Digital Signal Processing Reference
In-Depth Information
2
Discrete-Ti
me Signals and Systems
2.1
Brief Theory of Discrete-Time Signals and Systems
In the previous chapter, we defined the concepts of discrete-time signals and
systems and noted that an important class of these systems is the LTI (linear
time invariant) system. It is well known that linearity implies superposition,
and time-invariance implies that properties of the system do not change with
time. Two of the most important concepts associated with discrete-time LTI
systems are
linear convolution
and
linear constant coefficient
difference equations.
Linear Convolution
Linear convolution
is a natural process of LTI systems. It defines the input-
output relation of the system and is defined as:
yn
()
=
xn
()* ()
hn
(2.1)
where the * symbol denotes the convolution process,
x
(
n
) is the system input,
y
of the system. The
impulse response is the output of the system, when the input
x
(
n
) is the sytem output, and
h
(
n
) is the
impulse response
(
n
) =
δ
(
n
),
the unit impulse.
The actual convolution process is defined as:
∞
∑
yn
()
=
xkhn k
()(
−
)
(2.2)
k
=−∞
for all values of
n.
Convolution can be performed using the
sliding tape
method
, as shown below or, more practically, by using computer software,
as will be described in Section 2.3.
Example
Determine the linear convolution of two discrete-time sequences,
x
(
n
) and
h
(
n
), given by:
17
