Introduction - Introduction to Digital Signal Processing and Filter Design - page 10

δ (n) δ (n − 3) δ (n + 3)

− 2 − 1 0 123 n − 1 0 123 n − 3 − 2 − 1123 0 n

( a )

( b ) ( c )

Figure 1.5 Unit pulse functions δ(n), δ(n − 3 ) ,and δ(n + 3 ) .

value of one at n 0 and zero at all other values of integer n , whereas the unit

impulse function δ(t) is defined entirely in a different way.
When the unit pulse function is delayed by k samples, it is described by

=

1

n = k

δ(n − k) = (1.7)

0 n = k

and it is plotted in Figure 1.5b for k = 3. When δ(n) is advanced by k = 3, we

get δ(n + k) , and it is plotted in Figure 1.5c.

1.3.3 Constant Sequence

This sequence x(n) has a constant value for all n and is therefore defined by
x(n) = K ;−∞ <n< ∞ .

1.3.4 Unit Step Function

The unit step function u(n) is defined by

1

n ≥ 0

u(n) =

(1.8)

0 n< 0

and it is plotted in Figure 1.6a.
When the unit step function is delayed by k samples, where k is a positive
integer, we have

1

n ≥ k

u(n − k) =

(1.9)

0 n<k