Digital Signal Processing Reference
In-Depth Information
For
n = 0,
h[0] = 2.5(1 - 0.6) = 1.
For
n G 1,
h[n] = 2.5(1 - 0.6
n+ 1
- 1 + 0.6
n
)
= 2.5(0.6
n
)(1 - 0.6) = 0.6
n
.
Hence,
h[n] = 0.6
n
u[n],
which is the given function.
■
In this section, the properties of memory, invertibility, causality, and stability
are investigated for discrete-time LTI systems. If a system is memoryless, its im-
pulse response is given by with
K
constant. The impulse response
h
[
n
] of a causal system is zero for A system is stable if its impulse response is
absolutely summable, as in (10.36). Invertibility is discussed, but no mathematical
test is developed.
h[n] = Kd[n],
n 6 0.
10.4
DIFFERENCE-EQUATION MODELS
In Sections 10.1 through 10.3, certain properties of LTI discrete-time systems are
developed. We now consider the most common model for systems of this type. LTI
discrete-time systems are usually modeled by
linear difference equations with con-
stant coefficients
. We emphasize that
models
of physical systems are being consid-
ered, not the physical systems themselves. A common discrete-time LTI physical
system is a digital filter. Digital filters are implemented either by digital hardware
that is constructed to solve a difference equation or by a digital computer that is
programmed to solve a difference equation. In either case, the difference-equation
model is usually accurate, provided that the computer word length is sufficiently
long that numerical problems do not occur.
In this section, we consider difference-equation models for LTI discrete-time
systems. Then, two methods are given for solving linear difference equations with
constant coefficients; the first is a classical procedure and the second is an iterative
procedure.
In Example 10.1 and Figure 10.3, we considered a discrete-time system with the dif-
ference equation
y[n] = (x[n] + x[n - 1] + x[n - 2])
>
3.
(10.43)
