Digital Signal Processing Reference
In-Depth Information
that be sampled every
H
seconds, resulting in the number sequence
with
n
an integer.
Let
x(t)
x(nH),
y(t)
be the following integral of
x(t):
t
y(t) =
L
x(t)dt.
(9.2)
0
The integral of
x(t)
from
t = 0
to
t = nH
in Figure 9.2 can be expressed as the inte-
gral for
t = 0
to
t = (n - 1)H
plus the integral from
(n - 1)H
to
nH
. Thus, in (9 .2),
nH
y(t) ƒ
t =nH
= y(nH) =
L
x(t)dt
0
(n- 1)H
nH
=
L
x(t)dt +
L
x(t)dt
0
(n- 1)H
L y[(n - 1)H] + Hx[(n - 1)H].
(9.3)
Ignoring the approximations involved, we expressed this equation as
y(nH) = y[(n - 1)H] + Hx[(n - 1)H].
(9.4)
However, is only an approximation to the integral of at
In the notation for discrete-time signals discussed earlier, (9.4) is expressed as
y(nH)
x(t)
t = nH.
y[n] = y[n - 1] + Hx[n - 1].
(9.5)
An equation of this type is called a
difference equation
. A general
N
th-order
linear
difference equation
with
constant coefficients
is of the form
Á
y[n] = b
1
y[n - 1] + b
2
y[n - 2] +
+ b
N
y[n - N]
Á
+ a
0
x[n] + a
1
x[n - 1] +
+ a
N
x[n - N],
(9.6)
where the coefficients and are constants. Replacing
n
with
we can also express this difference equation as
a
i
b
i
, i = 1, 2, Á , N,
(n + N),
y[n + N] = b
1
y[n + N - 1] + b
2
y[n + N - 2] +
Á
+ b
N
y[n]
+ a
0
x[n + N] + a
1
x[n + N - 1] +
Á
+ a
N
x[n].
(9.7)
The formats of both (9.6) and (9.7) are used in specifying difference equations. In
this chapter, we consider discrete-time signals of the type of
x
[
n
] and
y
[
n
] in (9.6)
and (9.7) and discrete systems described by difference equations. However, we do
not limit the difference equations to being linear.
EXAMPLE 9.1
Difference-equation solution
As an example of the solution of a difference equation, consider the numerical integration of
a unit step function
u(t)
by the use of Euler's rule in (9.5). The continuous unit step function
is defined as
