ABSTRACT
Two-dimensional discrete signals are obtained by sampling two-dimensional continuous signals. A point in the x-y sampling grid is (n1Δx,n2Δy), and the
sampled signal is f(n1Δx,n2Δy) or simply f(n1,n2) in the range 0 ≤ n1 ≤ N1 – 1; 0 ≤ n2 ≤ N2 – 1. The sampled signal can be represented by the matrix function:
f
f f f N
f f f N =
( ) ( ) −( ) ( ) ( ) 0 0 0 1 0 1
1 0 1 1 1 2, , ,
, , ,
1 0 1 1 1 1
−( )
−( ) −( ) − −( )
f N f N f N N, , ,
(6.1)
Each element of the matrix f can also be termed as a pixel, or picture element, which gives a total of N1 × N2 pixels in the entire image. Some common examples of two-dimensional discrete signals are:
2-D Impulse Function
δ n n n n
, ,
, ( ) =
= =
for
otherwise
(6.2)
2-D Unit Step Function
u n n n n
1 21 0 0
, , ,
, ( ) =
≥ ≥
for
otherwise
(6.3)
Example
Define the following functions:
1. δ n n1 23 5− −( ), 2. u n n1 23 5− −( ), Solution
δ n n for n n1 2 1 23 5 1 3 5
− −( ) = = =
=
, , , ;
, otherwise
u n n n n( , ) , , ;1 2 1 23 5 1 3 5− − = ≥ ≥
=
for
0, otherwise
6.2.2 Two-Dimensional Discrete Systems
A system with two-dimensional discrete space input and output signals is termed as a 2-D discrete system, as shown in Figure 6.2. The relationship between the output and input of a 2-D discrete system is given by:
g n n T f n n1 2 1 2, ,( ) = ( ) (6.4) where T is the system operator. If the system is LSI (Linear Shift Invariant), then we have the 2-D convolution relation:
g n n f k k h n k n k kk
, , ,( ) = ( ) − −( )∑∑ (6.5) or,
g n n f n n h n n1 2 1 2 1 2, , ,( ) = ( )∗∗ ( ) (6.6) 2-D discretespace output
where the symbol ∗∗ represents the 2-dimensional discrete convolution, and h(n1,n2) is the 2-dimensional impulse response of the system. The 2-D impulse response is defined as the output of the system, when the input f(n1,n2) = δ(n1,n2), the 2-D impulse function, defined in equation (6.2).
