# One-Step Discrete Fourier Transform-Based Sinusoid Frequency Estimation under Full-Bandwidth Quasi-Harmonic Interference

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## Abstract

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## 1. Introduction

**b**) in Figure 1 illustrates an accurate analysis of those contours by taking as input the same spectral information that is used in the represented spectrogram. The Matlab command file allowing us to replicate this figure is available on GitHub (https://github.com/Anibal-Ferreira/demo_AccSinFreqEst (URL accessed on 30 August 2023)).

## 2. DFT-Based Frequency Estimation

#### 2.1. The Estimation Problem

#### 2.2. Estimation Constraints

- Excludes iterative procedures and is computationally light;
- Maximizes the estimation accuracy and robustness when other interfering sinusoids co-exist in the signal, in addition to noise.

- Multiplication of a signal (or data) vector $x\left[n\right]$ by a window function that is represented by $h\left[n\right]$ (an operation also known as tapering);
- DFT computation, typically by means of an FFT;
- Peak picking in the DFT magnitude spectrum and frequency estimation of a target sinusoid by using a simple interpolation algorithm (or formula), taking several samples from the DFT magnitude spectrum.

#### 2.3. Degrees of Harmonic Interference and Test Settings

#### 2.4. Windows, Selectivity and Leakage

#### 2.5. Performance Criterion

## 3. Tested Frequency Estimators

#### 3.1. Rectangular Window-Based Estimators

#### 3.2. Hanning Window-Based Estimators

#### 3.3. Sine Window-Based Estimators

## 4. Test Results and Main Conclusions

#### 4.1. Results with No Harmonic Interference

#### 4.2. Results with Mild Quasi-Harmonic Interference

#### 4.3. Results with Strong Quasi-Harmonic Interference

#### 4.4. Main Conclusions

- First, systematic bias affects all frequency estimators in a similar way, varying between 0.1% and 0.3% of the bin width when the SNR is the range −10 dB to +10 dB, and vanishes rapidly for higher SNR, such that it can be considered negligible, except in the case of the Macleod98(R) and Jacobsen07(R) estimators, whose systematic bias can be as high as 0.1% of the DFT bin width under strong quasi-harmonic interference.
- Second, in terms of RMSE, it is clear that more severe quasi-harmonic interference conditions degrade the performance of all frequency estimators, but this degradation is not the same for all estimators. This fact is explained by the intrinsic robustness of each estimator, which depends not only on the window that is associated with the estimator, but also on their estimation approach dealing with spectral magnitude information only, or a combination of spectral magnitude and phase. For example, the Jacobsen07(H) estimator uses spectral magnitude only, and the Macleod98(H) estimator uses both spectral magnitude and phase, which is what gives it an ‘intrinsic leakage rejection’ capability [30].
- Third, the relative performance of the same estimator depends not only on the SNR (as expected), but is also highly influenced by the severity of the quasi-harmonic interference. For example, it is quite interesting to observe that the Jacobsen07(H) estimator is the worst-performing estimator when the test conditions do not involve harmonic interference, it belongs to the second group of worst-performing estimators under mild quasi-harmonic interference, and it belongs to the group of best-performing estimators under strong quasi-harmonic interference. This reflects the fact that all estimators suffer a stronger performance degradation when the test conditions become more severe, but that degradation affects different estimators differently. The Jacobsen07(H) estimator appears to be an exception, as its performance is quite consistent across test cases. Thus, it may be concluded that operational conditions dictate if a given estimator has a better or worse relative performance. For example, the Macleod98(R) estimator exhibits the best relative performance across test cases for very low SNR levels (because it approaches better the CRLB), but exhibits the worst relative performance across test cases for moderate and high SNR levels (because its performance curve saturates to higher RMSE values). Results also suggest that if a given estimator shows a good performance under no harmonic interference, it may perform poorly when subject to strong quasi-harmonic interference. This is the case of the Jacobsen07(R) estimator.
- Fourth, when the quasi-harmonic interference is strong, its impact on the frequency estimation performance for the majority of the estimators considered in this paper is quite significant, which confirms that it has a dominant effect in limiting the performance.
- Finally, our results suggest that when a frequency estimator shows a relative better performance across test cases at high SNR, that is obtained at the cost of a relatively worse performance across test cases at low SNR. That is clearly the case for the Macleod98(H) estimator when harmonic interference is mild or strong.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DFT | Discrete Fourier transform |

CRLB | Cramér–Rao lower bound |

ESPRIT | Estimation of Signal Parameters Via Rotational Invariance Techniques |

FFT | Fast Fourier transform |

MUSIC | Multiple Signal Classification |

ODFT | Odd discrete Fourier transform |

RMSE | Root Mean Squared Error |

SNR | Signal-to-Noise Ratio |

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**Figure 1.**Illustrative spectrogram of two FM sinusoids in close proximity (

**a**) and corresponding accurate frequency estimation (brown and blue stars in

**b**plot). The ground truth frequency contours are represented by the green solid line in (

**b**).

**Figure 2.**Illustration of three severity levels of quasi-harmonic interference affecting the estimation of a target sinusoid frequency: no interference (

**top**panel), and mild and strong quasi-harmonic interference, when the target sinusoid frequency corresponds approximately to the second (

**middle**panel) and fourth (

**bottom**panel) harmonic of an existing quasi-harmonic structure, respectively. The gray/black areas denote that the quasi-harmonic interference is AM/FM-modulated and the $\alpha $ index sets the maximum deviation in magnitude and fundamental frequency.

**Figure 3.**Magnitude frequency response of the rectangular window (dotted line), of the sine window (dashed line), of the Hanning window (solid line), and of the Gaussian window (dash-dotted line). The abscissae axis can also be read as a DFT-bin scale.

**Figure 4.**Illustration of the projection of a target sinusoid whose frequency is ${\omega}_{\mathsf{\ell}}=\frac{2\pi}{N}(\mathsf{\ell}+{\mathrm{\Delta}}_{\mathsf{\ell}})$ on the frequency response of three adjacent ODFT sub-bands (or channels): channel $k=\mathsf{\ell}-1$ (dash-dotted line), channel $k=\mathsf{\ell}$ (solid line), and channel $k=\mathsf{\ell}+1$ (dashed line). As $0.0\le {\mathrm{\Delta}}_{\mathsf{\ell}}<1.0$, the magnitude of the $\mathsf{\ell}{\scriptstyle \mathrm{th}}$ ODFT channel is a local maximum.

**Figure 5.**Illustrative spectrogram of a short excerpt of singing containing vibrato (

**a**) and corresponding accurate frequency estimation of the frequencies of the first 11 harmonics ((

**b**) plot). The different colors in (

**b**) represent the frequency trajectories of individual harmonics.

**Figure 6.**Estimation error (as % of the DFT bin width) for representative frequency estimators in the absence of noise and harmonic interference. Results are presented for a single real-valued sinusoid, when $N=512$, and when $\mathsf{\ell}=N/4$. Fractional Frequency denotes ${\mathrm{\Delta}}_{\mathsf{\ell}}$.

**Figure 7.**Bin width-normalized systematic bias for all tested frequency estimators as a function of the SNR and in absence of harmonic interference.

**Figure 8.**Bin width-normalized RMSE for all tested frequency estimators as a function of the SNR and in absence of harmonic interference. The CRLB is also represented as a reference.

**Figure 9.**Bin width-normalized systematic bias for all tested frequency estimators as a function of the SNR and under mild quasi-harmonic interference conditions.

**Figure 10.**Bin width-normalized RMSE for all tested frequency estimators as a function of the SNR and under mild quasi-harmonic interference conditions. The CRLB is also represented as a reference.

**Figure 11.**Bin width-normalized systematic bias for all tested frequency estimators as a function of the SNR and under strong quasi-harmonic interference conditions.

**Figure 12.**Bin width-normalized RMSE for all tested frequency estimators as a function of the SNR and under strong quasi-harmonic interference conditions. The CRLB is also represented as a reference.

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## Share and Cite

**MDPI and ACS Style**

Silva, J.M.; Oliveira, M.A.; Saraiva, A.F.; Ferreira, A.J.S.
One-Step Discrete Fourier Transform-Based Sinusoid Frequency Estimation under Full-Bandwidth Quasi-Harmonic Interference. *Acoustics* **2023**, *5*, 845-869.
https://doi.org/10.3390/acoustics5030049

**AMA Style**

Silva JM, Oliveira MA, Saraiva AF, Ferreira AJS.
One-Step Discrete Fourier Transform-Based Sinusoid Frequency Estimation under Full-Bandwidth Quasi-Harmonic Interference. *Acoustics*. 2023; 5(3):845-869.
https://doi.org/10.3390/acoustics5030049

**Chicago/Turabian Style**

Silva, João Miguel, Marco António Oliveira, André Ferraz Saraiva, and Aníbal J. S. Ferreira.
2023. "One-Step Discrete Fourier Transform-Based Sinusoid Frequency Estimation under Full-Bandwidth Quasi-Harmonic Interference" *Acoustics* 5, no. 3: 845-869.
https://doi.org/10.3390/acoustics5030049