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===Convolution Theorem in Multiple Dimensions===
For one-dimensional signals, the [[Convolution theorem|Convolution Theorem]] states that the [[Fourier transform]] of the convolution between two signals is equal to the product of the Fourier Transforms of those two signals. Thus, convolution in the time ___domain is equal to multiplication in the frequency ___domain. Mathematically, this principle is expressed via the following:<math display="block">y(n)=h(n)*x(n)\longleftrightarrow Y(\omega)=H(\omega)X(\omega)</math>This principle is directly extendable to dealing with signals of multiple dimensions.<math display="block">y(n_1,n_2,...,n_M)=h(n_1,n_2,...,n_M)*\overset{M}{\cdots}*x(n_1,n_2,...,n_M) \longleftrightarrow Y(\omega_1,\omega_2,...,\omega_M)=H(\omega_1,\omega_2,...,\omega_M)X(\omega_1,\omega_2,...,\omega_M)</math> This property is readily extended to the usage with the [[Discrete Fourier transform]] (DFT) as follows (note that linear convolution is replaced with circular convolution where <math>\otimes</math> is used to denote the circular convolution operation of size <math>N</math>):
<math>y(n)=h(n)\otimes x(n)\longleftrightarrow Y(k)=H(k)X(k)</math>
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