Any rectangularly periodic array with horizontal and vertical periods N₁ and N₂ can be expressed as a finite sum of harmonically related complex sinusoids.

Each sum involves only a finite number of samples

If the second sum is used to define , that second sequence is seen to be rectangularly periodic with horizontal and vertical periods N₁ and N₂, respectively.

There is a one-to-one relationship between a sequence with periods N₁ and N_2, and a finite extent sequence with support on [0, N₁-1]x[0, N₂-1].

where

The outside sequences are of finite extent. The direct relationship between them is the DFT.

The DFT corresponds to (rectangular) samples of the discrete-time Fourier transform

At location n₂ = 0 and n₃ = 1, this function is a line impulse in 3-D that varies over integer values of n₁. Recall that d(n₂, n₃-1) = 1 d(n₂) d(n₃-1).

First evaluate the sum with respect to n₃

Now evaluate with respect to n₂

• Linearity:
• Circular shifts:

• Symmetry: If , i.e. the sequence is real-valued, then
• Duality: If , then
• Circular Convolution:
• The convolution is circular with respect to both variables
• If the DFT support N₁ x N₂ is large enough to contain the linear convolution, then circular convolution = linear convolution
• If the extent of x is N₁ x N₂ and h is M₁ x M₂, then the linear convolution of x and h is P₁ x P₂ where P₁ = N₁ + M₁ – 1 and P₂ = N₂ + M₂ – 1.
• The DFT size must be at least P₁ x P₂