1.3.2    Properties

    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

(a).

(b)

 

  

  

Property 6. Differentiation

(a)

               (b) 

 

 

 

Property 7. Initial Value and DC Value Theorem

(a)  

(b)   

 

  

  

  

  

Property 8. Parsevalís Theorem

              (a)   

              (b)

 

  

  

Property 9. Symmetry Properties

(a)

                (b)

                (c)

                (d)        

(e)

     

(f)

              (g)

 

  

  

  

  

Property 10. Uniform Convergence

For a stable x(n1,n2) the Fourier transform of h(n1,n2) uniformly converges.