An optimal multitaper time-frequency estimator for multi-component transient signals

Multi-component transient signals show up in many different areas, most commonly in communication and radar but are also often found in measurements of human related signals, such as electrical responses of the heart and the brain. Transient signals are also typical in animal acoustic applications, e.g., dolphin echo locations and bird song syllables. The usual basis functions in linear modeling of transient signals are Karhunen-Loève, Laguerre or Hermite functions (HF) where these basis functions are applied to reduce noise disturbances or to extract features for classification.

To further extract information on transient signals, time-frequency (TF) analysis is a useful tool. The Wigner-Ville distribution (WVD) is the optimal choice from TF concentration viewpoint. However, the cross-terms that arise for multi-component signals as well as for disturbances, cause severe problems and many different TF distributions have been invented, usually with the aim to maintain concentration and suppress cross-terms. A computationally efficient algorithm that corresponds to a specific TF distribution can be found using a multitaper spectrogram. In this context the HF have been shown to be useful as multitapers, as they are optimal in the aspect of concentration and orthogonality in the TF domain, see [1] and references therein. These properties have made them to become often used for robust spectrogram estimation of non-stationary signals and processes.

In the multitaper estimator, the different spectrograms could be given weights and these weights could further be optimized according to some criterion. In [2], the theoretical expressions for the weights of an least squares optimal HF multitaper spectrogram estimate of a Gaussian function (the zeroth-order HF) were derived. The resulting estimator was shown to have better concentration properties and ability to suppress noise, than other TF estimators.

In this submission, the idea of optimal HF multitaper spectrograms are explored further. The aim is to find the optimally weighted HF multitaper spectrograms for more general transient signals. The transient signals are linearly expanded as sums of HF and the optimization of the weights will be based on the recently derived theoretical expressions of TF-operations between higher-order HF, [3]. The proposed multitaper estimators will be evaluated by simulations and compared to other estimators.

[1] S. Aviyente and W. J. Williams, “Multitaper Marginal Time-Frequency Distributions,” Signal Processing, 86, pp. 279–295, 2006.
[2] M. Hansson-Sandsten, ”Matched Gaussian Multitaper Spectrogram,” 21st European Signal Processing Conference (EUSIPCO), Marrakech, Morocko, 2013.

[3] C. W. Korevaar et. al., ”Closed-Form Expressions for Time-Frequency Operations Involving Hermite Functions ,” IEEE Trans. on Signal Processing, 64, 6, pp. 1383-1390, 2016.