Digital Signal Processing Reference
In-Depth Information
where and are arbitrary constants. A system is linear if it satisfies the principle
of superposition. No physical system is linear under all operating conditions. How-
ever, a physical system can be tested with the use of (9.75) to determine ranges of
operation for which the system is approximately linear.
An example of a linear operation (system) is that of multiplication by a con-
stant
K
, described by
a
1
a
2
y[n] = Kx[n].
An example of a nonlinear system is the opera-
tion of squaring a signal,
y[n] = x
2
[n].
For inputs of
x
1
[n] and x
2
[n],
the outputs of the squaring system are
x
1
[n] : y
1
[n] = x
2
[n]
and
x
2
[n] : y
2
[n] = x
2
[n].
(9.76)
However, the input
(x
1
[n] + x
2
[n])
produces the output
x
1
[n] + x
2
[n]
: (x
1
[n] + x
2
[n])
2
= x
2
[n] + 2x
1
[n]x
2
[n]
+ x
2
[n] = y
1
[n] + y
2
[n] + 2x
1
[n]x
2
[n].
(9.77)
A linear time-invariant (LTI) system is a linear system that is also time invari-
ant. LTI systems, for both continuous-time and discrete-time systems, are empha-
sized in this topic.
An important class of LTI discrete-time systems are those that are modeled by
linear difference equations with constant coefficients. An example of this type of
system is the Euler integrator described earlier in this section:
y[n] - y[n - 1] = Hx[n - 1].
In this equation,
x
[
n
] is the input,
y
[
n
] is the output, and the numerical-integration
increment
H
is constant.
The general forms of an
n
th-order linear difference equation with constant
coefficients are given by (9.6) and (9.7). Equation (9.6) is repeated here:
y[n] = b
1
y[n - 1] + b
2
y[n - 2] +
Á
+ b
N
y[n - N]
+ a
0
x[n] + a
1
x[n - 1] +
Á
+ a
N
x[n - N].
This difference equation is said to be of order
N
. The second version of this general
equation is obtained by replacing
n
with
(n + N).
[See (9.7).]
