The Schwartz inequality guarantees that ranges between 0 and 1. Our findings reveal interesting temporal interaction between contralateral and ipsilateral motor (MIc and MIi) cortices under the multisource interference task (MSIT). In the end, we apply the ST-based time-varying coherence to the MEG data. The numerical comparison with measures defined on the GT-spectrogram is also presented. The performance of the proposed ST-based measures is demonstrated using a pair of synthetic time series. The multiscale characteristic and the nonnegativity make the ST an effective tool to investigate the time-varying linear connection between two signals. Third, we define the time-varying coherence function using the ST-spectrogram. We derive the kernel function of the ST-spectrogram that can be used to investigate the characteristics of ST-based time-varying spectra in a simple way. As a bilinear TFR, the spectrogram of the ST can be studied as an extended Cohen's class distribution. Second, we propose a new time-varying spectrum based on the ST. We then show that the time-varying spectrum defined by the Cohen's class distributions coincides with the definition of the locally stationary time series. First, we revisit the definition of locally stationary time series to understand the desirable characteristics to define the time-varying spectra. More specifically, the main contributions of this paper are the following. In this paper, we establish a general framework to estimate time-varying spectra using the Cohen's class distribution functions and apply it for a magnetoencephalography (MEG) study using the ST, a particular Cohen's class distribution. The ST has gained popularity in the signal processing community because of its easy interpretation and fast computation. Utilizing a Gaussian frequency-localization window of frequency-dependent window width, the ST provides a time-frequency representation whose resolution varies inversely proportional to the frequency variable. The Stockwell transform (ST), proposed by geophysicists in 1996, is a hybrid of the Gabor transform (GT) and wavelet transform. However, the time-scale distribution provided by the wavelet transform may not be straightforwardly converted to a distribution in time-frequency domain. This is because the multiscale resolution provided by wavelet transforms offers a more accurate description of the nonstationary characteristics of a signal. In Section 2.3, we present such a connection in the context of Priestley's definition of time-varying spectrum.įollowing the development of wavelet theory over the last two decades, transforms that provide the multiresolution TFRs have been receiving growing attention in the field of time-frequency analysis. However, there is no explicit explanation in the literature about the general connection of the evolutionary spectrum and the Cohen's class representations. Specific Cohen's class distribution functions have been directly used to estimate evolutionary spectra in the past. In 1966, Cohen discovered that all the bilinear TFRs can be categorized as Cohen's class distributions whose properties are fully determined by their corresponding kernel functions. That is, time-varying spectra can be estimated through a variety of time-frequency representations (TFRs) with different advantageous features. His work links the theory of time series analysis to that of time-frequency analysis. In 1965, Priestley defined the class of locally stationary time series and proposed the theory of evolutionary spectra to study their time-varying characteristics. This leads to the development of time-varying spectrum. The temporal information, missed by Fourier analysis, needs to be addressed in order to better understand the dynamics of brain functionality. The functional interactions associated with cognitive and behavioral events are dynamic and transient. However, the brain is a complex, nonstationary, massively interconnected dynamic system. The traditional spectrum analysis, built on the theory of Fourier analysis, relies on the assumption that the underlying time series are stationary. ![]() Particularly, the coherence function, which estimates the linear relationship between two simultaneous time series as a function of frequency, is widely used to measure brain functional connectivity. Since brain activities are characterized by multiple oscillators from different frequency bands, spectrum analysis has become a popular tool to noninvasively investigate the mechanisms of the brain functions. Understanding the underlying mechanism is useful not only for learning brain functionality, but also for guiding treatments of mental or behavioral diseases. ![]() ![]() Previous studies in neuroscience have shown that cortico-cortical interactions play a crucial role in the performance of cognitive tasks.
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