Introduction - Introduction to Digital Signal Processing and Filter Design - page 9

We denote the DT sequence by x(n) and also the value of a sample of the
sequence at a particular value of n by x(n) . If a sequence has zero values for
n< 0, then it is called a causal sequence . It is misleading to state that the
causal function is a sequence defined for n 0, because, strictly speaking, a DT

sequence has to be defined for all values of n . Hence it is understood that a causal
sequence has zero-valued samples for

≥

−∞ <n< 0. Similarly, when a function

is defined for N 1 ≤ n ≤ N 2 , it is understood that the function has zero values for

−∞ <n<N 1 and N 2 <n< ∞ . So the sequence x 1 (n) in Equation (1.2) has

3. The discrete-time sequence
x 2 (n) given below is a causal sequence. In this form for representing x 2 (n) ,itis
implied that x 2 (n)

zero values for 2 <n< ∞ and for −∞ <n< −

= 0for −∞ <n< 0andalsofor4 <n< ∞ :

1
↑

− 20 . 40 . 30 . 4000

x 2 (n) = (1.3)

The length of a finite sequence is often defined by other authors as the number
of samples, which becomes a little ambiguous in the case of a sequence like x 2 (n)
given above. The function x 2 (n) is the same as x 3 (n) given below:

1
↑

− 20 . 40 . 30 . 4000000

x 3 (n) = (1.4)

But does it have more samples? So the length of the sequence x 3 (n) would be
different from the length of x 2 (n) according to the definition above. When a
sequence such as x 4 (n) given below is considered, the definition again gives an
ambiguous answer:

0
↑

00 . 40 . 3 . 4 x 4 (n) = (1.5)

The definition for the length of a DT sequence would be refined when we
define the degree (or order) of a polynomial in z − 1 to express the z transform of

a DT sequence, in the next chapter.
To model the discrete-time signals mathematically, instead of listing their
values as shown above or plotting as shown in Figure 1.2, we introduce some
basic DT functions as follows.

1.3.2 Unit Pulse Function

The unit pulse function δ(n) is defined by

1

n = 0

δ(n) = (1.6)

0 n = 0

and it is plotted in Figure 1.5a. It is often called the unit sample function and also
the unit impulse function . But note that the function δ(n) has a finite numerical