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*To*: AUDITORY@xxxxxxxxxxxxxxx*Subject*: Re: Robust method of fundamental frequency estimation.*From*: Arturo Camacho <acamacho@xxxxxxxxxxxx>*Date*: Sat, 3 Feb 2007 07:05:17 -0500*Delivery-date*: Sat Feb 3 07:08:05 2007*In-reply-to*: <2728.209.251.130.178.1170460061.squirrel@webmail.cise.ufl.edu>*List-archive*: <http://lists.mcgill.ca/scripts/wa.exe?LIST=AUDITORY>*List-help*: <http://lists.mcgill.ca/scripts/wa.exe?LIST=AUDITORY>, <mailto:LISTSERV@LISTS.MCGILL.CA?body=INFO AUDITORY>*List-owner*: <mailto:AUDITORY-request@LISTS.MCGILL.CA>*List-subscribe*: <mailto:AUDITORY-subscribe-request@LISTS.MCGILL.CA>*List-unsubscribe*: <mailto:AUDITORY-unsubscribe-request@LISTS.MCGILL.CA>*References*: <20070131171152.74585.qmail@web26301.mail.ukl.yahoo.com> <2a17c00a0701311310i538f446ana9c8532934e57529@mail.gmail.com> <2728.209.251.130.178.1170460061.squirrel@webmail.cise.ufl.edu>*Reply-to*: acamacho@xxxxxxxxxxxx*Sender*: AUDITORY - Research in Auditory Perception <AUDITORY@xxxxxxxxxxxxxxx>*User-agent*: SquirrelMail/1.5.1 [CVS]

Members of the list, I forgot to mention a seemingly insignificant detail in the description of CEP and AC, but it causes a huge problem to these algorithms. I should have described CEP as "Same as SHR but ADDS AN EXTRA PULSE AT ZERO and instead of pulses uses a cosine to transition from 1 to -1". This extra lobe at DC is what makes CEP and AC to have always a maximum at infinity (i.e. at a pitch period of zero). Arturo > Dear members, > > > I just want to add two points to what Yi-Wen said: > > >> Dear list, >> >> >> >> Just want to draw your attention to a good summary on various >> auto-correlation based pitch determination methods, >> >> Arturo Camacho and John G. Harris, "A biological inspired pitch >> determination algorithm", Fourth Joint Meeting of ASA and ASJ, Honolulu, >> Nov. 2006. >> >> >> Contact arturo@xxxxxxxxxxxx if interested. >> >> >> Best regards, >> Yi-Wen >> > > First, in that presentation we not only did a summary of pitch estimation > algorithms (PEA) but also pointed out some pitfalls they have. Second, > we did it not only for autocorrelation based algorithms, but also for many > other algorithms we considered to be ?classical?. Although some of these > algorithms were initially proposed using a time-domain approach, all of > them can also be formulated using the spectrum of the signal, and that is > the approach we took. We expressed those algorithms as the selection of > the pitch candidate (PC) that maximizes an integral transform of a > function of the spectrum. > > Below is a summary of our findings. For each algorithm, we give a short > DESCRIPTION, then the FUNCTION applied to the spectrum, the KERNEL of the > integral transform, and finally a PROBLEM of the algorithm. Sometimes you > will find that the algorithm also have problems presented before or > problems that will be presented later. Notice that the order we present > the algorithms is such that each subsequent algorithm does not exhibit > the problem mentioned for the previous algorithm. A final note about > semantics, to make the writing short in the descriptions, when we say > spectrum we mean MAGNITUDE of the spectrum. > > HARMONIC PRODUCT SPECTRUM (HPS) > ------------------------------- > DESCRIPTION: multiplies the spectrum at multiples of the PC, or > equivalently, adds the log of the spectrum at multiples of the PC. > FUNCTION: log > KERNEL: periodic sum of pulses > PROBLEM: If any harmonic of the pitch is missing, the log is minus > infinity and therefore the integral is also minus infinity. > > SUB-HARMONIC SUMMATION (SHS) > ---------------------------- > DESCRIPTION: adds the spectrum at multiples of the PC. > FUNCTION: none > KERNEL: periodic sum of pulses > PROBLEM: Any subharmonic of the pitch has the same score as the pitch. > > > SUB-HARMONIC SUMMATION with decay > --------------------------------- > DESCRIPTION: Same as SHS but uses a decaying factor to give less weight to > high order harmonics. FUNCTION: none > KERNEL: decaying periodic sum of pulses > PROBLEM: The same score it produces for a pulse train at the pitch is > produced for white noise at each PC. Therefore, not only it produces an > infinite number of pitch estimates for white noise but also they have the > same strength as a pulse train. > > SUBHARMONIC-TO-HARMONIC RATIO (SHR) > ----------------------------------- > DESCRIPTION: Same as SHS but subtracts the spectrum at the middle points > between harmonics. Uses log spectrum, though. FUNCTION: log > KERNEL: periodic sum of positive pulses plus half-period-shifted sum of > negative pulses PROBLEM: Like all the algorithms presented above, it does > not work for inharmonic signals > > HARMONIC SIEVE (HS) > ------------------- > DESCRIPTION: Same as SHS but instead of pulses it uses rectangles > FUNCTION: none > KERNEL: sum of rectangles > PROBLEM: weighting applied to spectrum is too sharp. A slight shift in a > component may take it in or out of the rectangle, possibly changing the > estimated pitch drastically. > > CEPSTRUM (CEP) > ------------- > DESCRIPTION: Same as SHR but instead of pulses uses a cosine to transition > from 1 to -1. FUNCTION: log > KERNEL: cosine > PROBLEM: uses the log (see HPS) > > > UNBIASED AUTOCORRELATION (UAC) > ------------------------------ > DESCRIPTION: Same as CEP but squares the spectrum > FUNCTION: square > KERNEL: cosine > PROBLEM: If signal is periodic then UAC is also periodic. Therefore there > are infinite number of maximums. Taking the first local maximum (excluding > maximum at zero) does not work either. Try a signal with first four > harmonics with magnitudes 1,6,1,1. At high enough levels its pitch > corresponds to the fundamental frequency, however, the first maximum in > the UAC corresponds to the second harmonic. > > BIASED AUTOCORRELATION (BAC) > ------------------------------ > DESCRIPTION: Same as UAC but a bias is applied such that a weight of one > is applied to a period of 0 and decays linearly to zero for a period T, > where T is the size of the window. FUNCTION: square > KERNEL: cosine > PROBLEM: Like UAC, the squaring of the spectrum gives to much emphasis to > salient harmonics. This feature combined with the bias may cause problems. > For example, for the 1,6,1,1 signal, the bias can make the score of the > second harmonic higher than the score of the fundamental (take for example > the fundamental period as T/4) > > END OF LIST > =========In ISCAS 2007 we will be presenting an algorithm that avoids the > problems presented here. It will be published in the proceedings of the > conference. From the order we presented here the algorithms it is easy to > infer what the algorithm looks like. > > Arturo > > > -- > __________________________________________________ > > > Arturo Camacho > PhD Student > Computer and Information Science and Engineering > University of Florida > > > E-mail: acamacho@xxxxxxxxxxxx > Web page: www.cise.ufl.edu/~acamacho > __________________________________________________ > > > -- __________________________________________________ Arturo Camacho PhD Student Computer and Information Science and Engineering University of Florida E-mail: acamacho@xxxxxxxxxxxx Web page: www.cise.ufl.edu/~acamacho __________________________________________________

**Follow-Ups**:**Re: Robust method of fundamental frequency estimation.***From:*Arturo Camacho

**References**:**Robust method of fundamental frequency estimation.***From:*Roisin Loughran

**Re: Robust method of fundamental frequency estimation.***From:*Yi-Wen Liu

**Re: Robust method of fundamental frequency estimation.***From:*Arturo Camacho

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