Hilbert Spectral Analysis

	Hilbert Spectral Analysis Hilbert spectral analysis is a signal analysis method applying the Hilbert transform to compute the instantaneous frequency of signals according to :\omega=\frac.\, After performing the Hilbert transform on each signal, we can express the data in the following form: :X(t)=\sum_^{n}a_j(t)\exp\left(i\int\omega_j(t)dt\right).\, This equation gives both the amplitude and the frequency of each component as functions of time. It also enables us to represent the amplitude and the instantaneous frequency as functions of time in a three-dimensional plot, in which the amplitude can be contoured on the frequency-time plane. This frequency-time distribution of the amplitude is designated as the Hilbert amplitude spectrum, or simply Hilbert spectrum. Hilbert spectral analysis method is an important part of Hilbert–Huang transform The Hilbert–Huang transform (HHT) is a way to decompose a signal into so-called intrinsic mode functions (IMF) along with a trend, and obtain instantaneous fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Hilbert Transform In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, of a real variable and produces another function of a real variable . This linear operator is given by convolution with the function 1/(\pi t) (see ). The Hilbert transform has a particularly simple representation in the frequency domain: It imparts a phase shift of ±90° ( radians) to every frequency component of a function, the sign of the shift depending on the sign of the frequency (see ). The Hilbert transform is important in signal processing, where it is a component of the analytic representation of a real-valued signal . The Hilbert transform was first introduced by David Hilbert in this setting, to solve a special case of the Riemann–Hilbert problem for analytic functions. Definition The Hilbert transform of can be thought of as the convolution of with the function , known as the Cauchy kernel. Because is not integrable across , the integ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Instantaneous Frequency Instantaneous phase and frequency are important concepts in signal processing that occur in the context of the representation and analysis of time-varying functions. The instantaneous phase (also known as local phase or simply phase) of a ''complex-valued'' function ''s''(''t''), is the real-valued function: :\varphi(t) = \arg\, where arg is the complex argument function. The instantaneous frequency is the temporal rate of change of the instantaneous phase. And for a ''real-valued'' function ''s''(''t''), it is determined from the function's analytic representation, ''s''a(''t''): :\begin \varphi(t) &= \arg\ \\ pt &= \arg\, \end where \hat(t) represents the Hilbert transform of ''s''(''t''). When ''φ''(''t'') is constrained to its principal value, either the interval or , it is called ''wrapped phase''. Otherwise it is called ''unwrapped phase'', which is a continuous function of argument ''t'', assuming ''s''a(''t'') is a continuous function of ''t''. Unless o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Hilbert Spectrum The Hilbert spectrum (sometimes referred to as the Hilbert amplitude spectrum), named after David Hilbert, is a statistical tool that can help in distinguishing among a mixture of moving signals. The spectrum itself is decomposed into its component sources using independent component analysis. The separation of the combined effects of unidentified sources (blind signal separation) has applications in climatology, seismology, and biomedical imaging. Conceptual summary The Hilbert spectrum is computed by way of a 2-step process consisting of: * Preprocessing a signal separate it into intrinsic mode functions using a mathematical decomposition such as singular value decomposition (SVD) or empirical mode decomposition (EMD); * Applying the Hilbert transform to the results of the above step to obtain the instantaneous frequency spectrum of each of the components. The Hilbert transform defines the imaginary part of the function to make it an analytic function (sometimes referred to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Hilbert–Huang Transform The Hilbert–Huang transform (HHT) is a way to decompose a signal into so-called intrinsic mode functions (IMF) along with a trend, and obtain instantaneous frequency data. It is designed to work well for data that is nonstationary and nonlinear. In contrast to other common transforms like the Fourier transform, the HHT is an algorithm that can be applied to a data set, rather than a theoretical tool. The Hilbert–Huang transform (HHT), a NASA designated name, was proposed by Norden E. Huang et al. (1996, 1998, 1999, 2003, 2012). It is the result of the empirical mode decomposition (EMD) and the Hilbert spectral analysis (HSA). The HHT uses the EMD method to decompose a signal into so-called intrinsic mode functions (IMF) with a trend, and applies the HSA method to the IMFs to obtain instantaneous frequency data. Since the signal is decomposed in time domain and the length of the IMFs is the same as the original signal, HHT preserves the characteristics of the varying freque ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]