1.1.1         Examples of Sequences


Certain sequences and classes of  sequences play a particularly important role in 2-D signal processing. These are 


          step sequences,

          exponential sequences,

          separable sequences,  and

          periodic sequences.


Impulses        ;


The impulse or unit sample sequence, denoted by  d(n1,n2) is defined as  ;



The sequence  d(n1,n2) , sketched in Figure 1.4 plays a similar role to the impulse d(n) in 1-D signal processing.







Figure 1.4 :  Impulse d(n1,n2).



Any sequence x(n1,n2) can be represented as a linear combination of shifted impulses as follows


x(n1,n2)=.......+ x(-1,-1) . d(n1+1,n2+1) + x(0,-1) . d(n1,n2+1) + x(1,-1) . d(n1-1,n2+1)

          ....+ x(-1,0) . d(n1+1,n2) + x(0,0) . d(n1,n2) + x(1,0) . d(n1-1,n2)

+.....+ x(-1,1) . d(n1+1,n2-1) +..........................

=                                 (1.2)

Line impulses constitute a class of impulses which do not have any counterparts in 1-D. An example of line impulse is the 2-D sequence dT(n1) , which is sketched in Figure 1.5 and is defined as ;























Another example for line impulse is dT(n2)








Another example for line impulse is dT((n1 - n2)
















Step Sequences   ;


The unit step sequence , denoted by u(n1,n2)




The sequence  u(n1,n2) , which is sketched in Figure 1.6 , is related to dT(n1,n2) as







Some step sequences have no counterparts in 1-D. An example is the 2-D sequence dT(n1) , which is sketched in Figure 1.7 and is defined as




Other examples include dT(n2) and dT(n1-n2) , which are defined similarly to dT(n1).















Exponential Sequences   ;


Exponential sequences are of the type        are important for system analysis



Separable Sequences    ;


A 2-D sequence x(n1,n2) is said to be a separable sequence if it can be expressed as


                              x(n1,n2) = f(n1) . g(n2)                                                        (1.7)


where f(n1) is a function of only n1 and g(n2) is a function of only n2 . Although it is posible to view f(n1) and  g(n2) as 2-D sequences , it is more convenient to consider them to be 1-D sequences . For that reason, wwe use the notations  f(n1)  and   g(n2)  , rather than  fT(n1)   and  gT(n2).


    The impulse d(n1,n2)  can be expressed as


                          d(n1,n2)= d(n1) . d(n2)                                                  (1.8)


where     d(n1) and  d(n2are 1-D impulses. The unit step sequence u(n1,n2)  is also a separable sequence since  u(n1,n2)  can be expressed as


                               u(n1,n2)=u(n1) . u(n2)                                                         (1.9)


where   u(n1)  and  u(n2are 1-D unit step sequence. Another example  of a separable sequence is


   which can be written as            .



    Separable sequences form a very special class of  2-D sequences. A typical 2-D sequence is not a separable sequence. As an illustration, consider a sequence x(n1,n2which is ;




A general sequence x(n1,n2) of this type has N1N2 degrees of freedom. If x(n1,n2) is a separable sequence, x(n1,n2) is completely specified by some   f(n1which is zero outside 0 n1 N1-1  and some g(n2) which is zero outside 0 n2 N2-1  and consequently has only N1 + N2 degrees of freedom.


    Despite the fact that separable sequences constitute a very special class of 2-D sequences, they play an important role in 2-D signal processing. In those cases where the results that apply to 1-D sequences do not expand to general 2-D sequences in a straightforward manner,they often do for separable 2-D sequences.


Periodic Sequences   ;


    A sequence x(n1, n2) is said to be periodic with a period of N1 x N2 if x(n1, n2) satisfies the following condition:


x(n1, n2) = x(n1 + N1, n2) = x(n1, n2 + N2)     for all (n1, n2)                         (1.10)


where N1 and N2 are positive integers.


For example, cos (∏n1+∏ n2) is a periodic sequence with a period of 2 x 4, since

   for all (n1,n2).

The sequence cos (n1 + n2) is not periodic, however, since cos (n1 + n2) cannot be expressed as  cos ((n1 + N1) + n2) = cos (n1 + (n2 + N2)) for all (n1,n2)  for any nonzero integers N1 and N2.  A periodic sequence is often denoted by adding a ~ (tilde). For example, , to distinguish it from an aperiodic sequence.


Equation (1.10) is not the most general representation of a 2-D periodic sequence. As an illustration, consider the sequence x(n1,n2) shown in Figure 1.8. Even though x(n1, n2) can be considered a periodic sequence with a period of 3 x 2 it cannot be represented as such a sequence by using (1.10). Specifically. x(n1,n2) x(n1 + 3, n2)  for all (n1,n2). It is possible to generalize (1.10) to incorporate cases such as that in Figure 1.8. However, we will use (1.10) to define a periodic sequence, since it is sufficient for our purposes, and sequences such as that in Figure 1.8 can be represented by (1.10) by increasing N1 and/or N2. For example, the sequence in Figure 1.8 is periodic with a period of 6 x 2 using (1.10).