The Obtainable Uncertainty for the Frequency Evaluation of Tones with Different Spectral Analysis Techniques
Abstract
:1. Introduction
2. Considered Methods
2.1. Non-Parametric Methods
2.1.1. IFFT-2p
2.1.2. IFFT-3p
2.1.3. IFFTc
2.2. Parametric Methods
2.2.1. MUSIC
2.2.2. ESPRIT
2.2.3. IWPA
3. Residual Errors
- For each distance, the best performance is obtained by the ESPRIT method, that exhibits the lowest error at any distance between the tones since the error due to the frequency quantization is negligible.
- When the distance between tones is small (), the non-parametric approaches detect only one tone and the errors on the detected tone are significant (comparable with ). Even if the IWPA method is able to estimate both tones and its errors are lower than those of the parametric approach, the error is still high.
- The tone distance slightly influences the algorithms based on the autocorrelation (MUSIC and ESPRIT): only for lower than one bin is the MUSIC algorithm affected by a highest residual error.
- The performance of IWPA and IFFT are comparable, but for small tone distances, the IWPA gives better estimations—vice versa occurs for larger distances .
- The IFFTc algorithm for tone distance greater than 8 bin gives results comparable with MUSIC: errors of the order of 10-6 are measured for both tones.
4. Repeatability under Noisy Conditions
4.1. Sensitivity to the First Tone Distance
4.2. Sensitivity to the Tone–Amplitude Ratio
4.3. Sensitivity to the Number of Samples
5. Uncertainty Evaluation
6. Concluding Remarks
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Tone 1 | Tone 2 | ||||||||||||
IFFT2p | IFFTc | IFFT3p | IWPA | ESPRIT | MUSIC | IFFT2p | IFFTc | IFFT3p | IWPA | ESPRIT | MUSIC | ||
3 | 4 | 6.7 × 10 | 1.5 × 10 | 2.4 × 10 | 1.1 × 10 | 1.1 × 10 | 1.6 × 10 | - | - | - | 1.1 × 10 | 9.6 × 10 | 1.6 × 10 |
4 | 5 | 3.0 × 10 | 6.0 × 10 | 9.1 × 10 | 1.1 × 10 | 1.1 × 10 | 1.8 × 10 | 6.6 × 10 | 5.8 × 10 | 6.7 × 10 | 1.0 × 10 | 9.3 × 10 | 1.8 × 10 |
5 | 6 | 1.6 × 10 | 1.0 × 10 | 4.3 × 10 | 1.1 × 10 | 1.1 × 10 | 1.9 × 10 | 1.2 × 10 | 3.6 × 10 | 5.5 × 10 | 1.0 × 10 | 9.9 × 10 | 1.9 × 10 |
6 | 7 | 9.5 × 10 | 3.4 × 10 | 2.3 × 10 | 1.1 × 10 | 1.0 × 10 | 1.6 × 10 | 7.7 × 10 | 3.2 × 10 | 2.6 × 10 | 1.0 × 10 | 9.9 × 10 | 1.7 × 10 |
7 | 12 | 3.5 × 10 | 5.6 × 10 | 6.7 × 10 | 1.1 × 10 | 1.0 × 10 | 1.7 × 10 | 2.9 × 10 | 2.8 × 10 | 7.4 × 10 | 1.0 × 10 | 9.4 × 10 | 1.7 × 10 |
12 | 20 | 7.8 × 10 | 1.8 × 10 | 9.4 × 10 | 1.1 × 10 | 1.0 × 10 | 1.7 × 10 | 6.5 × 10 | 2.2 × 10 | 9.8 × 10 | 9.7 × 10 | 9.3 × 10 | 1.7 × 10 |
Tone 1 | Tone 2 | ||||||||||||
IFFT2p | IFFTc | IFFT3p | IWPA | ESPRIT | MUSIC | IFFT2p | IFFTc | IFFT3p | IWPA | ESPRIT | MUSIC | ||
3 | 4 | 6.8 × 10 | 2.3 × 10 | 2.4 × 10 | 1.3 × 10 | 1.1 × 10 | 1.6 × 10 | 5.3 × 10 | 2.3 × 10 | 3.5 × 10 | 1.0 × 10 | 1.2 × 10 | 1.6 × 10 |
4 | 5 | 3.0 × 10 | 3.8 × 10 | 9.1 × 10 | 1.2 × 10 | 1.1 × 10 | 1.8 × 10 | 2.4 × 10 | 4.1 × 10 | 1.2 × 10 | 1.0 × 10 | 1.2 × 10 | 1.8 × 10 |
5 | 6 | 1.6 × 10 | 9.9 × 10 | 4.3 × 10 | 1.2 × 10 | 1.1 × 10 | 1.9 × 10 | 1.3 × 10 | 1.2 × 10 | 5.2 × 10 | 1.0 × 10 | 1.2 × 10 | 1.9 × 10 |
6 | 7 | 9.5 × 10 | 3.4 × 10 | 2.3 × 10 | 1.2 × 10 | 1.1 × 10 | 1.6 × 10 | 7.8 × 10 | 5.3 × 10 | 2.6 × 10 | 1.0 × 10 | 1.3 × 10 | 1.7 × 10 |
7 | 12 | 3.6 × 10 | 6.0 × 10 | 7.0 × 10 | 1.1 × 10 | 1.1 × 10 | 1.7 × 10 | 3.1 × 10 | 3.0 × 10 | 7.9 × 10 | 9.3 × 10 | 1.2 × 10 | 1.7 × 10 |
12 | 20 | 1.9 × 10 | 1.9 × 10 | 1.5 × 10 | 1.1 × 10 | 1.1 × 10 | 1.6 × 10 | 1.6 × 10 | 1.4 × 10 | 1.5 × 10 | 9.7 × 10 | 1.3 × 10 | 1.6 × 10 |
Tone 1 | Tone 2 | ||||||||||||
IFFT2p | IFFTc | IFFT3p | IWPA | ESPRIT | MUSIC | IFFT2p | IFFTc | IFFT3p | IWPA | ESPRIT | MUSIC | ||
3 | 4 | 6.8 × 10 | 2.3 × 10 | 2.4 × 10 | 4.0 × 10 | 1.1 × 10 | 1.6 × 10 | 5.4 × 10 | 2.3 × 10 | 3.5 × 10 | 1.0 × 10 | 1.1 × 10 | 1.6 × 10 |
4 | 5 | 3.0 × 10 | 3.8 × 10 | 9.1 × 10 | 4.2 × 10 | 1.0 × 10 | 1.8 × 10 | 2.4 × 10 | 4.1 × 10 | 1.2 × 10 | 1.0 × 10 | 1.1 × 10 | 1.8 × 10 |
5 | 6 | 1.6 × 10 | 9.9 × 10 | 4.3 × 10 | 2.9 × 10 | 1.1 × 10 | 1.9 × 10 | 1.3 × 10 | 1.1 × 10 | 5.2 × 10 | 1.0 × 10 | 1.1 × 10 | 1.9 × 10 |
6 | 7 | 9.5 × 10 | 3.4 × 10 | 2.3 × 10 | 3.0 × 10 | 1.1 × 10 | 1.7 × 10 | 7.8 × 10 | 4.0 × 10 | 2.6 × 10 | 1.0 × 10 | 1.1 × 10 | 1.7 × 10 |
7 | 12 | 3.6 × 10 | 7.3 × 10 | 7.0 × 10 | 2.2 × 10 | 1.1 × 10 | 1.7 × 10 | 3.1 × 10 | 8.5 × 10 | 7.9 × 10 | 9.3 × 10 | 1.0 × 10 | 1.7 × 10 |
12 | 20 | 7.4 × 10 | 3.0 × 10 | 9.4 × 10 | 1.6 × 10 | 1.1 × 10 | 1.7 × 10 | 6.5 × 10 | 2.4 × 10 | 9.7 × 10 | 9.6 × 10 | 1.0 × 10 | 1.7 × 10 |
Case 1 | Case 2 | Case 3 | Case 4 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
, , | , , | , , | , , | ||||||||||
SNR = 40 dB | SNR = 80 dB | SNR = 10 dB | SNR = 60 dB | ||||||||||
IFFT3p | ESPRIT | IFFTc | IFFT3p | ESPRIT | IFFTc | IFFT3p | ESPRIT | IFFTc | IFFT3p | ESPRIT | |||
meas. | 1.03 × 10 | 1.72 × 10 | 6.29 × 10 | 6.99 × 10 | 8.88 × 10 | 4.25 × 10 | 8.98 × 10 | 1.03 × 10 | 5.43 × 10 | 2.84 × 10 | 3.16 × 10 | 1.70 × 10 | |
exp. | 1.03 × 10 | 1.72 × 10 | 6.28 × 10 | 6.99 × 10 | 7.52 × 10 | 4.24 × 10 | 8.97 × 10 | 1.03 × 10 | 5.43 × 10 | 2.84 × 10 | 3.16 × 10 | 1.70 × 10 | |
meas. | 2.96 × 10 | 3.52 × 10 | 1.74 × 10 | 1.58 × 10 | 2.05 × 10 | 1.01 × 10 | 1.28 × 10 | 1.44 × 10 | 6.92 × 10 | 5.26 × 10 | 6.52 × 10 | 3.37 × 10 | |
exp. | 2.98 × 10 | 3.27 × 10 | 1.74 × 10 | 1.58 × 10 | 1.22 × 10 | 1.01 × 10 | 1.28 × 10 | 1.20 × 10 | 6.93 × 10 | 5.26 × 10 | 6.50 × 10 | 3.37 × 10 |
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Dello Iacono, S.; Di Leo, G.; Liguori, C.; Paciello, V. The Obtainable Uncertainty for the Frequency Evaluation of Tones with Different Spectral Analysis Techniques. Metrology 2022, 2, 216-229. https://doi.org/10.3390/metrology2020013
Dello Iacono S, Di Leo G, Liguori C, Paciello V. The Obtainable Uncertainty for the Frequency Evaluation of Tones with Different Spectral Analysis Techniques. Metrology. 2022; 2(2):216-229. https://doi.org/10.3390/metrology2020013
Chicago/Turabian StyleDello Iacono, Salvatore, Giuseppe Di Leo, Consolatina Liguori, and Vincenzo Paciello. 2022. "The Obtainable Uncertainty for the Frequency Evaluation of Tones with Different Spectral Analysis Techniques" Metrology 2, no. 2: 216-229. https://doi.org/10.3390/metrology2020013