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.
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) +..........................
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(n2) are 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(n2) are 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,n2) which 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(n1) which 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) ¹ 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).