Digital Signal Processing Reference
In-Depth Information
We can now write the equation for the balance of our savings account in a
closed form, as shown below. This expression does not need the calculation of
n
+ 1 terms in order for us to evaluate it.
n
+
1
1
−
1
01
y
(
n
)
=
100
⋅
1
−
1
01
(5.19)
We can also define the value of the infinite geometric sum, as shown below:
∞
=
k
IGS
=
a
k
0
(5.20)
By simply allowing
n
in (5.18) to approach infinity, we can see that IGS can
be specified as
1
⎧
⎨
,
for
a
<
1
IGS
=
1
−
a
⎩
undefined
otherwise
(5.21)
One way to completely define a discrete-time system is by using its difference
equation. In general, the difference equation describing the output for a discrete-
time system can be written as
M
N
∑
∑
y
(
n
)
=
a
⋅
x
(
n
−
k
)
−
b
⋅
y
(
n
−
k
)
k
k
k
=
0
k
=
1
(5.22)
We would have complete knowledge of the system if we know the
coefficients
a
k
and
b
k
. As indicated in (5.22), the output
y
(
n
) is a function of past
and present values of the input
x
(
n
) and past values of the output. A system such
as this is a
recursive
system. A system that has its output described only by past
and present values of the input is a
nonrecursive
system. We will see in the
chapters to come that these definitions effectively divide the types of digital filters
to be designed into two groups as well.
Notice that we have defined the output of our system in terms of only past and
present values of the input. Such a system is referred to as a
causal
system. Any
real-time system, of course, must be causal since we cannot determine the output
of a system based on input or output values we have not yet seen. However,
systems that are not real-time can be
noncausal
. For example, any system that can
draw its input from stored data can determine the output of the system at time
n
by
