#
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

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**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 (${e}_{\mathcal{K}}^{\mathrm{w}.\mathrm{c}.}$, ${e}_{\mathcal{X}}^{\mathrm{w}.\mathrm{c}.}$, $\mathrm{E}[\left(\right)open="|"\; close="|">{e}_{\mathcal{K}}\left(\mathcal{K}\right)$, $\mathrm{E}[\left(\right)open="|"\; close="|">{e}_{\mathcal{X}}\left(\mathcal{K}\right)$) 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 | ${\mathit{e}}_{\mathcal{K}}^{\mathbf{w}.\mathbf{c}.}$ | ${\mathit{e}}_{\mathcal{X}}^{\mathbf{w}.\mathbf{c}.}$ | $\mathit{E}[\left(\right)open="|"\; close="|">{\mathit{e}}_{\mathcal{K}}\left(\mathcal{K}\right)]$ | $\mathit{E}[\left(\right)open="|"\; close="|">{\mathit{e}}_{\mathcal{X}}\left(\mathcal{K}\right)]$ | 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 × ${10}^{-4}$ | $\mathrm{E}[\left(\right)open="|"\; close="|">{e}_{\mathcal{K}}\left(\mathcal{K}\right)$ |

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

**Table 2.**The values of p that minimize $\mathrm{E}[\left(\right)open="|"\; close="|">{e}_{\mathcal{K}}\left(\mathcal{K}\right)$ 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