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| | 48 yr old Agricultural Scientist Speller from Big Trout Lake, loves to spend some time beach, tires for sale and canoeing. Last month very recently visited Roman Walls of Lugo.<br><br>Also visit my website - [http://ow.ly/AhDNp tires traction] |
| {{Redirect|Cepstral|the software company based in Pennsylvania|Cepstral (company)}}
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| A '''cepstrum''' {{IPAc-en|ˈ|k|ɛ|p|s|t|r|ə|m}} is the result of taking the [[Inverse Fourier transform]] (IFT) of the [[logarithm]] of the estimated [[power spectrum|spectrum]] of a signal. There is a [[complex number|complex]] cepstrum, a [[real number|real]] cepstrum, a power cepstrum, and phase cepstrum.
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| The power cepstrum in particular finds applications in the analysis of human speech.
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| The name "cepstrum" was derived by reversing the first four letters of "spectrum". Operations on cepstra are labelled ''quefrency analysis'', ''liftering'', or ''cepstral analysis''.<!-- please don't change spellings without looking them up -->
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| ==Origin and definition==
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| The ''power cepstrum'' was defined in a 1963 paper by Bogert et al.<ref>B. P. Bogert, M. J. R. Healy, and [[J. W. Tukey]]: "The Quefrency Alanysis<!-- Quefrency Alanysis is the original and correct spelling --> of Time Series for Echoes: Cepstrum, Pseudo Autocovariance, Cross-Cepstrum and Saphe Cracking". ''Proceedings of the Symposium on Time Series Analysis'' (M. Rosenblatt, Ed) Chapter 15, 209-243. New York: Wiley, 1963.</ref> The power cepstrum of a signal is defined as the squared magnitude of the inverse Fourier transform of the logarithm of the squared magnitude of the Fourier transform of a signal.<ref>{{cite book| author = Norton, Michael Peter | coauthors = Karczub, Denis| title = Fundamentals of Noise and Vibration Analysis for Engineers| url = http://books.google.com/?id=jDeRCSqtev4C| date = November 17, 2003| publisher = Cambridge University Press| isbn = 0-521-49913-5 }}</ref>
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| :power cepstrum of signal <math>=\left|\mathcal{F}^{-1}\left\{\mbox{log}(\left|\mathcal{F}\left\{ f(t) \right\}\right|^2)\right\}\right|^2</math>
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| A short-time cepstrum analysis was proposed by Schroeder and Noll for application to pitch determination of human speech.<ref name="schroeder64">
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| A. Michael Noll and Manfred R. Schroeder, "Short-Time 'Cepstrum' Pitch Detection," (abstract) Journal of the Acoustical Society of America, Vol. 36, No. 5, p. 1030</ref><ref name="noll64">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.</ref><ref name="noll67">A. Michael Noll (1967), “Cepstrum Pitch Determination,” Journal of the Acoustical Society of America, Vol. 41, No. 2, pp. 293-309.</ref>
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| The ''complex cepstrum'' was defined by Oppenheim in his development of homomorphic system theory.<ref>A. V. Oppenheim, "Superposition in a class of nonlinear systems" (Ph.D. dissertation), Res. Lab. Electronics, Massachusetts Institute of Technology, Cambridge, MA, 1965.</ref> and is defined as the Fourier transform of the logarithm (with unwrapped phase) of the Fourier transform of the signal. This is sometimes called the spectrum of a spectrum.
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| : complex cepstrum of signal {{math|1== FT(log(FT(the signal))+''j''2π''m''}}) (where {{mvar|m}} is the integer required to properly [[phase unwrapping|unwrap]] the angle or [[imaginary part]] of the complex log function)
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| The ''real cepstrum'' uses the [[logarithm]] [[function (mathematics)|function]] defined for real values. The real cepstrum is related to the power via the relationship (4 * real cepstrum)^2 = power cepstrum, and is related to the complex cepstrum as real cepstrum = 0.5*(complex cepstrum + time reversal of complex cepstrum).
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| [[File:Cepstrum signal analysis.png|right|thumb|Steps in forming cepstrum from time history]]
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| The complex cepstrum uses the [[complex logarithm]] [[function (mathematics)|function]] defined for complex values.
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| The phase cepstrum is related to the complex cepstrum as phase spectrum = (complex cepstrum - time reversal of complex cepstrum).^2
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| The complex cepstrum holds information about magnitude and phase of the initial spectrum, allowing the reconstruction of the signal. The real cepstrum uses only the information of the magnitude of the spectrum.
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| Many texts define the process as FT → abs() → log → IFT, i.e., that the cepstrum is the "inverse Fourier transform of the log-magnitude Fourier spectrum".<ref>{{cite book| author = Curtis Roads| title = The computer music tutorial| url = http://books.google.com/?id=nZ-TetwzVcIC| date = February 27, 1996| publisher = MIT Press| isbn = 978-0-262-68082-0 }}</ref><ref>{{cite book| author = John G. Proakis| coauthors = Dimitris G. Manolakis| title = Digital signal processing| url = http://books.google.com/?id=H_5SAAAAMAAJ| year = 2007| publisher = Pearson Prentice Hall| isbn = 978-0-13-187374-2 }}</ref>
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| The ''kepstrum'', which stands for "Kolmogorov equation power series time response", is similar to the cepstrum and has the same relation to it as statistical average has to expected value, i.e. cepstrum is the empirically measured quantity while kepstrum is the theoretical quantity.<ref>
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| "Use of the kepstrum in signal analysis", M.T.Silvia and W.A.Robinson, Geoexploration, Volume 16, Issues 1-2, April 1978, Pages 55-73.
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| </ref><ref> | |
| "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
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| </ref>
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| ==Applications==
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| The cepstrum can be seen as information about rate of change in the different spectrum bands. It was originally invented for characterizing the seismic [[Echo (phenomenon)|echoes]] resulting from [[earthquake]]s and [[bomb]] explosions. It has also been used to determine the fundamental frequency of human speech and to analyze [[radar]] signal returns. Cepstrum pitch determination is particularly effective because the effects of the vocal excitation (pitch) and vocal tract (formants) are additive in the logarithm of the power spectrum and thus clearly separate.<ref name=noll67/>
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| The autocepstrum is defined as the cepstrum of the [[autocorrelation]]. The autocepstrum is more accurate than the cepstrum in the analysis of data with echoes.
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| The cepstrum is a representation used in [[homomorphic filtering|homomorphic signal processing]], to convert signals (such as a source and filter) combined by [[convolution]] 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 [[mel scale]]. The result is called the [[mel-frequency cepstrum]] or MFC (its coefficients are called mel-frequency cepstral coefficients, or MFCCs). It is used for voice identification, [[pitch detection algorithm|pitch detection]] and much more. The cepstrum is useful in these applications because the low-frequency periodic excitation from the [[vocal cord]]s and the [[formant]] filtering of the [[vocal tract]], which convolve in the [[time domain]] and multiply in the [[frequency domain]], are additive and in different regions in the quefrency domain.
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| ==Cepstral concepts==
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| The [[Dependent and independent variables|independent variable]] of a cepstral graph is called the '''quefrency'''. The quefrency is a measure of time, though not in the sense of a signal in the [[time domain]]. 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 pitch 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 pitch. Note that a pure sine wave should not be used to test the cepstrum for its pitch determination from quefrency as a pure sine wave does not contain any harmonics. Rather, a test signal containing harmonics should be used (such as the sum of at least two sines where the second sine is some harmonic (multiple) of the first sine).
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| ==Liftering==
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| Playing further on the anagram theme, a filter that operates on a cepstrum might be called a ''lifter''. A low pass lifter is similar to a low pass filter in the [[frequency domain]]. It can be implemented by multiplying by a window in the quefrency domain and when converted back to the frequency domain, resulting in a smoother signal.
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| ==Convolution==
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| A very important property of the cepstral domain is that the [[convolution]] of two signals can be expressed as the addition of their complex cepstra:
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| : <math>x_1 * x_2 \rightarrow x'_1 + x'_2</math>
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| ==References==
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| {{Reflist}}
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| ==Further reading==
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| * D. G. Childers, D. P. Skinner, R. C. Kemerait, "[http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1455016 The Cepstrum: A Guide to Processing]," ''Proceedings of the IEEE'', Vol. 65, No. 10, October 1977, pp. 1428–1443.
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| * "[http://www.phon.ucl.ac.uk/courses/spsci/matlab/lect10.html Speech Signal Analysis]"
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| * "[http://www.advsolned.com/example_speech_coding.html Speech analysis: Cepstral analysis vs. LPC] www.advsolned.com"
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| [[Category:Frequency domain analysis]]
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| [[Category:Signal processing]]
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48 yr old Agricultural Scientist Speller from Big Trout Lake, loves to spend some time beach, tires for sale and canoeing. Last month very recently visited Roman Walls of Lugo.
Also visit my website - tires traction