Special 2-D Sequences

• Unit Impulse $\delta (n_1,n_2) = \left\{ \begin{array}{cl} 1 & n_1=n_2=0 \\ 0 & {\rm otherwise} \end{array} \right .$

  Figure 2: 2-D Unit impulse

  

  2-D unit impulse is separable since $\delta(n_1,n_2)=\delta(n_1)\delta(n_2)$.

• Line Impulses
  Vertical line impulse: $x(n_1,n_2) = \delta(n_1)$ is separable since $x(n_1,n_2) = \delta(n_1) \times 1$

  Figure 3: 2-D vertical line impulse

  

  Horizontal line impulse: $x(n_1,n_2) = \delta(n_2)$ is separable since $x(n_1,n_2) = 1 \times \delta(n_2)$

  Figure 4: 2-D horizontal line impulse

  

  Diagonal line impulses are non-separable.

  Figure 5: Diagonal line impulse: $\delta (n_1+n_2)$

  

  Figure 6: diagonal line impulse: $\delta (n_1-n_2)$

  

  A general diagonal line impulse can be written as: $\delta(P n_1 + Q n_2)$ where $P$ and $Q$ are integers. The slope of the impulses is $-\frac{P}{Q}$.

• Unit step sequence:
  
  $$u(n_1,n_2)= \left \{ \begin{array}{cl} 1 & n_1 \ge 0 \; {... ...nd} \; n_2 \ge 0 \\ 0 & {\rm otherwise} \end{array} \right.$$
  
  

  The unit step function is separable since $u(n_1,n_2)=u(n_1)u(n_2)$.

  Figure 7: 2-D unit step