Digital Signal Processing Reference
In-Depth Information
In this section, we consider the response of LTI systems to complex-exponential
inputs, which leads us to the concept of transfer functions. Using the transfer-function
approach, we can easily find the system steady-state response to sinusoidal inputs. As
a final point, the relationship between the transfer function of a system and its
impulse response is derived.
In this chapter, we consider discrete-time linear time-invariant (LTI) systems. First,
it is shown that discrete-time signals can be represented as a sum of weighted dis-
crete impulse functions. This representation allows us to model an LTI system in
terms of its impulse response.
The modeling of a system by its impulse response is basic to the analysis and
design of LTI systems. The impulse response gives a complete input-output de-
scription of an LTI system. It is shown that through the convolution summation, the
input
x
[
n
], the impulse response
h
[
n
], and the output
y
[
n
] are related by
q
q
y[n] =
a
x[k]h[n - k] =
a
x[n - k]h[k].
k=-
q
k=-
q
The importance of the impulse response of an LTI system cannot be overempha-
sized. It is also shown that the impulse response of an LTI system can be derived
from its step response. Hence, the input-output description of a system is also con-
tained in its step response.
Next, some general properties of LTI systems are discussed. These properties
include memory, invertibility, causality, and stability.
The most popular method for modeling LTI systems is by ordinary linear dif-
ference equations with constant coefficients. This method is used for physical sys-
tems that can be modeled accurately by these equations. A linear time-invariant
digital filter is an LTI discrete-time system and, in general, is modeled very accu-
rately by a linear difference equation with constant coefficients.
A general procedure is given for solving linear difference equations with con-
stant coefficients. This procedure leads to a test that determines stability for causal
discrete-time LTI systems.
Next, a procedure for representing system models by simulation diagrams is
developed. Two simulation diagrams, the direct forms I and II, are given. However,
it should be realized that an unbounded number of simulation diagrams exist for a
given LTI system. In many applications, the simulation diagrams are called pro-
gramming forms.
As the final topic, the response of an LTI system to a sinusoidal input signal is
derived. This derivation leads to the transfer-function description of an LTI system.
It is shown in Chapter 11 that the transfer function allows us to find the response of
an LTI system to any input signal. Hence, the transfer function is also a complete
input-output description of an LTI system.
See Table 10.2.
