Certain sequences and classes of sequences play a particularly important role in 2D signal processing. These are
· impulses,
· step sequences,
· exponential sequences,
· separable sequences, and
· periodic sequences.
Impulses ;
The impulse or unit sample sequence, denoted by d(n_{1},n_{2}) is defined as ;
_{} (1.1)
The sequence d(n_{1},n_{2}) , sketched in Figure 1.4 plays a similar role to the impulse d(n) in 1D signal processing.
Figure 1.4 : Impulse d(n_{1},n_{2}).
Any sequence x(n_{1},n_{2}) can be represented as a linear combination of shifted impulses as follows
x(n_{1},n_{2})=.......+ x(1,1) . d(n_{1}+1,n_{2}+1) + x(0,1) . d(n_{1},n_{2}+1) + x(1,1) . d(n_{1}1,n_{2}+1)
· ....+ x(1,0) . d(n_{1}+1,n_{2}) + x(0,0) . d(n_{1},n_{2}) + x(1,0) . d(n_{1}1,n_{2})
+.....+ x(1,1) . d(n_{1}+1,n_{2}1) +..........................
=_{} (1.2)
Line impulses constitute a class of impulses which do not have any counterparts in 1D. An example of line impulse is the 2D sequence d_{T}(n_{1}) , which is sketched in Figure 1.5 and is defined as ;




_{} (1.3)
Another example for line impulse is d_{T}(n_{2})
_{}
Another example for line impulse is d_{T}((n_{1 } n_{2})




_{}
Step Sequences ;
The unit step sequence , denoted by u(n_{1},n_{2})
_{} (1.4)
The sequence u(n_{1},n_{2}) , which is sketched in Figure 1.6 , is related to d_{T}(n_{1},n_{2}) as
_{} (1.5a)
or
_{} (1.5b)
Some step sequences have no counterparts in 1D. An example is the 2D sequence d_{T}(n_{1}) , which is sketched in Figure 1.7 and is defined as
_{} (1.6a)
Other examples include d_{T}(n_{2}) and d_{T}(n_{1}n_{2}) , which are defined similarly to d_{T}(n_{1}).
_{} (1.6b)
and
_{} (1.6c)
_{ }
_{ }
_{ }
_{ }
_{ }
Exponential Sequences ;
Exponential sequences are of the type _{} are important for system analysis
Separable Sequences ;
A 2D sequence x(n_{1},n_{2}) is said to be a separable sequence if it can be expressed as
x(n_{1},n_{2}) = f(n_{1}) . g(n_{2}) (1.7)
where f(n_{1}) is a function of only n_{1 }and_{ }g(n_{2}) is a function of only n_{2 .} Although it is posible to view f(n_{1}) and g(n_{2}) as 2D sequences , it is more convenient to consider them to be 1D sequences . For that reason, wwe use the notations f(n_{1}) and g(n_{2}) , rather than f_{T}(n_{1}) and g_{T}(n_{2}).
The impulse d(n_{1},n_{2}) can be expressed as
d(n_{1},n_{2})= d(n_{1}) . d(n_{2}) (1.8)
where d(n_{1}) and d(n_{2}) are 1D impulses. The unit step sequence u(n_{1},n_{2}) is also a separable sequence since u(n_{1},n_{2}) can be expressed as
u(n_{1},n_{2})=u(n_{1}) . u(n_{2}) (1.9)
where u(n_{1}) and u(n_{2}) are 1D unit step sequence. Another example of a separable sequence is
_{} which can be written as _{} .
Separable sequences form a very special class of 2D sequences. A typical 2D sequence is not a separable sequence. As an illustration, consider a sequence x(n_{1},n_{2}) which is ;
_{}
A general sequence x(n_{1},n_{2}) of this type has N_{1}N_{2} degrees of freedom. If x(n_{1},n_{2}) is a separable sequence, x(n_{1},n_{2}) is completely specified by some f(n_{1}) which is zero outside 0 £n_{1}£ N_{1}1 and some g(n_{2}) which is zero outside 0 £n_{2}£ N_{2}1 and consequently has only N_{1} + N_{2} degrees of freedom.
Despite the fact that separable sequences constitute a very special class of 2D sequences, they play an important role in 2D signal processing. In those cases where the results that apply to 1D sequences do not expand to general 2D sequences in a straightforward manner,they often do for separable 2D sequences.
Periodic Sequences ;
A sequence x(n_{1}, n_{2}) is said to be periodic with a period of N_{1 }x N_{2 }if x(n_{1}, n_{2}) satisfies the following condition:
x(n_{1}, n_{2}) = x(n_{1 }+ N_{1}, n_{2}) = x(n_{1}, n_{2 }+ N_{2}) for all (n_{1}, n_{2}) (1.10)
where N_{1 }and N_{2 }are positive integers.
For example, cos (∏n_{1}+_{}∏ n_{2}) is a periodic sequence with a period of 2 x 4, since
_{} for all (n_{1},n_{2}).
The sequence cos (n_{1 }+ n_{2}) is not periodic, however, since cos (n_{1 }+ n_{2}) cannot be expressed as cos ((n_{1 }+ N_{1}) + n_{2}) = cos (n_{1 }+ (n_{2} + N_{2})) for all (n_{1},n_{2}) for any nonzero integers N_{1 }and N_{2}. 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 2D periodic sequence. As an illustration, consider the sequence x(n_{1},n_{2}) shown in Figure 1.8. Even though x(n_{1}, n_{2}) 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(n_{1},n_{2}) ¹ x(n_{1 }+ 3, n_{2})_{ }for all (n_{1},n_{2}). 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 N_{1 }and/or N_{2}. For example, the sequence in Figure 1.8 is periodic with a period of 6 x 2 using (1.10).



