We can derive a number of useful properties from the Fourier transform pair in (1.35). Some of the more important properties, often useful in practice, are listed in Table 1.1. Most are essentially straightforward extensions of 1-D Fourier trans≠form properties. The only exception is Property 4, which applies to separable sequences. If a 2-D sequence x(n1, n2) can be written as x1(n1)x2(n2), then its Fourier transform, is given by X1(w1)X2(w2), where X1(w1) and X2(w2) represent the 1-D Fourier transforms of x1(n1) and x2(n2) respectively. This prop≠erty follows directly from the Fourier transform pair of (1 .35). Note that this property is quite different from Property 3, the multiplication property. In the multiplication property, both x(n1, n2) and y(n1, n2) are 2-D sequences. in Property 4, x1(n1) and x2(n2) are 1-D sequences, and their product x1(n1)x2(n2) forms a 2-D sequence.
TABLE 1.1 : PROPERTIES OF FOURIER TRANSFORM
Property 1. Linearity
Property 2. Convolution
Property 3. Multiplication
Property 4. Separable Sequence
Property 5. Shift of a Sequence and a Fourier Transform
Property 6. Differentiation
Property 7. Initial Value and DC Value Theorem
Property 8. Parsevalís Theorem
Property 9. Symmetry Properties
Property 10. Uniform Convergence
For a stable x(n1,n2) the Fourier transform of h(n1,n2) uniformly converges.