DAFx 2021

This is the companion webpage for the DAFx 2021 article M. Caetano & P. Depalle. "On the Estimation of Sinusoidal Parameters via Parabolic Interpolation of Scaled Magnitude Spectra" In: Proceedings of the 24^th International Conference on Digital Audio Effects (DAFx20in21), Vienna, Austria, September 8-10, 2021. ([Caetano & Depalle 2021]). Here you will find additional results that could not be included in the article due to a lack of space.

Figures and Tables

Original Values for Optimal Power Scale `p`

The table below corresponds to Table 1 in Sec 2.2 of [Caetano & Depalle 2021].

Table 1: Optimum value of the power scaling factor `p` for twelve windows of four different sizes `M` in samples. Values reproduced from *[Werner & Germain 2016]*.
`M`	512	1024	2048	4096
Bartlett	0.2253	0.22535	0.22538	0.22539
Hann	0.22903	0.22911	0.22915	0.22917
Hanning	0.22903	0.22911	0.22915	0.22917
Blackman	0.13056	0.13057	0.13058	0.13058
Blackman-Harris	0.08552	0.08553	0.08553	0.08554
Hamming	0.18505	0.18575	0.18611	0.18628
Bartlett-Hann	0.21635	0.21642	0.21645	0.21647
Gaussian	0.12024	0.12074	0.12099	0.12112
Kaiser-Bessel	0.28214	0.28270	0.28298	0.28312
Nuttall	0.08153	0.08155	0.08157	0.08157
Dolph-Chebychev	0.08403	0.08403	0.08404	0.08404
Tukey	0.50592	0.50609	0.50618	0.50622

Below are the figures and the table with the final model parameters for all 12 windows tested.

Maximizing the Line Fit

Coefficient of determination as a function of the variable kappa for the Bartlett window.

Coefficient of determination as a function of the variable kappa for the Hann window.

Coefficient of determination as a function of the variable kappa for the Hanning window.

Coefficient of determination as a function of the variable kappa for the Hamming window.

Coefficient of determination as a function of the variable kappa for the Blackman window.

Coefficient of determination as a function of the variable kappa for the Blackman-Harris window.

Coefficient of determination as a function of the variable kappa for the Bartlett-Hann window.

Coefficient of determination as a function of the variable kappa for the Gaussian window.

Coefficient of determination as a function of the variable kappa for the Kaiser-Bessel window.

Coefficient of determination as a function of the variable kappa for the Nuttall window.

Coefficient of determination as a function of the variable kappa for the Dolph-Chebychev window.

Coefficient of determination as a function of the variable kappa for the Tukey window.

Final Line and Curve Fits

Curve fit for the Bartlett window — Resulting curve fit for the Bartlett window.

Curve fit for the Hann window — Resulting curve fit for the Hann window.

Curve fit for the Hanning window — Resulting curve fit for the Hanning window.

Curve fit for the Hamming window — Resulting curve fit for the Hamming window.

Curve fit for the Blackman window — Resulting curve fit for the Blackman window.

Curve fit for the Blackman-Harris window — Resulting curve fit for the Blackman-Harris window.

Curve fit for the Bartlett-Hann window — Resulting curve fit for the Bartlett-Hann window.

Curve fit for the Gaussian window — Resulting curve fit for the Gaussian window.

Curve fit for the Kaiser-Bessel window — Resulting curve fit for the Kaiser-Bessel window.

Curve fit for the Nuttall window — Resulting curve fit for the Nuttall window.

Curve fit for the Dolph-Chebychev window — Resulting curve fit for the Dolph-Chebychev window.

Resulting linefit for the Tukey window — Resulting curve fit for the Tukey window.

Table with Final Model Parameters

The table below corresponds to Table 2 in Sec 3.1 of [Caetano & Depalle 2021].

Table 2: Line fit parameters and coefficient of determination `R²` for the twelve windows under investigation.
Window	`κ`	`a`	`b`	`R²`
Bartlett	0.2254	-0.81377	-0.37339	0.99689
Hann	0.22919	-0.69315	-1.0288	1.00000
Hanning	0.22919	-0.69315	-1.0288	1.00000
Blackman	0.1306	-0.29599	-5.5907	0.90000
Blackman-Harris	0.16554	-7.4991×10^-5	-0.72627	0.90000
Hamming	0.18645	-0.70779	1.481	0.99993
Bartlett-Hann	0.21649	-0.6291	-1.6925	0.99828
Gaussian	0.12125	-0.67356	1.2765	0.99999
Kaiser-Bessel	0.28326	-0.69315	0.70529	1.00000
Nuttall	0.081601	-0.29599	-4.4272	0.90000
Dolph-Chebychev	0.085039	-3.9824×10^-3	-4.3969	0.80000
Tukey	0.50626	-0.7241	-0.78342	0.99972

How much Zero-Padding for ZP-Log-PI to Outperform Pow-PI

Maximum Amplitude Error

Amplitude estimation error as a function of the variable L for the Bartlett window — Maximum amplitude estimation error as a function of the zero-padding factor for the Bartlett window.

Amplitude estimation error as a function of the variable L for the Hann window — Maximum amplitude estimation error as a function of the zero-padding factor for the Hann window.

Amplitude estimation error as a function of the variable L for the Hanning window — Maximum amplitude estimation error as a function of the zero-padding factor for the Hanning window.

Amplitude estimation error as a function of the variable L for the Hamming window — Maximum amplitude estimation error as a function of the zero-padding factor for the Hamming window.

Amplitude estimation error as a function of the variable L for the Blackman window — Maximum amplitude estimation error as a function of the zero-padding factor for the Blackman window.

Amplitude estimation error as a function of the variable L for the Blackman-Harris window — Maximum amplitude estimation error as a function of the zero-padding factor for the Blackman-Harris window.

Amplitude estimation error as a function of the variable L for the Bartlett-Hann window — Maximum amplitude estimation error as a function of the zero-padding factor for the Bartlett-Hann window.

Amplitude estimation error as a function of the variable L for the Gaussian window — Maximum amplitude estimation error as a function of the zero-padding factor for the Gaussian window.

Amplitude estimation error as a function of the variable L for the Kaiser-Bessel window — Maximum amplitude estimation error as a function of the zero-padding factor for the Kaiser-Bessel window.

Amplitude estimation error as a function of the variable L for the Nuttall window — Maximum amplitude estimation error as a function of the zero-padding factor for the Nuttall window.

Amplitude estimation error as a function of the variable L for the Dolph-Chebychev window — Maximum amplitude estimation error as a function of the zero-padding factor for the Dolph-Chebychev window.

Amplitude estimation error as a function of the variable L for the Tukey window — Maximum amplitude estimation error as a function of the zero-padding factor for the Tukey window.

Maximum Frequency Error

Frequency bin estimation error as a function of the variable L for the Bartlett window — Maximum frequency bin estimation error as a function of the zero-padding factor for the Bartlett window.

Frequency bin estimation error as a function of the variable L for the Hann window — Maximum frequency bin estimation error as a function of the zero-padding factor for the Hann window.

Frequency bin estimation error as a function of the variable L for the Hanning window — Maximum frequency bin estimation error as a function of the zero-padding factor for the Hanning window.

Frequency bin estimation error as a function of the variable L for the Hamming window — Maximum frequency bin estimation error as a function of the zero-padding factor for the Hamming window.

Frequency bin estimation error as a function of the variable L for the Blackman window — Maximum frequency bin estimation error as a function of the zero-padding factor for the Blackman window.

Frequency bin estimation error as a function of the variable L for the Blackman-Harris window — Maximum frequency bin estimation error as a function of the zero-padding factor for the Blackman-Harris window.

Frequency bin estimation error as a function of the variable L for the Bartlett-Hann window — Maximum frequency bin estimation error as a function of the zero-padding factor for the Bartlett-Hann window.

Frequency bin estimation error as a function of the variable L for the Gaussian window — Maximum frequency bin estimation error as a function of the zero-padding factor for the Gaussian window.

Frequency bin estimation error as a function of the variable L for the Kaiser-Bessel window — Maximum frequency bin estimation error as a function of the zero-padding factor for the Kaiser-Bessel window.

Frequency bin estimation error as a function of the variable L for the Nuttall window — Maximum frequency bin estimation error as a function of the zero-padding factor for the Nuttall window.

Frequency bin estimation error as a function of the variable L for the Dolph-Chebychev window — Maximum frequency bin estimation error as a function of the zero-padding factor for the Dolph-Chebychev window.

Frequency bin estimation error as a function of the variable L for the Tukey window — Maximum frequency bin estimation error as a function of the zero-padding factor for the Tukey window.

Table with Minimum Zero-Padding Factor

The table below corresponds to Table 3 in Sec. 4.1 of [Caetano & Depalle 2021].

Table 3: Minimum zero-padding factor `L` where **ZP-Log-PI** results in an estimation error lower than that of **Pow-PI**.
Window	min `L` for amplitude error ε_a	min `L` for frequency bin error ε_ν
Bartlett	2	7
Hann	2	6
Hanning	2	6
Blackman	3	13
Blackman-Harris	4	> 16
Hamming	2	6
Bartlett-Hann	2	6
Gaussian	3	9
Kaiser-Bessel	1	1
Nuttall	4	> 16
Dolph-Chebychev	4	> 16
Tukey	1	1

The Impact of Zero-Padding on Log-Scaled Parabolic Interpolation

Estimation error as a function of the variable L for the Bartlett window — Maximum estimation error as a function of the zero-padding factor for the Bartlett window.

Estimation error as a function of the variable L for the Hann window — Maximum estimation error as a function of the zero-padding factor for the Hann window.

Estimation error as a function of the variable L for the Hanning window — Maximum estimation error as a function of the zero-padding factor for the Hanning window.

Estimation error as a function of the variable L for the Hamming window — Maximum estimation error as a function of the zero-padding factor for the Hamming window.

Estimation error as a function of the variable L for the Blackman window — Maximum estimation error as a function of the zero-padding factor for the Blackman window.

Estimation error as a function of the variable L for the Blackman-Harris window — Maximum estimation error as a function of the zero-padding factor for the Blackman-Harris window.

Estimation error as a function of the variable L for the Bartlett-Hann window — Maximum estimation error as a function of the zero-padding factor for the Bartlett-Hann window.

Estimation error as a function of the variable L for the Gaussian window — Maximum estimation error as a function of the zero-padding factor for the Gaussian window.

Estimation error as a function of the variable L for the Kaiser-Bessel window — Maximum estimation error as a function of the zero-padding factor for the Kaiser-Bessel window.

Estimation error as a function of the variable L for the Nuttall window — Maximum estimation error as a function of the zero-padding factor for the Nuttall window.

Estimation error as a function of the variable L for the Dolph-Chebychev window — Maximum estimation error as a function of the zero-padding factor for the Dolph-Chebychev window.

Estimation error as a function of the variable L for the Tukey window — Maximum estimation error as a function of the zero-padding factor for the Tukey window.

The Impact of Zero-Padding on Power-Scaled Parabolic Interpolation

Table with Maximum Amplitude and Frequency Bin Estimation Error for Power Scaling

The table below corresponds to Table 4 in Sec. 4.2 of [Caetano & Depalle 2021].

Table 4: Maximum amplitude and frequency estimation errors for power scaling parabolic interpolation when `N`=`M`=512.
Window	Amplitude error	Frequency error
Bartlett	1.193×10^-3	5.878×10^-4
Hann	9.642×10^-4	5.726×10^-4
Hanning	9.642×10^-4	5.726×10^-4
Blackman	6.691×10^-5	8.194×10^-5
Blackman-Harris	1.092×10^-5	1.941×10^-5
Hamming	1.177×10^-3	4.099×10^-4
Bartlett-Hann	9.588×10^-4	5.377×10^-4
Gaussian	2.198×10^-4	1.599×10^-4
Kaiser-Bessel	2.207×10^-1	1.753×10^-1
Nuttall	1.202×10^-5	1.803×10^-5
Dolph-Chebychev	1.689×10^-5	1.823×10^-5
Tukey	7.782×10^-2	2.130×10^-2

Sound Examples

Below are a few sound examples comparing both scalings used in the article. The figures show the spectrogram of each sound. The SRER is given by $SRER ≝ 20 \log_{10} \frac{RMS [x (n)]}{RMS [ε (n)]}$ , where $x (n)$ represents the waveform of the original sound and $ε (n)$ represents the waveform of the residual, defined as $ε (n) ≝ x (n) - s (n)$ , where $s (n)$ is the waveform of the sinusoidal model. Download a zip file with the audio and figures below.

Singing Voice and Musical Instruments

Log Scaling

Original

Sinusoidal component

Residual component.

Residual component normalized to -16dB RMS

Power Scaling

Original

Sinusoidal component

Residual component

Residual component normalized to -16dB RMS

Log Scaling

Waveform of an A4 note played fortissimo on a Violin

Original

Sinusoidal component

Residual component

Residual component normalized to -16dB RMS

Power Scaling

Original

Sinusoidal component

Residual component

Residual component normalized to -16dB RMS

Log Scaling

Waveform of a Female Singer performing a breathy arpeggio

Original

Sinusoidal component

Residual component.

Residual component normalized to -16dB RMS

Power Scaling

Original

Sinusoidal component

Residual component

Residual component normalized to -16dB RMS

Reference

[Werner & Germain 2016] K.J. Werner & F.G. Germain "Sinusoidal Parameter Estimation Using Quadratic Interpolation around Power-Scaled Magnitude Spectrum Peaks" Applied Sciences. 2016; 6(10):306. DOI:app6100306

Additional Results

Figures and Tables

Original Values for Optimal Power Scale p

Maximizing the Line Fit

Final Line and Curve Fits

Table with Final Model Parameters

How much Zero-Padding for ZP-Log-PI to Outperform Pow-PI

Maximum Amplitude Error

Maximum Frequency Error

Table with Minimum Zero-Padding Factor

The Impact of Zero-Padding on Log-Scaled Parabolic Interpolation

The Impact of Zero-Padding on Power-Scaled Parabolic Interpolation

Table with Maximum Amplitude and Frequency Bin Estimation Error for Power Scaling

Sound Examples

Singing Voice and Musical Instruments

Reference

Original Values for Optimal Power Scale `p`