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Concept
Multidimensional transform
Summary
In mathematical analysis and applications, multidimensional transforms are used to analyze the frequency content of signals in a domain of two or more dimensions.
One of the more popular multidimensional transforms is the Fourier transform, which converts a signal from a time/space domain representation to a frequency domain representation. The discrete-domain multidimensional Fourier transform (FT) can be computed as follows:
where F stands for the multidimensional Fourier transform, m stands for multidimensional dimension. Define f as a multidimensional discrete-domain signal. The inverse multidimensional Fourier transform is given by
The multidimensional Fourier transform for continuous-domain signals is defined as follows:
Similar properties of the 1-D FT transform apply, but instead of the input parameter being just a single entry, it's a Multi-dimensional (MD) array or vector. Hence, it's x(n1,...,nM) instead of x(n).
if , and then,
if , then
if , then
if , and
then,
or,
If , then
If , then
If , then
If , then
if , and then,
if , then
A special case of the Parseval's theorem is when the two multi-dimensional signals are the same. In this case, the theorem portrays the energy conservation of the signal and the term in the summation or integral is the energy-density of the signal.
A signal or system is said to be separable if it can be expressed as a product of 1-D functions with different independent variables. This phenomenon allows computing the FT transform as a product of 1-D FTs instead of multi-dimensional FT.
if , ,
, and if
then
so
A fast Fourier transform (FFT) is an algorithm to compute the discrete Fourier transform (DFT) and its inverse. An FFT computes the DFT and produces exactly the same result as evaluating the DFT definition directly; the only difference is that an FFT is much faster. (In the presence of round-off error, many FFT algorithms are also much more accurate than evaluating the DFT definition directly).There are many different FFT algorithms involving a wide range of mathematics, from simple complex-number arithmetic to group theory and number theory.
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