Chapter 3 Proposed Work
3.1 Time-Frequency Analysis
In signal processing, time-frequency analysis is composed of techniques that resolve signals in both time and frequency domains simultaneously, using a variety of time-frequency representations. The most practical motivation of time-frequency analysis is that classical Fourier analysis considers the signal as a periodic or infinite function, while signals are not like that in practice.
3.1.1 Gabor transform
One of the most basic forms of time-frequency analysis is the short-time Fourier transform (STFT), which divides a longer time signal into shorter pieces of equal length
and computes the Fourier transform on each piece of signal respectively. Hence the result reveals the frequency spectrum of each piece and the changing spectra as a function of time. The continuous STFT can be described as
𝑋(𝑡, 𝑓) = ∫−∞∞ 𝑤(𝑡 − 𝜏)𝑥(𝜏)𝑒−𝑗2𝜋𝑓𝜏𝑑𝜏, (3.1) where w(t) is the window function or the mask function. Converting it into the discrete form by t = nΔt, f = mΔf and τ = pΔt, the equation changes to
𝑋(𝑛Δ𝑡, 𝑚Δ𝑓) = ∑∞𝑝=−∞𝑤((𝑛 − 𝑝)Δ𝑡)𝑥(𝑝Δ𝑡)𝑒−𝑗2𝜋𝑝𝑚Δ𝑡Δ𝑓Δ𝑡. (3.2) If we choose the Gaussian function as the window function, the transform is so-called the Gabor transform (GT). The generalized Gabor transform is shown as follows:
𝐺𝑥(𝑡, 𝑓) = √𝜎4 ∫−∞∞ 𝑒−𝜎𝜋(𝜏−𝑡)2 𝑥(𝜏)𝑒−𝑗2𝜋𝑓𝜏𝑑𝜏. (3.3) Suppose that w(t) ≈ 0 for |t| > B = QΔt, the generalized Gabor transform can be
rewritten as discrete form
𝐺𝑥(𝑛Δ𝑡, 𝑚Δ𝑓) = √𝜎4 ∑𝑛+𝑄𝑝=𝑛−𝑄𝑒−𝜎𝜋((𝑛−𝑝)Δ𝑡)2𝑥(𝑝Δ𝑡)𝑒−𝑗2𝜋𝑝𝑚Δ𝑡Δ𝑓Δ𝑡. (3.4) Here, we use unbalanced sampling in the implementation to lower the computation time and the complexity. 𝐵 = 1.9143/√𝜎 is suggested for decayed edge of the Gaussian function. Among all window functions, the Gaussian function has advantages that the area in time-frequency distribution is minimal, which means the Gabor transform has better clarity than others on both time domain and frequency domain simultaneously.
Furthermore, the Gabor transform has symmetric properties on time domain and
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frequency domain since the Gaussian function is the eigenfunction of the Fourier transform.
3.1.2 Wigner distribution function
The Wigner distribution function (WDF) is another commonly used transform in time-frequency analysis, which is first proposed for quantum corrections to classical statistical mechanics. The Wigner distribution function is defined as
𝑊𝑥(𝑡, 𝑓) = ∫ 𝑥 (𝑡 +𝜏
2) 𝑥∗(𝑡 +𝜏
2) 𝑒−𝑗2𝜋𝑓𝜏𝑑𝜏
∞
−∞ , (3.5)
where x*(t) is the conjugate function of the signal. Converting it into the discrete form by t = nΔt, f = mΔf and τ’ = τ/2 = pΔt, the equation changes to
𝑊𝑥(𝑛Δ𝑡, 𝑚Δ𝑓) = 2 ∑∞𝑝=−∞𝑥((𝑛 + 𝑝)Δ𝑡) 𝑥∗((𝑛 − 𝑝)Δ𝑡) 𝑒−𝑗4𝜋𝑝𝑚Δ𝑡Δ𝑓Δ𝑡. (3.6) Here, we use unbalanced sampling in the implementation to lower the computation time and the complexity. The most important advantage of the Wigner distribution function is that the clarity is higher comparing to the case of the STFT due to the signal autocorrelation function. It reduces to the spectral density function at all times t for stationary processes, which is the motivation for it, while it is still equivalent to the non-stationary autocorrelation function. There are also some good properties other transforms do not have. However, the Wigner distribution function is not a linear transform, which implies that the transform of the sum of two functions will not equal
to the sum of the transforms of two functions. The disadvantage of the cross term occurs when the signal has more than one component. It also needs more computation time rather than the STFT. The Wigner distribution function can be generalized to Cohen’s class distribution as a more powerful method of time-frequency analysis.
3.1.3 Gabor-Wigner transform
The Gabor-Wigner transform (GWT) [31] refers to the combination of the Gabor transform and the Wigner distribution function, which combines the advantages of both transforms. The basic idea is to use the Gabor transform as a filter to mask off the cross term of the Wigner distribution function, while the high clarity of the Wigner distribution function is preserved. There are a variety of definitions of the Gabor-Wigner transform and four examples are given as follows:
𝐶𝑥(𝑡, 𝑓) = 𝐺𝑥(𝑡, 𝑓) ∙ 𝑊𝑥(𝑡, 𝑓), (3.7) 𝐶𝑥(𝑡, 𝑓) = min{|𝐺𝑥(𝑡, 𝑓)|2, |𝑊𝑥(𝑡, 𝑓)|}, (3.8) 𝐶𝑥(𝑡, 𝑓) = 𝑊𝑥(𝑡, 𝑓) ∙ {|𝐺𝑥(𝑡, 𝑓)| > 𝑡ℎ𝑟}, (3.9) 𝐶𝑥(𝑡, 𝑓) = 𝐺𝑥𝛼(𝑡, 𝑓) ∙ 𝑊𝑥𝛽(𝑡, 𝑓). (3.10) Moreover, the Gabor-Wigner transform also preserves many good properties from the Gabor transform and the Wigner distribution function, such as the rotation relation with the fractional Fourier transform (FrFT), which is helpful for analyzing the
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characteristics of targets and modulating signals.
In our work, we have to decide N first, where N = 1/ΔtΔf, Δt is the time interval and Δf is the frequency interval in the implementation of time-frequency analysis. The
choice of N affects the fineness of the frequency axis and the computation time. Δt can be obtained by the reciprocal of the sampling frequency, and thus Δf can also be obtained. Before the time-frequency analysis, we convert the signal to the analytic signal and modulate it by a quarter of the sampling frequency, which makes the observation easier. Here, we use (3.10) as the definition of the Gabor-Wigner transforms in order to maintain the flexibility. Fig. 3-1 shows the time-frequency analysis of Cow signal, which is the mooing sound from a cow and is composed of several harmonics.
The alignment of the frequency axis must be completed since the frequency range of the Gabor transform and that of the Wigner distribution function are not identical.
The frequency range of the Wigner distribution function is about half of that of the Gabor transform in order to avoid the aliasing effect. After the combination of two transforms, we set a threshold thrgwt to filter the noise that may be created by the setting issue of the transform parameters. The value of the threshold is given by
𝑡ℎ𝑟𝑔𝑤𝑡 = (3 ∑ ∑ 𝐶𝑥(𝑛Δ𝑡,𝑚Δ𝑓) segmentation of the figure in time-frequency analysis will be done.
(a) (b) (c)
Fig. 3-1 Time-frequency analysis of Cow signal. (a) Gabor transform (b) Wigner distribution function (c) Gabor-Wigner transform.
3.1.4 Segmentation
The result of time-frequency analysis is viewed as a figure and dilated with an elliptical kernel. The dilation is able to connect neighbor components belonging to the same part that may be disconnected accidentally. Then we label connected components by bwlabel function, which gives the same numbers to pixels in each connected component individually. We set another threshold thrseg to exclude small area components that probably come from the noise. The value of the threshold, which is associated with the concept of the uncertainty principle, is given in the following:
𝑡ℎ𝑟𝑠𝑒𝑔 = ⌈𝐶𝑠𝑒𝑔
Δ𝑡Δ𝑓⌉, (3.12)
where Cseg is a constant. Fig. 3-2 displays the results of the processing of the time-frequency analysis. Afterwards, the labels are rearranged from the component with
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most pixels to that with the least for convenience, as shown in Fig. 3-3.
(a) (b) (c)
Fig. 3-2 Processing of the time-frequency analysis of Cow signal. (a) GWT thresholded by thrgwt (b) dilation of thresholded figure with an elliptical kernel (c) labeled signal thresholded by thrseg.
Fig. 3-3 Segmentation of the time-frequency analysis of Cow signal.