems" Ph.D. diss., Res. Lab. Electronics, M.I.T. 1965.
2434:
R.B. Randall: Cepstrum
Analysis and Gearbox Fault Diagnosis, Brüel&Kjaer Application Notes 233-80, Edition 2
2701:
Oppenheim, A.V.; Schafer, R.W. (2004). "Dsp history - From frequency to quefrency: a history of the cepstrum".
1276:
It is derived from the complex cepstrum (defined below) by discarding the phase information (contained in the
2566:
1225:
1061:{\displaystyle C_{p}=\left|{\mathcal {F}}^{-1}\left\{2\log |{\mathcal {F}}|\right\}\right|^{2},{\text{ or}}}
363:
237:
transformation to frequency domain, where the final independent variable, the quefrency, has a time scale.
381:
193:
164:. It may be pronounced in the two ways given, the second having the advantage of avoiding confusion with
2118:
explosions. It has also been used to determine the fundamental frequency of human speech and to analyze
523:
2781:
367:
1168:{\displaystyle C_{p}=4\cdot \left|{\mathcal {F}}^{-1}\left\{\log |{\mathcal {F}}|\right\}\right|^{2},}
2511:
Taschenbuch der
Informatik – Band III – Anwendungen und spezielle Systeme der Nachrichtenverarbeitung
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114:
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1534:
1510:
1185:
Further, it shall be noted, that the final squaring operation in the formula for the power spectrum
478:
336:
253:
complex cepstrum: The logarithm is taken from the spectrum, which is calculated via
Fourier analysis
2449:
Beckhoff information system: TF3600 TC3 Condition
Monitoring: Gearbox monitoring (online, 4.4.2020)
2327:] of Time Series for Echoes: Cepstrum, Pseudo Autocovariance, Cross-Cepstrum and Saphe Cracking
2282:
machine vibration analysis based on harmonic patterns (gearbox faults, turbine blade failures, ...)
1388:{\displaystyle C_{r}={\mathcal {F}}^{-1}\left\{\log({\mathcal {|{\mathcal {F}}\{f(t)\}|}})\right\}}
2410:
2019:{\displaystyle f(t)={\mathcal {F}}^{-1}\left\{b^{\left({\mathcal {F}}\{C_{c}\}\right)}\right\},}
2142:
463:
depends on the user. In some articles the base is not specified, others prefer base 10 or
566:
2746:
2656:
Childers, D.G.; Skinner, D.P.; Kemerait, R.C. (1977). "The cepstrum: A guide to processing".
2130:
2090:
Playing further on the anagram theme, a filter that operates on a cepstrum might be called a
1913:{\displaystyle C_{c}={\mathcal {F}}^{-1}\left\{\log _{e}({\mathcal {|F|}})+i\varphi \right\}}
421:
2710:
2665:
2630:
2295:
2087:. The autocepstrum is more accurate than the cepstrum in the analysis of data with echoes.
1188:
189:
2496:, (PDF) in: Mechanical Systems and Signal Processing, Volume 97, December 2017 (Elsevier).
2126:(formants) are additive in the logarithm of the power spectrum and thus clearly separate.
467:. The choice of the base has no impact on the basic calculation rules, but sometimes base
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8:
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20:
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184:
The concept of the cepstrum was introduced in 1963 by B. P. Bogert, M. J. Healy, and
1497:{\displaystyle C_{c}={\mathcal {F}}^{-1}\left\{\log({\mathcal {F}}\{f(t)\})\right\}}
2738:
2718:
2693:
2673:
2166:
2095:
229:
37:
27:
2483:
A. V. Oppenheim, R. W. Schafer, "Digital Signal
Processing", 1975 (Prentice Hall).
2514:
2369:
2084:
2069:. The quefrency is a measure of time, though not in the sense of a signal in the
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dealing with reflection inference (radar, sonar applications, earth seismology)
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and Noll in the 1960s for application to pitch determination of human speech.
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medical applications in analysis of electroencephalogram (EEG) and brain waves
1639:{\displaystyle \log({\mathcal {F}})=\log({\mathcal {|F|\cdot e^{i\varphi }}})}
257:
The following abbreviations are used in the formulas to explain the cepstrum:
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2722:
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509:
216:
The cepstrum is the result of following sequence of mathematical operations:
201:
2677:
2433:
2664:(10). Institute of Electrical and Electronics Engineers (IEEE): 1428–1443.
2494:"A history of cepstrum analysis and its application to mechanical problems"
171:
134:
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of two signals can be expressed as the addition of their complex cepstra:
1284:). It has a focus on periodic effects in the amplitudes of the spectrum:
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2123:
2070:
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phase spectrum = (complex cepstrum − time reversal of complex cepstrum).
2709:(5). Institute of Electrical and Electronics Engineers (IEEE): 95–106.
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452:
197:
126:
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2367:
2169:, are additive and in different regions in the quefrency domain.
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is sometimes called unnecessary and therefore sometimes omitted.
2532:
947:{\displaystyle \log |{\mathcal {F}}|^{2}=2\log |{\mathcal {F}}|}
250:
power cepstrum: The logarithm is taken from the "power spectrum"
125:. The method is a tool for investigating periodic structures in
2462:"Real cepstrum and minimum-phase reconstruction - MATLAB rceps"
2333:(M. Rosenblatt, Ed) Chapter 15, 209-243. New York: Wiley, 1963.
868:
Due to this formula, the cepstrum is also sometimes called the
221:
19:"Cepstral" redirects here. For the text-to-speech company, see
2266:
The concept of the cepstrum has led to numerous applications:
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1706:
1607:
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512:, which is calculated from real- and imaginary part using the
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in the signal, or to the occurrence of harmonic frequencies (
2368:
Norton, Michael Peter; Karczub, Denis (November 17, 2003).
2115:
2094:. A low-pass lifter is similar to a low-pass filter in the
246:
The cepstrum is used in many variants. Most important are:
94:
76:
46:
2371:
Fundamentals of Noise and
Vibration Analysis for Engineers
2176:
An important property of the cepstral domain is that the
61:
2513:(in German). Vol. 3 (3 ed.). Berlin, Germany:
1815:
Therefore: The complex cepstrum can be also written as:
2655:
2505:
234:
computation of the logarithm of the spectral amplitude
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2509:; Weber, Wolfgang; Heinemann, Traute, eds. (1974) .
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2417:, Vol. 65, No. 10, October 1977, pp. 1428–1443.
2331:
Proceedings of the Symposium on Time Series Analysis
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estimation of speaker fundamental frequency (pitch)
2083:The autocepstrum is defined as the cepstrum of the
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471:leads to simplifications (see "complex cepstrum").
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2251:{\displaystyle x_{1}*x_{2}\mapsto x'_{1}+x'_{2}.}
1531:is complex the log-term can be also written with
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2409:D. G. Childers, D. P. Skinner, R. C. Kemerait, "
2477:
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2321:B. P. Bogert, M. J. R. Healy, and J. W. Tukey,
2294:A short-time cepstrum analysis was proposed by
200:). Mathematically it deals with the problem of
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175:Steps in forming cepstrum from time history
2754:Speech analysis: Cepstral analysis vs. LPC
2444:
2442:
2336:
2606:
2604:
2129:The cepstrum is a representation used in
599:The "cepstrum" was originally defined as
2048:) is related to the complex cepstrum as
362:: The abbreviation can stand i.e. for a
170:
2439:
2769:
2636:
2601:
1266:{\displaystyle C_{p}=4\cdot C_{r}^{2}}
1178:which provides a relationship to the
211:
148:. Operations on cepstra are labelled
303:Signal, which is a function of time
133:has applications in the analysis of
2569:. Support.ircam.fr. January 1, 1990
2411:The Cepstrum: A Guide to Processing
2056:
2033:is the base of the used logarithm.
1398:
405:{\displaystyle {\mathcal {F}}^{-1}}
204:of signals in the frequency space.
13:
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2567:"Introduction - Discrete Cepstrum"
2065:of a cepstral graph is called the
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548:{\displaystyle \left|x\right|^{2}}
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2133:, to convert signals combined by
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742:Sometimes it is also defined as:
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414:Inverse of the fourier transform
113:) is the result of computing the
2761:A tutorial on Cepstrum and LPCCs
2323:The Quefrency Alanysis [
36:
2703:IEEE Signal Processing Magazine
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2276:speech analysis and recognition
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603:by the following relationship:
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2374:. Cambridge University Press.
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2131:homomorphic signal processing
364:continuous Fourier transform
7:
2141:. The result is called the
16:Concept in Fourier analysis
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368:discrete Fourier transform
18:
2777:Frequency-domain analysis
459:. The choice of the base
179:
115:inverse Fourier transform
2723:10.1109/msp.2004.1328092
2161:, which convolve in the
576:{\displaystyle \varphi }
241:
2678:10.1109/proc.1977.10747
2658:Proceedings of the IEEE
2415:Proceedings of the IEEE
443:{\displaystyle \log(x)}
2747:Speech Signal Analysis
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2143:mel-frequency cepstrum
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2756:", www.advsolned.com
2631:Manfred R. Schroeder
2629:A. Michael Noll and
2517:. pp. 272–274.
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2165:and multiply in the
2063:independent variable
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87:
82:
81:
78:
75:
72:
67:
66:
63:
60:
57:
54:
51:
48:
45:
42:
28:Fourier analysis
2797:
2796:
2792:
2791:
2790:
2788:
2787:
2786:
2767:
2766:
2652:
2650:Further reading
2647:
2646:
2641:
2637:
2628:
2624:
2618:
2614:
2609:
2602:
2597:
2593:
2588:
2586:
2582:
2572:
2570:
2565:
2564:
2560:
2548:
2547:
2538:
2537:
2525:
2515:Springer Verlag
2504:
2500:
2492:R.B. Randall:,
2491:
2487:
2482:
2478:
2473:
2469:
2460:
2459:
2455:
2447:
2440:
2432:
2421:
2408:
2389:
2382:
2366:
2337:
2320:
2309:
2304:
2264:
2236:
2220:
2207:
2203:
2194:
2190:
2188:
2185:
2184:
2147:pitch detection
2110:resulting from
2104:
2085:autocorrelation
2059:
1993:
1989:
1980:
1979:
1978:
1974:
1973:
1969:
1965:
1956:
1950:
1949:
1948:
1931:
1928:
1927:
1886:
1878:
1877:
1876:
1864:
1860:
1859:
1855:
1846:
1840:
1839:
1838:
1829:
1825:
1823:
1820:
1819:
1783:
1775:
1774:
1773:
1761:
1757:
1742:
1738:
1726:
1722:
1709:
1701:
1700:
1699:
1687:
1683:
1671:
1670:
1658:
1654:
1652:
1649:
1648:
1622:
1618:
1610:
1602:
1601:
1600:
1579:
1578:
1567:
1564:
1563:
1539:
1538:
1536:
1533:
1532:
1515:
1514:
1512:
1509:
1508:
1462:
1461:
1451:
1447:
1438:
1432:
1431:
1430:
1421:
1417:
1415:
1412:
1411:
1401:
1370:
1346:
1345:
1340:
1339:
1338:
1328:
1324:
1315:
1309:
1308:
1307:
1298:
1294:
1292:
1289:
1288:
1257:
1252:
1233:
1229:
1227:
1224:
1223:
1196:
1192:
1190:
1187:
1186:
1156:
1141:
1135:
1134:
1129:
1122:
1118:
1109:
1103:
1102:
1101:
1100:
1096:
1095:
1080:
1076:
1074:
1071:
1070:
1053:
1044:
1029:
1023:
1022:
1017:
1007:
1003:
994:
988:
987:
986:
985:
981:
980:
971:
967:
965:
962:
961:
957:and therefore:
939:
933:
932:
927:
909:
904:
903:
897:
896:
891:
883:
880:
879:
849:
829:
800:
799:
798:
794:
793:
789:
782:
778:
772:
771:
770:
766:
765:
756:
752:
750:
747:
746:
720:
700:
671:
670:
669:
665:
664:
660:
653:
649:
640:
634:
633:
632:
631:
627:
626:
617:
613:
611:
608:
607:
597:
568:
565:
564:
558:Absolute square
539:
528:
527:
525:
522:
521:
482:
480:
477:
476:
423:
420:
419:
393:
387:
386:
385:
383:
380:
379:
341:
340:
338:
335:
334:
312:
309:
308:
277:
274:
273:
244:
214:
182:
123:signal spectrum
84:
69:
39:
35:
24:
17:
12:
11:
5:
2795:
2785:
2784:
2779:
2765:
2764:
2757:
2750:
2743:
2698:
2651:
2648:
2645:
2644:
2635:
2622:
2612:
2600:
2591:
2580:
2558:
2523:
2498:
2485:
2476:
2467:
2453:
2438:
2419:
2387:
2380:
2335:
2306:
2305:
2303:
2300:
2284:
2283:
2280:
2277:
2274:
2271:
2263:
2260:
2259:
2258:
2247:
2243:
2239:
2235:
2231:
2227:
2223:
2219:
2215:
2210:
2206:
2202:
2197:
2193:
2103:
2102:Interpretation
2100:
2058:
2055:
2054:
2053:
2046:phase spectrum
2042:phase cepstrum
2027:
2026:
2015:
2011:
2005:
2001:
1996:
1992:
1988:
1983:
1977:
1972:
1968:
1962:
1959:
1953:
1947:
1944:
1941:
1938:
1935:
1921:
1920:
1908:
1904:
1901:
1898:
1895:
1889:
1885:
1881:
1875:
1872:
1867:
1863:
1858:
1852:
1849:
1843:
1837:
1832:
1828:
1813:
1812:
1801:
1798:
1795:
1792:
1786:
1782:
1778:
1772:
1769:
1764:
1760:
1756:
1753:
1748:
1745:
1741:
1737:
1734:
1729:
1725:
1721:
1718:
1712:
1708:
1704:
1698:
1695:
1690:
1686:
1682:
1679:
1674:
1669:
1666:
1661:
1657:
1646:
1635:
1628:
1625:
1621:
1617:
1613:
1609:
1605:
1599:
1596:
1593:
1590:
1587:
1582:
1577:
1574:
1571:
1542:
1518:
1505:
1504:
1492:
1488:
1485:
1482:
1479:
1476:
1473:
1470:
1465:
1460:
1457:
1454:
1450:
1444:
1441:
1435:
1429:
1424:
1420:
1400:
1397:
1396:
1395:
1383:
1379:
1373:
1369:
1366:
1363:
1360:
1357:
1354:
1349:
1343:
1337:
1334:
1331:
1327:
1321:
1318:
1312:
1306:
1301:
1297:
1278:imaginary part
1274:
1273:
1260:
1255:
1251:
1247:
1244:
1241:
1236:
1232:
1199:
1195:
1176:
1175:
1164:
1159:
1154:
1149:
1144:
1138:
1132:
1128:
1125:
1121:
1115:
1112:
1106:
1099:
1094:
1091:
1088:
1083:
1079:
1068:
1052:
1047:
1042:
1037:
1032:
1026:
1020:
1016:
1013:
1010:
1006:
1000:
997:
991:
984:
979:
974:
970:
955:
954:
942:
936:
930:
926:
923:
920:
917:
912:
907:
900:
894:
890:
887:
866:
865:
852:
847:
842:
837:
832:
827:
823:
820:
817:
814:
811:
808:
803:
797:
792:
788:
785:
781:
775:
769:
764:
759:
755:
737:
736:
723:
718:
713:
708:
703:
698:
694:
691:
688:
685:
682:
679:
674:
668:
663:
659:
656:
652:
646:
643:
637:
630:
625:
620:
616:
601:power cepstrum
596:
595:Power cepstrum
593:
590:
589:
583:
572:
561:
560:
555:
542:
537:
534:
531:
518:
517:
506:Absolute value
503:
491:
488:
485:
473:
472:
450:
439:
436:
433:
430:
427:
416:
415:
412:
399:
396:
390:
376:
375:
357:
344:
331:
330:
327:
316:
305:
304:
301:
290:
287:
284:
281:
270:
269:
266:
255:
254:
251:
243:
240:
239:
238:
235:
232:
213:
210:
181:
178:
131:power cepstrum
15:
9:
6:
4:
3:
2:
2794:
2783:
2780:
2778:
2775:
2774:
2772:
2762:
2758:
2755:
2751:
2748:
2744:
2740:
2736:
2732:
2728:
2724:
2720:
2716:
2712:
2708:
2704:
2699:
2695:
2691:
2687:
2683:
2679:
2675:
2671:
2667:
2663:
2659:
2654:
2653:
2639:
2632:
2626:
2616:
2607:
2605:
2595:
2584:
2573:September 16,
2568:
2562:
2554:
2542:
2534:
2530:
2526:
2524:3-540-06242-4
2520:
2516:
2512:
2508:
2502:
2495:
2489:
2480:
2471:
2463:
2457:
2450:
2445:
2443:
2435:
2430:
2428:
2426:
2424:
2416:
2412:
2406:
2404:
2402:
2400:
2398:
2396:
2394:
2392:
2383:
2381:0-521-49913-5
2377:
2373:
2372:
2364:
2362:
2360:
2358:
2356:
2354:
2352:
2350:
2348:
2346:
2344:
2342:
2340:
2332:
2328:
2324:
2318:
2316:
2314:
2312:
2307:
2299:
2297:
2292:
2289:
2281:
2278:
2275:
2272:
2269:
2268:
2267:
2245:
2241:
2237:
2233:
2229:
2225:
2221:
2217:
2208:
2204:
2200:
2195:
2191:
2183:
2182:
2181:
2179:
2174:
2170:
2168:
2164:
2160:
2156:
2152:
2148:
2144:
2140:
2136:
2132:
2127:
2125:
2121:
2117:
2113:
2109:
2099:
2097:
2093:
2088:
2086:
2081:
2079:
2074:
2072:
2068:
2064:
2051:
2050:
2049:
2047:
2043:
2038:
2034:
2032:
2013:
2009:
2003:
1994:
1990:
1975:
1970:
1966:
1960:
1957:
1945:
1939:
1933:
1926:
1925:
1924:
1906:
1902:
1899:
1896:
1870:
1865:
1861:
1856:
1850:
1847:
1835:
1830:
1826:
1818:
1817:
1816:
1799:
1796:
1793:
1767:
1762:
1758:
1754:
1746:
1743:
1739:
1732:
1727:
1723:
1719:
1693:
1688:
1684:
1680:
1664:
1659:
1655:
1647:
1626:
1615:
1594:
1591:
1588:
1572:
1569:
1562:
1561:
1560:
1558:
1490:
1477:
1471:
1455:
1452:
1448:
1442:
1439:
1427:
1422:
1418:
1410:
1409:
1408:
1406:
1381:
1332:
1329:
1325:
1319:
1316:
1304:
1299:
1295:
1287:
1286:
1285:
1283:
1279:
1258:
1253:
1249:
1245:
1242:
1239:
1234:
1230:
1222:
1221:
1220:
1218:
1217:real cepstrum
1213:
1197:
1193:
1183:
1182:(see below).
1181:
1180:real cepstrum
1162:
1157:
1152:
1147:
1126:
1123:
1119:
1113:
1110:
1097:
1092:
1089:
1086:
1081:
1077:
1069:
1050:
1045:
1040:
1035:
1014:
1011:
1008:
1004:
998:
995:
982:
977:
972:
968:
960:
959:
958:
924:
921:
918:
915:
910:
888:
885:
878:
877:
876:
873:
871:
850:
845:
840:
835:
830:
825:
815:
809:
795:
790:
786:
783:
779:
767:
762:
757:
753:
745:
744:
743:
740:
721:
716:
711:
706:
701:
696:
686:
680:
666:
661:
657:
654:
650:
644:
641:
628:
623:
618:
614:
606:
605:
604:
602:
588:
587:complex value
584:
570:
563:
562:
559:
556:
540:
535:
532:
529:
520:
519:
515:
511:
510:complex value
507:
504:
489:
486:
483:
475:
474:
470:
466:
462:
458:
454:
451:
434:
428:
425:
418:
417:
413:
397:
394:
378:
377:
373:
369:
365:
361:
358:
333:
332:
328:
314:
307:
306:
302:
285:
279:
272:
271:
267:
265:Abbreviation
264:
263:
258:
252:
249:
248:
247:
236:
233:
231:
227:
223:
219:
218:
217:
209:
205:
203:
202:deconvolution
199:
195:
191:
187:
173:
169:
167:
163:
159:
155:
151:
147:
143:
138:
136:
132:
129:spectra. The
128:
124:
120:
117:(IFT) of the
116:
112:
108:
102:
33:
29:
22:
2706:
2702:
2661:
2657:
2638:
2625:
2615:
2594:
2583:
2571:. Retrieved
2561:
2510:
2501:
2488:
2479:
2470:
2456:
2414:
2370:
2330:
2326:
2322:
2293:
2285:
2265:
2262:Applications
2175:
2171:
2128:
2105:
2091:
2089:
2082:
2077:
2075:
2066:
2060:
2041:
2039:
2035:
2030:
2028:
1922:
1814:
1556:
1506:
1404:
1402:
1275:
1216:
1214:
1184:
1179:
1177:
956:
874:
869:
867:
741:
738:
600:
598:
468:
464:
460:
456:
268:Explanation
256:
245:
215:
206:
183:
165:
161:
157:
153:
149:
145:
141:
139:
135:human speech
130:
110:
109:, adjective
106:
31:
25:
2549:|work=
2178:convolution
2163:time domain
2159:vocal tract
2151:vocal cords
2135:convolution
2124:vocal tract
2112:earthquakes
2071:time domain
372:z-transform
226:time domain
190:reflections
186:J. W. Tukey
2771:Categories
2302:References
2731:1053-5888
2686:0018-9219
2551:ignored (
2541:cite book
2296:Schroeder
2214:↦
2201:∗
2139:mel scale
2067:quefrency
1958:−
1903:φ
1871:
1848:−
1800:φ
1768:
1747:φ
1733:
1694:
1665:
1627:φ
1616:⋅
1595:
1573:
1456:
1440:−
1333:
1317:−
1246:⋅
1127:
1111:−
1093:⋅
1015:
996:−
925:
889:
787:
658:
642:−
571:φ
453:Logarithm
429:
395:−
329:Cepstrum
224:from the
198:overtones
158:liftering
140:The term
127:frequency
119:logarithm
105:; plural
2533:73-80607
2242:′
2226:′
2153:and the
2078:kepstrum
1055: or
194:partials
166:kepstrum
146:spectrum
142:cepstrum
111:cepstral
32:cepstrum
2739:1162306
2711:Bibcode
2694:6108941
2666:Bibcode
2436:. (PDF)
2155:formant
2044:(after
1280:of the
228:to the
107:cepstra
2737:
2729:
2692:
2684:
2531:
2521:
2378:
2108:echoes
2092:lifter
2029:where
222:signal
180:Origin
30:, the
2735:S2CID
2690:S2CID
2620:2017.
2120:radar
242:Types
160:, or
2727:ISSN
2682:ISSN
2575:2022
2553:help
2529:LCCN
2519:ISBN
2376:ISBN
2288:sEMG
2116:bomb
2114:and
2076:The
2061:The
2040:The
1403:The
1215:The
366:, a
152:(or
2719:doi
2674:doi
2413:",
2325:sic
1862:log
1759:log
1724:log
1685:log
1656:log
1592:log
1570:log
1507:As
1453:log
1330:log
1124:log
1012:log
922:log
886:log
784:log
655:log
455:of
426:log
156:),
83:-,-
26:In
2773::
2733:.
2725:.
2717:.
2707:21
2705:.
2688:.
2680:.
2672:.
2662:65
2660:.
2603:^
2545::
2543:}}
2539:{{
2527:.
2441:^
2422:^
2390:^
2338:^
2329:,
2310:^
1559::
516:.
196:,
168:.
137:.
2763:"
2759:"
2752:"
2749:"
2745:"
2741:.
2721::
2713::
2696:.
2676::
2668::
2577:.
2555:)
2535:.
2464:.
2451:.
2384:.
2246:.
2238:2
2234:x
2230:+
2222:1
2218:x
2209:2
2205:x
2196:1
2192:x
2031:b
2014:,
2010:}
2004:)
2000:}
1995:c
1991:C
1987:{
1982:F
1976:(
1971:b
1967:{
1961:1
1952:F
1946:=
1943:)
1940:t
1937:(
1934:f
1907:}
1900:i
1897:+
1894:)
1888:|
1884:F
1880:|
1874:(
1866:e
1857:{
1851:1
1842:F
1836:=
1831:c
1827:C
1797:i
1794:+
1791:)
1785:|
1781:F
1777:|
1771:(
1763:e
1755:=
1752:)
1744:i
1740:e
1736:(
1728:e
1720:+
1717:)
1711:|
1707:F
1703:|
1697:(
1689:e
1681:=
1678:)
1673:F
1668:(
1660:e
1634:)
1624:i
1620:e
1612:|
1608:F
1604:|
1598:(
1589:=
1586:)
1581:F
1576:(
1557:e
1541:F
1517:F
1491:}
1487:)
1484:}
1481:)
1478:t
1475:(
1472:f
1469:{
1464:F
1459:(
1449:{
1443:1
1434:F
1428:=
1423:c
1419:C
1382:}
1378:)
1372:|
1368:}
1365:)
1362:t
1359:(
1356:f
1353:{
1348:F
1342:|
1336:(
1326:{
1320:1
1311:F
1305:=
1300:r
1296:C
1259:2
1254:r
1250:C
1243:4
1240:=
1235:p
1231:C
1198:p
1194:C
1163:,
1158:2
1153:|
1148:}
1143:|
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1131:|
1120:{
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