Gabor Wavelet Transform
A Gabor Wavelet Transform is a short-time Fourier transform that ...
- Context:
- It can range from being a Continuous Gabor Transform to being a Discrete Gabor Transform.
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- Counter-Example(s):
- See: Gabor Filter, Timeseries Transform, Sine Wave, Gaussian Function, Window Function, Time-Frequency Analysis.
References
2014
- (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/Gabor_transform Retrieved:2014-7-25.
- The Gabor transform, named after Dennis Gabor, is a special case of the short-time Fourier transform. It is used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time. The function to be transformed is first multiplied by a Gaussian function, which can be regarded as a window function, and the resulting function is then transformed with a Fourier transform to derive the time-frequency analysis. The window function means that the signal near the time being analyzed will have higher weight. The Gabor transform of a signal x(t) is defined by this formula: :[math]\displaystyle{ G_x(t,f) = \int_{-\infty}^\infty e^{-\pi(\tau-t)^2}e^{-j2\pi f\tau}x(\tau)\,d\tau }[/math]
The Gaussian function has infinite range and it is impractical for implementation. However, a level of significance can be chosen (for instance 0.00001) for the distribution of the Gaussian function. :[math]\displaystyle{ \begin{cases} \lt P\gt e^{-{\pi}a^2} \ge 0.00001; & \left| a \right| \le 1.9143 \\ \lt P\gt e^{-{\pi}a^2} \lt 0.00001; & \left| a \right| \gt 1.9143 \lt P\gt \end{cases} }[/math]
Outside these limits of integration [math]\displaystyle{ \left| a \right| \gt 1.9143 }[/math], the Gaussian function is small enough to be ignored. Thus the Gabor transform can be satisfactorily approximated as :[math]\displaystyle{ G_x(t,f) = \int_{-1.9143+t}^{1.9143+t} e^{-\pi(\tau-t)^2} e^{-j2\pi f\tau} x(\tau) \, d\tau }[/math]
This simplification makes the Gabor transform practical and realizable.
The window function width can also be varied to optimize the time-frequency resolution tradeoff for a particular application by replacing the [math]\displaystyle{ {-{\pi}(\tau-t)^2} }[/math] with [math]\displaystyle{ {-{\pi}\alpha (\tau-t)^2} }[/math] for some chosen alpha.
- The Gabor transform, named after Dennis Gabor, is a special case of the short-time Fourier transform. It is used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time. The function to be transformed is first multiplied by a Gaussian function, which can be regarded as a window function, and the resulting function is then transformed with a Fourier transform to derive the time-frequency analysis. The window function means that the signal near the time being analyzed will have higher weight. The Gabor transform of a signal x(t) is defined by this formula: :[math]\displaystyle{ G_x(t,f) = \int_{-\infty}^\infty e^{-\pi(\tau-t)^2}e^{-j2\pi f\tau}x(\tau)\,d\tau }[/math]
1993
- (Qian & Chen, 1993) ⇒ Shie Qian, and Dapang Chen. (1993). “Discrete Gabor Transform.” In: Signal Processing, IEEE Transactions on, 41(7).