Digital Filter Design–II: Applications
B. Preetham Kumar · 2016
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).