Re: Robust method of fundamental frequency estimation.

[Arturo Camacho <acamacho@xxxxxxxxxxxx>]

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
__________________________________________________