Adaptive Gabor representation

The Gabor expansion of signal s(t) is defined by this formula:

${\displaystyle s(t)=\sum _{m=-\infty }^{\infty }\sum _{n=-\infty }^{\infty }C_{m,n}h(t-mT)e^{jnt\Omega }}$

where h(t) is the Gaussian elementary function:

${\displaystyle h(t)=\left({\frac {\alpha }{\pi }}\right)^{\frac {1}{4}}e^{\left(-{\frac {\alpha }{2}}t^{2}\right)}}$

Once the Gabor elementary function is determined, the Gabor coefficients ${\displaystyle C_{m,n}}$can be obtained by the inner product of s(t) and a dual function ${\displaystyle \gamma (t)}$

${\displaystyle C_{m,n}=\int s(t)\gamma ^{*}(t-mT)e^{-jnt\Omega }\,dt.}$

${\displaystyle T}$ and ${\displaystyle \Omega }$ denote the sampling steps of time and frequency and satisfy the criteria

${\displaystyle T\Omega \leqq 2\pi \,}$

Gabor transform simply computes the Gabor coefficients ${\displaystyle C_{m,n}}$ for the signal s(t).

If the elementary equation (${\displaystyle h_{p}\left(t\right)}$) is designed to have a unit energy. Then the energy contain in the residual at the pth stage can be determined by the residual at p+1th stage plus (${\displaystyle B_{p}}$). That is,

${\displaystyle \left\|s_{p}(t)\right\|^{2}=\left\|s_{p+1}(t)\right\|^{2}+\left|B_{p}\right|^{2},}$

${\displaystyle \left\|s(t)\right\|^{2}=\sum _{p=o}^{\infty }\left|B_{p}\right|^{2},}$

similar to the Parseval's theorem in Fourier analysis.

The selection of elementary function is the main task in adaptive signal decomposition. It is natural to choose a Gaussian-type function to achieve the lower bound for the inequality:

${\displaystyle h_{p}(t)=\left({\frac {\alpha }{\pi }}\right)^{\frac {1}{4}}e^{-{\frac {\alpha }{2}}(t-T_{p})^{2}}e^{jt\Omega _{p}},}$

where ${\displaystyle \left(T_{p},\Omega _{p}\right)}$ is th mean and ${\displaystyle \alpha _{p}^{-1}}$ is the variance of Gaussian at ${\displaystyle \left(T_{p},\Omega _{p}\right)}$. And

${\displaystyle s\left(t\right)=\sum _{p}B_{p}h_{p}(t)=\sum _{p}B_{p}\left({\frac {\alpha }{\pi }}\right)^{\frac {1}{4}}e^{-{\frac {\alpha }{2}}(t-T_{p})^{2}}e^{jt\Omega _{p}}}$

is called the adaptive Gabor representation.

Changing the variance value will change the duration of the elementary function (window size), and the center of the elementary function is no longer fixed. By adjusting the center point and variance of the elementary function, we are able to match the signal's local time-frequency feature. The better performance of the adaptation is achieved at the cost of matching process. The trade-off between different window length now become the trade-off between computation time and performance.