On Asymptotic Efficiency of the <i>M</i><sub>2</sub><i>M</i><sub>4</sub> Signal-to-Noise Estimator for Deterministic Complex Sinusoids
The moment-based <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>M</mi><mn>2</mn></msub><msub><mi>M</mi><mn>4</mn></msub>&l...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-07-01
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| Series: | Sensors |
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| Online Access: | https://www.mdpi.com/1424-8220/21/15/4950 |
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| author | Gianmarco Romano |
| author_facet | Gianmarco Romano |
| author_sort | Gianmarco Romano |
| collection | DOAJ |
| description | The moment-based <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>M</mi><mn>2</mn></msub><msub><mi>M</mi><mn>4</mn></msub></mrow></semantics></math></inline-formula> signal-to-noise (SNR) estimator was proposed for a complex sinusoidal signal with a <i>deterministic</i> but unknown phase corrupted by additive Gaussian noise by Sekhar and Sreenivas. The authors studied its performances only through numerical examples and concluded that the proposed estimator is asymptotically efficient and exhibits finite sample super-efficiency for some combinations of signal and noise power. In this paper, we derive the analytical asymptotic performances of the proposed <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>M</mi><mn>2</mn></msub><msub><mi>M</mi><mn>4</mn></msub></mrow></semantics></math></inline-formula> SNR estimator, and we show that, contrary to what it has been concluded by Sekhar and Sreenivas, the proposed estimator is neither (asymptotically) efficient nor super-efficient. We also show that when dealing with deterministic signals, the covariance matrix needed to derive asymptotic performances must be explicitly derived as its known general form for random signals cannot be extended to deterministic signals. Numerical examples are provided whose results confirm the analytical findings. |
| first_indexed | 2024-03-10T09:09:22Z |
| format | Article |
| id | doaj.art-9dc170d2fdc6474cbe7352f5cf41011d |
| institution | Directory Open Access Journal |
| issn | 1424-8220 |
| language | English |
| last_indexed | 2024-03-10T09:09:22Z |
| publishDate | 2021-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Sensors |
| spelling | doaj.art-9dc170d2fdc6474cbe7352f5cf41011d2023-11-22T06:08:13ZengMDPI AGSensors1424-82202021-07-012115495010.3390/s21154950On Asymptotic Efficiency of the <i>M</i><sub>2</sub><i>M</i><sub>4</sub> Signal-to-Noise Estimator for Deterministic Complex SinusoidsGianmarco Romano0Department of Engineering, University of Campania “L. Vanvitelli”, 81031 Aversa, CE, ItalyThe moment-based <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>M</mi><mn>2</mn></msub><msub><mi>M</mi><mn>4</mn></msub></mrow></semantics></math></inline-formula> signal-to-noise (SNR) estimator was proposed for a complex sinusoidal signal with a <i>deterministic</i> but unknown phase corrupted by additive Gaussian noise by Sekhar and Sreenivas. The authors studied its performances only through numerical examples and concluded that the proposed estimator is asymptotically efficient and exhibits finite sample super-efficiency for some combinations of signal and noise power. In this paper, we derive the analytical asymptotic performances of the proposed <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>M</mi><mn>2</mn></msub><msub><mi>M</mi><mn>4</mn></msub></mrow></semantics></math></inline-formula> SNR estimator, and we show that, contrary to what it has been concluded by Sekhar and Sreenivas, the proposed estimator is neither (asymptotically) efficient nor super-efficient. We also show that when dealing with deterministic signals, the covariance matrix needed to derive asymptotic performances must be explicitly derived as its known general form for random signals cannot be extended to deterministic signals. Numerical examples are provided whose results confirm the analytical findings.https://www.mdpi.com/1424-8220/21/15/4950signal-to-noise ratio (SNR) estimationmethod of momentsasymptotic varianceestimator’s efficiencysuper-efficiencyCramer-Rao bound |
| spellingShingle | Gianmarco Romano On Asymptotic Efficiency of the <i>M</i><sub>2</sub><i>M</i><sub>4</sub> Signal-to-Noise Estimator for Deterministic Complex Sinusoids Sensors signal-to-noise ratio (SNR) estimation method of moments asymptotic variance estimator’s efficiency super-efficiency Cramer-Rao bound |
| title | On Asymptotic Efficiency of the <i>M</i><sub>2</sub><i>M</i><sub>4</sub> Signal-to-Noise Estimator for Deterministic Complex Sinusoids |
| title_full | On Asymptotic Efficiency of the <i>M</i><sub>2</sub><i>M</i><sub>4</sub> Signal-to-Noise Estimator for Deterministic Complex Sinusoids |
| title_fullStr | On Asymptotic Efficiency of the <i>M</i><sub>2</sub><i>M</i><sub>4</sub> Signal-to-Noise Estimator for Deterministic Complex Sinusoids |
| title_full_unstemmed | On Asymptotic Efficiency of the <i>M</i><sub>2</sub><i>M</i><sub>4</sub> Signal-to-Noise Estimator for Deterministic Complex Sinusoids |
| title_short | On Asymptotic Efficiency of the <i>M</i><sub>2</sub><i>M</i><sub>4</sub> Signal-to-Noise Estimator for Deterministic Complex Sinusoids |
| title_sort | on asymptotic efficiency of the i m i sub 2 sub i m i sub 4 sub signal to noise estimator for deterministic complex sinusoids |
| topic | signal-to-noise ratio (SNR) estimation method of moments asymptotic variance estimator’s efficiency super-efficiency Cramer-Rao bound |
| url | https://www.mdpi.com/1424-8220/21/15/4950 |
| work_keys_str_mv | AT gianmarcoromano onasymptoticefficiencyoftheimisub2subimisub4subsignaltonoiseestimatorfordeterministiccomplexsinusoids |