In signal processing, nonlinear multidimensional signal processing (NMSP) covers all signal processing using nonlinear multidimensional signals and systems. Nonlinear multidimensional signal processing is a subset of signal processing (multidimensional signal processing). Nonlinear multi-dimensional systems can be used in a broad range such as imaging, teletraffic, communications, hydrology, geology, and economics. Nonlinear systems cannot be treated as linear systems, using Fourier transformation and wavelet analysis. Nonlinear systems will have chaotic behavior, limit cycle, steady state, bifurcation, multi-stability and so on. Nonlinear systems do not have a canonical representation, like impulse response for linear systems. But there are some efforts to characterize nonlinear systems, such as Volterra and Wiener series using polynomial integrals as the use of those methods naturally extend the signal into multi-dimensions. Another example is the Empirical mode decomposition method using Hilbert transform instead of Fourier Transform for nonlinear multi-dimensional systems. This method is an empirical method and can be directly applied to data sets. Multi-dimensional nonlinear filters (MDNF) are also an important part of NMSP, MDNF are mainly used to filter noise in real data. There are nonlinear-type hybrid filters used in color image processing, nonlinear edge-preserving filters use in magnetic resonance image restoration. Those filters use both temporal and spatial information and combine the maximum likelihood estimate with the spatial smoothing algorithm.
Nonlinear analyser A linear frequency response function (FRF) can be extended to a nonlinear system by evaluation of higher order transfer functions and impulse response functions by Volterra series. Suppose we have a time series y ( t ) {\displaystyle y(t)} , which is decomposed y ( t ) {\displaystyle y(t)} into components of various order
y ( t ) = y 0 + y 1 ( t ) + y 2 ( t ) + ⋯ + y n ( t ) . {\displaystyle y(t)=y_{0}+y_{1}(t)+y_{2}(t)+\cdots +y_{n}(t).}
Each component is defined as
y n ( t ) = ∫ − ∞ + ∞ ⋯ ∫ − ∞ + ∞ h n ( τ 1 , τ 2 , ⋯ , τ n ) ∏ i = 1 n x ( t − τ i ) d τ i {\displaystyle 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 {\displaystyle n=1} , y 1 ( t ) {\displaystyle y_{1}(t)} is the linear convolution. h n ( τ 1 , τ 2 , ⋯ , τ n ) {\displaystyle h_{n}(\tau _{1},\tau _{2},\cdots ,\tau _{n})} is the generalized impulse response of order n {\displaystyle n} . The 1D Fourier transform of y n ( t ) {\displaystyle y_{n}(t)} is
Y ( n ) ( ω ) = ∫ − ∞ + ∞ [ ∫ − ∞ + ∞ ⋯ ∫ − ∞ + ∞ h n ( τ 1 , ⋯ , τ n ) ∏ i = 1 n x ( t − τ i ) d τ i ] exp ( − j ω t ) d t . {\displaystyle Y_{(n)}(\omega )=\int _{-\infty }^{+\infty }[\int _{-\infty }^{+\infty }\cdots \int _{-\infty }^{+\infty }h_{n}(\tau _{1},\cdots ,\tau _{n})\prod _{i=1}^{n}x(t-\tau _{i})d\tau _{i}]\exp(-j\omega t)dt.}
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