... online¹

http://#1

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...#1²

http://#2

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...tex2html_comment_mark^3.1

To be precise, it is enough that the width of the frequency band the signal uses (the bandwidth) be less than the Nyquist frequency. This is exploited in digital telephony, where the transmitted signal is restricted to between c.a. 300 Hz and 4300 Hz, thus a sampling rate of 8000 Hz is sufficient.

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... way.^3.2

To discover the relationship between physically measurable features of sound and the human perception of sound is the aim of psychoacoustics , see [Zwi82].

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...tex2html_comment_mark^3.3

In mathematical terms, this ``purity'' means the sinusoids of different frequencies form an orthogonal basis of a vector space, the Fourier space. The time-domain is in fact another vector space, with the dirac functions  $\delta(n - n_0)$ for $n_0 \in {\bf Z}$ and the Fourier transform is simply a change of basis

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...tex2html_comment_mark^3.4

Just as, for the continuous case, the Laplace transform is the generalization of the continuous Fourier transform.

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... ^3.5

Some scolars prefer the name linear shift-invariant systems   for discrete signals, since there is no inherent notion of time in a sequence of numbers.

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...tex2html_comment_mark^3.6

When the sound is represented as a sequence of samples in time-domain, changing the duration will alter the pitch and vice versa, e.g. a transposition of an octave (twice the original frequency) will speed the sound up by a factor of two. With nowadays fast DSP chips, this can be compensated for, but the sound quality suffers proportionally to the amount of transposition or tempo change.

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...tex2html_comment_mark^3.6

I have deliberately avoided the term smoothness  here, since it could be understood both as regularity and steadyness. However, in the later chapters, the term smoothness will be used meaning regularity, since steadyness is guaranteed with the methods chosen.

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... jumps).^3.7

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...tex2html_comment_mark^3.8

They can, however, lead to problems in formant detection, when they are close to the center frequency of a (broad) formant, they seem to split it into two formants.

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...tex2html_comment_mark^4.1

In fact, we apply an inverse Fourier transform because then we get back the right units and quantities of time and amplitude. Comparing equations (2.9) and (2.10), it can be seen that the Fourier transform differs from the inverse Fourier transform only in a scaling factor and the sign of the exponent.

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... used.^5.1

A word of warning, though: When choosing too small a number n of points for the envelope in relation to the highest frequency of half the sampling rate, severe aliasing of the spectral envelope can result. This is the same effect as sampling a signal with a sampling rate lower than twice the highest frequency in the signal. Here, the signal is the spectral envelope to be represented, while the number of points corresponds to the sampling frequency (their distance is the sampling period). Of course, this problem is greatly alleviated using a logarithmic frequency scale also for representation.

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...tex2html_comment_mark^5.2

Nevertheless, splines can be of some use for spectral envelope estimation when they are used to pre-smooth the spectrum before an estimation method is applied to find the parameters of a filter model. In [TAW97], the authors present how this can be applied to low-order LPC-analysis.

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...tex2html_comment_mark^6.1

Of course, a spectral envelope is only defined between 0 Hz and half the sampling frequency.

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...$\omega_j$:^7.1

In general, the Fourier spectrum is complex. The cepstrum, however, is evaluated from the magnitude spectrum which is the absolute value of the spectrum. Therefore, the recreation of the spectrum has no imaginary part.

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... class.^8.1

For the design of a complex software system it is important to express the decision between a reference and containment, especially because it makes a statement about liftimes of objects. In our case, however, where the system is rather a batch system with a linear flow of information, it is advantageous to abstract from that decision for reason of simplicity.

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... error^8.2

Because the discipline required to check all return codes may cause problems to unexperience programmers and is not adequate for a quick test program, where you don't want to write an error check for every second statement, there is also this option to have the library abort on error, although it is not a good behaviour for a library in general.

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... ^8.3

A method is a member function , i.e. a function that is a member of a class and can therefore be called only for an object of that class and its subclasses.

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... space^9.1

Timbral parameters being e.g. dynamics, pitch, harmonicity, noise, etc. They are completely user-defined.

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....^10.1

Contrary to spline representation (section 4.4), where there are inflection and extrema points, the wavelet analysis yields information about the curve behaviour at multiple resolutions.

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...tex2html_comment_mark^11.1

If the number of samples is a power of two, the Fast Fourier Transform (FFT) algorithm will be used, otherwise a Discrete Fourier Transform (DFT) will be performed.

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... online^12.1

http://mediatheque.ircam.fr/articles/textes/Dechelle94a/

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...
Abstract^12.2

http://www.ircam.fr/equipes/analyse-synthese/listePublications/articlesRodet/ICASS P93HMM/ICASSP93HMMabstract.html

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... online^12.3

http://www.ircam.fr/equipes/analyse-synthese/listePublications/articlesRod et/ICMC94CASTRAT

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...
Contents^12.4

http://www.ims.uni-stuttgart.de/phonetik/aims.html

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... online^12.5

http://www.ircam.fr/equipes/analyse-synthese/listePublications/articlesRod et/ICSPAT92/ICSPAT92.html

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... online^12.6

http://www.ircam.fr/equipes/analyse-synthese/pm/

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... online^12.7

http://www.ircam.fr/equipes/analyse-synthese/listePublications/articlesRod et/TFTS96/tfts96.ps.gz

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...
Abstract^12.8

http://www.ircam.fr/equipes/analyse-synthese/listePublications/articlesRodet/ICMC9 6HRMP/abstract.txt

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... PostScript^12.9

http://www.ircam.fr/equipes/analyse-synthese/listePublications/articlesRodet/ICMC9 6HRMP/ICMC96HRMP.ps.gz

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... tutorial^12.10

http://www.spd.eee.strath.ac.uk/ interact/ztransform

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... from^12.11

http://man104nfs.ucsd.edu/ mpuckett/

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... from^12.12

http://man104nfs.ucsd.edu/ mpuckett/

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... online^12.13

http://www.ircam.fr/equipes/analyse-synthese/listePublications/articlesRodet/ISMA9 5/ISMA95.html

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... PostScript^12.14

http://www.ircam.fr/equipes/analyse-synthese/listePublications/articlesRodet/ISMA9 5/ISMA95.ps.gz

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...
Abstract^12.15

http://www.ircam.fr/equipes/analyse-synthese/listePublications/articlesRodet/ICMC9 7/ICMC97DiphoneAbstract.html

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... PostScript^12.16

http://www.ircam.fr/equipes/analyse-synthese/listePublications/articlesRodet/ICMC9 7/ICMC97Diphone.ps.gz

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... tutorial^12.17

http://svr-www.eng.cam.ac.uk/ ajr/SA95/SpeechAnalysis.html

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...
Abstract^12.18

http://www.ircam.fr/equipes/analyse-synthese/listePublications/articlesRodet/TFTS9 7/TFTS97abstract.html

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... PostScript^12.19

http://www.ircam.fr/equipes/analyse-synthese/listePublications/articlesRodet/TFTS9 7/TFTS97.ps.gz

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... online^12.20

http://www.ircam.fr/equipes/analyse-synthese/additive/index-e.html

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... online^12.21

http://www.ircam.fr/equipes/analyse-synthese/listePublications/articlesRod et/CHANT85/Chant85Tout.html

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... documentation^12.22

http://www.rational.com/uml/documentation.html

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... online^12.23

http://www.ircam.fr/equipes/analyse-synthese/libchant/index.html

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... online^12.24

http://www.ircam.fr/equipes/analyse-synthese/sdif/index.html

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... online^12.25

http://www.ircam.fr/equipes/analyse-synthese/udi/udi.html

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