User:Tjhuston225/Multidimensional Signal Processing

Multidimensional Signal Processing

In signal processing, multidimensional signal processing covers all signal processing done using multidimensional sampling. While multidimensional signal processing is a subset of signal processing, it is unique in the sense that it deals specifically with data that can only be adequately detailed using more than one dimension. Examples of this are image processing and multi-sensor radar detection. Multidimensional signals are part of multidimensional systems, and as such are generally more complex than classical, single dimension signal processing. Processing in m-D (multi-dimension) requires more complex algorithms to handle calculations such as the Fast Fourier Transform due to more degrees of freedom. In some cases, m-D signals and systems can be simplified into single dimension signal processing methods, utilizing assumptions such as symmetry.

Typically, multidimensional signal processing is directly associated with digital signal processing because its complexity warrants the use of computer modelling and computation.

Sampling

Main article: Multidimensional sampling

Multidimensional sampling requires different analysis than typical 1-D sampling. Single dimension sampling is executing by selecting points along a continuous line and storing the values of this data stream. In the case of multidimensional sampling, the data is selected utilizing a lattice, which is a "pattern" based on the sampling vector of the m-D data set. These vectors can be single dimensional or multidimensional depending on the data and the application.

Multidimensional sampling is similar to classical sampling as it must adhere to the Nyquist–Shannon sampling theorem. It is affected by aliasing and considerations must be made for eventual reconstruction.

Fourier Analysis

Main articles: Fourier Analysis, Multidimensional Transform, and Fast Fourier Transform

A multidimensional signal can be represented in terms of sinusoidal components. This is typically done with a type of Fourier transform. The M-D Fourier transform transforms a signal from a time domain representation to a frequency domain representation of the signal. In the case of digital processing, a discrete time Fourier transform is utilized to transform a sampled time domain representation into a frequency domain representation:

$X(k_{1},k_{2},\dots ,k_{m})=\sum _{n_{1}=-\infty }^{\infty }\sum _{n_{2}=-\infty }^{\infty }\cdots \sum _{n_{m}=-\infty }^{\infty }x(n_{1},n_{2},\dots ,n_{m})e^{-jk_{1}n_{1}}e^{-jk_{2}n_{2}}\cdots e^{-jk_{m}n_{m}}$

where X stands for the multidimensional discrete Fourier transform, x stands for the sampled time domain signal, m stands for the number of dimensions in the system, n are time samples and k are frequency samples. Computational complexity is usually the main concern when implementing any Fourier transform. For multidimensional signals, the complexity can be reduced by a number of different methods. The computation may be simplified if there is independence between variables of the multidimensional signal. In general, Fast Fourier Transforms (FFTs), utilize efficiencies of the system to reduce the number of computations by a substantial factor. While there are a number of different implementations of this algorithm for m-D signals, two often used variations are the vector-radix FFT and the row-column FFT.

Filtering

Main article: Filter (signal processing)

A 2-D filter (left) defined by its 1-D prototype function (right) and a mapping function.

Filtering is an important part of any signal processing application. Similar to typical single dimension signal processing applications, there are varying degrees of complexity within filter design for a given system. M-D systems utilize digital filters in many different applications. The actual implementation of these m-D filters can pose a design problem depending on whether the multidimensional polynomial is factorable. Typically, a prototype filter is designed in a single dimension and that filter is extrapolated to m-D using a mapping function. Both FIR and IIR filters can be utilized in this manner, depending on the application.

Applicable Fields

• Image processing
• Towed array sonar
• X-ray computed tomography