Non-linear multi-dimensional signal processing

Nonlinear multi-dimensional signal processing

In signal processing, nonlinear multidimensional signal processing (NMSP) covers all signal processing using nonlinear multidimensional signals and systems. While multidimensional signal processing is a subset of signal processing (multidimensional signal processing). Nonlinear systems cannot be treated as linear system, using Fourier transformation and wavelet analysis. Nonlinear systems will have chaotic behavior, limit circle, steady state, bifurcation, multi-stability and so on. As the complicated of real nonlinear system, there didn't have canonical representation, like impose-response for linear systems.But there are some effects have to representation nonlinear system. Volterra and Wiener series using polynomial integral instead of linear convolution to representation nonlinear systems as the using of this methods naturally extended the signal into multi-dimensional. Empirical mode decomposition method using Hulbert transform instead of Fourier Transform apply to nonlinear multi-dimensional system. Multi-dimensional nonlinear filter (MDNF) is also an important part of NMSP, MDNF is always be used to filter noise in real data.There are nonlinear-type hybird filters using in color image, using multi-dimensional

Nonlinear analyser

Extension a linear frequency response function (FRF) to a nonlinear system by evaluation of higher order transfer functions and impulse response functions by Volterra series. Then extension modal analysis to nonlinear system and applicability. suppose we have time series $y(t)$, decomposition $y(t)$ into components of various order

$y(t)=y_{0}+y_{1}(t)+y_{2}(t)+\cdots +y_{n}(t)$,

each component is defined as

$y_{n}(t)=\int _{-\infty }^{+\infty }\cdots \int _{-\infty }^{+\infty }h_{n}(\tau _{1},\tau _{2},\cdots ,\tau _{n})\displaystyle \prod _{i=1}^{n}x(t-\tau _{i})d\tau _{i}$ ,

for $n=1$, we can identify the linear described by linear convolution. The $h_{n}(\tau _{1},\tau _{2},\cdots ,\tau _{n})$ is the generalized impulse response of order $n$. The above formula is using delay time series to reconstruction nonlinear system. However, we can also using multi-dimensional signal instead of the delay time series.

Transfer function

Applying the $n$th dimensional FT to $h_{n}(\tau _{1},\tau _{2},\cdots ,\tau _{n})$ obtain the transfer function

$H_{n}(\omega ,\cdots ,\omega )=\int _{-\infty }^{+\infty }\cdots \int _{-\infty }^{+\infty }h_{n}(\tau _{1},\cdots ,\tau _{n})\exp(-j\omega _{1}\tau _{1}-\cdots -j\omega _{n}\tau _{n})d\tau _{1}\cdots d\tau _{n}$

A nonlinear multi-dimensional (frequency) analyser represents