1. Introduction
The recent advances in technology have paved the way for deployment of reliable biological sensors in clinical practice. A wide variety of such sensors have been developed to measure biosignals that reflect various underlying physiological phenomena. For example, gyroscope and accelerometers are employed for pathological and physiological tremor signal measurement [
1], accelerometers are employed for cardiac mechanical vibrations monitoring [
2], infrared sensors are employed for respiration motion monitoring [
3], and common electrodes are employed for brain and heart electrical activity measurement [
4,
5]. In order to adequately interpret the signals and make useful observations, a proper understanding of the involved phenomena and their influence on the signals is necessary. However, most of the physiological signals are non-stationary due to the complex nature of the biological systems. Very subtle changes in the time-frequency characteristics of these signals can potentially correspond to an underlying condition. Therefore, analysis of biosignals require high resolution time-frequency decomposition methods to effectively detect these subtle changes.
The frequency domain information of a signal is usually obtained by applying the well-known Fourier transform. However, its strong assumption on the stationarity of the signal is often violated by the signals that are collected from biological or biomedical systems. To handle the non-stationarity of such signals, time-frequency decomposition methods are usually employed for obtaining time-frequency mapping of such signals [
4,
5,
6,
7]. The time-varying characteristics are thus analyzed in both time and frequency domains. Commonly employed time-frequency decomposition methods can be categorized into parametric and non-parametric depending on whether a signal model is required [
8]. Fourier transform and Wavelet transform based methods are the typical parametric time-frequency decomposition methods, whereas the autoregressive model is the most popular choice for parametric time-frequency decomposition [
5,
6]. For non-uniformly sampled data, the windowed Lomb periodogram is usually employed [
9,
10].
If the signal is uniformly sampled, and its characteristics change slowly over time, we can safely assume that the stationarity holds for a short time interval. The Fourier transform is then applied to the portion of the signal, which is extracted from the original signal by multiplying it with a properly selected window function. This procedure is known as STFT [
11]. Although the global stationary requirement of the signal is relaxed by assuming piecewise stationarity, the optimal window length is often difficult to be determined.
In STFT, the temporal and spectral resolutions are proportional to each other. The product of spectral resolution and temporal resolution is bounded by a fixed value [
12]. If the window length is fixed, a trade-off between temporal and spectral resolutions must be made. The imbalance between spectral and temporal resolutions may cause a leakage effect in the obtained spectrogram [
11,
12]. Moreover, the window function needs to be tailored according to the frequency range of interest to achieve the desired temporal and spectral resolution. The CWT solves this problem by decomposing the signal into a basis of dilated and shifted version of a pre-defined mother wavelet [
13]. By shrinking the mother wavelet, CWT attains a better trade-off in temporal and spectral resolution as compared to STFT.
Apart from the non-parametric time-frequency decomposition methods such as STFT and CWT, the time-frequency mapping of a signal can also be obtained by fitting a parametric model to the signal and then transforming the estimated coefficients of the model into its corresponding time-frequency plane. As shown in [
14], the time-frequency mapping can be obtained from the estimated coefficients of the autoregressive (AR) model. To account for the time-varying characteristics of the signal, the AR coefficients are estimated through adaptive algorithms such as least mean square (LMS) and recursive least mean square (RLS) [
15]. With the assumption that the AR process is driven by the Gaussian noise, the state-space form of the AR model can be formulated with the Kalman filter as the optimal estimator [
16]. The Kalman smoother is also employed for achieving better results in offline analysis [
15,
17]. To further refine the estimation of the AR model, the expectation maximization algorithm is employed for identification of the state transition matrix [
17] and particle filter to account for the non-Gaussian noise case [
18]. The temporal resolution of AR based time-frequency decomposition is guaranteed by the employed adaptive algorithm, whereas the spectral resolution is strongly affected by the AR model order. As the commonly applied order selection methods rely on the estimation error, the selected order is only optimal for either reconstruction or prediction and does not offer better time-frequency mapping [
14].
To mitigate the sensitivity of spectral resolution on model order, the frequency characteristics of the signal need to be considered in the signal model. One such model is BMFLC [
19,
20]. The BMFLC divides the frequency band of interest with a fixed frequency gap and adopts a truncated Fourier series as the signal model. The estimated Fourier coefficients for each frequency component can thus be used to form the time-frequency mapping of the signal. It was shown in [
21] that the BMFLC with adaptive filter algorithms, such as least-mean square and Kalman filter, was successful for time-frequency decomposition of motion induced Electroencephalography (EEG) signal in real time. The estimated time-frequency mapping can be fine-tuned with the help of a smoother procedure. The comparison study in [
21] also suggests that the BMFLC with Kalman smoother (BMFLC-KS) provides better temporal and spectral resolutions than STFT.
In the scenario where two frequency components lie closely to each other in spectral domain [
22], BMFLC requires a frequency gap that is at least equal to the distance between the two frequency components to differentiate them [
21]. However, the dimension of states in BMFLC increases with the increased spectral resolution (i.e., smaller frequency gap), which further increases computational complexity. Moreover, the high-dimensionality also causes the adaptive filter paired with BMFLC to fail at providing an accurate time-frequency decomposition [
21]. In this work, we address this problem by incorporating sparse linear regression with BMFLC.
Recently the sparse linear regression model has found numerous applications in signal processing [
23,
24,
25]. The sparse linear regression uses
norm to regularize the regression coefficients. It has been proven to generate a better signal model for non-stationary signals [
26]. In [
27], the sparse linear regression model is used for estimating the frequency-hopping signal in communication applications. In [
28], the AR coefficients are expanded onto a redundant set of basis functions, which simplifies the identification of time-varying AR coefficients into time-invariant case. Then, a sparse-aware regression method [
29] is employed to find the most informative one in the redundant model in order to improve the overall estimation performance. Its application to phase retrieval of sparse signal is shown in [
30].
The large dimension of states in BMFLC caused by increased spectral resolution necessitates the imposition of sparsity on the model of BMFLC under the assumption that only a few coefficients change at any time instant. As BMFLC inherits linearity from the Fourier series, the BMFLC model can thus be modeled in the form of sparse linear regression model. Similar to [
22], the super-resolution can be achieved by estimating the amplitude from a redundant set of frequencies.
In this work, we model the BMFLC in the form of a sparse linear model and impose constraint on the model coefficients. The convex optimization algorithm is then employed to estimate model coefficients. The estimation accuracy of the model is compared with BMFLC Kalman smoother (BMFLC-KS). The time-frequency decomposition performance of the proposed model is compared with STFT, CWT and BMFLC-KS on four synthetic signals. An energy ratio metric is also employed to demonstrate the effectiveness of the proposed model.
4. Discussion
Although, by construction, the performance of BMFLC based methods depends on the prior knowledge of the frequency characteristics of the signal, and the results obtained with the chirp signal show that the proposed model was able to tolerate the discrepancy to a certain extent and provide a reasonable time-frequency mapping. However, as our results suggest, if the frequency mismatch is suspected in the signal, BMFLC-KS should be employed instead of sparse-BMFLC.
The proposed sparse-BMFLC relies on the optimization algorithm to estimate the amplitude and the frequency in the model. However, the sparse-BMFLC can provide better temporal and spectral resolution than the other methods in comparison, the proposed method is computationally more expensive as compared to other existing methods. In comparison, STFT and CWT do not require much computational power for estimation of a large number of frequency components. Hence, the proposed method is more suitable for band-limited signals when more detailed and accurate time-frequency decomposition is required for analysis.