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Gabor transform

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Special case of the short-time Fourier transform

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:

G x ( τ , ω ) = ∫ − ∞ ∞ x ( t ) e − π ( t − τ ) 2 e − j ω t d t {\displaystyle G_{x}(\tau ,\omega )=\int _{-\infty }^{\infty }x(t)e^{-\pi (t-\tau )^{2}}e^{-j\omega t}\,dt}
Magnitude of Gaussian function.

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.

{ e − π a 2 ≥ 0.00001 ; | a | ≤ 1.9143 e − π a 2 < 0.00001 ; | a | > 1.9143 {\displaystyle {\begin{cases}e^{-{\pi }a^{2}}\geq 0.00001;&\left|a\right|\leq 1.9143\\e^{-{\pi }a^{2}}<0.00001;&\left|a\right|>1.9143\end{cases}}}

Outside these limits of integration ( | a | > 1.9143 {\displaystyle \left|a\right|>1.9143} ) the Gaussian function is small enough to be ignored. Thus the Gabor transform can be satisfactorily approximated as

G x ( τ , ω ) = ∫ − 1.9143 + τ 1.9143 + τ x ( t ) e − π ( t − τ ) 2 e − j ω t d t {\displaystyle G_{x}(\tau ,\omega )=\int _{-1.9143+\tau }^{1.9143+\tau }x(t)e^{-\pi (t-\tau )^{2}}e^{-j\omega t}\,dt}

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 − π ( t − τ ) 2 {\displaystyle {-{\pi }(t-\tau )^{2}}} with − π α ( t − τ ) 2 {\displaystyle {-{\pi }\alpha (t-\tau )^{2}}} for some chosen α {\displaystyle \alpha } .

Inverse Gabor transform

The Gabor transform is invertible. Because it is over-complete, the original signal can be recovered in a variety of ways. For example, the "unwindowing" approach can be used for any τ 0 ∈ ( − ∞ , ∞ ) {\displaystyle \tau _{0}\in (-\infty ,\infty )} :

x ( t ) = e π ( t − τ 0 ) 2 1 2 π ∫ − ∞ ∞ G x ( τ 0 , ω ) e j ω t d ω {\displaystyle x(t)=e^{\pi (t-\tau _{0})^{2}}{\frac {1}{2\pi }}\int _{-\infty }^{\infty }G_{x}(\tau _{0},\omega )e^{j\omega t}\,d\omega }

Alternatively, all of the time components can be combined:

x ( t ) = ∫ − ∞ ∞ 1 2 π ∫ − ∞ ∞ G x ( τ , ω ) e j ω t d ω d τ {\displaystyle x(t)=\int _{-\infty }^{\infty }{\frac {1}{2\pi }}\int _{-\infty }^{\infty }G_{x}(\tau ,\omega )e^{j\omega t}\,d\omega \,d\tau }

Properties of the Gabor transform

The Gabor transform has many properties like those of the Fourier transform. These properties are listed in the following tables.

Signal Gabor transform Remarks
x ( t ) {\displaystyle x(t)\,} G x ( τ , ω ) = ∫ − ∞ ∞ x ( t ) e − π ( t − τ ) 2 e − j ω t d t {\displaystyle G_{x}(\tau ,\omega )=\int _{-\infty }^{\infty }x(t)e^{-\pi (t-\tau )^{2}}e^{-j\omega t}\,dt}
1 a ⋅ x ( t ) + b ⋅ y ( t ) {\displaystyle a\cdot x(t)+b\cdot y(t)\,} a ⋅ G x ( τ , ω ) + b ⋅ G y ( τ , ω ) {\displaystyle a\cdot G_{x}(\tau ,\omega )+b\cdot G_{y}(\tau ,\omega )\,} Linearity property
2 x ( t − t 0 ) {\displaystyle x(t-t_{0})\,} G x ( τ − t 0 , ω ) e − j ω t 0 {\displaystyle G_{x}(\tau -t_{0},\omega )e^{-j\omega t_{0}}\,} Shifting property
3 x ( t ) e j ω 0 t {\displaystyle x(t)e^{j\omega _{0}t}\,} G x ( τ , ω − ω 0 ) {\displaystyle G_{x}(\tau ,\omega -\omega _{0})\,} Modulation property
Remarks
1 ∫ − ∞ ∞ | G x ( τ , ω ) | 2 d ω = ∫ − ∞ ∞ | x ( t ) | 2 e − 2 π ( t − τ ) 2 d t ≈ ∫ τ − 1.9143 τ + 1.9143 | x ( t ) | 2 e − 2 π ( t − τ ) 2 d t {\displaystyle \int _{-\infty }^{\infty }\left|G_{x}(\tau ,\omega )\right|^{2}\,d\omega =\int _{-\infty }^{\infty }\left|x(t)\right|^{2}e^{-2\pi (t-\tau )^{2}}dt\approx \int _{\tau -1.9143}^{\tau +1.9143}\left|x(t)\right|^{2}e^{-2\pi (t-\tau )^{2}}dt} Power integration property
2 ∫ − ∞ ∞ ∫ − ∞ ∞ G x ( τ , ω ) G y ∗ ( τ , ω ) d ω d τ = ∫ − ∞ ∞ x ( t ) y ∗ ( t ) d τ {\displaystyle \int _{-\infty }^{\infty }\int _{-\infty }^{\infty }G_{x}(\tau ,\omega )G_{y}^{*}(\tau ,\omega )\,d\omega \,d\tau =\int _{-\infty }^{\infty }x(t)y^{*}(t)\,d\tau } Energy sum property
3 { ∫ − ∞ ∞ | G x ( τ , ω ) | 2 d ω < e − 2 π ( t − t 0 ) 2 ∫ − ∞ ∞ | G x ( τ 0 , ω ) | 2 d ω ; if  x ( t ) = 0  for  t > t 0 ∫ − ∞ ∞ | G x ( τ , ω ) | 2 d τ < e − ( ω − ω 0 ) 2 ∫ − ∞ ∞ | G x ( τ , ω 0 ) | 2 d τ ; if  X ( ω ) = F T [ x ( t ) ] = 0  for  ω > ω 0 {\displaystyle {\begin{cases}\displaystyle \int _{-\infty }^{\infty }\left|G_{x}(\tau ,\omega )\right|^{2}d\omega <e^{-2\pi (t-t_{0})^{2}}\int _{-\infty }^{\infty }\left|G_{x}(\tau _{0},\omega )\right|^{2}\,d\omega ;&{\text{if }}x(t)=0{\text{ for }}t>t_{0}\\\displaystyle \int _{-\infty }^{\infty }\left|G_{x}(\tau ,\omega )\right|^{2}\,d\tau <e^{-(\omega -\omega _{0})^{2}}\int _{-\infty }^{\infty }\left|G_{x}(\tau ,\omega _{0})\right|^{2}\,d\tau ;&{\text{if }}X(\omega )=FT=0{\text{ for }}\omega >\omega _{0}\end{cases}}} Power decay property
4 ∫ − ∞ ∞ G x ( τ , ω ) e j ω t d ω = 2 π e − π τ 2 x ( 0 ) {\displaystyle \int _{-\infty }^{\infty }G_{x}(\tau ,\omega )e^{j\omega t}\,d\omega =2\pi e^{-\pi \tau ^{2}}x(0)} Recovery property

Application and example

Time/frequency distribution.

The main application of the Gabor transform is used in time–frequency analysis. Take the following function as an example. The input signal has 1 Hz frequency component when t ≤ 0 and has 2 Hz frequency component when t > 0

x ( t ) = { cos ⁡ ( 2 π t ) for  t ≤ 0 , cos ⁡ ( 4 π t ) for  t > 0. {\displaystyle x(t)={\begin{cases}\cos(2\pi t)&{\text{for }}t\leq 0,\\\cos(4\pi t)&{\text{for }}t>0.\end{cases}}}

But if the total bandwidth available is 5 Hz, other frequency bands except x(t) are wasted. Through time–frequency analysis by applying the Gabor transform, the available bandwidth can be known and those frequency bands can be used for other applications and bandwidth is saved. The right side picture shows the input signal x(t) and the output of the Gabor transform. As was our expectation, the frequency distribution can be separated into two parts. One is t ≤ 0 and the other is t > 0. The white part is the frequency band occupied by x(t) and the black part is not used. Note that for each point in time there is both a negative (upper white part) and a positive (lower white part) frequency component.

Discrete Gabor-transformation

A discrete version of Gabor representation

y ( t ) = ∑ m = − ∞ ∞ ∑ n = − ∞ ∞ C n m ⋅ g n m ( t ) {\displaystyle y(t)=\sum _{m=-\infty }^{\infty }\sum _{n=-\infty }^{\infty }C_{nm}\cdot g_{nm}(t)}

with g n m ( t ) = s ( t − m τ 0 ) ⋅ e j Ω n t {\displaystyle g_{nm}(t)=s(t-m\tau _{0})\cdot e^{j\Omega nt}}

can be derived easily by discretizing the Gabor-basis-function in these equations. Hereby the continuous parameter t is replaced by the discrete time k. Furthermore, the now finite summation limit in Gabor representation has to be considered. In this way, the sampled signal y(k) is split into M time frames of length N. According to Ω ≤ 2 π τ 0 {\displaystyle \Omega \leq {\tfrac {2\pi }{\tau _{0}}}} , the factor Ω for critical sampling is Ω = 2 π N {\displaystyle \Omega ={\tfrac {2\pi }{N}}} .

Similar to the DFT (discrete Fourier transformation) a frequency domain split into N discrete partitions is obtained. An inverse transformation of these N spectral partitions then leads to N values y(k) for the time window, which consists of N sample values. For overall M time windows with N sample values, each signal y(k) contains K = N ⋅ {\displaystyle \cdot } M sample values: (the discrete Gabor representation)

y ( k ) = ∑ m = 0 M − 1 ∑ n = 0 N − 1 C n m ⋅ g n m ( k ) {\displaystyle y(k)=\sum _{m=0}^{M-1}\sum _{n=0}^{N-1}C_{nm}\cdot g_{nm}(k)}

with g n m ( k ) = s ( k − m N ) ⋅ e j Ω n k {\displaystyle g_{nm}(k)=s(k-mN)\cdot e^{j\Omega nk}}

According to the equation above, the N ⋅ {\displaystyle \cdot } M coefficients C n m {\displaystyle C_{nm}} correspond to the number of sample values K of the signal.

For over-sampling Ω {\displaystyle \Omega } is set to Ω ≤ 2 π N = 2 π N ′ {\displaystyle \Omega \leq {\tfrac {2\pi }{N}}={\tfrac {2\pi }{N^{\prime }}}} with N′ > N, which results in N′ > N summation coefficients in the second sum of the discrete Gabor representation. In this case, the number of obtained Gabor-coefficients would be M ⋅ {\displaystyle \cdot } N′ > K. Hence, more coefficients than sample values are available and therefore a redundant representation would be achieved.

Scaled Gabor transform

As in short time Fourier transform, the resolution in time and frequency domain can be adjusted by choosing different window function width. In Gabor transform cases, by adding variance σ {\displaystyle \sigma } , as following equation:

The scaled (normalized) Gaussian window denotes as:

W gaussian ( t ) = e − σ π t 2 {\displaystyle W_{\text{gaussian}}(t)=e^{-\sigma \pi t^{2}}}

So the Scaled Gabor transform can be written as:

G x ( t , f ) = σ 4 ∫ − ∞ ∞ e − σ π ( τ − t ) 2 e − j 2 π f τ x ( τ ) d τ {\displaystyle G_{x}(t,f)={\sqrt{\sigma }}\textstyle \int _{-\infty }^{\infty }\displaystyle e^{-\sigma \pi (\tau -t)^{2}}e^{-j2\pi f\tau }x(\tau )d\tau \qquad }

With a large σ {\displaystyle \sigma } , the window function will be narrow, causing higher resolution in time domain but lower resolution in frequency domain. Similarly, a small σ {\displaystyle \sigma } will lead to a wide window, with higher resolution in frequency domain but lower resolution in time domain.

Time-causal analogue of the Gabor transform

When processing temporal signals, data from the future cannot be accessed, which leads to problems if attempting to use Gabor functions for processing real-time signals. A time-causal analogue of the Gabor filter has been developed in based on replacing the Gaussian kernel in the Gabor function with a time-causal and time-recursive kernel referred to as the time-causal limit kernel. In this way, time-frequency analysis based on the resulting complex-valued extension of the time-causal limit kernel makes it possible to capture essentially similar transformations of a temporal signal as the Gabor function can, and corresponding to the Heisenberg group, see for further details.

See also

References

  1. E. Sejdić, I. Djurović, J. Jiang, “Time-frequency feature representation using energy concentration: An overview of recent advances,” Digital Signal Processing, vol. 19, no. 1, pp. 153-183, January 2009.
  2. ^ Lindeberg, T. (23 January 2023). "A time-causal and time-recursive scale-covariant scale-space representation of temporal signals and past time". Biological Cybernetics: 1–39. arXiv:2202.09209. doi:10.1007/s00422-022-00953-6.
  • D. Gabor, Theory of Communication, Part 1, J. Inst. of Elect. Eng. Part III, Radio and Communication, vol 93, p. 429 1946 (http://genesis.eecg.toronto.edu/gabor1946.pdf)
  • Jian-Jiun Ding, Time frequency analysis and wavelet transform class note, the Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan, 2007.
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