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A quadratic form for the TF representation with the Wigner–Ville distribution can also be obtained using the signal in the frequency domain as:
(4.68)
The Wigner–Ville distribution is real and has very good resolution in both the time‐ and frequency‐domains. Also it has time and frequency support properties, i.e. if x(t) = 0 for | t | > t0 , then XWV (t, ω) = 0 for | t | > t0 , and if X(ω) = 0 for | ω | > ω 0, then XWV (t, ω) = 0 for | ω | > ω 0. It has also both time‐marginal and frequency‐marginal conditions of the form:
(4.69)
and
(4.70)
If x(t) is the sum of two signals x 1(t) and x 2(t), i.e. x(t) = x 1(t) + x 2(t), the Wigner–Ville distribution of x(t) with respect to the distributions of x 1(t) and x 2(t) will be:
(4.71)
where Re{.} denotes the real part of a complex value and
It is seen that the distribution is related to the spectra of both auto‐ and cross‐correlations. A pseudo‐Wigner–Ville distribution (PWVD) is defined by applying a window function, w(τ), centred at τ = 0 to the time‐based correlations, i.e.:
Figure 4.8 (a) A segment of a signal consisting of two modulated components, (b) ambiguity function for x 1(t) only, and (c) the ambiguity function for x(t) = x 1(t) + x 2(t).
In order to suppress the undesired cross‐terms the two‐dimensional Wigner–Ville (WV) distribution may be convolved with a TF‐domain window. The window is a two‐dimensional lowpass filter, which satisfies the time and frequency‐marginal (uncertainty) conditions, as described earlier. This can be performed as:
(4.74)
where
(4.75)
and φ(τ, ν) is often selected from a set of well known waveforms called Cohen's class. The most popular member of the Cohen's class of functions is the bell‐shaped function defined as:
A graphical illustration of such a function can be seen in Figure 4.9. In this case the distribution is referred to as a Choi–Williams distribution.
The application of a discrete‐time form of the Wigner–Ville distribution to BSS will be discussed later in this chapter and its application to seizure detection has been explained in the chapter devoted to seizure. To improve the distribution a signal‐dependent kernel may also be used [24].
Figure 4.9 Illustration of
4.6 Empirical Mode Decomposition
Empirical mode decomposition (EMD) may be considered as a multiresolution signal decomposition technique. EMD is an adaptive time–space analysis method suitable for processing nonstationary and nonlinear time series. EMD partitions a series into ‘modes’ namely, intrinsic mode functions (IMFs) in the time domain. Like Fourier and WTs EMD does not follow any physical concept. However, the modes may provide insight into various signals contained within the data and have distinct frequency bands/components.
EMD is based on The Hilbert–Huang transform (HHT) which is a way to decompose a signal into IMFs along with a trend and obtain IF data. Its difference from other common transforms like the Fourier transform, is that the HHT is more like an algorithm (an empirical approach) that can be applied to a data set, rather than a theoretical tool.
IMF represents a simple oscillatory mode as a counterpart to the simple harmonic function, but instead of constant amplitude and frequency in a simple harmonic component, an IMF can have variable amplitude and frequency along the time axis.
The procedure of extracting an IMF is called sifting. The sifting process is as follows [25, 26]:
1 Identify all the local extrema in the test data.
2 Connect all the local maxima by a cubic spline line as the upper envelope.
3 Repeat the procedure for the local minima to produce the lower envelope.
The upper and lower envelopes should cover all the data between them. Their mean is m 1. The difference between the data and m 1 is the first component d 1:
(4.77)
Ideally, d 1 should satisfy the definition of an IMF, since the construction of d 1 described above should have made it symmetric and having all maxima positive and all minima negative. After the first round of sifting, a crest may become a local maximum. New extrema generated in this way actually reveal the proper modes lost in the initial examination. In the subsequent sifting process, d 1 can only be treated as a proto‐IMF. In the next step,
