Introduction to systems - Continuous and Discrete Time Signals and Systems - page 93

Since y [ k ] = y 1 [ k ] + y 2 [ k ], the response of the overall system is given by

y [ k ] = 2 x [ k ] − 3 x [ k − 1] − x [ k − 2] .

2.3.3 Feedback configuration

The feedback configuration is shown in Fig. 2.19(c), where the output of system
S 1 is fed back, processed by system S 2 , and then subtracted from the input signal.
Such systems are difficult to analyze in the time domain and will be considered
in Chapter 6 after the introduction of the Laplace transform.

2.4 Summary

In this chapter we presented an overview of CT and DT systems, classifying
the systems into several categories. A CT system is defined as a transformation
that operates on a CT input signal to produce a CT output signal. In contrast, a
DT system transforms a DT input signal into a DT output signal. In Section 2.1,
we presented several examples of systems used to abstract everyday physical
processes. Section 2.2 classified the systems into different categories: linear
versus non-linear systems; time-invariant versus variant systems; memoryless
versus dynamic systems; causal versus non-causal systems; invertible versus
non-invertible systems; and stable versus unstable systems. We classified the
systems based on the following definitions.

(1) A system is linear if it satisfies the principle of superposition.
(2) A system is time-invariant if a time-shift in the input signal leads to
an identical shift in the output signal without affecting the shape of the
output.
(3) A system is memoryless if its output at t = t 0 depends only on the value
of input at t = t 0 and no other value of the input signal.
(4) A system is causal if its output at t = t 0 depends on the values of the input
signal in the past, t ≤ t 0 , and does not require any future value ( t > t 0 )of
the input signal.
(5) A system is invertible if its input can be completely determined by observing
its output.
(6) A system is BIBO stable if all bounded inputs lead to bounded outputs.

An important subset of systems is described by those that are both linear and
time-invariant (LTI). By invoking the linearity and time-invariance properties,
such systems can be analyzed mathematically with relative ease compared with
non-linear systems. In Chapters 3-8, we will focus on linear time-invariant CT
(LTIC) systems and study the time-domain and frequency-domain techniques
used to analyze such systems. DT systems and the techniques used to analyze
them will be presented in Part III, i.e. Chapters 9-17.