Estimating Frequency, Amplitude And Phase Of Two Sinusoids With Very Close Frequencies

This paper presents an algorithm to estimate the parameters of two closely spaced sinusoids, providing a frequency resolution that is more than 800 times greater than that obtained by using the Discrete Fourier Transform (DFT). The strategy uses a highly optimized grid search approach to accurately estimate frequency, amplitude and phase of both sinusoids, keeping at the same time the computational effort at reasonable levels. The proposed method has three main characteristics: 1) a high frequency resolution; 2) frequency, amplitude and phase are all estimated at once using one single package; 3) it does not rely on any statistical assumption or constraint. Potential applications to this strategy include the difficult task of resolving coincident partials of instruments in musical signals.35740747Benaroya, L., Bimbot, F., Gribonval, R., Audio source separation with a single sensor (2006) IEEE Tran. on Audio, Speech, and Language Proc., 14 (1), pp. 191-199. , JanKay, S., Marple, S.L., Spectrum analysis-A modern perspective (1981) Proc. IEEE, 9, pp. 1380-1419. , Nov(1982) Proceedings of IEEE, 70. , Special Issue on Spectral Estimation, SeptKay, S.M., (1988) Modern Spectral Estimation: Theory and Application, , Prentice-Hall Signal Processing Series, Englewood Cliffs, NJ, USAMarple, S.L., (1987) Digital Spectral Analysis with Applications, , Prentice-Hall Signal Processing Series, Englewood Cliffs, NJ, USAStoica, P., Moses, R., (1997) Introduction to Spectral Analysis, , Prentice-Hall, Upper Saddle River, NJ, USAKitsios, K., Spanias, A., Welfert, B., Loizou, P., An Adaptive Modified Covariance Algorithm For Spectral Analysis (1996) Proc. of 8th IEEE Signal Processing Workshop on Statistical Signal and Array Processing, pp. 56-59. , in Corfu, Greece, JunBresler, Y., Macovski, A., Exact maximum likelihood parameter estimation of superimposed exponential signals in noise (1986) IEEE Trans. Acoust., Speech and Signal Processing, ASSP-34 (5), pp. 1081-1089. , OctStoica, P., Moses, R.L., Friedlander, B., Söderström, T., Maximum Likelihood estimation of the parameters of multiple sinusoids from noisy measurements (1989) IEEE Trans. Acoust., Speech and Signal Processing, 37 (3), pp. 378-391. , MarchStoica, P., Sharman, C., Novel eigenanalysis method for direction estimation (1990) IEE Proceedings, 137 (1), pp. 19-26. , FebRife, D.C., Boorstyn, R.R., Multiple tone parameter estimation from discrete-time observation (1976) Bell System Technical Journal, 55 (9), pp. 1389-1410. , NovStoica, P., Händel, P., Söderström, T., Approximate maximum likelihood frequency estimation (1994) Automatica, 30 (1), pp. 131-145. , JanuaryMacleod, M.D., Joint detection and high resolution ML estimation of multiple sinusoids in noise (2001) Proc. of 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), 5, pp. 3065-3068. , in, Salt Lake City, MayHwang, J.-K., Chen, Y.-C., Superresolution Frequency Estimation by Alternating Notch Periodogram (1993) IEEE Transaction on Signal Processing, 41 (2), pp. 727-741. , FebruaryStoica, P., Sharman, K.C., Maximum Likelhood Methods for Directionof-Arrival Estimation (1990) IEEE Trans. Acoust., Speech and Signal Processing, 38 (7), pp. 1132-1143. , JulyBurg, J.P., Maximum Entropy Spectral Analysis (1967) Proceedings of the 37th Annual International Meeting of the Society Exploration Geophysicists, , in Oklahoma City, USADepalle, P., Helie, T., Extraction of spectral peak parameters using a short-time Fourier transform modeling and no sidelobe windows (1997) Proc. of the IEEE Workshop on Applications of Signal Processing to Audio and Acoustics, pp. 19-22. , in New Paltz, USA, OctoberTufts, D., Kumaresan, R., Estimation of Frequencies of Multiple Sinusoids: Making Linear Prediction Perform Like Maximum Likelihood (1982) Proceedings of the IEEE, 70 (9), pp. 975-989. , SeptSchmidt, R.O., (1981) A signal subspace approach to multiple emitter location and spectral estimation, , Ph.D. dissertation, Stanford University, Stanford, CAWang, H., Kaveh, M., On the Performance of Signal-Subspace Processing-Part I: Narrow-Band Systems (1986) IEEE Trans. Acoust., Speech and Signal Processing, 34 (5), pp. 1201-1209. , OctCadzow, J.A., Multiple Source Location-The Signal Subspace Approach (1990) IEEE Trans. Acoust., Speech and Signal Processing, 38 (7), pp. 1110-1125. , JulyRoy, R., Kailath, T., ESPRIT-Estimation of Signal Parameters Via Rotational Invariance Techniques (1989) IEEE Trans. on Acoustics, Speech and Signal Processing, 37 (7). , JulyViberg, M., Ottersten, B., Sensor array processing based on subspace fitting (1991) IEEE Trans. on Signal Processing, 39 (5), pp. 1110-1121. , MayViberg, M., Ottersten, B., Kailath, T., Detection and estimation in sensor arrays using weighted subspace fitting (1991) IEEE Trans. on Signal Processing, 39 (11), pp. 2436-2449. , NovWilkes, D.M., Cadzow, J.A., The effects of phase on high-resolution frequency estimators (1993) IEEE Transaction on Signal Processing, 41 (3), pp. 1319-1330. , MarchFeyh, G., Inverse Eigenvalue Problem for Sinusoidal Frequency Estimation (1997) Proc. of 1997 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), 5, pp. 3985-3988. , in Munich, Germany, AprilGe, F.-X., Zhang, H., Yang, J., Peng, Y.-N., Super-resolution frequency estimation of the sinusoidal signals with unknown lowpass envelopes (2003) Proc. of 2003 IEEE Radar Conference, p. 273. , in Huntsville, USA, MayChristensen, M.G., Jakobsson, A., Jensen, S.H., Joint High-Resolution Fundamental Frequency and Order Estimation (2007) IEEE Transactions on Audio, Speech, and Language Processing, 15 (5), pp. 1635-1644. , JulyCapon, J., High-resolution frequency-wavenumber spectrum analysis (1969) Proc. IEEE, 57, pp. 1408-1418. , AugustNishimura, S., Adaptive detection and enhancement of closely spaced sinusoids using multirate techniques (1993) Proc. of 1993 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), 3, pp. 400-403. , in Minneapolis, USA, AprilByrnes, C.I., Georgiou, T.T., Lindquist, A., A New Approach to Spectral Estimation: A Tunable High-Resolution Spectral Estimator (2000) IEEE Transactions on Signal Processing, 48 (11), pp. 3189-3205. , NovemberLambrecht, C.B., Karrakchou, M., Wavelet packets-based highresolution spectral estimation (1995) Signal Processing, 47 (2), pp. 135-144. , NovemberLi, H.-T., Djuric, P.M., A novel approach to detection of closely spaced sinusoids (1996) Signal Processing, 51 (2), pp. 93-104. , JuneO'Shea, P., A High-Resolution Spectral Analysis Algorithm for Power-System Disturbance Monitoring (2002) IEEE Transactions on Power Systems, 17 (3). , AugustHainsworth, S., Macleod, M., On Sinusoidal Parameter Estimation (2003) Proc. of the 6th International Conference on Digital Audio Effects, , in London, UK, SeptemberSarkar, I., Fam, A.T., The interlaced chirp Z transform (2006) Signal Processing, 86 (9), pp. 2221-2232. , SeptemberBesson, O., Stoica, P., Nonlinear least-squares approach to frequency estimation and detection for sinusoidal signals with arbitrary envelope (1999) Digital Signal Processing, 9 (1), pp. 45-56. , JanuaryTolonen, T., (1999) Methods for Separation of Harmonic Sound. Sources using Sinusoidal Modeling, , presented at the 106th AES Convention, paper 4958, Munich, GermanyRoy, R.H., (1987) ESPRIT-Estimation of signal parameters via rotational invariance techniques, , Ph.D. dissertation, Stanford University, Stanford, CAStoica, P., Soderstrom, T., Statistical analysis of MUSIC and ESPRIT estimates of sinusoidal frequencies (1991) Proc. International Conference on Acoustics, Speech, and Signal Processing, 5, pp. 3273-3276. , in Toronto, Canada, AprilBesson, O., Stoica, P., Analysis of MUSIC and ESPRIT frequency estimates for sinusoidal signals with lowpass envelopes (1996) IEEE Transactions on Signal Processing, 44 (9), pp. 2359-2364. , SeptemberStoica, P., Li, H., Li, J., Amplitude estimation of sinusoidal signals: Survey, new results, and an application (2000) IEEE Transactions on Signal Processing, 48 (2), pp. 338-352. , FebruaryKay, S.M., (1993) Fundamentals of Statistical Signal Processing-Estimation Theory, , Prentice-Hall Signal Processing Series, Englewood Cliffs, NJ, US