#
Sinusoidal Parameter Estimation Using Quadratic Interpolation around Power-Scaled Magnitude Spectrum Peaks^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Problem Statement

## 3. Estimating Sinusoidal Parameters from 3 Bins of Spectrum

## 4. Proposed Method: XQIFFT

#### 4.1. Logarithmically-Scaled QIFFT

#### 4.2. Power-Scaled QIFFT (XQIFFT)

#### 4.3. Invariance Properties of QIFFT Methods

## 5. Estimation Error, Properties, and Reduction

#### 5.1. Error Definition

#### 5.2. Interpolation Bias

#### 5.3. Choice of the Power-Scaling Factor p

#### 5.4. Correction Functions

#### 5.5. Sensitivity to Noise

## 6. Results

#### 6.1. Optimal p as a Funtion of Window Length

#### 6.2. Bin Estimate MSE as a Function of SNR (Hann)

#### 6.3. Bin Estimate MSE as a Function of SNR (Twelve Windows)

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Belega, D.; Petri, D. Frequency estimation by two- or three-point interpolated Fourier algorithms based on cosine windows. Signal Process.
**2015**, 117, 115–125. [Google Scholar] [CrossRef] - Candan, Ç. A method for fine resolution frequency estimation from three DFT samples. IEEE Signal Process. Lett.
**2011**, 18, 351–354. [Google Scholar] [CrossRef] - Liao, Y. Phase and Frequency Estimation: High-Accuracy and Low-Complexity Techniques. Master’s Thesis, Worcester Polytechnic Institute, Worcester, MA, USA, 2011. [Google Scholar]
- Smith, J.O., III. Spectral Audio Signal Processing; W3K Publishing: Standford, UK, 2011. [Google Scholar]
- Serra, X. Musical sound modeling with sinusoids plus noise. In Musical Signal Processing; Roads, C., Pope, S., Picialli, A., Poli, G.D., Eds.; Routledge: London, UK, 1997; pp. 91–122. [Google Scholar]
- Schmidt, R.O. Multiple emitter location and signal parameter estimation. IEEE Trans. Antennas Propag.
**1986**, 34, 276–280. [Google Scholar] [CrossRef] - Roy, R.; Kailath, T. ESPRIT—Estimation of signal parameter via rotational invariance techniques. IEEE Trans. Acoust. Speech Signal Process
**1989**, 37, 984–995. [Google Scholar] [CrossRef] - Macleod, M.D. Fast nearly ML estimation of the parameter of real of complex single tones or resolved multiple tones. IEEE Trans. Signal Process
**1998**, 46, 141–148. [Google Scholar] [CrossRef] - McAulay, R.; Quatieri, T.F. Speech analysis/synthesis based on a sinusoidal representation. IEEE Trans. Acoust. Speech Signal Process
**1986**, 34, 744–754. [Google Scholar] [CrossRef] - Serra, X. A System for Sound Analysis/Transformation/Synthesis Based on a Deterministic Plus Stochastic Decomposition. Ph.D. Thesis, Stanford University, Stanford, CA, USA, October 1989. [Google Scholar]
- Serra, X.; Smith, J.O., III. Spectral modeling synthesis: A sound analysis/synthesis system based on a deterministic plus stochastic decomposition. Comput. Music J.
**1990**, 14, 12–24. [Google Scholar] [CrossRef] - Smith, J.O., III; Serra, X. PARSHL: A program for the analysis/synthesis of inharmonic sounds based on a sinusoidal representation. In Proceedings of the International Computer Music Conference (ICMC), Champaign–Urbana, IL, USA, 23–26 August 1987.
- Maher, R.C.; Beauchamp, J.W. Fundamental frequency estimation of musical signals using a two-way mismatch procedure. J. Acoust. Soc. Am.
**1994**, 95, 2254–2263. [Google Scholar] [CrossRef] - Duda, K. DFT interpolation algorithm for Kaiser–Bessel and Dolpha–Chebyshev windows. IEEE Trans. Instrum. Meas.
**2011**, 60, 784–790. [Google Scholar] [CrossRef] - Candan, Ç. Fine resolution frequency estimation from three DFT samples: Case of windowed data. Signal Process.
**2015**, 114, 245–250. [Google Scholar] [CrossRef] - Agrež, D. Weighted multipoint interpolated DFT to improve amplitude estimation of multifrequency signal. IEEE Trans. Instrum. Meas.
**2002**, 51, 287–292. [Google Scholar] [CrossRef] - Offelli, C.; Petri, D. Interpolation techniques for real-time multifrequency waveform analysis. IEEE Trans. Instrum. Meas.
**1990**, 39, 106–111. [Google Scholar] [CrossRef] - Candan, Ç. Analysis and further improvement of fine resolution frequency estimation method from three DFT samples. IEEE Signal Process. Lett.
**2013**, 20, 913–916. [Google Scholar] [CrossRef] - Jacobsen, E.; Kootsookos, P. Fast, accurate frequency estimators. IEEE Signal Process. Mag.
**2007**, 24, 123–125. [Google Scholar] [CrossRef] - Werner, K.J. The XQIFFT: Increasing the accuracy of quadratic interpolation of spectral peaks via exponential magnitude spectrum weighting. In Proceedings of the 41st International Computer Music Confernece (ICMC), Denton, TX, USA, 25 September–1 October 2015.
- Auger, F.; Flandrin, P. Improving the readability of time-frequency and time-scale representations by the reassignment method. IEEE Trans. Signal Process.
**1995**, 43, 1068–1089. [Google Scholar] [CrossRef] - Degani, A.; Dalai, M.; Leonardi, R.; Miglorati, P. Time-frequency analysis of musical signals using the phase coherence. In Proceedings of the International Conference on Digital Audio Effects (DAFx-13), Maynooth, Ireland, 2–5 September 2013.
- Fitz, K.; Haken, L. On the use of time-frequency reassignment in additive sound modeling. J. Audio Eng. Soc.
**2002**, 50, 879–893. [Google Scholar] - Fitz, K.R. The Reassigned Bandwidth-Enhanced Method of Additive Synthesis. Ph.D. Thesis, University of Illinois at Urbana–Champaign, Urbana, IL, USA, 1999. [Google Scholar]
- Keiler, F.; Marchand, S. Survey on extraction of sinusoids in stationary sounds. In Proceedings of the International Conference on Digital Audio Effects (DAFx-02), Hamburg, Germany, 26–28 September 2002; pp. 51–58.
- Nam, J.; Mysore, G.J.; Ganseman, J.; Lee, K.; Abel, J.S. A super-resolution spectrogram using coupled PLCA. In Proceedings of the Conference of the International Speech Communication Association (Interspeech), Makuhari, Japan, 26–30 September 2010; Volume 11, pp. 1696–1699.
- Liao, J.-R.; Chen, C.-M. Phase correction of discrete Fourier transform coefficicents to reduce frequency estimation bias of single tone complex sinusoid. Signal Process.
**2014**, 94, 108–117. [Google Scholar] [CrossRef] - McLeod, P. Fast, Accurate Pitch Detection Tools for Music Analysis. Ph.D Thesis, University of Otago, Dunedin, New Zealand, 30 May 2008. [Google Scholar]
- Aboutanios, E.; Mulgrew, B. Iterative frequency estimation by interpolation on Fourier coefficients. IEEE Trans. Signal Process.
**2005**, 53, 1237–1242. [Google Scholar] [CrossRef] - Abe, M.; Smith, J.O., III. CQIFFT: Correcting Bias in a Sinusoidal Parameter Estimator Based on Quadratic Interpolation of FFT Magnitude Peaks; STAN-M 117; CCRMA, Department of Music, Stanford University: Stanford, CA, USA, 2004. [Google Scholar]
- Abe, M.; Smith, J.O., III. Design Criteria for the Quadratically Interpolated FFT Method (I): Bias Due to Interpolation; STAN-M 114; CCRMA, Department of Music, Stanford University: Stanford, CA, USA, 2004. [Google Scholar]
- Goto, Y. Highly accurate frequency interpolation of apodized FFT magnitude-mode spectra. Appl. Spectrosc.
**1998**, 52, 134–138. [Google Scholar] [CrossRef] - Smith, J.O., III; Serra, X. PARSHL: An Analysis/Synthesis Program for Non-Harmonic Sounds Based on a Sinusoidal Representation; STAN-M 43; CCRMA, Department of Music, Stanford University: Stanford, CA, USA, 1985. [Google Scholar]
- McIntyre, M.C.; Dermott, D.A. A new fine-frequency estimation algorithm based on parabolic regression. In Proceedings of the International Conference on Acoustics, Speech, and Signal Processing (ICASSP), San Francisco, CA, USA, 23–26 March 1992; Volume 2, pp. 541–544.
- Archer, B.; Weisstein, E.W. Lagrange interpolating polynomial. MathWorld—A Wolfram Web Resource. Available online: http://mathworld.wolfram.com/LagrangeInterpolatingPolynomial.html (accessed on 18 October 2016).
- Quinn, B.G. Estimating frequency by interpolation using Fourier coefficients. IEEE Trans. Signal Process.
**1994**, 42, 1264–1268. [Google Scholar] [CrossRef] - Berman, G. Minimization by successive approximation. SIAM J. Numer. Anal.
**1966**, 3, 123–133. [Google Scholar] [CrossRef] - Kiefer, J. Sequential minimax search for a maximum. Proc. Am. Math. Soc.
**1953**, 4, 503–506. [Google Scholar] [CrossRef] - Mathews, J.H.; Fink, K.D. Numerical Methods Using Matlab, 4th ed.; Pearson: Harlow, UK, 2004. [Google Scholar]
- Ortega, J.M.; Rheinboldt, W.C. Iterative Solution of Nonlinear Equations in Several Variables; Academic Press: New York, NY, USA, 1970. [Google Scholar]
- Moin, P. Fundamentals of Engineering Numerical Analysis; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Blackman, R.B.; Tukey, J.W. The Measurement of Power Spectra from the Point of View of Communications Engineering, 2nd ed.; Dover Publications, Inc.: New York, NY, USA, 1959. [Google Scholar]
- Harris, F.J. On the use of windows for harmonic analysis with the discrete Fourier transform. Proc. IEEE
**1978**, 66, 51–83. [Google Scholar] [CrossRef] - Rife, D.C.; Boorstyn, R.R. Single tone parameter estimation from discrete-time observations. IEEE Trans. Inf. Theory
**1974**, 20, 591–598. [Google Scholar] [CrossRef] - Ha, Y.H.; Pearce, J.A. A new window and comparison to standard windows. IEEE Trans Acoust. Speech Signal Process.
**1989**, 37, 298–301. [Google Scholar] [CrossRef] - Kaiser, J.F.; Schafer, R.W. On the use of the i
_{0}-sinh window for spectrum analysis. IEEE Trans. Acoust. Speech Signal Process.**1980**, 28, 105–107. [Google Scholar] [CrossRef] - Nuttall, A.H. Some windows with very good sidelobe behavior. IEEE Trans. Acoust. Speech Signal Process.
**1981**, 29, 84–91. [Google Scholar] [CrossRef] - Bloomfield, P. Fourier Analysis of Time Series: An Introduction, 2nd ed.; John Wiley & Sons: New York, NY, USA, 2000. [Google Scholar]

**Figure 2.**Mean and worst-case of absolute value of errors for each method as a function of bin offset. (

**a**) nearest bin method; (

**b**) MQIFFT; (

**c**) LQIFFT; (

**d**) XQIFFT, p = 0.1; (

**e**) nearest bin method; (

**f**) MQIFFT; (

**g**) LQIFFT; (

**h**) XQIFFT, p = 0.1.

**Figure 3.**Mean and worst-case for absolute value of errors for the QIFFT method on a length-4096 Hann window, including bin error and detail on bin error, magnitude error and detail on magnitude error. Horizontal lines indicate the worst case and mean errors for the nearest bin, MQIFFT, and LQIFFT methods for comparison. Shaded regions indicate the range of the detail in corresponding figures to the right. The thick lines indicate the XQIFFT. (

**a**) Bin error; (

**b**) Bin error (zoom); (

**c**) Magnitude error (zoom); (

**d**) Magnitude error (zoom).

**Figure 4.**Correction curves. (

**a**) MQIFFT; (

**b**) LQIFFT; (

**c**) XQIFFT, p = 0.1; (

**d**) MQIFFT; (

**e**) LQIFFT; (

**f**) XQIFFT, p = 0.1.

**Figure 5.**Bin index error bias for the MQIFFT (${\circ}$), the LQIFFT (${\times}$), and the XQIFFT (${\xb7}$, $p=0.2276$) at various SNR: 0, 10, 20, and 30 dB.

**Figure 6.**Bin index error variance for the MQIFFT (${\circ}$), the LQIFFT (${\times}$), and the XQIFFT (${\xb7}$, $p=0.2276$) at various SNR: 0, 10, 20, and 30 dB.

**Figure 7.**Comparison of the estimation variance under the unbiased hypothesis for the MQIFFT (${\circ}$), the LQIFFT (${\times}$), and the XQIFFT (${\xb7}$, $p=0.2276$) against the Cramér–Rao bound at various SNRs.

**Figure 8.**Value of p to minimize four different metrics (${e}_{\mathcal{K}}^{\mathrm{w}.\mathrm{c}.}$, ${e}_{\mathcal{X}}^{\mathrm{w}.\mathrm{c}.}$, $\mathrm{E}\left[\left|{e}_{\mathcal{K}}\left(\mathcal{K}\right)\right|\right]$, $\mathrm{E}\left[\left|{e}_{\mathcal{X}}\left(\mathcal{K}\right)\right|\right]$) for different length Hann windows.

**Figure 10.**Testing different window types. (

**a**) Hann window; (

**b**) Bartlett–Hann window; (

**c**) Bartlett window; (

**d**) Hamming window; (

**e**) Blackman window; (

**f**) Blackman–Harris window; (

**g**) Gaussian window; (

**h**) DPSS window; (

**i**) Kaiser–Bessel window; (

**j**) Nuttall window; (

**k**) Chebyshev window; (

**l**) Tukey window (50%).

**Table 1.**Comparing ${e}_{\mathcal{K}}\left(\mathcal{K}\right)$ and ${e}_{\mathcal{X}}\left(\mathcal{K}\right)$ for various estimation approaches on a length-4096 Hann window. XQIFFT cases which heuristically minimize worst case and mean error bin and magnitude errors are shown. Numbers are given to five significant figures and minimized values are shaded.

Case | ${\mathit{e}}_{\mathcal{K}}^{\mathbf{w}.\mathbf{c}.}$ | ${\mathit{e}}_{\mathcal{X}}^{\mathbf{w}.\mathbf{c}.}$ | $\mathit{E}\left[\left|{\mathit{e}}_{\mathcal{K}}\left(\mathcal{K}\right)\right|\right]$ | $\mathit{E}\left[\left|{\mathit{e}}_{\mathcal{X}}\left(\mathcal{K}\right)\right|\right]$ | Minimizes |
---|---|---|---|---|---|

nearest | 5.0000 × ${10}^{-1}$ | 1.5110 × ${10}^{-1}$ | 2.5000 × ${10}^{-1}$ | 5.1688 × ${10}^{-2}$ | n/a |

MQIFFT | 5.2764 × ${10}^{-2}$ | 6.6237 × ${10}^{-2}$ | 3.4221 × ${10}^{-2}$ | 2.5601 × ${10}^{-2}$ | n/a |

LQIFFT | 1.5997 × ${10}^{-2}$ | 3.7932 × ${10}^{-1}$ | 1.0392 × ${10}^{-2}$ | 1.3121 × ${10}^{-2}$ | n/a |

XQIFFT, $p=0.23086$ | 2.4484 × ${10}^{-4}$ | 9.5196 × ${10}^{-4}$ | 1.5693 × ${10}^{-4}$ | 2.0239 × ${10}^{-4}$ | ${e}_{\mathcal{K}}^{\mathrm{w}.\mathrm{c}.}$ |

XQIFFT, $p=0.23437$ | 4.4380 × ${10}^{-4}$ | 4.7735 × ${10}^{-4}$ | 2.3462 × ${10}^{-4}$ | 2.5251 × ${10}^{-4}$ | ${e}_{\mathcal{X}}^{\mathrm{w}.\mathrm{c}.}$ |

XQIFFT, $p=0.22917$ | 3.1861 × ${10}^{-4}$ | 1.1803 × ${10}^{-3}$ | 1.4645 × ${10}^{-4}$ | 2.0637 × ${10}^{-4}$ | $\mathrm{E}\left[\left|{e}_{\mathcal{K}}\left(\mathcal{K}\right)\right|\right]$ |

XQIFFT, $p=0.23039$ | 2.6445 × ${10}^{-4}$ | 1.0149 × ${10}^{-3}$ | 1.5203 × ${10}^{-4}$ | 2.0170 × ${10}^{-4}$ | $\mathrm{E}\left[\left|{e}_{\mathcal{X}}\left(\mathcal{K}\right)\right|\right]$ |

**Table 2.**The values of p that minimize $\mathrm{E}\left[\left|{e}_{\mathcal{K}}\left(\mathcal{K}\right)\right|\right]$ for the twelve length-4096 windows (highlighted column) used for tests in Figure 10, as well as length-512, -1024, and -2048 versions of the same windows. p is rounded to five decimal points.

Window | Length-512 | Length-1024 | Length-2048 | Length-4096 |
---|---|---|---|---|

Hann | $0.22903$ | $0.22911$ | $0.22915$ | $0.22917$ |

Bartlett–Hann | $0.21635$ | $0.21642$ | $0.21645$ | $0.21647$ |

Bartlett | $0.22530$ | $0.22535$ | $0.22538$ | $0.22539$ |

Hamming | $0.18505$ | $0.18575$ | $0.18611$ | $0.18628$ |

Blackman | $0.13056$ | $0.13057$ | $0.13058$ | $0.13058$ |

Blackman–Harris | $0.08552$ | $0.08553$ | $0.08553$ | $0.08554$ |

Gaussian | $0.12024$ | $0.12074$ | $0.12099$ | $0.12112$ |

DPSS/Slepian ($NW=3$) | $0.11144$ | $0.11144$ | $0.11144$ | $0.11144$ |

Kaiser–Bessel ($\beta =0.5$) | $0.28214$ | $0.28270$ | $0.28298$ | $0.28312$ |

Nuttall | $0.08153$ | $0.08155$ | $0.08157$ | $0.08157$ |

Dolph–Chebyshev (sidelobes at $-100$dB) | $0.08403$ | $0.08403$ | $0.08404$ | $0.08404$ |

Tukey window (50% taper) | $0.50592$ | $0.50609$ | $0.50618$ | $0.50622$ |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Werner, K.J.; Germain, F.G. Sinusoidal Parameter Estimation Using Quadratic Interpolation around Power-Scaled Magnitude Spectrum Peaks. *Appl. Sci.* **2016**, *6*, 306.
https://doi.org/10.3390/app6100306

**AMA Style**

Werner KJ, Germain FG. Sinusoidal Parameter Estimation Using Quadratic Interpolation around Power-Scaled Magnitude Spectrum Peaks. *Applied Sciences*. 2016; 6(10):306.
https://doi.org/10.3390/app6100306

**Chicago/Turabian Style**

Werner, Kurt James, and François Georges Germain. 2016. "Sinusoidal Parameter Estimation Using Quadratic Interpolation around Power-Scaled Magnitude Spectrum Peaks" *Applied Sciences* 6, no. 10: 306.
https://doi.org/10.3390/app6100306